16. V. Volterra, Lec , ons sur les e´quations inte´grales et les e´quat- ions integro-differentielles, Gauthier-Villars, Paris, 1913. 17. V. Volterra, Lec , ons sur la the´orie mathe´matique de la lutte pour la vie, Gauthier-Villars, Paris, 1931. 18. F. Bloom, Ill-Posed Problems for Integrodifferential Equations in Mechanics and Electromagnetic Theory, SIAM, Philadel- phia, 1981. INTERMEDIATE-FREQUENCY AMPLIFIERS H. R. WALKER Pegasus Data Systems, Inc. Edison, New Jersey The intermediate-frequency (IF) amplifier is the circuitry used to process the information bearing signal between the first converter, or mixer, and the decision making cir- cuit, or detector. It can consist of a very few or a great many component parts. Generally, it consists of an ampli- fying stage or device to provide gain, plus a bandpass fil- ter, or filters, to limit the frequency band to be passed. The signal to be processed can be audio, video, digital, or pulsed, using amplitude modulation, frequency modula- tion, phase modulation, or combinations thereof. Several examples are shown in Figs. 13–19. Bandpass IF amplifiers are also used in radio trans- mitters to limit the occupied bandwidth of the transmitted signal. Certain modulation methods create a very broad frequency spectrum, which can interfere with adjacent channels. Regulatory agencies, such as the FCC (Federal Communications Commission), require that these out-of- band signals be reduced below a certain permissible level, so they must undergo processing through a bandwidth- limiting filter and amplifier at the transmitter. For each application there are certain design restric- tions or rules that must be followed to achieve optimum results. 1. GENERAL IF AMPLIFIER FUNCTIONS AND RESTRICTIONS The five basic IFamplifier functions and requirements are as follows: 1. Image Rejection. The mixer stages in a superhet- erodyne receiver can convert any frequency below or above the local oscillator frequency to an intermediate frequency. Only one of these frequencies is desired. The intermediate frequency must be chosen so that undesir- able frequencies or images are removed by the RF ampli- fier filter (prefiltering) and are rejected by the mixer. This may mean that two or three different intermediate fre- quencies must be used within the same receiver. The in- termediate frequencies in common use range from 0 Hz to approximately 2.0GHz. 2. Selectivity. Selectivity is required to reject as much as possible of any adjacent channel interfering signal. Generally this means attempting to obtain a bandpass filter characteristic as close to that of the ideal filter as possible that will pass the necessary Nyquist bandwidth (the baseband bandwidth from 0Hz to the highest frequency to be passed) without introducing harmful amplitude or phase distortion. 3. Gain. Gain is required to amplify a weak signal to a useful level for the decisionmaking circuit. This gain must be provided by means of a stable amplifier that introduces a minimum of noise, so as not to degrade the receiver noise figure. All circuit input and output impedances should be properly matched for optimum power transfer and circuit stability. 4. Automatic Gain Control. The amplifier gain must vary automatically with signal strength so that the deci- sionmaking circuit receives a signal of as nearly constant level as possible. The stages of the IFamplifier must not be overdriven, or go into limiting, until after the last band- pass filter, to prevent ‘‘splattering,’’ or broadening and dis- tortion of the signal. 5. Linearity. The amplifier should be linear in phase or amplitude to prevent distortion of the recovered informa- tion. AM receivers should be linear in amplitude, while FM or PM receivers should be linear in phase. Some mod- ulation methods can tolerate more linearity distortion than others. 2. SELECTING THE INTERMEDIATE FREQUENCY Image rejection and signal selectivity are the primary rea- sons for selecting an intermediate frequency. Most cur- rently manufactured bandpass filters of the crystal, or resonator type, have become standardized so that the de- signer can obtain off-the-shelf components at reasonable cost for these standard frequencies. The standard AM broadcast receiver utilizes a 455-MHz IF filter because extensive experience has shown that this will reject all but the strongest images. Assume the desired signal is at 600 kHz. A local oscillator operating at 1055kHz will have an image frequency at 1510 kHz, which the RF in- put filter can easily reject. Similarly, an FM receiver op- erating at 90.1MHz with an intermediate frequency of 10.7 MHz will have an image at 111.5 MHz, which will be rejected by the RF amplifier. In both of these cases, a sin- gle intermediate frequency can be used. A receiver operating at 450 MHz will require two In- termediate frequencies obtained by using first and second mixers, as in Fig. 16. The first IFamplifier may consist of a relatively broadband filter operating at 10.7 or 21.4 MHz, followed by a second converter and IF stage operating at 455 kHz. The first IF filter is narrow enough to reject any 455-kHz images, and the second IF filter is a narrowband filter that passes only the desired signal bandwidth. If the 455-kHz filter had been used as the first IF filter, the 450-MHz RF filter, which is relatively broad, would not have eliminated the image frequency, which is 455kHz above or below the local oscillator (LO) frequency. Television receivers use a video intermediate frequency of approximately 45 MHz, since this permits a relatively broad RF filter to pass the broadband TV signal, while still INTERMEDIATE-FREQUENCY AMPLIFIERS 2175 Previous Page rejecting the images. The video signal from the IF ampli- fier is AM, with an FM sound carrier riding on it. Televi- sion sound is generally obtained from a beat, or difference frequency between the video and sound carriers, which is at 4.5MHz. Satellite receivers use a broadband first intermediate frequency covering a frequency block from 900 MHz to 2.1 GHz. This is done by means of a low-noise block (LNB) converter. The second mixer is made tunable so that any frequency in the block can be converted to the second in- termediate frequency, which is usually fixed at 70 or 140MHz. The second intermediate frequency, which drives the detector, has a narrower bandwidth to reduce noise and reject adjacent channel interference. Crystal, ceramic resonator, and SAW filters are massed produced at relatively low cost for the frequencies men- tioned above, so that most consumer products employ one or more of the abovementioned standard frequencies and standard mass-produced filters. 3. SELECTIVITY Carson’s rule, and the Nyquist sampling theorem on which it is based, state that a certain bandwidth is re- quired to transmit a signal undistorted. The necessary bandwidth for an AM signal is given as follows: BW¼2f m ð1Þ Thus an AM broadcast receiver will require 10 kHz of bandwidth to pass a 5kHz ¼f m audio tone. ( f m ¼Fre- quency of modulation.) In data transmission systems, the frequency f m corresponding to the data rate f b , is f m ¼ 1 2 f b . The data clock frequency is twice the frequency of the data in ones and zeros. This means that a baud rate f b of 9,600 bits per second (bps) will require a bandwidth of 9.6kHz. For FM, the necessary bandwidth required for trans- mission is BW¼2ðf m þDf Þ ð2Þ A 15-kHz audio tone ( ¼f m ) and an FM transmitter being deviated with a modulation index of 5 will require 2 (15 þ (15 Â5)) ¼180kHz of bandwidth. Df is (5 Â15) and f m is 15 kHz. Narrowband FM, or phase modulation (PM) (with a modulation index of o0.7), is somewhat different in that the bandwidth actually required is the same as that for AM. This is due to the fact the higher J n Bessel products are missing [Eq. (1) applies]. These values are for double-sideband transmission. Single-sideband transmission will require half as much bandwidth. The required baseband bandwidth is the same as the value for f m . This is also known as the Nyquist bandwidth, or the minimum bandwidth that can carry the signal undistorted at baseband. Ideally, the IF filter, or the equivalent baseband filter, need pass only this bandwidth and no more. This requires the use of an ‘‘ideal’’ bandpass or lowpass filter, which does not exist in practice, but can be approached by various means. The filter must be as narrow as conditions permit to reduce the noise bandwidth and any adjacent channel interference, since noise power rises linearly with increas- ing filter bandwidth [14]: S o N o ¼b 2 bit rate filter BW S i N i ð3aÞ S o N o ¼modulation gain loss Âprocessing gain S i N i ð3bÞ These two equations show a generalized relationship be- tween the signal-to-noise ratio (SNR) at the receiver input and the SNR at the receiver output. The term b 2 repre- sents a gain, or loss, in power due to the modulation meth- od. In FM or PM it is the modulation angle. The term [(bit rate)/(filter bandwidth)] is generally known as processing gain. Narrowing the bandwidth improves the S o /N o ratio, but this improvement is not always available, depending on the modulation method. The Nyquist bandwidth rules state that it should be (symbol rate)/BW¼1. Pulse modulation, as in radar (radio detection and ranging), generally requires a much broader filter band- width than the other modulation methods. A condition called envelope delay or group delay must also be ob- served. This is discussed later along with the transfer functions of the filters. For optimum results, the filter bandwidth (Df) must be equal to [1/(pulsewidth)]. If the filter bandwidth is too narrow, the amplitude detected is reduced and the SNR is adversely affected. In this case, the processing gain is ideally ¼1 [14]. S o N o ¼ðprocessing gainÞ S i N i ¼ E b N o ð4Þ 4. GAIN The IF amplifier must provide sufficient gain to raise a weak signal at the RF input to the level required, or de- sired, by the decisionmaking circuit or detector. This re- ceiver gain can vary from 0 up to 130 dB, most of which is usually provided by the IFamplifier. The RFamplifier and mixer circuits preceding the IF amplifier usually provide Z20 dB of gain so that the IF amplifier generally contrib- utes little to the receiver noise figure. (See NOISE FIGURE article elsewhere in this encyclopedia.) Amplifiers with very high gain have a tendency to oscillate; hence two different intermediate frequencies may be used to reduce the gain on any one frequency, or more of the gain may be obtained from the RF section. Gain is provided by an amplifying device, such as a transistor, or vacuum tube (in older equipment). These devices have input and output impedances of a complex nature that must be matched to the filtering circuits for best power transfer, stability, and lowest noise. Current practice is often to use a ‘‘gain stage,’’ which consists of multiple amplifying devices in an integrated circuit 2176 INTERMEDIATE-FREQUENCY AMPLIFIERS package. These packages often contain the mixer stages and detectors as well. 5. AUTOMATIC GAIN CONTROL Receivers must respond to a wide range of input levels while maintaining a nearly constant level at the detector or decisionmaking circuit. The user or operator does not wish to manually adjust the gain to obtain a constant sound or picture level when changing stations. This func- tion is performed by detecting the output level of the IF amplifier and correcting it by means of a feedback circuit that adjusts the gain to keep the level as constant as pos- sible. Since this detected level can vary rapidly, it is passed through a lowpass filter [usually an RC (resis- tance Âcapacitance) pair] to integrate or slow down the changes, then amplified by a DC (direct-current) amplifier and applied to an IF amplifier circuit or gain stage that has variable gain characteristics. Some receivers, such as those used in an automobile, require relatively rapid act- ing AGC circuits, while fixed receivers can use a much slower AGC time constant. Dual-gate field-effect transis- tors use the second gate to control the gain. Bipolar or single-gate field-effect transistors vary the gain by means of a bias voltage or current applied to the input terminal along with the signal. Special integrated circuit gain stag- es for IF amplification are available, such as the Motorola MC1350, which both amplify and provide a variable gain control function. 6. FILTERS FOR IF AMPLIFIERS Except for block conversions, which convert wide frequen- cy bandwidths, such as those used on satellite receivers, IF amplifiers in general use a narrow bandpass, or a low- pass filter, to limit the bandwidth to the Nyquist band- width. Block conversion, on the other hand, can use a highpass–lowpass filter pair, where the bandwidth to be passed lies between the high and low cutoff frequencies. The traditional bandpass filter requires one or more resonant elements. Although the actual resonator may be a coil and capacitor, ceramic resonator, or SAW filter, the principles are basically the same. Digital filters, which do not use resonators, have been employed more recently. These will be discussed later in brief. They are discussed in more detail elsewhere in this encyclopedia. The inductance/capacitance resonator was the first used, and is still a comparison standard. Figures 1a and 1b show series resonant circuits, and Fig. 1b shows a par- allel resonant circuit. These circuits will pass a signal at the resonant peak and reject a signal off resonance. Re- sistances R s and R p are naturally occurring losses that reduce the circuit efficiency. Figure 2 shows the universal resonance curve, which is applicable to both series and parallel resonant circuits. It is important to note that the signal rejection never goes to a zero level in the area of interest, but reaches an asymptotic value between 0.1 and 0.2 or about À17 dB. If it is necessary to reject a signal on the shoulders of this curve by 60 dB, then four cascaded stages of this filter must be used to obtain the necessary rejection. Note also that there is a nonlinear phase shift that reaches a maximum in the area of interest, then changes to 7701. When stages are cascaded, this phase shift is multiplied by the number of stages. A nonlinear phase shift can cause distortion in FM receivers. The phase shift curve plotted is for a parallel resonant circuit. The phase reverses for a series circuit. The phase at any point on the curve is obtained by plotting horizontally from the vertical amplitude/phase scale: a ¼Q (cycles off resonance/resonant frequency). A frequency f 0 at which the response of a parallel res- onant LC filter is a maximum, that is, the point at which the parallel impedance is a maximum, is defined as a pole. A frequency at which the impedance is a minimum, as in the series LC circuit, is defined as a zero. Thus the as- sumed four cascaded stages above would constitute a four- pole filter, since it contains four resonant poles. The fre- quency of resonance is given by Eq. (5); this is the fre- quency at which [X c ¼1/ ÀjoC] and [X L ¼joL] are equal: f 0 ¼ 1 2pðLCÞ 1=2 ð5Þ The bandwidth that an analog LC filter can pass is altered by the circuit efficiency, or circuit Q, given in Eqs. (6). C R p R s R p C C L L L (a) (b) (c) Figure 1. Series (a,b) and parallel (c) resonant circuits. 1.0 0.6 –3dB –6dB Phase lag Amplitude Phase lead 2.5 2.0 1.5 1.0 .5 .5 1.0 1.5 2.0 2.5 α +75° +50° 45° 60° 0° 25° 50° 75° 0.4 0.3 0.2 0.1 0.8 0.9 Figure 2. Universal resonance curve (BT¼bandwidthÂbit period). INTERMEDIATE-FREQUENCY AMPLIFIERS 2177 Generally the bandwidth is specified as the bandwidth between the À3dB points, where the phase shift is 7451. Q¼ X c R s for a series circuit ð6aÞ Q¼ R p X c for aparallel circuit ð6bÞ Q¼ f 0 3dBBW ð6cÞ For simplicity in analyzing the following circuits, the Q determining R will be assumed to be a parallel resistance R p across the inductance. Figure 3 shows a typical IF amplifier stage as used in earlier transistor radios [1,2]. In this circuit R p (the total shunting resistive load) is actually three resistances in parallel; one is the equivalent R p of the coil itself (repre- senting the coil losses), another is the input resistance of the following stage, as reflected, and the third is the out- put resistance of the driving transistor as reflected. It cannot be assumed that the resulting coil Q, and hence the selectivity of the circuit, is that of the unloaded coil and capacitor alone. Dual-gate field effect transistors have the highest shunting resistance values, bipolar transistors the lowest. The gain can be varied by increasing or decreasing the bias voltage V b applied to the input terminal. Manufacturers of amplifying devices often provide the impedances, or admittances of their products on their data sheets. Formerly this was done in the form of h parame- ters. The more common practice today is to provide the information in the form of S parameters. These values can be converted to impedances and admittances, but the manual process is rather complicated. An easier method is to use the various software programs (see Available Software section at end of this article) to make the con- version. Matrix algebra, h and S parameters are discussed elsewhere in this encyclopedia and also in the Refs. 3 and 4 in this article. Unfortunately, S parameters for bandpass filters are rarely available. Figure 4a shows the equivalent circuit of the transistor as the tuned LC ‘‘sees’’ it. The transistor amplifies a cur- rent, which is passed through a relatively low driving re- sistance R s , to the outside. At the same time, the attached LC sees an equivalent shunting resistance R c and capac- itance C c , which must be added in parallel to R p , L, and C. The input to the following stage, assumed to be an identical transistor, will have a relatively low shunting resistance R i , and capacitance C i , which must be added. Unless the added capacitances are large compared to the resonant C, they merely add to it without greatly detuning the circuit. When tuned, the total C plus L will determine the frequency and the resulting total R 0 p will determine the Q of the LC circuit, and hence the bandwidth. Thus the complex components can be tuned out and the remain- ing design problem consists of matching the real or resis- tive part of the input and output impedances to the best advantage. The desired end result is to couple the output of the driving stage to the input of the following stage with the least loss by matching the differing impedances. An addi- tional desired result is to narrow the band of frequencies passed by means of a filter. These objectives are accom- plished by transforming the input and output impedances to a higher or lower shunting impedance that maintains the desired bandpass characteristic of the filter. A low driving or load impedance can be stepped up to become a very high impedance, which maintains the circuit Q at the desired value. Impedance matching enables the designer to change the actual impedance to a different apparent value, which V cc V b Input Output Figure 3. Typical IF amplifier stage. R c Rp Xc XC′ i XC′ c XL Rc′ Ri′ R s R i C c C 12 C i (a) (b) Figure 4. Equivalent circuit of transistor as seen by tuned LC. 2178 INTERMEDIATE-FREQUENCY AMPLIFIERS is optimum for the circuit. Figure 5 shows how impedanc- es are matched by transformer action. A transformer with a 3 : 1 turns ratio is shown as an example. The output im- pedance relative to the input impedance is given by Eq. (7), where N i and N o are the input and output numbers of turns on the winding. Z i Z o ¼ ffiffiffiffiffiffi N i N o ¸ ð7Þ Thus 90 O at the input is seen as 10 O at the output with a 3 : 1 stepdown turns ratio. The automatic transformer (tapped coil in Fig. 5) has the same relationship. When all the reactances and resistances from the tuned circuit and the transistor input and output as modified by the stepup/stepdown process of the impedance-matching networks are added, the network in Fig. 4b results. Cal- culation of the resonant frequency and circuit Q from these reactances and resistances in parallel is complicated unless they are converted to admittances. Software is available at reasonable cost to perform these calculations (see Available Software section at end of this article). Stock, or mass-produced IF transformers, which are used to provide bandpass filtering as well as impedance matching, seldom have the desired turns ratio to match the impedances properly. An additional Z-matched circuit using capacitors enables the available transformers to match almost any impedance while preserving the circuit Q. This capacitor divider circuit is often used instead of a tapped coil or transformer as shown in Fig. 6. The formulas used to calculate the matching conditions using capacitors are more complex than those used for transformer coupling, since there are more variables. In this circuit R i is assumed to be lower than R p . Although R p is the equivalent parallel resistance of the LC circuit in Fig. 6, it could also be the reduced resistance or reflected R p2 at a transformer tap. N in these equations is equal to the loaded resonator Q, or to a lower arbitrary value if total shunting R p is lowered by transformer action as in Fig. 6, or if the component ratios become unwieldy [12]: X C2 ¼ R i R i ðN 2 þ1Þ R p À1 _ _ 1=2 ð8Þ X C1 ¼ R p N N 2 þ1 1 À R i NX C2 _ _ ð9Þ X C2 % ffiffiffiffiffiffiffiffiffiffiffi R i R p Q ¸ ð10Þ X C1 % R p Q ¼X L ð11Þ Equations (8) and (9) calculate the reactances of the two capacitors. Note that NX L is the same as QX L . Starting with a value of N¼Q, find X C1 ; then X C2 : If N is large in Eq. (8), the equations reduce to the approximate values in Eqs. (10) and (11). Unless Q is less than 10, these approximate equations are accurate enough for general use. As an example, let R i ¼100 O and R p ¼10,000O with Q¼100. Then, using Eq. (11), X C2 be- comes 10 O and X C1 becomes 100O. C 2 is approximately 10 times larger than C 1 . Note the similarity of this ratio to Eq. (7). If a transformer is involved, N becomes much smaller and the full formulas (8) and (9) should be used. Equations (8)–(10) apply for R i oR p and N4(R p / R i À1) 1/2 . 7. DOUBLE-TUNED CIRCUITS When two identical LC circuits are coupled together as shown in Fig. 7, a number of responses are possible as shown in Fig. 8. The amplitude response depends on the coupling coefficient K. Undercoupling results in a two-pole filter with the sharpest selectivity. Critical coupling re- sults in the narrowest bandwidth with the highest gain. Transitional coupling is slightly greater than critical cou- pling and results in a flat-topped response with a wider 90 ohms 10 ohms 10 ohms Figure 5. Impedance matching by transformer action. R p R i C 2 C 1 Figure 6. Lowering of total shunting by transformer action. C c C c M C 1 C 1 C 1 (a) (b) (c) Figure 7. Coupling of identical LC circuits. INTERMEDIATE-FREQUENCY AMPLIFIERS 2179 bandwidth. Overcoupling results in a double-humped re- sponse with sharper skirts and broad bandwidth. The cou- pling coefficient can be calculated using Eqs. (12). Equation (12a) applies to mutual inductive coupling and (12b)–(12d), to capacitive coupling. K ¼ M ffiffiffiffiffiffiffiffiffiffiffi L 1 L 2 p ð12aÞ K c ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffi Q 1 Q 2 _ ð12bÞ K ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C c C c þC 1 ¸ ð12cÞ K ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 1 C c þC 1 ¸ ð12dÞ Equation (12a) calculates the coupling coefficient for two identical LC tuned circuits that are coupled together by leakage inductance (Fig. 7a), often obtained by using shielded coils with holes in the sides of the shield cans to allow the magnetic fields to interact. The size of the hole determines the value of the mutual inductance M. Since this is difficult to control, a coupling capacitor is often used as shown in Figs. 7b and 7c. The critical coupling value is given by Eq. (12b). The coupling coefficients for Figs. 7b and 7c are given in Eqs. (12c) and (12d). The amplitude response curves in Fig. 8 do not yield any information as to the phase shifts that take place through the filter. In AM circuits, phase is generally of little concern, with most attention paid to the amplitude ripple and linearity. In FM circuits, nonlinear phase shift or a related term, differential group delay, becomes more of a problem and efforts are made to keep the phase shift as linear as possible. In data transmission circuits using phase modulation, or amplitude modulation, any nonlin- earity must be avoided. For these reasons, the coupling coefficients are carefully adjusted and cascaded IF ampli- fier stages are used to get the desired transfer function for the IF amplifier. 8. CASCADING IF AMPLIFIER STAGES AND FILTERS All filtering actions that take place between the RF input of the receiver and the decisionmaking circuit are parts of the IF amplifier bandpass filter. Since the final decision- making circuit is at baseband, or 0 Hz, all filtering prior to the decisionmaking circuit is part of the IF bandpass fil- tering, which should be treated as a whole. A single LC circuit seldom has the desired bandpass characteristic for an IF amplifier. Cascading IF amplifier stages with differing coupling and Q values enables the designer to obtain the desired transfer response. One com- bination of LC filters uses an overcoupled double-tuned stage followed by a single-tuned stage with a lower Q. The result is a three-pole filter with relatively steep skirt slopes. Cascading these stages results in filters with re- sponses resembling Butterworth, Chebyshev, elliptical, or equal-ripple filters, which are noted for their rejection of adjacent channel interference (see Figs. 9 and 10). When additional filtering is required at baseband, simple RC filters, lowpass LC filters, or digital finite impulse response (FIR) filters are used. These and other filters are discussed in greater detail elsewhere in this encyclopedia. 9. CRYSTAL AND CERAMIC FILTERS Figure 10a shows the equivalent circuit of a crystal or a ceramic resonator. These devices have both a pole and a (a) (b) Figure 9. Curves resulting from cascad- ing IF amplifier stages. A B C D Figure 8. Results of LC circuit coupling: critical (curve A); tran- sitional (curve B); overcoupled (curve C); undercoupled (curve D). 2180 INTERMEDIATE-FREQUENCY AMPLIFIERS zero that are located relatively close to each other in fre- quency. Quartz crystals have Q values ranging from 2000 to 10,000 depending on the mechanical loading of the crystal. Ceramic resonators usually have Q values be- tween 100 and 2000. The higher the Q, the narrower the filter bandpass. When two of these devices are connected as shown in Fig. 10b, the result is a bandpass filter with steep skirts as shown in Figure 11. These resonators are used in pairs to create a two-pole filter, which can then be combined in a single container with other pairs, to create a filter with as many as eight or more poles. They usually have excellent adjacent-channel rejection characteristics. When using these devices, care must be taken to care- fully match the specified impedance. Any impedance mis- match can seriously alter the response curve of the filter. The impedance-matching techniques discussed previously will enable the designer to obtain a very close match, which will optimize the circuit performance. Typical input and output impedances range from 50 to 4000O. Crystal filter manufacturers often build in transformer or other tuned matching circuits so that the user does not need to provide a matching circuit outside the crystal filter. SAW (surface acoustic wave) filters utilize a crystal os- cillating in a longitudinal mode with many fingers or taps placed along the surface. They can be made with very broad bandpass characteristics, which makes them well suited for TV IF amplifiers, spread-spectrum IF filters, and other uses requiring a wide RF bandwidth. They have losses, which are typically about 8–20 dB, so they must have amplifiers with adequate gain ahead of them if the receiver noise figure is not to be degraded. They are not suitable for use in ultranarrowband or low-frequency ap- plications. The group delay quoted in the specifications is usually the differential group delay and not the actual group delay, which is much higher. 10. BASEBAND IF FILTERING IF bandpass filters with specific response characteristics are sometimes very difficult to obtain, whereas the desired characteristic is easily and inexpensively obtainable at baseband. This concept is often applied to transmitters where a sharp-baseband-cutoff filter can be obtained using simple components, such as the switched filter. An 8-pole equivalent at baseband becomes a 16-pole filter at the modulation intermediate frequency. For example, a sharp- cutoff filter for voice with a 4-kHz audio cutoff results in a bandpass filter 8kHz wide at RF after modulation, with the same sharp cutoff. The same cutoff characteristics at RF would be almost impossible to obtain in a crystal filter, which would also be very costly and beyond the manufac- turing budget for a low-cost transmitter such as a cordless telephone. By using baseband filtering, a poor-quality RF filter that only rejects the opposite image can be used. Similarly, a wideband, or poor-quality IF filter, can be used ahead of a detector, if the undesired signal components can be filtered off after detection at baseband, by using a sharp-cutoff filter. Switched-capacitor filters are available as packaged in- tegrated circuits that can be used at baseband and some lower intermediate frequencies. They have internal oper- ational amplifiers with a switched feedback capacitor, the combinations of which determine the filter characteristics. Since they are dependent on the speed of the operational amplifiers and the values of the feedback capacitors, they seldom function much above 100kHz. They can be config- ured as Bessel, equal-ripple, and Butterworth filters. Typ- ical of this type of filter are the LTC1060 family manufactured by Linear Technology Corporation (a) and the MAX274 from Maxim (b) [see items (a) and (b) in Available Software list at end of this article]. 1 As Bessel filters they perform well out to about 0.7 times the cutoff bandwidth, after which the phase changes rapidly and the Bessel characteristic is lost. Digital signal processing (DSP) at baseband is widely used to reduce the component count and size of the base- band filters in very small radio receivers, such as cordless and cellular telephones. Almost any desired filter response can be obtained from DSP and FIR filters without using inductors and capacitors, which would require factory tuning (c,d). Separate FIR filters have a flat group delay response and are the best choice for FM or PM filtering, or filters at baseband. Commercially available software design Figure 11. Steep-skirted bandpass filter. (a) (b) Series Parallel Figure 10. Equivalent circuit of a crystal or ceramic reasonator. 1 In the remainder of this article, all lowercase letters in paren- theses refer to entries in the Available Software list following the Bibliography. Numbers in brackets refer to Bibliography entries (references) as usual. INTERMEDIATE-FREQUENCY AMPLIFIERS 2181 packages permit the design of trial circuits to investigate phase shift and group delay (e,f). Unfortunately, digital filtering of any type is frequency- limited. The filter must use a sampling rate that is much higher than the frequency to be passed. To use a digital filter, such as a FIR filter, or DSP as a bandpass filter at 10.7 MHz, requires an analog-to-digital converter (ADC) operating at 160 MHz or higher. Filtering at baseband means the sampling rate can be much lower. 11. AMPLIFYING DEVICES FOR IF AMPLIFIERS Transistors in one form or another have become the stan- dard for IF amplifiers. The single bipolar or field-effect transistor used as an individual component, was formerly the preferred device. For very-high-Q circuits, the dual- gate FET performs best, since it is the most stable and offers the lowest shunt resistance. Single-gate FET devic- es often have too much drain to gate capacitance for good stability. Modern bipolar transistors usually have good stability, but lower shunt resistances than dual-gate FETs. Stability is discussed later in this section. Monolithic amplifiers [MMIC (monolithic microwave integrated circuit) devices] are stable and have good gain, but the shunt load impedance is too low for most bandpass filters other than a crystal or SAW filter matched to 50 O. The most recent practice for IF amplifiers is to use in- tegrated circuit blocks containing more than one transis- tor in a gain stage. These are then packaged together in an integrated circuit with other circuit components to form an almost complete radio. Integrated circuits of this type are shown below. 12. TYPICAL CONSUMER IF AMPLIFIERS Consumer radio and TV equipment is mass-produced for the lowest possible cost consistent with reasonable quality. Manufacturers of integrated circuits now produce single- chip IF amplifiers that can be combined with mass-pro- duced stock filters to produce a uniform product with a minimum of adjustment and tuning on the assembly line. In the examples that follow, some circuit components in- side and outside the IC have been omitted to emphasize the IF amplifier sections. Figure 12 shows a single-chip AM receiver that uses the Philips TDA1072 [7] integrated circuit and ceramic IF filters at 455kHz. The input impedance of the ceramic fil- ter is too low to match the output impedance of the mixer, so a tuned matching transformer is used to both reduce the passed bandwidth (prefilter) and match the imped- ances. The input impedance of the IF amplifier was de- signed to match the average impedance of the ceramic filters available. This integrated circuit has a built in au- tomatic gain control that keeps the received audio output level relatively constant at 250mV as long as the input signal level to the chip exceeds 30 mV. Figure 13 shows a single-chip FM radio based on the Phillips NE605 integrated circuit (g) that uses ceramic IF filters at 10.7 MHz. The input and output impedance of the IF amplifier sections is approximately 1500 O, to match the ceramic filter impedance, so no matching transformer is required. The audio output is maintained level at 175 mV for all signal levels at the input level from À110 to 0 dBm. An automatic frequency control (AFC) voltage can be obtained from the quadrature detector output. This circuit can also be used for narrow-angle phase modula- tion if a crystal discriminator is used for a phase reference at the quadrature input. AGC is available from all FM integrated circuits so that the gain of the mixer and RF stages can be controlled at a level that does not allow these stages to be saturated by a strong incoming signal. Saturation, or nonlinearity before filtering, results in undesirable signal spreading. The NE605 has a ‘‘received-signal strength indicator’’ (RSSI) output that can be amplified and inverted if necessary to provide an AGC voltage, or current, for the RF amplifier and mixer. Figure 14 shows a TV IF amplifier using the Motorola MC44301/2 video IF integrated circuit (h) with a SAW filter at 45 MHz. The SAW filter bandpass is made V cc V cc z Match Ceramic filter RF Input Oscillator LC Ac output AF det. Mixer AGC det./amp IF amp TDA 1072 AGC Figure 12. Layout of a single-chip AM receiver. Audio out RSSI/AGC Oscillator LC RF input IF LIM Filter Filter NE605 Mixer RSSI level det. Quad det. Figure 13. Configuration of a single-chip FM radio. 2182 INTERMEDIATE-FREQUENCY AMPLIFIERS approximately 6 MHz wide to pass the video and sound. The circuit has both automatic frequency control (AFC) and automatic gain control (AGC) features built in. Unlike the IF amplifiers used for AM and FM audio broadcast applications, the TV IF amplifier includes a phase-locked loop (PLL) and synchronous detector that locks the fre- quency of an internal oscillator to the intermediate fre- quency. This locked, or synchronous, oscillator output is then mixed with the information-bearing portion of the signal to create a baseband signal. The system shown in Fig. 14 is one of a family of 0-Hz IF amplifiers, are becoming more popular in wireless de- signs, since they permit most or additional signal process- ing at baseband. In Fig. 14, the video and sound carriers are both passed by the SAW filter. They beat together at 4.5 MHz in the detector, providing a second IF stage with the sound information. This 4.5-MHz IF information is then filtered by a ceramic bandpass filter approximately 50 kHz wide to remove any video components, limited and detected as a standard FM signal to provide the TV sound. The video portion, consisting of signals from 15 kHz to approximately 4.25 MHz, is then further processed to separate the color information at 3.58 MHz from the black-and-white information. The video output level is de- tected to provide the AGC voltage. The phase-locked oscillator, operating at the interme- diate frequency, can also be used to provide automatic fre- quency control to the first mixer stage local oscillator. Figure 15 shows a dual-conversion receiver in a single integrated circuit for communications use, utilizing the Motorola MC13135 integrated circuit (h). When the receiver is operated at 450 or 850MHz, as was men- tioned above, single-conversion IF stages do not offer the necessary image rejection. This receiver is for narrow- band FM as opposed to wideband FM for entertain- ment purposes. The first IF filter is a low-cost ceramic filter at 10.7 MHz. The second filter is a multipole crystal or ceramic filter with a bandpass just wide enough to pass the signal with a small FM deviation ratio. Receivers of this type can be used with 12.5 and 25 kHz of chan- nel separation for voice-quality audio. Analog cellular telephones, aircraft, marine, police, and taxicab radios are typical examples. 13. DIRECT CONVERSION AND OSCILLATING FILTERS Direct conversion converts the RF frequency directly to baseband by using a local oscillator at the RF carrier fre- quency. The TV IFamplifier with the detector circuit given in Fig. 14 illustrates some of the reasons. Conversion to baseband can occur at the intermediate frequency or di- rectly from the RF frequency. There is a noticeable trend in integrated circuit design to utilize synchronous detection [5] with the carrier restored by means of a phase-locked loop, as shown in Fig. 14, or by means of regenerative IF amplifiers [6], to accomplish several desirable features that cannot be obtained from the classical circuits with square-law detectors. In the case of direct RF-to-baseband conversion, there is no IF stage in the usual sense, and all filtering occurs at baseband. For this reason direct-conversion receivers are referred to as zero-hertz (0-Hz) IF radios. Integrated cir- cuits for direct RF conversion are available that operate well above 2.1 GHz at the RF input. The Maxim 2820 (b) and the AMD1771 (i) are examples. DSP and FIR filters are the preferred lowpass filters at baseband, where they are referred to as ‘‘windows’’. It was discovered in the 1940s that the performance of a TV receiver could be improved by using a reconstructed synchronous or exalted carrier, as occurs in the TV IF amplifier depicted in Fig. 14. The carrier is reduced by vestigial sideband filtering at the transmitter and con- tains undesirable AM and PM signal components. By causing an oscillator to be locked to, or to be synchronized with the carrier, and then to be used by the detector, a significant improvement in the received signal can be achieved. Prior to using circuits of this type, the intercar- rier sound at 4.5MHz in earlier TV sets had a character- istic 60 Hz buzz due to the AM and PM on the carrier. By substituting the recovered synchronous carrier instead, this buzz was removed. Figure 14 illustrates an example. The earliest direct-conversion receivers using locked oscillators or synchronous detectors were built in the Saw FL Video det. Video out Audio out Phase shifter Phase det. Sound IF 4.5 MHz cer. fil 4.5 MHz det. VCO AGC amp IF amp Limiter IF input AFC/AFT out I Vcc Vcc Quadrature L-C VCO L-C Q Figure 14. Layout of a television IF amplifier. 10.7 MHz cer. fil 455 kHz cer. fil IF amp Limiter 1st mix 2nd mix RF input RSSI Quad det. MC13136 Quadrature L-C Vcc 1st LO. L-C 2nd LO. L-C RSSI out Audio out Figure 15. Configuration of a dual-conversion receiver in a sin- gle IC. INTERMEDIATE-FREQUENCY AMPLIFIERS 2183 1920s, when they were known as synchrodyne or homo- dyne receivers. The theory is relatively simple. A signal from the RF amplifier is coupled to an oscillator, causing a beat or difference frequency. As the frequencies of the two sources come closer together, the oscillator is pulled to match the incoming signal and locks to it. The lock range depends on the strength of the incoming signal. The two signals are then mixed to provide a signal at baseband, which can be further filtered by means of a lowpass filter. In this way, a relatively broad RF filter can be used, while the resulting AM signal bandwidth after detection and baseband filtering can be very narrow. The Q of the oscil- lator tank circuit rises dramatically with oscillation so that Q values of 6000–10,000 are not unusual and selec- tivity is greatly improved. AGC can be obtained from the audio signal to maintain an input signal level that is con- stant to ensure a good lock range. An undesirable charac- teristic is the whistle or squeal that occurs between stations. Later receivers used a squelch circuit to make the signal audible only after locking has occurred. High- quality receivers for entertainment and communications use were produced in the 1990s using this principle. They offer higher sensitivity, better fidelity, and more controlled response. Integrated circuits for receivers of this type (di- rect conversion) are now being produced for paging, wi-fi (wireless fidelity), direct-broadcast TV, and cellular and cordless telephones. The Maxim 2820 (b) and the AMD 1771 (i) are examples. Oscillating filters and phase-locked loops are similar in principle. An intermediate frequency is applied to a phase/ frequency detector that compares the intermediate fre- quency with the oscillator frequency. An error voltage is created that changes the oscillator frequency to match, or become coherent with, that of the incoming IF carrier fre- quency. In some cases the phase-locked loop signal is 901 out of phase with the carrier, so a phase shifter is used to restore the phase and make the signal from the oscillator coherent in phase with the incoming signal. (See Figs. 14 and 18, where phase-locked loops and phase shifters are employed.) Synchronous oscillators and phase-locked loops not only extend the lower signal-to-noise ratio but also have a bandwidth filtering effect. The noise bandwidth of the PLL filter is the loop bandwidth, while the actual signal filter bandwidth is the lock range of the PLL, which is much greater. Figure 16 shows the amplitude and linear phase response of a synchronous oscillator. The PLL is not always the optimum circuit for this use because its fre- quency/phase-tracking response is that of the loop filter. The locked oscillator [6] performs much better than the PLL since it has a loop bandwidth equal to the lock range without sacrificing noise bandwidth, although with some phase distortion. Some authors hold that the synchronous oscillator and locked oscillator are variations of the PLL in which the phase detection occurs in the nonlinear region of the oscillating device and the voltage-controlled oscil- lator (VCO) frequency change characteristic comes from the biasing of the oscillator. Both the PLL and the locked oscillator can introduce phase distortion in the detected signal if the feedback loop is nonlinear. A later circuit shown in Fig. 17 has two feedback loops and is considered to be nearly free of phase distortion [5]. This circuit has the amplitude/phase res- ponse given in Fig. 16. Phase-locked loops have been used for many years for FM filtering, amplification, and detection. They are in common use with satellite communications links for audio and video reception. A 74HC4046 phase-locked loop inte- grated circuit operating at 10.7 MHz (the FM intermediate frequency) can be used to make an FM receiver for broad- cast use [7]. The phase-locked loop extends the lower sig- nal-to-noise limit of the FM receiver by several decibels while simultaneously limiting bandwidth selectivity to the lock range of the PLL. The detected audio signal is taken from the loop filter. 14. AM STEREO (C-QUAM) AM stereo radio is another application of the phase-locked oscillator at the intermediate frequency. AM stereo radio is dependent on two programs being transmitted at the same time at the same frequency. They arrive at the re- ceiver detector circuitry through a common IF amplifier V cc L-C tank Input Level adjust Output Figure 17. Circuit with same amplitude and phase response as in Fig. 16 but with two feedback loops and markedly decreased phase distortion. Amplitude Noise BW Filter signal bandwidth Phase Figure 16. Amplitude and linear phase response of a synchro- nous oscillator. 2184 INTERMEDIATE-FREQUENCY AMPLIFIERS operating at 455kHz. The normal program heard by all listeners is the LþR program. The stereo information (LÀR) is transmitted at the same frequency, but in quad- rature phase to the LþR program. Quadrature, or or- thogonal transmission, is used because the orthogonal channels do not interfere with one another. Each program section requires a reference carrier, that is coherent with its own sideband data. The LþR program, which has a carrier, may use an ordinary square-law detector or a synchronous detector. This is the program heard over monaural radios. To obtain the LÀR program that is transmitted without a carrier, a phase-locked loop is used at the intermediate frequency to lock a voltage controlled oscillator to the carrier of the LþR program. This carrier is then shifted 901 in phase and becomes the reference carrier for the LþR segment. The output of the PLL has the proper phase for the LÀR detector, so no phase shifting is necessary. The LÀR detector is a coherent or synchronous detector that ignores the orthogonal LþR information. By adding, then inverting and adding, the left and right channels are separated. Figure 18 shows a simplified block diagram of the C-QUAM receiver. The Motorola MC1032X (h) series of integrated circuits are designed for AM stereo use. The MC10322 and MC10325 have most of the components required, includ- ing the IF amplifiers, for a complete AM stereo receiver in two integrated circuit packages. 15. SUBCARRIERS Subcarriers are used to carry two or more signals on the same carrier. They differ from the orthogonal signals used with C-QUAM in that they are carried as separate signals superimposed over the main carrier information, as in the video sound carrier shown in Fig. 14. In Fig. 14, a fre- quency-modulated subcarrier at 4.5MHz is carried on top of the main video signal information, which extends from 0 to 4.25 MHz. This is an example of an AM/FM subcar- rier. Nondigital satellites utilize a frequency-modulated video carrier with as many as 12 subcarriers at frequen- cies ranging from 4.5 to 8.0 MHz. Normal FM stereo broadcasting utilizes a FM/AM subcarrier at 38 kHz to carry the LÀR portion of the stereo program. FM stations frequently carry additional subcarriers at 67 and 92 kHz. These FM/FM subcarriers are used to carry background music, ethnic audio programs, and digital data. To detect a subcarrier, the signal is first reduced to baseband, then a bandpass filter is used that separates only the subcarrier frequencies. The subcarrier frequen- cies are then passed to a second detector, which must be of the type appropriate for the subcarrier modulation. This can be seen in Fig. 14, where a 4.5-MHz filter is used. This is followed by a limiter and quadrature detector, which is appropriate for the FM signal. In the case of a 67-kHz FM/ FM subcarrier, the filter is 15 kHz wide at 67 kHz. Detec- tion can be accomplished by a discriminator, quadrature detector, or PLL. 16. CELLULAR AND CORDLESS TELEPHONES Analog cellular telephones employ the circuits shown in Figs. 13 and 15. Digital telephones utilizing Gaussian minimum shift keying (GMSK) also use these circuits. Digital telephones using quadrature amplitude modula- tion (QAM) or phase shift keying (PSK) employ circuits similar to that used for C-QUAM with digital filtering and signal processing instead of audio filtering at base- band. The PLL used for digital receivers is a more complex circuit known as the Costas loop, which is neces- sary to restore a coherent carrier for digital data recovery. Some cellular phones are dual-mode; that is, they can transmit and receive analog voice or digital GMSK modulation using circuits similar to those shown in Figs. 13 and 15. 17. NEUTRALIZATION, FEEDBACK, AND AMPLIFIER STABILITY Earlier transistors and triode vacuum tubes had consid- erable capacitance between the output element (collector or plate) and the input side of the device (see Fig. 4). Feed- back due to this capacitance is multiplied by the gain of the stage so that enough signal from the output was often coupled back to the input to cause the stage to oscillate unintentionally, as opposed to the planned oscillation of the locked oscillator, synchronous oscillator, or PLL. To prevent this, feedback of an opposite phase was deliber- ately introduced to cancel the undesired feed back. A neu- tralized IF amplifier is shown in Fig. 19. Transistors and integrated circuits made since 1985 are rarely unstable and generally do not require neutralization unless seri- ously mismatched. A better solution than neutralization is usually to improve the matching of the components and the circuit layout. By carefully controlling the feedback, a regenerative IF amplifier can be constructed that operates on the verge of oscillation. This greatly increases the Q of the tuned circuit, thus narrowing the IF bandwidth. Circuits of this type were once used in communication receivers for L + R out L – R out Phase shifter Phase det. 455 kHz cer. fil I det. Q det. VCO AGC amp IF amp Limiter IF input AFC out I Vcc VCO L-C Q AGC Carrier MC10322 Figure 18. Simplified block diagram of the C-QUAM receiver. INTERMEDIATE-FREQUENCY AMPLIFIERS 2185 commercial and amateur use, where they were referred to as ‘‘Q multipliers.’’ The maximum stable gain (MSG) that can be achieved from a potentially unstable amplifier stage without neu- tralization is obtainable from the S parameters and can be calculated from Eq. (13). This equation assumes that the input and output impedances are matched and there is little or no scattering reflection at either the input or out- put. The stability factor K, usually given with the S pa- rameters, must be 41. A failure to match the impedances can result in an unstable amplifier, but does not necessar- ily do so. A higher gain can be obtained, but at the risk of instability. K ¼MSG¼ S 21 S 12 ð13Þ In addition to impedance mismatch, the most frequent cause of amplifier instability, or oscillation, is poor circuit- board layout or inadequate grounding and shielding, not the device parameters. The wiring, whether printed or handwired, forms inductive or capacitive coupling loops between the input and output terminals of the amplifying device. This is particularly noticeable when high-gain ICs such as the NE605 are used. These integrated circuits have IF gains of 4100dB and require very careful board layouts for best results. Undesirable feedback can greatly decrease the usable gain of the circuit. 18. SOFTWARE RADIO Digital radios, or radios based on digital signal processing (DSP), offer some technical advantages over their analog predecessors. Digital radios can be used not only for dig- ital modulation but also for AM and FM. One receiver can simultaneously detect both digital and analog modulation; thus they can be used for cellular telephones in environ- ments where multiple modulation standards are used. As a class, they belong to the 0-Hz intermediate-frequency group. The typical receiver consists of a conventional RF front end and a mixer stage that converts the signal to a lower frequency, as in the dual conversion radios discussed above (Fig. 16). The signal at this stage is broadband in nature, but not broadband enough to include the image frequencies. The signal is then fed to an analog-to-digital converter (ADC), which is sampled at several times f m . This converts the portion of interest of the signal to base- band (or 0 Hz) instead of a higher intermediate frequency. The actual filtering to remove unwanted interfering sig- nals then takes place at baseband, using digital filtering. Digital signal processing and decimation are covered else- where in this work. The ADC (c) performs the same func- tions as do the oscillating detectors shown above. Noise figure, amplification, and AGC considerations of the first IF amplifier are the same as those for a conven- tional receiver. The ADC and the DSP filters function best with a constant signal input level. The term ‘‘software radio’’ has been adopted because the tuning function is done in software by changing the sampling frequency at the ADC. The sampling frequency is obtained from a digitally controlled frequency synthe- sizer instead of tuned LC circuits. 19. SPREAD-SPECTRUM RADIOS The spread-spectrum receiver also uses a conventional front end with a wideband first IF stage. The same con- ditions apply as to software radios and dual-conversion receivers. The first IF stage must have the necessary bandwidth to accommodate the spread bandwidth, ampli- fy it with minimum added noise, and match the output to the despreading circuitry. Spread-spectrum technology is covered elsewhere in this encyclopedia. While usually as- sociated with digital reception, spread-spectrum technol- ogy can also be used for analog audio. 20. ORTHOGONAL FREQUENCY-DIVISION MULTIPLEXING (OFDM) AND CODED OFDM (COFDM) These modulation methods could be considered similar to spread-spectrum techniques, or to methods requiring dual conversion, in that they use a very broad spectrum as a first level, followed by a narrowband filter to extract an individual channel. SAW filters are generally used at RF, while second-stage processing can use digital filtering, as in the software radio, or be done at baseband. 21. TRANSFER FUNCTIONS The amplitude response, plotted relative to frequency of a filter, is usually given in terms of the transfer function H( f ). Some typical transfer functions are as follows. For the LC filter of Fig. 2 H LC ðf Þ ¼ expÀ Qot 2 ð14Þ The LaPlace transform equivalent is HðsÞ ¼ K s 2 þBs þo 2 0 ð15Þ V cc Figure 19. Configuration of a neutralized IF amplifier. 2186 INTERMEDIATE-FREQUENCY AMPLIFIERS A similar curve obtainable with digital filters is the Gauss- ian filter: H Gauss ðf Þ ¼ expÀ1:38 1 ðBTÞ 2 _ _ ð16Þ A generalized Nyquist IF filter bandpass spectrum is seen in Fig. 20. In Fig. 20 the centerline represents either the carrier frequency, or 0 Hz. The portion of the spectrum to the right of the centerline is the baseband response, while both ex- tremes represent the RF double-sideband response with the carrier at the center. The B region is the baseband re- sponse of an ‘‘ideal’’ filter, which does not exist in practice. Practical filters have a rolloff, or excess bandwidth, shown in a. Outside the desired bandpass, there is a ‘‘comeback’’ in region C. The ‘‘ideal’’ filter has no rolloff and no come- back. The region B is the required Nyquist bandwidth. Multilevel digital modulation methods such as quad- rature amplitude modulation (QAM) and multiple phase shift keying (MPSK) require filters that are free of ampli- tude and phase distortion within the Nyquist bandwidth, then having a rolloff a as abrupt as reasonably possible outside that distortion-free bandwidth. The optimum filter for this is considered to be the raised-cosine filter, so called because the region after the uniform response is half-cycle of a cosine wave squared (cosine raised to second power). The transfer function for the raised-cosine filter is as fol- lows. In the central bandpass region B, we obtain Hð f Þ ¼1 for jf j > ¼ Àf m ð1 ÀaÞ; or o¼f m ð1þaÞ ðor Æf m Þ ð17Þ When a ¼0, the filter is said to be the ‘‘ideal’’ filter. In the transition region a; since cos 2A¼cos 2 AÀ1 or 1 þcos 2A ¼cos 2 A¼0 elsewhere, we obtain the following forms of Eq. (17): 1. H( f) ¼cos 2 [(p|f|T)/2a) Àp(1 Àa)/4a], for Àf m (1Àa) o¼|f|o¼f m (1 þa) 2. H( f) ¼ 1 2 {1þcos [(p|f|T)/a) Àp(1Àa)/2a]} In practice, there is always some comeback as seen in re- gion C. Figure 20 shows the double-sided RF bandwidth when the center reference is the carrier. The right-hand side is the baseband bandwidth with the reference at 0 Hz. When used as a lowpass filter at baseband, the filter is referred to as a ‘‘window.’’ There are many rolloff curves associated with windows, which are realizable with DSPs or field- programmable gated arrays (FPGAs) used as FIR filters. Designing an RF bandpass filter with these rolloff curves is very difficult; therefore, the preferred practice is to do the filtering at baseband where numerous windowing curves are available. Some popular rolloff curves for FIR filters used as ‘‘win- dows’’ are the Bartlett, Blackman, Hamming, Hanning, Elanix, Truncated Sinx/x, and Kaiser. These are usually realized by changing the multipliers in the 2 of Eq. (17) [above; after text following Eq. (17)] form of the raised- cosine equation. For example, using this form of Eq. (17), the Hamming window has the equation H(f) ¼{0.54 þ 0.46cos[(p|f|T)/a) Àp(1Àa)/2a]}. The ‘‘ideal’’ filter shape (a ¼0) at baseband is called a ‘‘rectangular’’ window. 22. GROUP DELAY, ENVELOPE DELAY, AND RISE TIME The group delay for conventional filters is traditionally calculated to be [11]: T g ¼ DF 2pDf ð18Þ For LC or Gaussian filters (Fig. 2), this is T g ¼ 1 ð4Df Þ and T g ¼ QDF o ð19Þ Obviously, a very narrow bandwidth filter [Df] has a very large group delay, which will adversely affect pulse mod- ulation. There is an associated equation for the risetime of the conventional filter: T r ¼0.7/B, where B is the 3-dB band- width [Df] of the filter. This is the time interval from 10% to 90% on the RC curve. Bandwidth, risetime, and sam- pling rate are mathematically linked. A radar system with a narrow pulse must have a RC risetime that allows the pulse to pass. This necessarily means a very broad filter bandwidth and an accompany- ing high noise level. Two-level modulation methods, such as BPSK, QPSK, GMSK, NBFM, and NBPM (binary, quadrature, Gaussian minimum shift keying and narrowband frequency and phase modulation), can use a narrower-than-usual bandpass filter. The bandpass can be as low as 0.2[Df] in Eqs. (18) and (19). Refer to Fig. 2 and Eq. (4), which demonstrate a reduc- tion of the output level of the high-frequency portion of the signal (sidebands) that pass through the filter and a si- multaneously reduction of the noise power bandwidth. The result is an overall S o /N o improvement. The notation BT is used for this concept B¼bandwidth and T¼bit pe- riod ¼1/f b . The value of T is fixed, but B can be altered. The effect is to raise the processing gain in Eq. (4) by 1/BT. Certain newer modulation concepts (ultranarrowband) require a filter that does not conform to the group delay B +f m (1+) +f m -f m (1−) −f m C C Reference Figure 20. A simplified Nyquist filter bandpass spectrum (from Sklar [13] and Feher [14]). INTERMEDIATE-FREQUENCY AMPLIFIERS 2187 equation [Eq. (18)]. These so called zero-group-delay filters have a very narrow bandwidth with almost instantaneous pulse response at a single frequency. Figure 21 shows the circuit of a half-lattice, or bridge-type, filter, that has near- zero group delay to a single-pulsed frequency. At the par- allel resonant frequency of the crystal, the crystal appears to have a very high resistance and the signal passes via the phasing capacitor in the opposite bridge arm. This circuit has a frequency response similar to that of the uni- versal curve (Fig. 2) with shoulders that extend from 0 Hz to infinity. Therefore, it must be used together with pre- filters to narrow the total noise bandwidth. A small ca- pacitor or inductor can be used at z to extend the tuning range of the crystal [11]. 23. COMPUTER-AIDED DESIGN AND ENGINEERING Digital filters are easily designed using commercially available software packages and information provided by the IC manufacturers (d–g, l). For IF filter design using discrete components, the ad- mittances rather than the impedances are easiest to use, since most components are in parallel as shown in the equivalent circuit of Fig. 4b. Unfortunately, most available data are in the form of S parameters, which are very dif- ficult to convert manually to impedances or admittances. Parameters for the filters are rarely available, so calcu- lated values based on assumed input and output imped- ances must be used unless test equipment capable of measuring return losses or standing waves is available, in which case the S parameters can be measured or cal- culated. Smith and Linville charts have been used by some au- thors to design IF amplifiers, but these methods are not totally satisfactory for IF amplifier design, since a high-Q circuit has its plot near the outer edge of the circle and changes are difficult to observe. The network admittance values shown in Fig. 4 would be used. Computer-aided programs for linear or analog designs, such as the various ‘‘SPICE’’ programs are readily avail- able (j). Other programs which concentrate specifically on filter design (f, k–m) can simplify the filter design. They have outputs that then interface with the SPICE pro- grams if desired. Most semiconductor manufacturers pro- vide scattering parameters (S parameters) or SPICE input data on disk for use with these programs. Some design software sources are listed below (after the Bibliography). Some of the IF amplifier integrated circuit manufacturers also provide software specific to their products. BIBLIOGRAPHY (References 8, 9, 10, and 13 contains applicable software). 1. J. M. Petitt and M. M. McWhorter, Electronic Amplifier Cir- cuits, McGraw-Hill, New York, 1961. 2. W. Th. Hetterscheid, Transistor Bandpass Amplifiers, Philips Technical Library, N.V. Philips, Netherlands/Philips Semicon- ductors, 1964. 3. Roy Hejhall, RF Small Signal Design Using Two-Port Param- eters, Motorola Applications Note AN 215A. 4. F. Davis, Matching Network Designs with Computer Solu- tions, Motorola Applications Note AN 267. 5. V. Uzunoglu and M. White, Synchronous oscillators and co- herent phase locked oscillators, IEEE Trans. Circuits Syst. 36(7) (1989). 6. H. R. Walker, Regenerative IFamplifiers improve noise band- width, Microwaves RF Mag. (Dec. 1995, Jan. 1996). 7. R. E. Best, Phase Locked Loops, McGraw-Hill, New York, 1984. 8. R. W. Goody, P-Spice for Windows, Prentice-Hall, Englewood Cliffs, NJ, 2001. 9. M. E. Herniter, MicroSim P-Spice, Prentice-Hall, Englewood Cliffs, NJ, 2000. 10. J. Keown, Orcad PSpice and Circuit Analysis, Prentice-Hall, Englewood Cliffs, NJ, 2001. 11. W.-K. Chen, The Circuits and Filters Handbook, IEEE Press, New York, 1995. 12. ARRL Handbook, Amateur Radio Relay League, Newington, CT, 2000. 13. B. Sklar, Digital Communications, Prentice-Hall, Englewood Cliffs, NJ, 2001. (Contains the Elanix SysView design soft- ware on CD.) 14. K. Feher, Wireless Digital Communications, Prentice-Hall, Englewood Cliffs, NJ, 1995. AVAILABLE SOFTWARE The following companies are representative of those pro- viding packaged IFamplifiers as integrated circuits #, and those offering development software packages *. (a) #*Linear Technology Corporation, 720 Sycamore Drive, Mil- pitas, CA 95035 (www.linear-tech.com). (b) #Maxim Integrated Products, 120 San Gabriel Drive, Sunny- vale, CA 94086 (www.maxim-ic.com). (c) #*Analog Devices, One Technology Way, P.O. Box 9106, Nor- wood, MA 02062 (www.analog.com). (d) *#Texas Instruments, P.O. Box 954, Santa Clarita CA 91380 (www.ti.com/sc or www.ti.com/sc/expressdsp). (e) #*Altera Corp., 101 Innovation Drive, San Jose, CA 95134 (www.altera.com). Z Figure 21. Circuit of a half-lattice (bridge-type) filter with near- zero group delay to a single-pulsed frequency (from Chen [12]). 2188 INTERMEDIATE-FREQUENCY AMPLIFIERS (f) *#Xilinx, Inc., 2100 Logic Drive, San Jose, CA 95124 (www. xilinx.com). (g) *# Philips Semiconductors, 811 E. Arques Avenue, P.O. Box 3409; Sunnyvale, CA 94088 (www.semiconductors.philips. com). (h) *#Motorola Literature Distribution Center, P.O. Box 5405; Denver, CO 80217 (www.motorola.com/semiconductors/ or www.Design-net.com). (i) *#AMD, One AMD Place, P.O. Box 3453; Sunnyvale, CA 94088 (www.amd.com). (j) *MicroSim Corp., 20 Fairbanks, Irvine, CA 92618 (www. orcad.com). (k) *Eagleware Corp., 1750 Mountain Glen, Stone Mountain, GA 30087 (www.eagleware.com). (l) *Elanix Inc., 5655 Lindero Canyon Road, Suite 721, Westlake Village, CA 91362 (www.elanix.com). (m) *The Math Works (MatLab), 3 Apple Hill Drive, Natick, MA 01760-2098 (www.mathworks.com). (n) *Intusoft, P.O. Box 710, San Pedro, CA 90733 (www.intusoft. com). (o) *#Hewlett-Packard Company, P.O. Box 58199, Santa Clara, CA 95052 (www.hp.com). (p) *Z Domain Technologies, 555 Sun Valley Drive, Roswell, GA 30076 (www.zdt.com/Bdsp). (q) #Rockwell Semiconductor Systems, 4311 Jamboree Road, Newport Beach, CA 92660 (www.rockwell.com). (r) #RF Micro Devices, 7628 Thorndike Road, Greensboro, NC 27409-9421 (www.rfmd.com). INTERMODULATION JOSE ´ CARLOS PEDRO University of Aveiro Portugal 1. INTRODUCTION 1.1. What Is Intermodulation Distortion? Although the term intermodulation is used by some authors to describe a specific manifestation of nonlinear distortion, in this text we will adopt the wide-sense mean- ing of intermodulation as any form of nonlinear distortion, unless otherwise explicitly stated. So, it is convenient to start an introduction to intermodulation by saying a few words about distortion. In the field of telecommunication systems, distortion is understood as any form of signal impairment. In this way, distortion takes the broad sense of all differences between the received and the transmitted information signals, specifically, those added or signal-dependent perturba- tions. In the first set of added, or signal-independent, pertur- bations, we should include random noise and determinis- tic interferences. Typical examples of the former are the always present thermal noise or shot noise of electronic circuits. The second could be illustrated by some man- made (synthetic) repetitive impulsive noise or simply another telecommunications channel that shares the same transmission medium but that does not carry any useful information. The set of signal-dependent perturbations can also be divided into two major parts—linear distortion and non- linear distortion—according to whether what distin- guishes the received signal from its transmitted version is due to a linear or a nonlinear process. The reason for this organization stands in the easiness with which we correct linear distortion and the difficulty we have in dealing with nonlinearity. In fact, since linear distortion describes all differences in time-domain waveform, or frequency-domain spectrum, as the ones caused by any usual filter or dis- persive transmission medium, it can be corrected by an- other inverse filter, with a methodology usually known as pre- or postequalization. On the other hand, nonlinear distortion cannot be corrected this way, remaining nowa- days as a very tough engineering problem. So, from a purely theoretical point of view, what distinguishes linear distortion from nonlinear distortion is simply the essence of the mapping corresponding to the telecommunication system, from the signal source to the detected signal. If that mapping responds to scaled ver- sions of two different signals with two scaled versions of the responses to these two signals, when they are pro- cessed individually, we say that our transmission system obeys superposition, and is thus linear [1]. In any other case, we say that the system is a source of nonlinear distortion. Nonlinear distortion can, therefore, manifest itself in many different forms that range from the obvious signal clipping of saturated amplifiers, to the almost unnoticeable total harmonic distortion present in our high-fidelity audio amplifiers. Because nonlinear distortion is a property not shared by our more familiar linear systems, we could think of it as something visible only in some special-purpose systems or poorly designed circuits. Unfortunately, that is not the case. To a greater or lesser extent, nonlinear distortion is present in the vast majority of electronic systems. Because nonlinear distortion is associated with PN and PIN diodes or varactors [2], it is found in many control devices such as solid-state switches, controlled attenuators, phase shif- ters, or tunable filters—and, because of the recognized nonlinearity of magnetic-core inductors, it can also arise from other passive filters and diplexers. However, prob- ably more surprising, is the fact that it can even arise from devices usually assumed as linear. One example is the nonlinear distortion produced by some RF MEM (radiofrequency micromachined electro- mechanical) switches [3]. Another is passive intermodula- tion (PIM), which is frequently observed when loose connections, or junctions made of different metals or of similar but oxidized metals are subject to high power levels [4]. So, PIM is generated in many RF connectors, antennas, antenna pylons, wire fences, and other compo- nents. Finally, intermodulation can even arise from our supposedly linear electronic circuits as it is inherent to the operation of all electronic active transducers. To under- stand this, let us take the example of the general amplifier described in Fig. 1. Because our amplifier is a physical system, it must obey energy conservation, which implies that the sum of all INTERMODULATION 2189 forms of input power—either signal power P in or DC supply power P DC —must equal the sum of all forms of output power, whether it is signal power delivered to the load P out or dissipated power P diss such as heat or harmonic distortion: P in þP DC ¼P out þP diss ð1Þ On the other hand, we know that output signal power should be a scaled replica of the input signal power, defining, in this way, a certain amplifier power gain: G P P out P in ð2Þ However, (1) also implies that G P ¼1þ P DC ÀP diss P in ð3Þ which shows that no amplifier that relies on a real supply of finite power can keep its gain constant for any increasing input signal level. Sooner or later, it will have to show gain compression, presenting, therefore, nonlinearity. 1.2. Characterizing Intermodulation Distortion After this brief introduction to the concept of intermodula- tion distortion, let us now see in more detail which forms of distortion it describes. For that, we will assume a simple system represented by the following cubic model: yðtÞ ¼a 1 xðt Àt 1 Þ þa 2 xðt Àt 2 Þ 2 þa 3 xðt Àt 3 Þ 3 þ Á Á Á ð4Þ in which the input x(t) is nonlinearly transformed into an output y(t). Note that this system not only shows non- linearity as it also has memory, since it does not respond instantaneously to the input, but to certain past versions of it. This dynamic behavior is due to the presence of the delays t 1 , t 2 , and t 3 . 1.2.1. Single-Tone Distortion Characterization. Suppos- ing the input is initially composed of one amplitude A(t) and phase y(t) modulated RF carrier of frequency o c xðtÞ ¼AðtÞ cos½o c t þyðtÞ ð5Þ the output will be composed of three sets of terms, say, y 1 (t), y 2 (t), and y 3 (t), each one corresponding to a certain polynomial degree. Illustrations of the time-domain wave- forms and frequency-domain spectra of the input x(t) and output y(t) are depicted in Figs. 2a and 2b and in Fig. 3a and 3b, respectively. The first term of the output is given by y 1 ðtÞ ¼a 1 Aðt Àt 1 Þ cos½o c t þyðt Àt 1 Þ Àf 1 ð6Þ (where f 1 ¼o c t 1 ) and corresponds to the expected linear response. Note that it includes exactly the same frequency components already present at the input. The second term y 2 ðtÞ ¼ 1 2 a 2 Aðt Àt 2 Þ 2 þ 1 2 a 2 Aðt Àt 2 Þ 2 cos½2o c t þ2yðt Àt 2 Þ À2f 2 ð7Þ (where f 2 ¼o c t 2 ) involves baseband products whose fre- quency falls near DC and some other products whose frequencies are located around the second harmonic, 2o c . The first ones consist of second-order intermodulation products of the form o x ¼o 1 Ào 2 (in which o x is the resulting frequency, while o 1 and o 2 are any two distinct frequencies already present at the input), and describe the demodulation generally provided by even-order nonlinea- rities. When o 1 ¼o 2 , then o x ¼0, and the terms fall exactly at DC. So, they also describe the circuit’s DC bias shift. Because they are what is sought in AC to DC converters, in amplifiers they model the variation of the y(t) mean value from the quiescent point, to the mean value shown in presence of a significant RF excitation— the large-signal bias point. The second type of even-order products is again second-order intermodulation distortion whose frequency now falls at o x ¼o 1 þo 2 . For o 1 ¼o 2 , Signal source Signal load Power supply P diss P diss P diss P diss P diss P diss P diss P diss P diss P diss P diss P diss P diss P diss P diss P diss P diss P diss P dc P in P out V DD V GG Figure 1. Conceptual amplifier showing input/output power relations. 2190 INTERMODULATION 0 1000 3000 6000 8000 −300 −200 −100 0 Frequency (MHz) |S xx (f )| (dB) 2000 4000 7000 5000 (b) 1.0 1.1 1.2 1.3 1.4 1.5 −1 −0.5 0 0.5 1 x(t ) ( ) 1.05 1.15 1.25 1.35 1.45 (a) Time ( s) µ Figure 2. (a) Time-domain waveform of the in- put signal x(t); (b) corresponding frequency- domain spectrum. −10 −5 0 5 10 y(t ) ( ) 1.0 1.1 1.2 1.3 1.4 1.5 1.05 1.15 1.25 1.35 1.45 Time ( s) (a) 0 1000 2000 3000 4000 5000 6000 7000 8000 −300 −200 −100 0 Frequency (MHz) |S yy (f )| (dB) (b) µ Figure 3. (a) Time-domain waveform of the output signal y(t); (b) corresponding frequency- domain spectrum. INTERMODULATION 2191 then o x ¼2o 1 , and the products are known as second- order harmonic distortion. Finally, the third term is y 3 ðtÞ ¼ 3 4 a 3 Aðt Àt 3 Þ 3 cos½o c t þyðt Àt 3 Þ Àf 3 þ 1 4 a 3 Aðt Àt 3 Þ 3 cos½3o c t þ3yðt Àt 3 Þ À3f 3 ð8Þ (where f 3 ¼o c t 3 ) and also involves two different sets of products located near the input frequency band (or funda- mental band) o c and the third harmonic 3o c . As are the even-order products, the third-order pro- ducts falling near the third harmonic 3o c are classified as out-of-band products. Appearing at o x ¼o 1 þo 2 þo 3 , that is, out of the fundamental signal band in RF systems of narrow bandpass characteristics, these products seldom constitute a major source of nonlinear signal impairment as they can be easily filtered out. Note, however, that they may also constitute in-band products in ultra-wide-band systems such as cable television (CATV). Third-order products falling exactly over, or in the vicinity of o c , in which the resulting frequencies can be either o x ¼o 1 þo 2 Ào 3 , o x ¼2o 1 Ào 2 , or even o x ¼o 1 (whether they arise from the combination of three distinct, two equal and one different, or three equal input frequen- cies, respectively), are obviously called in-band products. Contrary to the products treated above, they cannot be eliminated by linear filtering, constituting the principal object of intermodulation distortion studies in microwave and wireless systems. In fact, some authors even reserve the term intermodulation distortion for this particular form of nonlinear signal perturbation. To analyze these in-band distortion products in more detail, we will now consider two different situations of system memory. In the first case, it is assumed that the time delays of (4) are due only to the active device’s reactive components or to the input and output matching networks. In this way, they may be comparable to the RF carrier period, but negligible when compared to the much slower modulation timescale. Therefore, the in-band pro- ducts can be approximated by 3 4 a 3 AðtÞ 3 cos½o c t þyðtÞ Àf 3 ð9Þ which shows that, although the system kept its dynamic behavior to the RF carrier, it became memoryless (i.e., responds instantaneously) to the modulation envelope. Since general amplitude and phase modulations have frequency components that start at DC, we have already seen that these products include spectral lines falling exactly over the ones already present at the input, and some other new components named as spectral regrowth. The third-order signal components that are coincident with the input are given by o x ¼o 1 þo 1 Ào 1 ¼o 1 þ (o 1 Ào 1 ) ¼o 1 and can be understood as being generated by mixing second-order products at DC with first-order (or linear) ones. Except for their associated gain, which is no longer a 1 , but 3 4 a 3 multiplied by the input amplitude- averaged power A 2 , these products are indistinguishable from the linear components of (6). They carry the same information content, and are, therefore, termed signal- correlated products. Although, in a strict sense, they should be considered as nonlinear distortion products (as their signal power rises at a slope of 3 dB/dB against the 1 dB/dB that characterizes truly linear components), from an information content viewpoint, they may also be con- sidered as linear products. In fact, since, for a constant- input-averaged power, they cannot be distinguished from the first-order components, it all happens as if the ampli- fier had remained linear but with a gain that changed from its small-signal value of G¼a 1 exp( Àjf 1 ) to an amplitude-dependent large-signal gain of G(A) ¼ a 1 exp( Àjf 1 ) þ(3/4)A 2 a 3 exp( Àjf 3 ). So, input amplitude signal variations [or amplitude modulation (AM)] produce different output amplitude variations, according to the so- called amplifier AM–AM conversion. But, since the gain is also characterized by a certain phase, it is obvious that input amplitude signal variations will also generate out- put phase variations. In conclusion, and as illustrated in Figs. 4a and 4b, the amplifier will show not only AM–AM but also AM–PM conversion. Figure 5 depicts a possible block diagram of a labora- tory setup intended to measure these static AM–AM and AM–PM characteristics [6]. As shown, it relies on a usual microwave vector network analyzer whose signal source is swept in power. As a curious aside from this analysis, we should point out that, although our nonlinearity manifests a signal amplitude-dependent gain, it is completely insensitive to P in (dBm) P out (dBm) −15 −10 −5 0 5 10 15 20 25 −5 0 5 10 15 20 25 AM-AM (a) −1° 0 1° 2° 3° 4° 5° ( °) φ P in (dBm) AM-PM −15 −10 −5 0 5 10 15 20 25 (b) Figure 4. (a) Amplifier’s AM–AM conversion; (b) AM–PM conversion. 2192 INTERMODULATION the input signal phase. In fact, as can be concluded from (9), the bandpass characteristics of our amplifier would be completely transparent to a phase-modulated signal of constant amplitude, in the sense that the phase informa- tion present at the output would be exactly equal to the phase information present at its input. In the second case, it is supposed that, beyond the usual time constants of the order of the RF carrier period, our system may even present time delays, t 1 0 , t 2 0 , and t 3 0 , comparable to the modulation period (e.g., determined by the bias circuitry, active-device charge carrier traps, self- heating). Such time constants are no longer irrelevant for the envelope evolution with time, and the system is said to present long-term or envelope memory effects. The in-band output distortion becomes 3 4 a 3 Aðt Àt 3 0 Þ 3 cos½o c t þyðt Àt 3 0 Þ Àf 3 0 ð10Þ and the output envelope will show a phase shift that is dynamically dependent on the rate of amplitude varia- tions. In this case, the output AM–AM or AM–PM is no longer static, and dynamic (or hysteretic) AM–AM and AM–PM conversions are observed, as shown in Figs. 6a and 6b. This shows that, if our nonlinear system only presents short-term memory effects, and thus is memoryless for the envelope, it may be characterized by a set of gain and AM-AM P in (dBm) P out (dBm) (a) −15 −10 −5 0 5 10 15 20 25 −5 0 5 10 15 20 25 AM-PM (°) φ P in (dBm) (b) −1° 0 1° 2° 3° 4° 5° −10 −5 0 5 10 15 20 25 −15 Figure 6. Typical hysteretic AM–AM (a) and AM–PM (b) characteristics shown by nonlinear dynamic amplifiers suffering from both short-term and long-term memory effects. 1.800 GHz dBm DC AMP Power and Frequency control Figure 5. AM–AM and AM–PM characteriza- tion setup based on a microwave vector network analyzer. INTERMODULATION 2193 phase shift tests made with a sinusoidal, or CW (contin- uous-wave), excitation with swept amplitude and, even- tually, with varying frequency. However, if the system is also dynamic to the envelope, then the observed AM–AM/ AM–PM varies with the speed of the input amplitude sweep, and such a test becomes questionable. Since each of the tested CW signals can be seen as a carrier modu- lated by a constant (DC) envelope, it becomes obvious that we cannot fully characterize a dynamic system using only these simple DC excitations. Moreover, it is clear that testing in-band intermodula- tion products with a CW signal will never be an easy task, as the output will only have signal-correlated components where o x ¼o c , which all overlap onto the usually much higher linear output. Obviously, in-band intermodulation characterization requires more complex stimuli. 1.2.2. Two-Tone Distortion Characterization. One way to increase the complexity of our test signal is to use a two-tone excitation: xðtÞ ¼A 1 cosðo 1 tÞ þA 2 cosðo 2 tÞ ð11Þ The in-band output components of (4) when subject to this new stimulus will be a 1 A 1 cos½o 1 t Àf 110 þa 1 A 2 cosðo 2 t Àf 101 Þ þ 3 4 a 3 A 2 1 A 2 cos½ð2o 1 Ào 2 Þt Àf 32À1 þ 3 4 a 3 A 3 1 þ 6 4 a 3 A 1 A 2 2 _ _ cos½o 1 t Àf 310 þ 6 4 a 3 A 2 1 A 2 þ 3 4 a 3 A 3 2 _ _ cosðo 2 t Àf 301 Þ þ 3 4 a 3 A 1 A 2 2 cos½ð2o 2 Ào 1 Þt Àf 3À12 ð12Þ Beyond the expected linear components arising at o 1 and o 2 , (12) is also composed of other third-order products at o 1 , o 2 , 2o 1 Ào 2 , and 2o 2 Ào 1 . They constitute again the signal-correlated (o 1 and o 2 ) and signal-uncorrelated (2o 1 Ào 2 and 2o 2 Ào 1 ) components. The terms at o 1 (o 2 ) that are dependent only on A 1 (A 2 ) constitute the AM–AM/AM–PM conversion discussed above. But now there are some new terms at o 1 (o 2 ) whose amplitude is also controlled by A 2 (A 1 ). They model two different, but obviously related, nonlinear effects. One is cross-modula- tion, a nonlinear effect in which amplitude modulation of one RF carrier is converted into amplitude modulation of the other; the other is known as desensitization, the loss of receiver sensitivity to one signal when in presence of an incoming strong perturbation (e.g., a jammer). The terms at 2o 1 Ào 2 and 2o 2 Ào 1 are spectral re- growth components that appear as sidebands located side by side to the fundamentals at a distance equal to their frequency separation o 2 Ào 1 . These in-band intermodula- tion distortion (IMD) sidebands rise at a constant slope of 3 dB per dB of input level rise, until higher-order compo- nents (in the case of our polynomial model, output con- tributions due to higher-degree terms) show up. Since first-order components rise at a slope of only 1 dB per dB, we could conceive of an extrapolated (never reached in practice) output power where the output IMD and funda- mentals would take the same value. As illustrated in Fig. 7, this is the so-called third-order intercept point IP 3 . Although meaningful only for small-signal regimes, where the fundamental and IMD components follow their idealized straight-line characteristics, IP 3 is still the most widely used (some times erroneously) intermodulation distortion figure of merit. Figure 8 shows a block diagram of the most popular laboratory setup used for two-tone intermodulation tests. It relies on a two-tone generator of high-spectral purity, and a high-dynamic range microwave spectrum analyzer. Although, for many years, two-tone intermodulation characterization has been restricted to these amplitude measurements, more recently we have seen an increasing interest to also identify the IMD components’ phase. The reason for this can be traced to the efforts devoted to extract behavioral models capable of representing the device’s IMD characteristics and to the design of amplifier linearizers that must be effective even when the main nonlinear device presents long-term memory effects. In fact, since most of the linearizers can be understood as auxiliary circuits capable of generating IMD components that will cancel the ones arising from the main amplifier, it is obvious that those linearizing circuits must be designed to meet both IMD amplitude and phase require- ments. Unfortunately, the first problem that arises when try- ing to measure the IMD components’ phase is that, despite phase is a relative entity, we have no phase reference for IMD. Contrary to what happens to the output fundamen- tals in which we can refer their phases to the phases at the input (usually arbitrarily assumed zero), the problem is that now there are no input components at the IMD frequencies. So, we first need to create a reference signal at that IMD frequency. That is usually done with a −10 20 −100 −80 −60 −40 −20 0 20 40 P Fund ( 2 ) 1dB/dB 3dB/dB P out (dBm) P in (dBm) 0 P IMD (2 2 – 1 ) IP 3i 10 30 IP 3 ω ω ω Figure 7. Typical fundamental and third-order intermodulation power versus input power plots. Note the definition of the extra- polated third-order intercept point IP 3 . 2194 INTERMODULATION reference nonlinearity; thus IMD phase measurement results become relative to the reference nonlinearity used in the setup. For example, in the setup depicted in Fig. 9, the reference nonlinearity is based in the nonlinear characteristic of broadband Schottky diodes, and the phase value is acquired from the variable phase shift necessary to balance the device under test (DUT) and reference arms. 1.2.3. Multitone Distortion Characterization. For com- pleteness, let us now briefly introduce intermodulation characterization under multitone excitations. A detailed analysis of this important and up-to-date subject can be found in various references [e.g. 5,6]. First, we will assume that our stimulus can be de- scribed as a sum of Q sinusoids of different frequencies: xðtÞ ¼ Q q¼1 A q cosðo q tÞ ¼ 1 2 Q q¼ÀQ A q e jo q t ð13Þ The output of a general power series such as (4) to the excitation of (13) will be yðtÞ ¼ N n¼1 y n ðtÞ ð14aÞ where each of the orders can be expressed as y n ðtÞ ¼ 1 2 n a n Q q ¼ÀQ A q e jo q t _ _ n ¼ 1 2 n a n Q q1 ¼ÀQ Á Á Á Q qn ¼ÀQ A q 1 Á Á Á A q n e jðo q 1 þÁÁÁ þo qn Þt ð14bÞ which contains various frequencies at o x ¼o q1 þ?þo qn , originating from many different mixing products. Since there is, in general, more than one mixing product—that is, more than one combination of input frequencies—falling at the same frequency, the calcula- tion of their output amplitude requires that first we are able to determine the number of those different combina- tions. One systematic way to do this is to recognize that their frequencies must obey [7] o n;m ¼o q 1 þ Á Á Á þo qn ¼m ÀQ o ÀQ þ Á Á Á þm À1 o À1 þm 1 o 1 þ Á Á Á þm Q o Q ð15Þ 2 1 1 2 1 2 AMP DC − 00.53 + 15.00 1.709 1.801 GHz GHz dBm dBm Isolators Low-Pass Filters Figure 8. The most popular laboratory setup used for two-tone intermodulation tests. DC AMP 1.709 GHz dBm Reference Nonlinearity Phase Shifter Attenuator DC Isolator Low-Pass Filters Isolator 1.801 GHz dBm Figure 9. Possible IMD phase measurement setup based on a reference nonlinearity, a spectrum analyzer, and an IMD cancellation loop. INTERMODULATION 2195 where Q q ¼ÀQ m q ¼m ÀQ þ Á Á Á þm À1 þm 1 þ Á Á Á þm Q ¼n ð16Þ defining the following mixing vector: v¼½m ÀQ Á Á Á m À1 m 1 Á Á Á m Q ð17Þ Then, the number of different ways of generating the same mixing vector is given by the multinomial coefficient [7]: t n;n ¼ n! m ÀQ ! . . . m À1 !m 1 ! . . . m Q ! ð18Þ These Q-tone distortion components allow a generaliza- tion of the two-tone signal-to-intermodulation distortion ratio [IMR; sometimes also known as carrier-to-IMD ratio (C/I)] to various multitone distortion figures of merit. One of these is defined as the ratio between the constant- amplitude output fundamental signals and the highest sideband IMD component (M-IMR). Another measure is the ratio between integrated fun- damental output power and integrated upper or lower sideband distortion. As this sideband spectral regrowth falls exactly over the location of a potentially present adjacent channel, it is called the adjacent-channel power ratio (ACPR). Finally, a measure of the ratio of the fundamentals to the signal-uncorrelated distortion components that fall exactly among the fundamental components is given by the so-called noise-power-ratio (NPR). The reason for this denomination comes from the fact that, although that figure of merit is being introduced in this text for a multitone excitation, it was traditionally measured with a bandlimited white-noise stimulus—a generalized multi- tone excitation with an infinite number of tones. Besides all these figures are measures of nonlinear effects that share a common physical origin, and it has not been easy to relate them, except for very particular situations. First, we [5] presented relations between various multitone distortion figures and IMR, obtained for a third-degree polynomial memoryless model. Then, Boulejfen et al. [8] extended those results for a fifth-degree polynomial. As a summary of these results, Fig. 10 pre- sents the ratio of IMR to the above-defined multi-tone distortion figures versus the number of tones Q for a memoryless cubic polynomial. A laboratory setup for multitone distortion tests is similar to the one already shown for two-tone tests, except, obviously, with respect to the signal generator [6]. How- ever, since a NPR test focuses on the distortion that falls exactly over the output fundamentals, something must be done to separate the desired distortion components from the much higher fundamental signals. The usual way to solve that problem consists in creating a very narrow measurement window within the input signal bandwidth. This is accomplished by either shutting down a few input tones—when a multitone signal generator is used—or introducing a notch filter between the bandlimited white-noise generator and the nonlinear device under test [6]. 2. CAD TOOLS FOR INTERMODULATION DISTORTION PREDICTION Because intermodulation distortion is a nonlinear effect, any attempt to predict its behavior by hand for even the most simple practical circuits or devices becomes extre- mely difficult, if not impossible. So, intermodulation dis- tortion prediction relies heavily on good device models and appropriate computer simulation algorithms. Unfortu- nately, these subjects are so vast that we have to restrict this text to a first guiding overview. So, we will concen- trate our discussion on a set of criteria for model quality (for this specific purpose) and give some hints concerning usual simulation tools. 2.1. Nonlinear Device Modeling for Distortion Analysis: General Considerations Starting with nonlinear device models, we can divide them into four general groups: (1) physical and empirical mod- els and (2) global and local models. Physical models are mathematical descriptions of the internal device operation that are drawn from the know- ledge of the device’s geometric and physical structure, and ACPR M-IMR NPR 0 10 20 30 40 50 60 70 80 90 100 −4 −2 0 2 4 6 8 10 Number of Tones, Q IMR/Q-Tone Distortion (dB) 110 120 130 IMR (6.0 dB) 1 4 4 3 IMR (1.3 dB) IMR (7.8 dB) 6 1 140 150 160 Figure 10. Ratio of two-tone IMR to NPR, M- IMR and ACPR versus the number of tones Q for a memoryless cubic polynomial. 2196 INTERMODULATION from the application of a certain set of basic physics laws. Although relying on extremely complex formulations that require an enormous number of parameters and are computationally expensive to evaluate, they can provide much better accuracy than the empirical models as they necessarily mimic the basic device operation. On the other hand, empirical models do not require any information about the internal structure of the device, relying com- pletely on input–output behavioral observations. Hence, they are also known as blackbox models or behavioral models. Typical examples of physical models are the Schottky diode equation and the device models described by a set of coupled partial-differential equations of electric potential, charge, and charge carrier mobility. Examples of purely behavioral models are the linear scattering matrix, table- based device models, or even the abovementioned AM– AM/AM–PM models. Local models can be distinguished from global models for their approximation range. Because they are beha- vioral in nature, they constitute two different compro- mises between the domain of fitting and the level of accuracy. While local models are very good in representing mild nonlinear behavior in the vicinity of some quiescent point, global models are conceived as valid for any possible operation regime, but at the expense of an increased error. It is therefore natural that they are also known as small- signal or large-signal models, respectively. The Gummel– Poon model of BJTs, the quadratic model of FETs, or an AM–AM/AM–PM representation are examples of global models, while the poor extrapolation capability usually associated with polynomial approximators tends to grant them a distinct local behavior. For example, the cubic polynomial that could be extracted from the third-order intercept point is necessarily a local model valid only for small-signal excitation levels. The polynomial example given above is not accidental as it plays a fundamental role in all nonlinear distortion analysis. In fact, for some unique reason we began this article using exactly the same polynomial of expression (4). As we then concluded, if the model is a polynomial, we have a direct and easy way to calculate the various intermodulation products to any signal that can be de- scribed as a sum of sinusoids. Furthermore, by simply selecting its coefficients, we can tailor the polynomial for very different approximation goals. To understand that, let us use an illustration example. Figures 11 and 12 depict the approximation of one typical transfer function characteristic by two different polynomials: a Taylor series and a Chebyshev polynomial series, both of 10th degree. As seen from Figs. 11a and 11b, the coefficients of the Chebyshev polynomial were selected so that the polyno- mial could produce an optimum approximation to the response of our original nonlinearity to a sinusoid of 1.5 V peak amplitude, centered at a quiescent point of 0 V (something close to what is found in typical class AB power amplification regimes). On the other hand, the coefficients of the Taylor series were taken as the appropriately scaled derivatives of the nonlinearity at the same quiescent voltage of 0V. It constitutes, therefore, the optimum polynomial approxi- mation to the nonlinear response to any signal of infini- tesimal input amplitude (see Fig. 11a). When excited by sinusoids of variable amplitude, the output DC component (Fig. 12a), the fundamental compo- nent (Fig 12b), and the second- and third-harmonic dis- tortion components (Figs. 12c and 12d, respectively) reveal that these two polynomial approximators present, indeed, very distinct properties. The Taylor series is clearly a local approximator that produces optimum results in the vicinity of the quiescent point, but then suffers from a catastrophic degradation when the excitation exceeds B0.3V of amplitude. On the −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1 0 1 2 3 4 5 6 (a) (b) f (x)( ) x (V) 0 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 Time (ns) f (x(t ))( ) 0 Figure 11. (a) Nonlinear memoryless transfer function, f(x) (- -) and its 10th-order Taylor series approximation around x ¼0 V (. . .) and Chebyshev polynomial optimized for a sinusoidal input amplitude of A¼1.5V (—); (b) time-domain waveform of the output of the transfer function f[x(t)] (- -) and of its 10th-order Taylor series approximation (. . .) and Chebyshev polynomial (—), when excited by a CW input of amplitude A¼1.5 V. INTERMODULATION 2197 contrary, the Chebyshev series is worse at those small- signal levels, but performs much better up to excitation amplitudes of 1.5 V. It behaves, therefore, as a global approximator. (In fact, the Chebyshev series is still a local approximator whose domain is no longer defined around the fixed quiescent point of 0 V, but around a new general- ized dynamic quiescent point imposed by an input sinu- soid of 1.5 V amplitude.) As any other mean-square error approximator, the Chebyshev polynomial wanders around the original nonlinearity (see Figs. 11a and 11b), obviously failing the higher-order derivatives of the function. This is why, contrary to the Taylor series, which, by construction, osculates these derivatives, the Chebyshev series does not show good small-signal distortion behavior. What we have just seen in this example is common to almost all empirical models and results in an important message as far as intermodulation distortion calculations are concerned. Simultaneously reproducing the device’s mild nonlinear details (local characteristics) and the gen- eral trends (global properties) is so difficult that we should –35 –30 –25 –20 –15 –10 –5 0 5 4 4.5 5 5.5 6 6.5 7 7.5 8 A(dB v ) –35 –30 –25 –20 –15 –10 –5 0 5 A(dB v ) F 0 (A) (dB) (a) (b) –30 –25 –20 –15 –10 –5 0 5 F 1 (A) (dB) –35 –30 –25 –20 –15 –10 –5 0 5 –80 –70 –60 –50 –40 –30 –20 –10 –80 –70 –60 –50 –40 –30 –20 –10 0 A (dB v ) A (dB v ) F 2 ( A) (dB) (c) (d) 3 F (A) (dB) –35 –30 –25 –20 –15 –10 –5 0 5 Figure 12. (a) DC component of the output of the transfer function f[x(t)] (- -) of its 10th-order Taylor series approximation (Á Á Á) and Chebyshev polynomial (—), when excited by a CW input of amplitude 0.015VoAo1.95V; (b) fundamental component of output of transfer function f[x(t)] (- -) of its 10th-order Taylor series approximation (Á Á Á) and Chebyshev polynomial (—), when excited by a CW input of amplitude 0.015VoAo1.95V; (c) second-harmonic component of output of transfer function f[x(t)] (- -) of its 10th-order Taylor series approximation (Á Á Á) and Chebyshev polynomial (— ), when excited by a CW input of amplitude 0.015VoAo1.95V; (d) third-harmonic component of output of unit transfer function f[x(t)] (- -) and its 10th-order Taylor series approximation (y) and Chebyshev polynomial (—), when excited by a CW input of amplitude 0.015VoAo1.95V. 2198 INTERMODULATION never trust an empirical model unless we have guarantees that it was specifically tested for nonlinear distortion. 2.2. Nonlinear Models for Distortion Analysis at the Circuit Level To perform intermodulation analysis at the circuit level, that is, to compute the distortion arising from a certain electronic circuit subject to a specific bandpass RF input signal stimulus, the device must be represented by some equivalent-circuit model [6]. This is the normal modeling requirement for using either time-marching algorithms, like the ones used by SPICE, or frequency-domain simu- lators, such as the harmonic-balance solvers. Such equiva- lent circuits have topologies and parameter sets usually supported from both physical and empirical data. Linear, or bias-independent, elements are usually ex- tracted from a broadband small-signal AC characteriza- tion. Nonlinear elements can be either voltage-controlled current sources (nonlinear device currents) i(v), voltage- controlled electric charge sources (nonlinear capacitances) q(v), or current-controlled magnetic flux sources (non- linear inductances) f(i). Each of these is assumed to be described by a static, or memoryless, function of its controlling variable(s), which can, again, be supported by both physical device knowledge or by empirical obser- vations. Mostly in this latter case, it is the selection of these functions that determines the quality of the model for nonlinear distortion predictions. A small mean-square error between measured and modeled data in the whole range of device operation guarantees good global proper- ties, but says nothing about local properties. To be able to also provide good predictability under small-signal re- gimes, the model must osculate at least the first three derivatives of the actual device function, which requires special model extraction procedures. Although those derivatives can be obtained from suc- cessive differentiation of measured i(v), q(v), or f(i) data, this is not recommended for at least two important reasons: (1) since most of the microwave transistors show low-frequency dispersion effects, differentiating DC data may not lead to the real AC behavior; and (2) the aggravation of measurement noise produced by numerical differentiation. If we rely on averages (data integration) to reduce random measurement errors, it is natural to expect an aggravation of those errors if we go backward, that is, numerically differentiating measurement data. So, the best way to obtain these device derivatives is to measure entities that directly depend on them; and one good example of those entities is exactly the harmonic or intermodulation distortion produced by the device under a CW or a two-tone excitation. As an example, the laboratory setup depicted in Fig. 13 uses exactly this principle to acquire the nine coefficients of the Taylor series expansion of the drain–source current of a FET: i ds ðv ds ; v ds Þ ¼G m v gs þG ds v ds þG m2 v 2 gs þG md v gs v ds þG d2 v 2 ds þG m3 v 3 gs þG m2d v 2 gs v ds þG md2 v gs v 2 ds þG d3 v 3 ds ð19Þ Exciting the FET at the gate side with a sinusoid of frequency o 1 and at the drain side with a sinusoid of frequency o 2 allows the extraction of G m from the output current component at o 1 , G ds from the component at o 2 , G m2 from the component at 2o 1 , G md from the component at o 1 þo 2 , G d2 from the component at 2o 2 , and so on. Unfortunately, the actual procedure is not that simple. Although the unilateral properties presented by micro- wave FETs at low frequencies guarantee that v gs will have only the o 1 component, the requirement that the device is terminated at the drain side by a nonnull impedance determines that v ds will have components at o 1 , at o 2 , and at all their mixing products. This impedes the ortho- gonal (or one-to-one) extraction just explained, demanding the solution of a 2 Â2 linear system for G m and G ds ; a 3 Â3 linear system for G m2 , G md , and G d2 ; and a 4 Â4 linear system for extracting G m3 , G m2d , G md2 , and G d3 [9]. Since the concept supporting this setup is general, it can be extended to other nonlinear current sources present in any nonlinear device equivalent-circuit model, or even to charge sources [10]. As an illustrative example, Fig. 14 shows all nine coefficients of (19) extracted with the setup of Fig. 13, FET LPF LPF Diplexer V DS V L (2) V GS ATTN ATTN ATN Spectrum analyzer Double power supply Vs(1) Figure 13. Laboratory setup used to extract the Taylor series coefficients of a bidimensional nonlinearity such as the i DS (v GS ,v DS ) of a FET. INTERMODULATION 2199 from a medium-power microwave GaAs MESFET biased in the saturation region. 2.3. Nonlinear Models for Distortion Analysis at the System Level Although system simulation for the modulated bandpass RF signals has already taken the first steps, system simulation at the complex envelope level is, by far, the most usual way to assess distortion performance of entire communication systems. It assumes that the amplitude/ phase-modulated RF signal of (5) can be given by xðtÞ ¼AðtÞ cos½o c t þyðtÞ ¼Re½AðtÞe jyðtÞ e jo c t ¼Re½ ~ xxðtÞe jo c t ð20Þ in which ~ xxðtÞ is the complex envelope—the lowpass equiva- lent signal of x(t) [11]—and that we are interested only in the system’s in-band characteristics. Thus, the object of the analysis ceases to be the real bandpass RF-modulated signal to become only the complex lowpass envelope. In this way, a significant improvement in simulation effi- ciency is achieved because time-domain simulations no longer need to be carried on with sampling rates imposed by the RF carrier and its harmonics, but only by the much slower envelope. So, the models required for these envel- ope-level system simulators are lowpass complex equiva- lent behavioral models of the original bandpass RF components [11]. They are, therefore, single-input/single- output maps, which may be either linear on nonlinear. Linear maps are easily implemented as gain factors in the memoryless case, or as finite or infinite impulse responses, FIR or IIR, digital filters [11,12], when in presence of dynamic elements. A linear dynamic complex envelope filter whose fre- quency response function is ~ HHðj ~ ooÞ can be directly derived from the corresponding circuit level filter HðjoÞ by simply going through the following bandpass–lowpass transfor- mation [11] ~ HHðj ~ ooÞ ¼H½jð ~ ooþo c Þuð ~ ooþo c Þ ð21Þ where u(o) is the unity step function. G m (mS) G m2 (mS/V), G m3 (mS/V 2 ) 200 150 100 50 0 –50 200 150 100 50 0 –50 V GS (V) 4.0 –3.5 –3.0 –2.5 –2.0 –1.5 –1.0 –0.5 G ds (mS) G md (mS/V), G m2d (mS/V 2 ) G d3 (mS/V 2 ) G d2 (mS/V), G md2 (mS/V 2 ) 50 40 30 20 0 –20 15 10 5 –5 10 –10 0 –0.4 1 0 –3 –0.3 2 –2 –1 3 –0.2 –0.1 0.1 0.2 0.3 0.4 0 V GS (V) 4.0 –3.5 –3.0 –2.5 –2.0 –1.5 –1.0 –0.5 V GS (V) 4.0 –3.5 –3.0 –2.5 –2.0 –1.5 –1.0 –0.5 (a) (b) (c) Figure 14. Taylor series coefficients of the bidimensional voltage-controlled i DS (v GS ,v DS ) current source of a GaAs MESFET for a constant V DS in the saturation zone: (a) G m (—), G m2 (–K–), and G m3 (– þ–); (b) G ds (—), G md (–K–) and G m2d (– þ–); (c) G d2 (—), G md2 (–K–), and G d3 (– þ–). 2200 INTERMODULATION 2.3.1. Memoryless AM–AM/AM–PM Models. In their most basic form, nonlinear complex envelope models simply try to describe the amplitude-dependent memory- less nonlinear effects observed for the amplitude and phase modulation content. They are the AM–AM/AM– PM models discussed above, and of which the quadra- ture Saleh model of Fig. 15 is one of the most widely known [13]. When modeled as a polynomial nonlinearity, this AM– AM/AM–PM model can include only odd-degree (2nþ1) terms involving n negative carrier frequencies plus (nþ1) positive ones [11] ~ yyðtÞ ¼ ðNÀ1Þ=2 n¼0 p 2nþ1 2 2n 2nþ1 nþ1 _ _ ~ xxðtÞj ~ xxðtÞj 2n ð22Þ where m r _ _ stands for the number of different combinations of r elements taken from a population of size m and p 2nþ1 are the polynomial coefficients, now having real and imaginary parts. 2.3.2. Dynamic AM–AM/AM–PM Models. As already seen in the introduction, when the system presents mem- ory not only to the RF signal (as indicated by the AM–PM effect) but also to the slowly varying lowpass envelope, this AM–AM/AM–PM model becomes unsatisfactory and a true dynamic model is required. For example, one possi- bility for such an extension could be to make the in-phase A I (.) and quadrature A Q (.) static nonlinear functions dependent not on the amplitude envelope but on some dynamic version of it. In this way, the AM–AM and AM–PM conversions would no longer be instantaneous functions of A(t), but, as shown in Fig. 16, become instan- taneous functions of an auxiliary dynamic variable ~ zzðtÞ, and thus dynamically varying with A(t). 2.3.3. Memoryless Nonlinearity: Linear Filter Cascade Models. Beyond the methods described above, several other approximated topologies have been tried for build- ing nonlinear dynamic models [14]. Some of those, like the two- or three-box models shown in Fig. 17, deserve men- tion because of their practical relevance. In fact, they somehow mimic the internal structure of typical RF devices (as microwave power amplifiers), which are usually constituted by a broadband (memoryless) non- linear active device sandwiched between two linear dy- namic input and output matching networks. As shown in Fig. 17, these two- or three-box nonlinear dynamic models can be cascades of a linear filter followed by the measured memoryless AM–AM/AM–PM nonlinear model (known as the Wiener model), be cascades of this AM–AM/AM–PM memoryless nonlinearity followed by a linear filter (the Hammerstein model), or even be consti- tuted by a combination of both (the Wiener–Hammerstein model). Other parallel combinations of memoryless non- linearities and linear filters also became popular when an optimal extraction procedure was shown to be practically possible [15]. 2.3.4. General Nonlinear Dynamic Models. Unfortu- nately, these three-box models become hopelessly inaccu- rate when the dynamic effects presented to the envelope are not due to the bandwidth limitations of the linear matching networks, but are intrinsically mixed with the nonlinearity [14]. That is the case, for example, with wireless power amplifiers whose nonlinear dynamic ef- fects cannot obviously arise from bandwidth limitations— the RF signal can have bandwidths as narrow as 1% or 0.01%, but from the active device self-heating or from reactive (to the envelope) bias paths. In such cases, more general nonlinear dynamic models, as the ones briefly explained in the following paragraphs, must be attempted. When the lowpass equivalent system is stable, contin- uous, and of fading memory (i.e., its response cannot keep memory from an infinitely remote past), mathematics operator theory has shown that its response ~ yyðtÞ to any input ~ xxðtÞ can be approximated, within any desired error margin, by ~ yyðsÞ ¼f NL ½ ~ xxðsÞ; ~ xxðs À1Þ; . . . ; ~ xxðs ÀQÞ ð23Þ where f NL (.) is a (Qþ1)-to-one static nonlinear function, s is the time instant in which the output is being calculated; ~ xxðs À1Þ; . . . ; ~ xxðs ÀQÞ are delayed, or past versions of the x(t ) y(t ) Nonlinear / Memoryless Nonlinear / Memoryless jA Q [A(t)]sin y [A(t)] |x(t)| |x(t )| x(t) ~ ~ ~ ~ ~ e j (t) A y [A(t )]e j [A(t )] A(t) A I [A(t)]cos y [A(t)] Figure 15. AM–AM and AM–PM memoryless lowpass equiva- lent behavioral model known as the Saleh quadrature model. Nonlinear/Memoryless e j (t ) A(t ) Linear Filter y(t ) ~ x(t ) ~ z(t ) ~ |x(t )| x(t) ~ ~ |x(t )| ~ A y [A(t ),z(t)]e j y [A(t ),z(t )] ~ ~ H z () Figure 16. An AM–AM/AM–PM model in which the amplifier is modeled as a dynamic gain function of the envelope amplitude. x(t ) y(t ) AM-AM/AM-PM O() H() Figure 17. A three-box, or Wiener–Hammerstein, lowpass equivalent model. INTERMODULATION 2201 input; ~ xxðsÞ; and Q is the system’s finite memory span. Indeed, expression (23) simply states that the system output at a certain instant can be calculated as the non- linear combination of the input at that instant and all its past versions within the memory span. There are basically two ways of implementing this nonlinear and dynamic input–output mapping, depending on whether f NL (.) is approximated by a (Qþ1)-to-one polynomial or by a neural network: polynomial filters [12] and artificial neural networks (ANNs) [16]. In the first case, (23) becomes ~ yyðsÞ ¼ N n¼1 ~ yy n ðsÞ ð24aÞ where ~ yy n ðsÞ ¼ Q q 1 ¼0 Á Á Á Q q 2nþ1 ¼0 ~ hh 2nþ1 ðq 1 ; . . . ; q 2nþ1 Þ ~ xxðs Àq 1 Þ Á Á Á ~ xxðs Àq nþ1 Þ ~ xxðs Àq nþ2 Þ Ã Á Á Á ~ xxðs Àq 2nþ1 Þ Ã ð24bÞ Such a dynamic polynomial formulation (also known as a Volterra filter [12]) presents two important advantages: 1. Its various output components can be traced to a particular coefficient or term. Therefore, it leads to useful concepts as nonlinear order and gives insights into parameter extraction. In fact, this immediately allows model implementations such as the ones depicted in Figs. 18a and 18b for the first- and third-order outputs, ~ yy 1 ðsÞ and ~ yy 3 ðsÞ, respectively. 2. The second advantage, shared with all polynomial approximators, is that the formulation is linear in the parameters (although obviously nonlinear in the inputs). Thus, it allows a direct model parameter extraction based on the solution of a system of simultaneous linear equations. Unfortunately, it also presents an important disadvan- tage. Like any other polynomial approximator, it is a local model. It is mostly this drawback that justifies the alternative ANN formulation. A single hidden-layer ANN can be expressed as [16] ~ uu k ðsÞ ¼ Q q ¼0 ½w k ðqÞ ~ xxðs ÀqÞ þb k ð25aÞ ~ yyðsÞ ¼b o þ K k ¼1 w o ðkÞf s ½ ~ uu k ðsÞ ð25bÞ in which the w k (q) and w o (k) are weighting factors and b o and b k are bias values, constituting the model parameter set. f s (.) are static single-input/single-output nonlinear functions (the so-called activating functions) of sigmoid shape. Because a sigmoid is an output-bounded function, an ANN is well behaved for all inputs. ... ... ... y 3 (s) ... ... ... ... ... ... ... ... ... x(s) (b) x 3 a 3,000 a 3,001 a 3,00Q a 3,011 a 3,01Q a 3,0QQ a 3,112 a 3,1QQ a 3,QQQ a 3,111 x 3 z −1 z −1 z −1 .. .. x 3 ~ ~ y 1 (s) x(s) (a) z −1 z −1 z −1 a 1,0 a 1,1 a 1,2 a 1,Q ... ... ~ ~ Figure 18. Implementation examples of first- (a) and third- (b) order kernels of a general polynomial filter. 2202 INTERMODULATION A direct implementation of a dynamic ANN is shown in Fig. 19. However, recognizing that (25a) constitutes a biased linear FIR filter, whose bias is b k and impulse response is w k (.), this dynamic ANN can also be imple- mented as a set of parallel branches of the Wiener type, as depicted in Fig. 20 [14]. Unfortunately, since all terms of the ANN are similar, there is no way to identify relations between the system’s output properties and any particular ANN terms. Furthermore, as the model is now also nonlinear for the w k (q) and b k parameters, the parameter extraction process must rely on some form of optimization. This optimization process, called ‘‘ANN training,’’ is known to give results that are highly dependent on the input–output training data. Moreover, there is no guarantee that the parameter set found is unique or even optimum, which can constitute a severe limitation to the model’s predictability. 2.4. A Glimpse of Nonlinear Simulation Algorithms for Distortion Prediction In circuit-level simulators [17], the mathematical repre- sentation of the circuit is built by substituting each electronic element with its constitutive relation [e.g., a linear resistor can be represented as Ohm’s law, i ¼v/R; a nonlinear resistor would be given by a voltage-controlled current source, i(v); while a capacitor would be given by a linear or nonlinear charge, q(v)] and then applying Kirchhoff ’s current and voltage laws to the complete circuit. This leads to a system of ordinary nonlinear differential equations (ODEs) in time such as i½yðtÞ þ dq½yðtÞ dt ¼xðtÞ ð26Þ where x(t) and y(t) stand for the time-domain waveform of the excitation and the state-variable vectors, respectively; i[y(t)] represents memoryless linear or nonlinear ele- ments, while q[y(t)] models memoryless linear or non- linear charges (capacitors) or fluxes (inductors). The objective of the simulation is to find the y(t) circuit solution vector given a known x(t) input excitation. On the other hand, system-level simulators are usually implemented as either event-driven or envelope-driven machines. In both cases the simulator treats the system in the time domain, computing a set of time samples of the information signal. Event-driven machines operate at a very high logic level, in which the information is simply a set of successive logic states. They are, therefore, state flow simulators, without enough subsystem description detail to allow distortion calculations. Envelope-driven simulators operate with the analogue complex envelope. Hence, they do not handle the true bandpass RF blocks but simply their complex lowpass equivalents. Nevertheless, since these blocks are still nonlinear dynamic blocks, the lowpass equivalent system mathematical representation will again be an ordinary differential equation similar to (26) with the only differ- ence that now both the excitation vector x(t) and the state variable vector y(t) are, in general, complex entities. So, except for the type of signals handled, an ODEsuch as (26) can be used to represent bandpass RF circuits, bandpass RF systems, or even complex lowpass equivalent systems. 2.4.1. Time-Domain Techniques. The most intuitive way to solve (26) is to covert it into a difference equation i½yðsÞ þ q½yðsÞ Àq½yðs À1Þ T s ¼xðsÞ ð27aÞ or i½yðsÞT s þq½yðsÞ ¼xðsÞT s þq½yðs À1Þ ð27bÞ in which T s is the sampling period, and then determine all time samples of y(t), y(s), starting from a known initial state y(0). Because we are integrating the nonlinear ODE in a set of discretized timesteps, this is known as timestep integration, and constitutes the basic approach adopted in all time-domain circuit simulators (time-marching machines) such as SPICE, or system simulators like Simulink 1 . z −1 z −1 z −1 u k b k w o (k) w k (q) f (u k ) u k b o y s ~ x s ~ Figure 19. Implementation of a nonlinear dynamic artificial neural network. + + + + + + + + + f [u 1 (s)] W 1 ( ) u 1 (s) [u k (s)] u k (s) f [u K (s)] u K (s) b 1 b o b k b k W k ( ) W k ( ) w o (1) w o (k) w o (k) y(s) ~ ~ ~ ~ ~ ~ ~ f x(s) ~ Figure 20. Alternative implementation of the model of Fig. 19, in which the ANN is rebuilt as a parallel combination of several biased linear filter/memoryless nonlinearity branches. 1 Simulink is a general-purpose system simulation package that is supported by the Matlab scientific computation software plat- form. INTERMODULATION 2203 Although timestep integration is still the nonlinear analysis method of wider acceptance, it suffers from several disadvantages in the RF distortion circuit simula- tion field. First, since it was conceived to compute the circuit’s transient response, while our interest normally resides in the steady state, it becomes quite inefficient as it has to wait until all transients have vanished. Also, by operating in the time domain, it cannot handle linear elements having a frequency-domain description, such as dispersive distributed transmission media. Finally, even if that drawback is circumvented (e.g., by approximating these elements by lumped networks of reduced order), the necessity of operating in the time domain, while the input and resulting signals are usually handled in frequency domain, would end up in all difficulties associated with the discrete Fourier transform (DFT), namely, spectral leak- age when transforming quasiperiodic multitone signals. Fortunately, some time-domain alternatives to the initial timestep integration method, like the ‘‘shooting Newton’’ [17], can bypass the transient response, therefore obviat- ing the waste of time needed to let it vanish. Furthermore, time-domain methods benefit from two important advantages: (1) since they rely on the SPICE simulator engine, they are well known and available in many electronic design automation tools; and (2) as they use time as a natural continuation parameter [17], they are especially suitable for supporting strong nonlinear regimes. Envelope-driven system-level simulators must handle the information envelopes, which are aperiodic by nature. So, timestep integration does not suffer from the inefficiency attributed to the calculation of the periodic steady-state response, becoming the obvious choice in solving (26). 2.4.2. Frequency-Domain Techniques. Frequency-do- main techniques no longer seek a set of time samples of the circuit output or the state variables’ waveforms but a spectral representation of them. In their most simple form, they assume that both the steady state of the excitation and the ODE solution are periodic in time, so that they can be expanded in a truncated DFT of (2Kþ1) frequency points. For example, the state variables vector would be represented by yðtÞ ¼ K k¼ÀK Yðko 0 Þe jko 0 t ð28Þ Since, in the frequency domain, time-domain derivatives are transformed into products by jo, substituting (28) into (26) leads to I½YðoÞ þjXQ½YðoÞ ¼XðoÞ ð29Þ which is a nonlinear algebraic function in the DFT coeffi- cients Y(ko 0 ). The orthogonality between different fre- quency components provided by the DFT determines that, despite its appearance, this is not a single equation but can be expanded in a set of (2Kþ1) equations, each of these must be fulfilled for its harmonic component; in other words, the LHS and RHS (left- and right-hand side) components must be in equilibrium, which is why (29) is known as the ‘‘harmonic-balance equation.’’ Since this harmonic-balance (HB) technique computes the periodic steady state directly, it circumvents most of the disadvantages attributed to time-marching techni- ques. Its only drawbacks are that, depending on the DFT, it can handle only moderate nonlinear regimes, where the y(t) can be described by a relatively small number of harmonics, and that it requires both the ex- citation and the vector of state variables to be periodic. As we have already seen in Section 1, the excitations used for intermodulation distortion analysis are often of the two- tone or multitone type. In general, the frequencies of these tones do not constitute any harmonic set (they cannot be made harmonics of a common fundamental), and the corresponding waveform is aperiodic. (Such multitone signals are actually said to be quasiperiodic waveforms.) One way to circumvent this problem consists in imagining that a multitone time-domain waveform is evolving, not in the natural time t, but in a number of artificial timescales equal to the number of nonharmonically related tones, t 1 ,y,t Q . For example, for a two-tone regime, the ODE in time becomes a multirate partial-differential equation (MPDE) in t 1 and t 2 : i½yðt 1 ; t 2 Þ þ @q½yðt 1 ; t 2 Þ @t 1 þ @q½yðt 1 ; t 2 Þ @t 2 ¼xðt 1 ; t 2 Þ ð30Þ Since y(t 1 , t 2 ) is now double-periodic in t 1 and t 2 , it admits a bidimensional Fourier expansion yðt 1 ; t 2 Þ ¼ K k 1 ¼ÀK K k 2 ¼ÀK Yðk 1 o 1 ; k 2 o 2 Þe jðk 1 o 1 t 1 þk 2 o 2 t 2 Þ ð31Þ which, substituted in (30), results in a new bidimensional HB equation. This is the technique known as the multi- dimensional discrete Fourier transform harmonic-balance (MDFT HB). 2.4.3. Time-Domain/Frequency-Domain Hybrid Techni- ques. When the excitation is a RF carrier of frequency o c , modulated by some independent baseband modulation signal, like the one expressed in (5), it can be again conceived as varying according to two independent time- scales: one, t 1 , with fast evolution, for the carrier; and another, t 2 , slower, for the modulation. So, the circuit can again be described by a bidimensional MPDE such as (30). If we now recognize that this regime is periodic for the carrier but aperiodic for the modulation, we immediately conclude that simulation efficiency would be maximized if we treated the carrier evolution in t 1 in the frequency domain, but kept the baseband evolution t 2 in time. This supposes a solution in which the vector of state variables is decomposed in a t 2 time-varying Fourier series yðt 1 ; t 2 Þ ¼ K k ¼ÀK Yðko c ; t 2 Þe jðkoct 1 Þ ð32Þ 2204 INTERMODULATION which, substituted in (30), leads to I½Yðko c ; t 2 Þ þjX c Q½Yðko c ; t 2 Þ þ @Q½Yðko c ; t 2 Þ @t 2 ¼Xðko c ; t 2 Þ ð33Þ Solving (33) for the envelope, with a timestep integration scheme, and for the carrier, with harmonic balance, leads to the following recursive HB equation: I½Yðko c ; sÞT s þjX c Q½Yðko c ; sÞT s þQ½Yðko c ; sÞ ¼Xðko c ; sÞT s þQ½Yðko c ; s À1Þ ð34Þ By handling the RF signal components in the frequency domain and the envelope in the time domain, (34) is particularly appropriate to bridge the gap between circuit and envelope-driven system simulation. In fact, we can conceive of a simulator in which all except a few circuits of a communication system are treated as system-level com- plex equivalent lowpass behavioral input–output blocks— for maximized computational efficiency—while the re- maining circuits are treated at the RF bandpass circuit level—for maximum accuracy. 2.4.4. Volterra Series. Although the Volterra series method is not very widely used outside the intermodula- tion prediction field, it plays a determinant role for the analysis and design of very-low-distortion circuits. In comparison with the previously mentioned methods, Volterra series no longer tries to find a solution in an iterative and numerical way, but seeks for an analytic solution of a polynomial approximation of the original circuit or system. In fact, it assumes that if the nonlinea- rities of the original circuit or system can be decomposed in a Taylor series around a certain fixed quiescent point iðyÞ ¼g 1 yþg 2 y 2 þg 3 y 3 ð35Þ qðyÞ ¼c 1 yþc 2 y 2 þc 3 y 3 ð36Þ then, the solution can be approximated by the following functional series in the time domain: yðtÞ ¼y 1 ðtÞ þy 2 ðtÞ þy 3 ðtÞ ¼ _ 1 À1 h 1 ðtÞxðt ÀtÞdt þ _ 1 À1 _ 1 À1 h 2 ðt 1 ; t 2 Þxðt Àt 1 Þxðt Àt 2 Þdt 1 dt 2 þ _ 1 À1 _ 1 À1 _ 1 À1 h 3 ðt 1 ; t 2 ; t 3 Þxðt Àt 1 Þ Âxðt Àt 2 Þxðt Àt 3 Þ dt 1 dt 2 dt 3 ð37Þ If the excitation can be expressed as a frequency-domain sum of complex exponentials (possibly, but not necessarily, harmonically related sinusoids) xðtÞ ¼ Q q ¼ÀQ Xðo q Þe joqt ð38Þ then we obtain a frequency-domain version of (37) yðtÞ ¼ Q q¼ÀQ H 1 ðoÞXðo q Þe joqt þ Q q 1 ¼ÀQ Q q 2 ¼ÀQ H 2 ðo q 1 ; o q 2 ÞXðo q 1 ÞXðo q 2 Þe jðoq 1 þoq 2 Þt þ Q q 1 ¼ÀQ Q q 2 ¼ÀQ Q q 3 ¼ÀQ H 3 ðo q 1 ; o q 2 ; o q 3 Þ ÂXðo q 1 ÞXðo q 2 ÞXðo q 3 Þe jðo q 1 þo q 2 þo q 3 Þt ð39Þ in which the h n (t 1 ,y,t n ) of (37) and the H n (o 1 ,y,o n ) of (39) are the nth-order impulse responses and the nth- order nonlinear transfer functions, respectively. Each of these sets can be obtained from the other by the direct application of a n-dimensional Fourier transform pair. The Volterra series method consists in determining the set of h n (.) or of H n (.) (as occurs with conventional linear systems, the frequency-domain version is usually pre- ferred), which then becomes a true nonlinear dynamic model of the system. In fact, note that if one knows all the H n (o 1 ,y,o n ) of a circuit or system, up to a certain order, one immediately knows its response up to that order [from (39)] to any multitone input represented by (38). To show how these nonlinear transfer functions can be determined, let us consider again the general circuit or system described by the ODE of (26). Substituting (35), (36), and (39) into (26), and assuming that the input is now a first-order elementary excitation of xðtÞ ¼e jot ð40Þ the orthogonality of the complex exponentials leads us to the conclusion that H 1 (o) must be given by H 1 ðoÞ ¼ 1 g 1 þjoc 1 ð41Þ In fact, this H 1 (o) is merely the usual transfer function of the linear circuit or system obtained from a linearization around the quiescent point. To obtain the second-order nonlinear transfer function, we would now assume that the system is excited by a second-order elementary excitation of xðtÞ ¼e jo 1 t þe jo 2 t ð42Þ Substituting (35), (36), (39), and (42) into (26), and collect- ing components in the second-order mixing product of INTERMODULATION 2205 o 1 þo 2 would lead to H 2 ðo 1 ; o 2 Þ ¼ À g 2 þjðo 1 þo 2 Þc 2 g 1 þjðo 1 þo 2 Þc 1 H 1 ðo 1 ÞH 1 ðo 2 Þ ð43Þ Similarly, the calculation of the third-order nonlinear transfer function assumes an input of xðtÞ ¼e jo 1 t þe jo 2 t þe jo 3 t ð44Þ and leads to H 3 ðo 1 ; o 2 ; o 3 Þ ¼ À 2 3 g 2 þjðo 1 þo 2 þo 3 Þc 2 g 1 þjðo 1 þo 2 þo 3 Þc 1 ½H 1 ðo 1 ÞH 2 ðo 2 ; o 3 Þ þH 1 ðo 2 Þ ÂH 2 ðo 1 ; o 3 Þ þH 1 ðo 3 ÞH 2 ðo 1 ; o 2 Þ À g 3 þjðo 1 þo 2 þo 3 Þc 3 g 1 þjðo 1 þo 2 þo 3 Þc 1 ÂH 1 ðo 1 ÞH 1 ðo 2 ÞH 1 ðo 3 Þ ð45Þ The terms À[g 2 þj(o 1 þo 2 )c 2 ]H 1 (o 1 )H 1 (o 2 ) in (43) and the terms À 2 3 ½g 2 þjðo 1 þo 2 þo 3 Þc 2 ½H 1 ðo 1 ÞH 2 ðo 2 ; o 3 Þ þH 1 ðo 2 ÞH 2 ðo 1 ; o 3 Þ þH 1 ðo 3 ÞH 2 ðo 1 ; o 2 Þ and À½g 3 þjðo 1 þo 2 þo 3 Þc 3 H 1 ðo 1 ÞH 1 ðo 2 ÞH 1 ðo 3 Þ in (45) are known as the elementary second-order and third-order nonlinear sources, respectively. In fact, com- paring (43) and (45) with (41), we immediately conclude that for the calculation of first-, second-, and third-order solutions, what we have been doing was to always analyze the same linearized version of the original ODE with the appropriate elementary nonlinear sources at o for first order, at o 1 þo 2 for second order, and at o 1 þo 2 þo 3 for third order. That is why the method of Volterra series analysis is known to solve a forced nth-order nonlinear dynamic problem, solving n times the same linear pro- blem, with the appropriate (1st, 2nd,y,nth)-order forcing functions, in a recursive way. As it is based on a poly- nomial approximation of the nonlinearities, the Volterra series is a local model restricted to small-signal levels, or mildly nonlinear regimes. In practice, it can be used only for calculating the distortion in nonsaturated mixers, small-signal amplifiers, or nonsaturated power amplifiers, that is, well below the 1-dB compression point. However, because it is a closed-form nonlinear model, it provides qualitative information on the nonlinear circuit or sys- tem’s operation, giving, for instance, insight into the physical origins of nonlinear distortion, and can be di- rectly used for circuit and system’s design. 3. INTERMODULATION DISTORTION IN SMALL-SIGNAL AMPLIFIERS First, let us clarify the meaning of ‘‘small-signal ampli- fiers,’’ as most of us would expect no appreciable nonlinear distortion from these circuits. This term is used to distin- guish low-noise or high-gain amplifiers from the essen- tially different power amplifiers, treated in the next section. While the small-signal amplifiers referred to here always supposedly operate in highly backed-off class A regimes, power amplifiers are operated close to satura- tion, usually even in strongly nonlinear modes as class AB, B, or C. So, now, one question comes to our minds: ‘‘What are the mechanisms capable of causing significant nonlinear distortion in small-signal amplifiers?’’ To advance with an answer, let us consider the case of the low-noise amplifier of a wireless communication receiver front end. This is a circuit that faces, beyond the very weak desired channel, many other incoming channels present in the same com- munication system’s band. For example, a low-noise front end of a handset can be simultaneously excited by the desired channel coming from a remote base station, and by another channel coming from a nearby transmitter hand- set. Since the ratio of distances between our receiver and the base station, and our receiver and the perturbing transmitter, can easily be on the order of several kilo- meters to one meter, the ratio of incoming powers can be higher than 10 8 to 1. Therefore, the signal-to-perturbation ratio can be as poor as À70 or À80 dB, and, if it is true that a desired signal of, say, À70 dBm cannot generate any significant nonlinear distortion, that is no longer the case for the þ10 dBm perturbation. Indeed, as seen in Section 1, this high-level perturba- tion can produce nonlinear distortion sidebands falling over the desired channel, cross-modulation, and desensi- tization. These effects are illustrated in Fig. 21 and can constitute a severe limitation to the performance of RF front ends. In fact, they allow the definition of a very important signal fidelity figure of merit, the dynamic range, which is the ratio between the amplitudes of the highest and lowest incoming detectable signals that still guarantee the specified signal-to-noise ratio SNR. The lowest-amplitude detectable signal—the receiver’s sensi- tivity S i —is the one that stands the desired SNR over the noise floor. The highest-amplitude detectable signal, P max , is defined as the one that generates a nonlinear distortion perturbation whose power equals the noise floor. So DR P max S i or DR dB P max dBm ÀS i dBm ð46Þ where DR is the dynamic range. Since signal excursions appearing at the nonlinear active device are always kept much smaller than the applied DC voltages and currents, the amplifier can be approximately described by a local model. For example, considering the ideal (low-frequency) situation in which the only nonlinear effects can be attributed to the i ds (v gs ,v ds ) current of a FET, the amplifier would be described by the equivalent circuit depicted in Fig. 22, 2206 INTERMODULATION while the nonlinearity would be represented by the Taylor series of (19). Although the model shown in Fig. 22 is very simplified, it will already give us some insight onto the mechanisms controlling IMD generation in these small-signal ampli- fiers. For that, we will first redraw this circuit as the one of Fig. 23 over which a Volterra series analysis will then be performed. Note that ports 1 and 2 in Fig. 23 handle the amplifier’s input and output, respectively, but were defined after the terminal admittances Y S and Y L were incorporated into the main circuit. port 3 serves to define v gs , one of the control voltages of the i ds nonlinearity, and port 4 serves to define v ds , the other control voltage. Furthermore, since Fig. 23 is the linearized equivalent-circuit version of the original circuit of Fig. 22, its only i ds (v gs ,v ds ) components are the first-order ones: G m v gs and G ds v ds . All the other nonlinear terms of (19) will behave as nonlinear sources that will be incorporated as independent current sources applied to port 4 [6,7]. Assuming that the equivalent Norton current excita- tion corresponds to a narrowband two-tone stimulus, this circuit can be represented by the following [Z] matrix and input and output boundary conditions: V 1 ðoÞ V 2 ðoÞ V 3 ðoÞ V 4 ðoÞ _ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ _ ¼ Z 11 Z 12 Z 13 Z 14 Z 21 Z 22 Z 23 Z 24 Z 31 Z 32 Z 33 Z 34 Z 41 Z 42 Z 43 Z 44 _ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ _ . I 1 ðoÞ I 2 ðoÞ I 3 ðoÞ I 4 ðoÞ _ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ _ ð47aÞ I 1 ðoÞ ¼I s ðoÞ : i s ðtÞ ¼I s 1 cosðo 1 tÞ þI s 2 cosðo 2 tÞ ð47bÞ I 2 ðoÞ ¼0 ð47cÞ I 3 ðoÞ ¼0 ð47dÞ If the two-tones are closely separated in frequency, the circuit reactances are similar for all in-band products. So, using o 0 as the center frequency [o 0 ¼(o 1 þo 2 )/2], Z i- ij (o 1 )EZ ij (o 2 )EZ ij (2o 1 Ào 2 )EZ ij (2o 2 Ào 1 )EZ ij (o 0 ) for any i, j ¼1,2,3,4. After determining the linear equivalent-circuit [Z] matrix, the nonlinear currents’ method of Volterra series analysis [6,7] proceeds by determining first-order control voltages V 3,1 (o 0 ) and V 4,1 (o 0 ) (q¼1,2) and first-order output voltage V 2,1 (o 0 ), from the excitation I s (o 0 ): V gs;1 ðo 1 Þ; V gs;1 ðo 2 Þ : V 3;1 ðo 0 Þ ¼Z 31 ðo 0 Þ I S ðo 0 Þ 2 ð48aÞ V ds;1 ðo 1 Þ; V ds;1 ðo 2 Þ : V 4;1 ðo 0 Þ ¼Z 41 ðo 0 Þ I S ðo 0 Þ 2 ð48bÞ V L;1 ðo 1 Þ; V L;1 ðo 2 Þ : V 2;1 ðo 0 Þ ¼Z 21 ðo 0 Þ I S ðo 0 Þ 2 ð49Þ From these first-order control variables, the second-order nonlinear current of i ds at o 1 Ào 2 Do and o 1 þo 2 So, SNR DR P Max S i N i S xx () Figure 21. Nonlinear distortion impairments in a mildly nonlinear receiver system: illustration of the concepts of receiver’s desensitization and dynamic range. + − + − + − + − i DS (v GS ,v DS ) v S (t ) v DS (t ) R 0 v o (t) R 0 Z L ( ) v GS (t ) V DS V GS Input matching network Output matching network Figure 22. Model of a mildly nonlinear amplifier for small-signal distortion studies. INTERMODULATION 2207 I 4,2 (o) should now be determined: I 4;2 ðDoÞ ¼ À½2G m2 jZ 31 ðo 0 Þj 2 þG md Z 31 ðo 0 ÞZ 41 ðo 0 Þ Ã þG md Z 31 ðo 0 Þ Ã Z 41 ðo 0 Þ þ2Gd 2 jZ 41 ðo 0 Þj 2 jI S j 2 4 ð50Þ I 4;2 ðSoÞ ¼ À2½G m2 Z 31 ðo 0 Þ 2 þG md Z 31 ðo 0 ÞZ 41 ðo 0 Þ þG d2 Z 41 ðo 0 Þ 2 I 2 S 4 ð51Þ Then, the linear circuit should be analyzed again for this new second-order current source, determining the second- order control voltages at the difference o 1 Ào 2 Do and sum o 1 þo 2 So frequencies, V 3,2 (o) and V 4,2 (o): V 3;2 ðDoÞ ¼Z 34 ðDoÞI 4;2 ðDoÞ ð52Þ V 3;2 ðSoÞ ¼Z 33 ðSoÞI 3;2 ðSoÞ þZ 34 ðSoÞI 4;2 ðSoÞ ð53Þ V 4;2 ðDoÞ ¼Z 44 ðDoÞI 4;2 ðDoÞ ð54Þ V 4;2 ðSoÞ ¼Z 43 ðSoÞI 3;2 ðSoÞ þZ 44 ðSoÞI 4;2 ðSoÞ ð55Þ The last step consists in calculating the third-order non- linear current of i ds at 2o 1 Ào 2 , I 4,3 (2o 1 Ào 2 ) from first- and second-order control voltages V 3,1 (o), V 3,2 (o), V 4,1 (o), and V 4,2 (o): I 4;3 ð2o 1 Ào 2 Þ ¼ À½2G m2 Z 31 ðo 0 Þ Ã Z 34 ð2o 0 ÞI 4;2 ð2o 0 Þ þG md Z 31 ðo 0 Þ Ã Z 44 ð2o 0 ÞI 4;2 ð2o 0 Þ þG md Z 41 ðo 0 Þ Ã Z 34 ð2o 0 ÞI 4;2 ð2o 0 Þ þ2G d2 Z 41 ðo 0 Þ Ã Z 44 ð2o 0 ÞI 4;2 ð2o 0 Þ I à S 2 À½2G m2 Z 31 ðo 0 ÞZ 34 ðDoÞI 4;2 ðDoÞ þG md Z 31 ðo 0 ÞZ 44 ðDoÞI 4;2 ðDoÞ þG md Z 41 ðo 0 ÞZ 34 ðDoÞI 4;2 ðDoÞ þ2G d2 Z 41 ðo 0 ÞZ 44 ðDoÞI 4;2 ðDoÞ I S 2 À 3G m3 ½ Z 31 ðo 0 ÞjZ 31 ðo 0 Þj 2 þG m2d Z 31 ðo 0 Þ 2 Z 41 ðo 0 Þ Ã þ2G m2d jZ 31 ðo 0 Þj 2 Z 41 ðo 0 Þ þG md2 Z 31 ðo 0 Þ Ã Z 41 ðo 0 Þ 2 þ2G md2 Z 31 ðo 0 ÞjZ 41 ðo 0 Þj 2 þ3G d3 Z 41 ðo 0 ÞjZ 41 ðo 0 Þj 2 I S jI S j 2 8 ð56Þ and then, finally, determine third-order output voltage at the IMD frequency 2o 1 Ào 2 : V 2;3 ð2o 1 Ào 2 Þ ¼Z 24 ðo 0 ÞI 4;3 ð2o 1 Ào 2 Þ ð57Þ Now we are in a position to calculate the amplifier’s signal-to-IMD ratio (IMR) by first determining output power at the fundamental P L ¼ 1 2 G L ðo 0 ÞjV L ðo 0 Þj 2 ¼ 1 2 G L ðo 0 ÞjZ 21 ðo 0 Þj 2 jI S j 2 ð58Þ and IMD components P L 3 ¼2G L ðo 0 ÞjV 2;3 ð2o 1 Ào 2 Þj 2 ð59Þ where G L (o) is the real part of load admittance Y L (o). A full expression for this IMR would be very complex. But, under the assumptions that internal feedback is negligible [Z 34 (o)E0] and that both second-harmonic and o 1 Ào 2 distortion will be very small (usually verified in small- signal amplifiers), it can be approximated by IMR %16 A v ðo 0 Þ Z D ðo 0 Þ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ 2 . j3G m3 þG m2d A v ðo 0 Þ Ã þ2G m2d A v ðo 0 Þ þG md2 A v ðo 0 Þ 2 þ2G md2 jA v ðo 0 Þj 2 þ3G d3 A v ðo 0 ÞjA v ðo 0 Þj 2 j À2 jV S j À4 ð60Þ where Z D (o)1/[G ds þY L (o)], A v (o) is the intrinsic voltage gain defined by A v (o)V ds (o)/V gs (o), and V S is the voltage amplitude of the signal source, V S (o) ¼Z S (o)I S (o). If we now study the variation of various third-order current components with V GS bias, as shown in Fig. 24 for a typical general-purpose small-signal MESFET, we con- clude that there are two very good points of IMD perfor- mance, the so-called small-signal IMD ‘‘sweet spots.’’ The first one is located at the FET’s threshold voltage [6] and thus has a very small associated power gain. The other is located in high-V GS regions, and although it may not correspond to very good noise figures, it is definitely useful for designing high-gain, highly linear small-signal ampli- fiers. Unfortunately, this latter small-signal IMD ‘‘sweet spot’’ is a peculiarity of only some GaAs MESFETs, and was never observed on HEMTs, MOSFETs or BJTs. C gs C gd R i R s L s I ds,1 V gs , ( ) V ds I 4 V 3 =V gs I 3 V 2 I 2 I S Y S I 1 V 1 V 4 =V ds Y L + − + − + − + − Figure 23. Linearized equivalent-circuit model of a FET-based mildly nonlinear amplifier used in Volterra series calculations. 2208 INTERMODULATION Turning now our attention to the IMR variation with source impedance, we can conclude that since P L can also be given by P L ¼ 1 2 G L ðo 0 ÞjA v ðo 0 Þj 2 jV S j 2 , Eq. (60) confirms the empirical observation that, for constant output power, third-order distortion in FET-based small-signal ampli- fiers is almost independent of input termination Z S (o). As far as the IMR variation with load impedance is concerned, Fig. 24 and (60) indicate that, for typical V GS bias, the nonlinear current contributions of G m3 and G m2d have effects on IMR that are important but, fortunately, opposite in sign. Since NIG m2d is proportional to voltage gain, and thus to Z L (o), this implies that a maximization of voltage gain can also be beneficial to IMR. This hypoth- esis was indeed fully confirmed by the measured and simulated IMR 3 load-pull data [9], showing that and optimum Z L (o) really exists and it tends to coincide with the one that maximizes small-signal voltage and power gains in MESFET-based small-signal amplifiers [6,9]. Since BJTs and HBTs have mildly nonlinear character- istics that are completely different from those of FETs, these results cannot be directly extrapolated to bipolar- based small-signal amplifiers. For example, while the most important nonlinearity source, i DS (v GS ,v DS ), is lo- cated at the FET’s output, in bipolars it is manifested in both the input, i B (v BE ,v CE ), and the output, i C (v BE ,v CE ) [6]. 4. INTERMODULATION DISTORTION IN HIGH-POWER AMPLIFIERS Let us now turn our attention to power amplifiers (PAs). Contrary to small-signal amplifiers where noise figure and gain are of primary concern, power amplifiers are de- signed for high output power P out and power-added effi- ciency (PAE). In a well-designed PA, maximum output power is determined by the loadline (load impedance termination) and available output signal excursion. Power-added effi- ciency is dependent mostly on the PA operation class (quiescent point) and on a convenient output voltage and current waveform shaping, specifically, selection of har- monic terminations. Therefore, it seems that little is left for optimizing intermodulation distortion. Fortunately, as we will see, that is not necessarily the case. Since real devices do not present abrupt turnon points, it is difficult to precisely define the PA operation class. So, to prevent any ambiguity in the following discussion, we will first define classes A, AB, B, and C. Taking into account the discussion in Ref. 6, we will adopt the follow- ing definitions: (1) the turnon bias V T is defined as the input quiescent point to which the turnon small-signal IMD ‘‘sweet spot’’ corresponds (see Fig. 24); (2) biasing the device below V T corresponds to class C (G m3 40); (3) biasing it exactly at V T corresponds to class B (G m3 ¼0); and (4) biasing it above V T will determine the usual class AB or class A (G m3 o0). The first design step to be taken when designing a PA is to decide whether precedence should be given to PAE or to IMD specs, as they generally lead to opposite design solutions. The traditional PA design rules state that a PA optimized for IMD requires unsaturated class A opera- tion; that is, the device should be biased and always kept comfortably inside the linear amplification zone (saturation region of FETs and the active region of BJTs or HBTs). On the other hand, a PA optimized for PAE is usually biased near class B or slightly into class C—that is, with a quiescent point where output voltage is halfway between knee voltage and breakdown, and output current is close to turnon—and then is allowed to be driven into satura- tion. This leads to saturated classes such as classes E and F [18]. Unfortunately, as such operation classes achieve their high efficiencies by operating the active device in an almost switching mode, their associated nonlinear distor- tion is also huge. In fact, recognizing that a switching power amplifier turns any waveform into a constant- amplitude square wave, it is easy to conclude that those class E or F PAs cannot be used when the amplitude of the RF-modulated signal also carries information content (modulation formats of non-constant-amplitude envelope). The basic goal when designing linear PAs is to get class B PAE with class A IMD—and, although this is seldom possible, there are some particular PA features that provide a means to escape from this apparent deadend. One that has been receiving a great deal of attention is the so-called large-signal IMD sweet spots [19]. Contrary to their small-signal counterparts studied above, which were associated to a particular quiescent point and found effective only at very-small-signal levels, these are pecu- liar points of the IMD–input power characteristic (see Fig. 25) where only a few decibels of output-power backoff (and thus a few percent of efficiency degradation) can lead to astonishingly high levels of IMD reduction. To understand how this curious effect takes place, we need to abandon our small-signal local model, since, for the signal levels where these large-signal IMD sweet spots are observed, the Taylor expansion of (19) presents an unac- ceptable error or simply may not converge. Instead, we are –40 –20 0 20 40 60 80 NIG m3 , NIG m2d , NIG md2 , NIG d3 , NI d3 (mS/V 2 ) –2 –1.8 –1.6 –1.4 –1.2 –1 –0.8 –0.6 –0.4 –0.2 0 V GS (V) –40 –20 0 20 40 60 80 Figure 24. Magnitude of total i ds (v gs ,v ds ) third-order current components NI d3 (—) and its various components due to G m3 NIG m3 (–K–), G m2d NIG m2d (– þ–), G md2 NIG md2 (Á Á Á), and G d3 NIG d3 (- - -) as a function of V GS bias. INTERMODULATION 2209 forced to rely on qualitative solutions of approximated global models. For that, we will transform the bidimen- sional dependence of (19) on the input and output control voltages, v GS (t) and v DS (t), into a one-dimensional model, generating in this way an equivalent single-input/single- output transfer function (TF), i DS [v GS (t)]. This assumes an output boundary condition imposed by load impedance, Z L (o), V ds (o) ¼V DC ÀZ L (o)I ds (o), where V DC is the applied output quiescent voltage, beyond knowledge of the active device nonlinear model i DS [v GS (t), v DS (t)]. In order to describe, with enough generality, the global nonlinearities of the device, we will also consider that turnon can be represented by an exponential of input voltage. This method is commonly adopted to represent subthreshold conduction in FETs, and is even more faith- ful for describing i C (v BE ) in common-base or common- emitter bipolar devices. Then, for increasing v GS voltages, it is assumed that the FET passes through a nearly quadratic zone, which, because of short-channel effects, tends to become linear for even higher v GS . This i DS (v GS ) behavior was shown to be well reproduced by the following empirical expression [20] i DS ðv GS Þ ¼b ½smtðv GS Þ 2 1 þy smtðv GS Þ ð61aÞ where smt(v GS ) is a smooth turnon function of v GS given by smtðv GS Þ ¼K V ln 1þe v GS K V _ _ ð61bÞ and b, y and K V are empirical scaling parameters. If we now take the output boundary into account, i DS (v DS ) will be almost unchanged unless v GS is so high that R L Á i DS becomes close to V DC . There, v DS is so small that the FET enters the triode region (the saturation region for a bipolar based PA). v GS rapidly looses control over i DS , and the TF saturates (the PA enters into strong compression). So, the global transfer characteristic i DS (v GS ) presents a distinct sigmoid shape. Assuming again a two-tone stimulus, several qualita- tive conclusions may be drawn for large-signal operation. One of the most important is that when the amplifier is driven into harder and harder nonlinear regimes, its gain goes into compression, which means that the phase of the in-band nonlinear distortion components must oppose those of the fundamentals. So, PA energy balance con- siderations derived in Section 1 show that the large-signal asymptotic phase value of the IMD sidebands, at 2o 1 Ào 2 and 2o 2 Ào 1 , must tend to a constant value of 1801 [19]. On the other hand, we also know that small-signal IMD phase is determined by the sign of the TF local derivatives, determined by the active device’s quiescent point. As seen above, G m3 is positive below V T (class C operation), is null exactly at V T (class B) and is negative above V T (classes AB and A). So, since small-signal IMD sign can be made positive, and tends to negative values in the large-signal asymptotic regime, the Bolzano–Weierstrass theorem guarantees the existence of at least one IMD null some- where in between. This will be observed as a more or less pronounced notch in an IMD–P in plot, constituting a large- signal IMD sweet spot. From these general conclusions it is clear that the existence of a large-signal IMD sweet spot depends on the small-signal IMD phase and on the physical effects that determine large-signal gain compression. So, each operation class will have its own particular IMD behavior. Under class C, V GS oV T , G m3 40, the PA presents gain expansion and a high IMD level with 01 phase. When the signal excursion reaches a strong nonlinearity as the gate–channel junction conduction, gate–channel break- down, or, more likely, the saturation–triode region transi- tion, the PA enters into compression and an IMD notch is observed. So, a large-signal IMD sweet spot should be expected for class C amplifiers when the signal excursion is at the onset of saturation, not far from the PA’s 1-dB compression point. Although the PAE is not yet at its maximum, it may present an interesting value. In class A, V GS 4V T , G m3 o0, the PA starts at small signal with almost unnoticeable gain compression and a very small level of IMD with 1801 phase. As this phase is maintained when the device enters strong compression, no IMD sweet spot will be generated. Thus, and unless the PA is biased above the small-signal IMD sweet spot found for high V GS bias in certain MESFETs [9], no large-signal IMD sweet spot will be observed. On the contrary, a sudden increase of IMD power at the on-set of saturation is the usual outcome of class A PAs. In class AB, where V GS is only slightly higher than V T and G m3 o0, the PA again shows a very shallow gain compression and a low-level IMD of 1801 phase. Hence, similar to what was concluded for class A operation, no IMD sweet spot should be expected. Nevertheless, depend- ing on the abruptness of turnon and succeeding lineariza- tion of the TF characteristic, it can be shown that a transition from 1801 to 01 IMD phase can occur at lower values of output power [20,21] generating an IMD sweet spot. At this stage, the circuit begins to behave as a class C PA, with 01 IMD phase and gain expansion. Conse- quently, a new IMD sweet spot will have to occur at large signal when gain compression will finally take place. In –80 V GS = –0.35V V GS = –0.41V V GS = –0.53V –90 –70 –60 –50 –40 –30 –20 –10 –15 –10 –5 0 5 P in (dBm) P nout (dBm) Figure 25. Different IMD versus P in patterns showing large- signal IMD ‘‘sweet spots’’ for a HEMT device. 2210 INTERMODULATION summary, depending on the actual device’s transfer char- acteristic and on the adopted quiescent point, class AB may be significantly different from class A in that it may even present two IMD sweet spots, one for small to moderate levels of P in and another for the onset of saturation. When the device is biased for class B (i.e., V GS ¼V T and G m3 ¼0), there is no small-signal IMD to be compensated by the large-signal distortion behavior. The PA presents very low levels of small-signal distortion (remember that it was biased at a small-signal IMD sweet spot) and then presents a sudden rise of distortion power at the onset of saturation. To illustrate the results of this analysis, Fig. 26 shows three IMR–P in patterns typically observed in MOSFET-based PAs biased for classes A, AB, and C, respectively. To close this qualitative analysis, let us draw some conclusions about the dependence of large-signal IMD sweet spots on impedance terminations. Starting with source impedance, it is intuitive to realize that, because the i DS (v GS ,v DS ) nonlinearity is located at the output, the large-signal IMD behavior will be mostly invariant with Z S (o), as was the case studied earlier for small-signal amplifiers. Note that this conclusion may not be extra- polated to bipolar based amplifiers, in which there is an input nonlinearity due to the base–emitter junction, and an output nonlinearity due to the active-to-saturation region transition [6,22]. As seen from the small-signal Volterra series analysis above, the dependence of i DS on v DS should also produce its own impact on the large-signal IMD sweet spot, via Z L (o). In fact, since these sweet spots were related to the output signal excursion that crosses the saturation-to-triode re- gion transition, and as the loadline slope determines that signal level (see Fig. 27), it should be expected that the P in for which the IMD sweet spot is observed will be strongly dependent on load termination. This is illustrated in Figs. 27a and 27b, where a shift of the simulated IMD sweet- spot position is evident when loadline slope 1/R L is varied. Furthermore, if the PA output is not perfectly matched, the intrinsic load impedance actually presented to the nonlinear current source may have a certain nonnull phase. The output-induced large-signal distortion compo- nents will no longer be in exact opposite phase with the small-signal ones, and the previously observed large- signal IMD sweet spots cease to be sharp dips of IMD power to become more or less smooth valleys. Further conclusions can also be drawn about the im- pact of out-of-band terminations on the large-signal IMD sweet spots [6,22]. As was seen above from the small- signal analysis, the presence of even-order mixing pro- ducts—which, as we have already seen, can be remixed with the fundamentals—will generate new odd-order pro- ducts. But, contrary to the small-signal case in which it was assumed that the quasilinear operation of the ampli- fier would determine a minor effect to these indirect odd- order products, that is no longer valid for a PA, and its analysis becomes again much more complex: 1. Efficiency considerations may have previously dictated a certain second-harmonic termination. Further, if in most usual situations we seek a squared output voltage waveform, that is, without even-order harmonics, there are situations (e.g., the I Max V K V DC V BR = V Max v DS i DS (v GS ,v DS ) P IMD (dBm) Q P in (dBm) −180 −40 −30 −20 −10 0 10 20 −160 −140 −120 −100 −80 −60 −40 −20 0 20 (a) (b) R L R L Q R L Figure 27. Impact of PA loadline slope 1/R L on large-signal IMD ‘‘sweet spots’’. IMR (dBc) 70 60 50 40 30 20 10 –10 –5 0 5 10 15 20 P out (dBm) Class C Class AB Class A Figure 26. IMR versus P out power plots of typical MOSFET- based PAs at the three operation classes studied: C ( Á Á Á Á ), AB (- - -) and A (—). INTERMODULATION 2211 so-called inverse class F [18,23]) in which those even harmonics are indeed maximized. 2. If, in small-signal amplifiers, there would be no difficulty in designing bias networks presenting a very low impedance to the modulation baseband (o 2 Ào 1 ), in PAs that is again incomparably more difficult. Indeed, as output currents may be on the order of several amperes, any parasitic resistance or inductance may immediately develop a nonnegligi- ble output voltage. 3. There will be even additional, contributing base- band reactances in PAs from more or less unex- pected physical origins. That is the case of trap- induced low-frequency dispersion presented by some microwave active devices [24], and dynamic self- heating, common to almost all PAs [25]. Depending on the phase of the out-of-band terminations, these new indirect odd-order products may have a phase that either reinforces or reduces the directly generated IMD. As far as the even-harmonics-induced products are concerned, since the modulation bandwidth (or the two- tone separation Do) is usually much smaller than the PA bandwidth, it may be assumed that Z L (2o 1 )EZ L (2o 2 )E Z L (2o 0 ). So, no important IMD behavior variation within the bandwidth should be expected; that is, the indirect odd-order distortion products may reduce, reinforce, or be in quadrature with the direct ones, but their impact will be the same along the whole modulation bandwidth. The situation regarding the baseband-induced pro- ducts is completely different. Now, Z L (Do) may vary significantly within the modulation bandwidth, especially if the bias networks present resonances. Therefore, it is likely that IMD power will vary within that bandwidth, and the amplifier will show (undesirable) long-term mem- ory effects. Moreover, the complex conjugate symmetry of load impedance requires that the imaginary part of Z L (o 2 Ào 1 ) have a sign opposite that of Z L (o 1 Ào 2 ). So, if some other odd-order products (e.g., the ones due to the presence of second harmonics) also have significant ima- ginary parts, their addition will even produce asymmetric IMD sidebands [22]. These strange IMD effects have received a lot of atten- tion more recently as their induced long-term memory immensely complicates the design of PA linearizers. For- tunately, since direct static IMD usually dominates this indirect dynamic distortion, those long-term memory ef- fects are seldom noticed. They would be evident only if the direct static odd-order products were reduced. Unfortu- nately, IMD sweet spots are, by nature, exactly one of these situations, and so the selection of these out-of-band impedances should not be overlooked during the PA de- sign and implementation phases. 5. INTERMODULATION DISTORTION IN MICROWAVE AND RF MIXERS A mixer can be viewed as a special kind of amplifier in which the bias supply no longer provides a constant voltage or current, but one that varies in time—the local oscillator. In the same way, an amplifier is a device where the constant quiescent point is perturbed by a certain dynamic signal; a mixer is a similar device where the local-oscillator (LO) time-varying ‘‘quiescent point’’ is perturbed by a dynamic radiofrequency (RF) excitation. Assuming that the mixer is operated in an unsaturated mode, as is the case of most practical interest, the RF signal level is much smaller than the LO level, and the mixer can be analyzed, for the RF signal, as a mild nonlinearity. Thus, it admits a low-degree polynomial expansion in the vicinity of the time-varying LO quiescent point. That constitutes the standard large-signal/small- signal analysis of mixers [7,26]. Mixer distortion analysis can thus follow exactly the one already carried out for small-signal amplifiers, with the exception that now we must start by determining the strong nonlinear regime imposed by the LO and, eventually, some DC bias. The voltage and current waveforms calculated in this way constitute the time-varying quiescent point. Despite the sinusoidal form of the LO excitation, the device’s strong nonlinearities will determine a periodic regime composed by the LO frequency o LO and its harmonics. So, referring to the illustrative case of the active FET mixer depicted in Fig. 28, the time-varying quiescent voltages that control the FET’s i DS (v GS ,v DS ) nonlinearity will be given by v GS ðtÞ ¼ K k ¼ÀK V gs ðko LO Þe jko LO t ð62aÞ v DS ðtÞ ¼ K k¼ÀK V ds ðko LO Þe jko LO t ð62bÞ Then, the nonlinearity must be approximated by a local polynomial model. For instance, a Taylor series such as v RF (t ) v IF (t ) v LO (t ) V GS RF/LO Diplexer + − V DS + − + − + − Figure 28. Simplified schematic of the active FET mixer used in the mixer distortion analysis. 2212 INTERMODULATION (19), in the vicinity of this time-varying LO quiescent point, [v GS (t), v DS (t)], where the small-signal component, i ds (v gs ,v ds ), is determined by the small-signal RF excita- tion. Since the coefficients of such Taylor series depend on the control voltages v GS (t) and v DS (t), they will also be time-variant: i ds ðv gs ; v ds Þ ¼G m ðtÞv gs ðtÞ þG ds ðtÞv ds ðtÞ þG m2 ðtÞv gs ðtÞ 2 þG md ðtÞv gs ðtÞv ds ðtÞ þG d2 ðtÞv ds ðtÞ 2 þG m3 ðtÞv gs ðtÞ 3 þG m2d ðtÞv gs ðtÞ 2 v ds ðtÞ þG md2 ðtÞv gs ðtÞv ds ðtÞ 2 þG d3 ðtÞv ds ðtÞ 3 ð63Þ As v GS (t) and v DS (t) are periodic, the coefficients of (63) are again periodic obeying a Fourier expansion of the form gðtÞ ¼ K k ¼ÀK Gðko LO Þe jko LO t ð64Þ Assuming that the RF signal is a two-tone signal v RF ðtÞ ¼V RF cosðo 1 tÞ þV RF cosðo 2 tÞ ð65Þ (in which we consider, without any lack of generality, that o 1 oo 2 oo LO ), the small-signal components of v GS (t), v DS (t) Àv gs (t), v ds (t) À are again two-tone signals. Substi- tuting (64) and (65) in (63) determines a small-signal current i ds (t), whose components obey ko LO Æm 1 o RF 1 Æm 2 o RF 2 (k is any integer number and m 1 ,m 2 A{ À3, À2, À1,0,1,2,3}, |m 1 |þ|m 2 |r3) and thus include the input tone frequencies at o RF 1;2 (k¼0), the intermediate fre- quencies IF at o IF 1;2 ¼o LO Ào RF 1;2 (k ¼1), its second and third harmonics at 2o IF 1;2 ¼2o LO À2o RF 1;2 (k ¼2), and 3o IF 1;2 ¼3o LO À3o RF 1;2 (k ¼3), respectively, and second- and third-order intermodulation products at o IF 2D ¼o IF 2 À o IF 1 ¼o RF 2 Ào RF 1 (k ¼0) and o IF 3 ¼2o IF 2 Ào IF 1 ¼o LO À ð2o RF 2 Ào RF 1 Þ (k ¼1), respectively. One surprising conclusion that may be drawn from this analysis is that, contrary to an amplifier in which a single Taylor coefficient determines both nth-order harmonics and intermodulation products, in a mixer, for example, the baseband second-order products are determined by the DC component of the Fourier expansion of a coefficient while the second harmonic is determined by the compo- nent at 2o LO . Similarly, it is the o LO Fourier component that determines the in-band third-order products, while the third harmonic is controlled by the Fourier component at 3o LO . Therefore, contrary to what happens in a mem- oryless amplifier, the behavior of the harmonics of a memoryless mixer may say nothing about the behavior of the corresponding in-band distortion products. A de- tailed analysis of the distortion arising in a mixer is quite laborious and requires a full small-signal/large-signal analysis using the conversion matrix formalism [6,7,26]. However, some qualitative insight can already be obtained if we consider the ideal situation of a unilateral gate mixer (total absence of feedback) where the input is tuned for o RF and o LO and the output is tuned for o IF . v gs (t) will have only o RF components, while v ds (t) will have only the resulting o IF components and its in-band distortion pro- ducts o IF 3 . In such an ideal case the FET’s i ds (t) current component at the IF fundamental frequencies will be given by I ds ðo IF Þ %G m1 V gs ðÀo IM Þ þG mÀ1 V gs ðo RF Þ þG ds 0 V ds ðo IF Þ ð66Þ where G m k and G ds k stand for the kth-order harmonic of the Fourier expansion of G m (t) and G ds (t), as expressed by (64), and V gs (o) and V ds (o) represent the v gs (t) and v ds (t) components at o. o IM is the so-called image frequency. Because it is symmetrically located near the RF compo- nents, taking o LO as the symmetry axis (since, in the present case, o RF ¼o LO Ào IF , then o IM ¼o LO þo IF ), it will be also converted to the IF output, thus constituting additive interference. If now the third-order intermodulation product compo- nents of i ds (t), I ds ðo IF 3 Þ, were calculated, we would have I ds3 ðo IF 3 Þ % G m3À1 V ð3Þ gs ð2o RF 1 Ào RF 2 Þ ð67Þ in which V ð3Þ gs ð2o RF 1 Ào RF 2 Þ stands for the terms at o IF 3 that result from the frequency-domain convolutions of V gs (o) * V gs (o) * V gs (o) or the time-domain products of v gs (t) 3 . Expressions (66) and (67) show that a mixer designed for high linearity, namely, one in which conver- sion gain is maximized and IMD is minimized, requires a (V GS ,V DS ) bias point and a LO drive level that maximize first-order Fourier component of the time-varying trans- conductance G m (t) and minimize first-order Fourier com- ponent of G m3 (t). Unfortunately, these are conflicting requirements since maximizing G m1 or G mÀ1 means searching for a G m (t) waveform of highest amplitude and odd symmetry, while reducing G m3 À1 implies reducing G m3 (t) swing and a G m3 (t) waveform of even symmetry, which, as we will see next, cannot be accomplished simultaneously. So, a compromise should be sought in terms of conversion gain and linearity optimization. To illustrate this simplified analysis, Fig. 29a shows three G m (t) waveforms for three distinct V GS bias points, Fig. 29b shows the corresponding G m3 (t) waveforms, and, finally, Fig. 29c shows the resulting conversion gain and IMD ratio ½I ds ðo IF Þ=I ds ðo IF 3 Þ for the whole range of V GS bias. As stated above, there is indeed a compromise between linearity and conversion gain. Although very high IMR values can be obtained for particular bias points, none of them coincides with the zone of highest conversion gain. In a typical sigmoidal G m (v GS ) (such that depicted in Fig. 14a), conversion efficiency is optimized when the FET is biased for maximum G m (v GS ) variation, that is, for the G m2 (v GS ) peak. Unfortunately, that maximized variation of G m (v GS ) is accompanied by an also nearly ideal odd symmetry of G m3 (v GS ), which is responsible for the INTERMODULATION 2213 observed IMD impairment. Furthermore, this simplified analysis also shows that, as was previously studied for amplifiers, IMD behavior of mixers strongly depends on the actual shape of the device’s nonlinearity. (For example, the very sharp peaks of IMR shown in Fig. 29c are due to the ideal symmetric sigmoidal model used for the simu- lated FET’s transconductance.) So, different devices will show quite distinct IMR patterns, impeding a straightfor- ward extrapolation of these active FET mixer results to diode mixers [6,27] or even resistive FET mixers [28,29]. 6. CONCLUSIONS This article showed that the study of nonlinear distortion mechanisms is a subject of fundamental interest that spreads through almost all microwave and RF signal processing circuits and systems. Involving various scien- tific disciplines that range from the physical level of the active-device modeling, to the circuit and system’s level of communication links, it requires a broad range of micro- wave knowledge. Hence, and despite the now more than 40 years of continued progress, intermodulation distortion is still an exciting and challenging field of strong active research both in industry and academia. Acknowledgements The author would like to express his gratitude to several of his colleagues and graduate students who contributed with some of the knowledge presented in this article. Of Time (ps) Time (ps) IMR (dB) V GS (V) Conversion gain (dB) G m (t ) (mS) (a) (b) (c) G m3 (t) (mS/ V 2 ) 10 0 100 200 300 400 500 600 700 800 900 1000 0 20 30 40 50 60 0 100 200 300 400 500 600 700 800 900 1000 25 20 −25 −20 −15 −10 −10 −20 −30 −40 −2.5 −2.0 −1.5 −1.0 −0.5 −5 15 10 5 0 10 20 100 90 80 70 60 50 40 30 0 Figure 29. (a) Time-domain waveforms of G m (t) for three different V GS bias points: V GS ¼ À1.5 V (- - -), V GS ¼ À1.0 V (—), and V GS ¼ À0.6V (Á Á Á); (b) corresponding time-domain waveforms of G m3 (t) for the same three V GS bias points; (c) conversion gain (—) and IMD ratio (– þ–) for the whole range of the FET’s V GS bias. 2214 INTERMODULATION these, Nuno B. Carvalho, Jose A. Garcia, and Christian Fager deserve a special mention. BIBLIOGRAPHY 1. L. Chua, C. A. Desoer, and E. S. Kuh, Linear and Nonlinear Circuits, McGraw-Hill, 1987. 2. R. H. Caverly, Distortion modeling of PIN diode switches and attenuators, IEEE Int. Microwave Symp. 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Zirath, Predic- tion of IMD in LDMOS transistor amplifiers using a new large-signal model, IEEE Trans. Microwave Theory Tech. MTT-50:2834–2842 (2002). 21. C. Fager, J. C. Pedro, N. B. Carvalho, H. Zirath, F. Fortes, and M. J. Rosa´ rio, A comprehensive analysis of IMD behavior in RF CMOS power amplifiers, IEEE J. Solid State Circ. JSSC- 39:24–34 (2004). 22. N. B. Carvalho and J. C. Pedro, A comprehensive explanation of distortion sideband asymmetries, IEEE Trans. Microwave Theory Tech. MTT-50:2090–2101 (2002). 23. L. R. Gomes and J. C. Pedro, Design rules for highly efficient power amplifiers driven by low voltage supplies, Proc. 29th European Microwave Conf., 1999, Vol. II, pp. 267–270. 24. J. M. Golio, M. G. Miller, G. N. Maracas, and D. A. Johnson, Frequency-dependent electrical characteristics of GaAs MES- FETs, IEEE Trans. Electron. Devices ED-37:1217–1227 (1990). 25. R. Anholt, Electrical and Thermal Characterization of MES- FETs, HEMTs, and HBTs, Artech House, Norwood, MA, 1995. 26. S. A. Maas, Microwave Mixers, Artech House, Norwood, MA, 1986. 27. S. A. Maas, Two-tone intermodulation in diode mixers, IEEE Trans. Microwave Theory Tech. MTT-35:307–314 (1987). 28. S. Maas, A GaAs MESFET mixer with very low intermodula- tion, IEEE Trans. Microwave Theory Tech. MTT-35:425–429 (1987). 29. J. A. Garcia, J. C. Pedro, M. L. de La Fuente, N. B. Carvalho, A. Mediavilla, and A. Tazo´n, Resistive FET mixer conversion loss and IMD optimization by selective drain bias, IEEE Trans. Microwave Theory Tech. MTT-47:2382–2392 (1999). INTERMODULATION MEASUREMENT MUHAMMAD TAHER ABUELMA’ATTI King Fahd University of Petroleum and Minerals Dhahran, Saudi Arabia 1. INTRODUCTION Virtually all electronic circuits and systems exhibit non- linear input–output transfer characteristic. Mixers, fre- quency multipliers, modulators, and square-law detectors represent examples of intentional class members, while linear power amplifiers, active filters, and microwave transmitters, in which nonlinearity represents an unde- sirable deviation of the system from ideal, linear opera- tion, are examples of unintentional members. Whenever a number of signals of differing frequencies pass through a nonlinear device, energy is transferred to frequencies that are sums and differences of the original frequencies. These are the intermodulation products (IMPs). In such cases, the instantaneous level of one sig- nal may effectively modulate the level of another signal; INTERMODULATION MEASUREMENT 2215 hence the term intermodulation. In a transmitting system, the results of excessive intermodulation are unwanted signals that may cause interference. In a receiver, internally generated intermodulation can hinder reception of the desired signals. It is interesting to note that the ear’s cochlea has a similar nonlinear response and produces sums and differences of the input frequen- cies in the same way, particularly with loud sounds [1]. It has also been found that passive components, nor- mally considered to be linear, can also generate IMPs. A variety of situations can arise in which nonlinear resis- tance junctions can be formed at metallic mating surfaces. Such junctions may result from salt or chemical deposi- tions or from corrosion. The result is sometimes known as the ‘‘rusty bolt effect’’ because rusted bolts in structures have been known to exhibit such nonlinearities. This phe- nomenon is referred to as passive intermodulation (PIM). Sources of PIM include waveguides, directional couplers, duplexers, and antennas [2–6]. Intermodulation may also occur at the amplifier–loud- speaker interface [7], or in general as a result of the non- linear interaction between the input signal of a two-port and a signal injected to the output port and propagating into the input via a feedback network [8]. Externally induced transmitter intermodulation, also known as reverse intermodulation, backward intermodulation, and antenna-induced intermodulation, is the mixing of a carrier frequency with one or more interfering signals in a transmitter’s final stage [9]. Moreover, lack of screening of open-wire transmission lines can result in significant coupling to adjacent lines frequently giving rise to inter- modulation products [10]. Furthermore, intermodulation may arise when an array of receiving antennas is illumi- nated with a transient impulsive electromagnetic plane wave [11]. In discussing the sources of IMPs it is convenient to divide nonlinear mechanisms yielding IMPs into two prin- cipal forms. The first is due to a nonlinear amplitude in- put/output characteristic (AM/AM), which causes amplitude compression with increasing input amplitude. The second mechanism occurs because of the variation of phase shift through the device, or the system, as the input amplitude is changed (AM/PM). Depending on the signal characteristics, sources of IMPs can be divided into two categories: (1) static nonlin- earity, depending solely on the amplitude of the signal, and (2) dynamic nonlinearity, depending not only on the amplitude but also on the time properties or frequency composition of the signal. Static nonlinearities usually encountered in electronic circuits and systems can be classified into clipping, cross- over, and soft nonlinearities [12] as shown in Fig. 1. Among the hard nonlinearities of clipping (which is sig- nificant near maximum input amplitudes) and crossover (significant mostly at small input amplitudes), the soft nonlinearity is usually the most important in the transfer characteristic of an electronic circuit. If the frequency con- tent or the time properties of the input signal affect the transfer characteristic of the circuit or the system, the re- sulting nonlinearities may be called dynamic. Intermodu- lation products resulting from dynamic nonlinearities are referred to as transient intermodulation (TIM), slew-in- duced distortion (SID), or dynamic intermodulation dis- tortion (DIM) [13–16]. 2. SIMPLE INTERMODULATION THEORY IMPs occur when two or more signals exist simultaneously in a nonlinear environment. In general, if N signals with frequencies f 1 to f N are combined in a static nonlinearity, the output will contain spectral components at frequencies given by N n¼1 k n f n where k n is a positive integer, a negative integer, or zero, and N n¼1 jk n j is the order of the IMP. Even with a small number of input signals N, a very large number of IMPs are generated. Fortunately, not all products are equally troublesome. Depending on the system involved, some of these IMPs can be neglected since they will be filtered out at some point. For example, most of the communication systems operate over a limited frequency band. Thus, IMPs falling out of the band will be attenuated. Moreover, amplitudes of the IMPs generally decrease with the order of the products, and high-order products can often be ne- glected. Low-order intermodulation components such as the second-order component f m Àf n and f m þf n and the third-order components occurring at frequencies 2f m Àf n and f m þf n Àf q are usually the most troublesome, having the largest magnitudes and/or lying close to the originat- ing frequencies, making their removal by filtering practi- cally difficult. However, a salient characteristic of PIM, as distinguished from the conventional IM counterpart, dis- cussed above, is that the PIMs causing trouble are of a high order, say, 11th–21st. Analysis of nonlinear systems differs from that of linear systems in several respects: (1) there is no single analyt- ical approach that is generally applicable (such as Fourier a a b b c c Input Output Figure 1. Different types of static nonlinearities: (a) clipping; (b) soft; (c) crossover. 2216 INTERMODULATION MEASUREMENT or Laplace transforms in linear systems); (2) closed-form analytical solutions of nonlinear equations are not ordi- narily possible; and (3) there is rarely sufficient informa- tion available to enable a set of equations that accurately model the system to be derived. These factors preclude the exact analytical determination of nonlinear effects, such as IMPs, in the general case. In order to get anything done at all, it is usually necessary to make various simplifying assumptions and then use an approximate model that will provide results of acceptable accuracy for the problem in hand. A simple approach, therefore, is to use frequency-do- main techniques that provide a separate solution for each frequency present in the output. In general, such methods are (1) centered around a description of the nonlinear mechanism by a continuous function type of characteris- tic, for example, a polynomial or a Fourier series repre- sentation of the output in terms of the input; and (2) based on the simplifying assumption that this characteristic does not vary with frequency, in other words, that it is a memoryless characteristic. Memoryless nonlinear circuits are oftenly modeled with a power series of the form V out ¼ N n¼0 k n V n i ð1Þ The first coefficient, k 0 , represents the DC offset in the circuit. The second coefficient, k 1 , is the gain of the circuit associated with linear circuit theory. The remaining coef- ficients, k 2 and above, represent the nonlinear behavior of the circuit. If the circuit were completely linear, all the coefficients except k 1 would be zero. The model can be simplified by ignoring the terms that come after the k 3 term. For soft nonlinearities, the size of k n decreases rapidly as n gets larger. For many applica- tions the reduced model of Eq. (2) is sufficient, since the second-order and third-order effects dominate. However, many devices, circuits, and systems present difficulties for the polynomial approximation: V out ¼k 0 þk 1 V i þk 2 V 2 i þk 3 V 3 i ð2Þ Assuming that the input signal is a two-tone of the form V i ¼V 1 cos o 1 t þV 2 cos o 2 t ð3Þ then combining Eqs. (2) and (3), yields V out ¼a 0 þb 1 cos o 1 t þc 1 cos o 2 t þb 2 cos 2o 1 t þc 2 cos 2o 2 t þb 3 cosðo 1 þo 2 Þt þc 3 cosðo 1 Ào 2 Þt þb 4 cos 3o 1 t þc 4 cos 3o 2 t þb 5 ðcosð2o 1 þo 2 Þt þ cosð2o 1 Ào 2 ÞtÞ þc 5 ðcosð2o 2 þo 1 Þt þ cosð2o 2 Ào 1 ÞtÞ ð4Þ where a 0 ¼k 0 þ k 2 2 ðV 2 1 þV 2 2 Þ b 1 ¼k 1 V 1 þ 3 4 k 3 V 3 1 þ 3 2 k 3 V 1 V 2 2 c 1 ¼k 1 V 2 þ 3 4 k 3 V 3 2 þ 3 2 k 3 V 2 1 V 2 b 2 ¼ 1 2 k 2 V 2 1 c 2 ¼ 1 2 k 2 V 2 2 b 3 ¼c 3 ¼k 2 V 1 V 2 b 4 ¼ 1 4 k 3 V 3 1 c 4 ¼ 1 4 k 3 V 3 2 b 5 ¼ 3 4 k 3 V 2 1 V 2 c 5 ¼ 3 4 k 3 V 1 V 2 2 For equal-amplitude input tones, Eq. (4) shows that the second-order terms, of amplitudes b 2 , c 2 , b 3 , c 3 will be in- creased 2 dB in amplitude when input tones are increased by 1dB. The third-order terms, of amplitudes b 4 , c 4 , b 5 , c 5 , are increased by 3 dB in amplitude when the input tones are increased by 1 dB. While Eq. (1) is adequate, and widely used, to predict the intermodulation performance of a wide range of de- vices, circuits, and systems, it seldom can be used. Exam- ples include, but are not restricted to, prediction of spectral regrowth in digital communication systems, tran- sient intermodulation and frequency-dependent nonlin- earities, and passive intermodulation. 3. SPECTRAL REGROWTH When a modulated signal passes through a nonlinear de- vice, its bandwidth is broadened by odd-order nonlinear- ities. This phenomenon, called spectral regrowth or spectral regeneration, is a result of mixing products (in- termodulation) between the individual frequency compo- nents of the spectrum [17]. The spectral regrowth can be classified in the two following categories: (1) in-band in- termodulations and (2) out-of-band intermodulations. The first cannot be eliminated by linear filtering and are re- sponsible for the signal-to-noise ratio degradation and, consequently, for the bit error rate (BER) degradation in digital communication systems. The second generates the interference between adjacent channels and can be fil- tered out at the nonlinear device output with certain out- put power penalty that is caused by the filter insertion losses. This spectral regrowth causes adjacent-channel INTERMODULATION MEASUREMENT 2217 interference (ACI), which is measured by the adjacent- channel power ratio (ACPR). The ACPR is the power in the main channel divided by the power in the lower plus upper adjacent channels. Con- sidering just the lower channel yields ACPR lower and the upper channel alone yields ACPR upper . Analog cellular ra- dio uses frequency or phase modulation, and the ACPR is adequately characterized by intermodulation distortion of discrete tones. Typically, third-order intermodulation product (IMP3) generation, in a two-tone test, is adequate to describe spectral regrowth. Thus, distortion in analog radio is accurately modeled using discrete-tone steady- state simulation. Digital radio, however, uses complex modulation, and adjacent-channel distortion has little re- lationship to intermodulation in a two-tone test [18,19]. A modulated input signal applied to radiofrequency (RF) electronics in digital radio is a sophisticated waveform re- sulting from coding, filtering, and quadrature generation. Neither can it be represented by a small number of dis- crete tones (or frequencies), nor can the waveform be rep- resented in a simple analytic form. Thus, in digital radio, ACPR is more difficult to predict than one- or two-tone responses since it depends not only on the intrinsic non- linear behavior of the device (e.g. amplifier) but also on the encoding method (i.e., the statistics of the input stream) and the modulation format being used. The only way the input stream can conveniently and accurately be repre- sented is by its statistics, and transforming these using an appropriate behavioral model provides accurate and effi- cient modeling of ACPR [20]. While in Ref. 20 the input signal is assumed Gaussian, digital communication sig- nals are often far from being Gaussian. In Ref. 21 the in- put is assumed stationary but not necessarily Gaussian. ACPR is, therefore, defined differently in the various wireless standards. The main difference is the way in which adjacent-channel power affects the performance of another wireless receiver for which the offending signal is cochannel interference [20]. In general the ACPR can be defined as [20] ACPR¼ _ f 4 f 3 Sðf Þdf _ f 2 f 1 Sðf Þdf ð5Þ where S(f) is the power spectral density (PSD) of a signal whose channel allocation is between frequencies f 1 and f 2 , and its adjacent channel occupies frequencies between f 3 and f 4 . Regulatory authorities impose strict constraints on ACPR and accurate methods of its determination are of par- ticular interest to those involved in wireless system design. 4. SIMPLE TRANSIENT INTERMODULATION THEORY To illustrate how TIM distortion arises, consider a differ- ential amplifier with negative feedback applied between the output and the inverting input and a voltage step ap- plied to the noninverting input. If the open-loop gain of the amplifier were flat and the time delay through it were zero, the voltage step would instantaneously propagate undis- torted through the amplifier, back through the feedback loop, and into the inverting input, where it would be sub- tracted from the input signal, and the difference signal, which is a voltage step occurring at the same time that the input voltage does, would be amplified by the amplifier. However, this is not the case when the open-loop gain of the amplifier is not flat and the time delay through it is not zero. When the voltage step occurs, the limited high-fre- quency response of the amplifier prevents the appearance of a signal at the amplifier output terminal until the inter- nal capacitors of the amplifier can charge or discharge. This causes the momentary absence of a feedback signal at the inverting input to the amplifier, possibly causing the am- plifier to severely overload until the feedback signal arrives. If the input signal to the differential amplifier is formed of a sine wave superimposed on a square wave, the am- plifier will exhibit the same response to the abrupt level changes in the square wave as it did to the voltage step discussed above. During the momentary absence of the feedback when the square wave changes level, the ampli- fier can either saturate or cut off. If this occurs, the sine wave momentarily disappears from the signal at the out- put terminal of the amplifier, or it momentarily decreases in amplitude. This happens because the saturated or cut- off amplifier appears as a short circuit or open circuit, re- spectively, to the sine wave, and this component of the input signal is interrupted from the output signal, thus resulting in TIM [16]. A point to be noted is that if the term were understood literally, this would imply transients of both high and low frequencies and/or high or low operating levels, in other words, all transients. In actual practice, however, TIM oc- curs only for signals with simultaneous high level and high frequencies—not lower levels or lower frequencies. The key parameter of such signals is that they are char- acterized by high signal slopes, not just high frequencies or high levels. Neither high frequencies nor high levels in themselves necessarily result in distortion, unless their combination is such that a high effective signal slope is produced. TIM is actually generated when the signal slope approaches or exceeds the amplifier slew rate. This can happen for either transient or steady-state signals. Thus, a more easily understood term to what actually happens would be one that relates both slew rate and signal slope. A more descriptive term to describe the mechanism would, therefore, be the slew-induced distortion (SID); other de- scriptive variations of this term are ‘‘slew rate distortion’’ or ‘‘slewing distortion’’ [22]. Because of the complexity of the mechanism resulting in TIM, especially handling the frequency dependence of the amplifier nonlinearity and incorporation of the feed- back, Eq. (1) cannot be used to predict the TIM perfor- mance of nonlinear devices, and recourse to other analytical techniques, for example, Volterra series or har- monic balance analysis, would be inevitable. 5. VOLTERRA SERIES ANDHARMONIC BALANCE ANALYSIS Volterra series describes a system with frequency-depen- dent nonlinearity in a way that is equivalent to the 2218 INTERMODULATION MEASUREMENT manner in which Taylor series approximates an analytic function. Depending on the amplitude of the exciting sig- nal, a nonlinear system can be described by a truncated Volterra series. Similar to the Taylor series representa- tion, for very high amplitudes the Volterra series diverges. Volterra series describe the output of a nonlinear system as the sum of the response of a first-order operator, a sec- ond-order one, a third-order one, and so on [23]. Every operator is described in either the time domain or the fre- quency domain with a kind of transfer function called a Volterra kernel. In Volterra series analysis the nonlinear circuit is treated purely as an AC problem. Assuming that none of the input signals are harmonically related, an iterative solution can be applied for circuits not operated under distortion saturation conditions. First the circuit is solved for the input signals. These results are then used to cal- culate the second-order distortion products, and these are treated as generators at a different frequency to the input signals and the network is again solved. This is then re- peated for higher-order distortion products. This leads to extremely fast calculation of distortion behavior. Simula- tion at higher power levels can be achieved by feeding back contributions from higher-order distortion products [24]. The use of Volterra series to characterize the output as a function of the input [25,26] can, therefore, provide closed-form expressions for all the distortion products of a frequency-dependent nonlinearity excited by a multisinu- soidal signal. However, techniques using Volterra series suffer from the disadvantage that a complex mathematical procedure is required to obtain a closed-form expression for the out- put amplitude associated with a single component of the output spectrum. Moreover, the problem of obtaining out- put products of orders higher than the third becomes pro- hibitively difficult unless it may be assumed that higher- order contributions vanish rapidly [27]. The Volterra se- ries approach is, therefore, most applicable to mild non- linearities where low-order Volterra kernels can adequately model the circuit behavior. With appropriate assumptions and simplifications, many useful features of the Volterra series technique can be used to find approx- imate expressions for TIM (SID). These are quite accurate for relatively small distortion conditions [28,29]. Alternatively, most RF and microwave circuit analysis are based on the harmonic balance analysis [30]. The har- monic balance technique works by processing the linear part of the circuit in the frequency domain and the non- linear part in the time domain. Computation in the fre- quency domain is very fast and efficient, especially for frequency-selective components such as transmission lines and resonant circuits. Computations in the time do- main are followed by Fourier transform. Harmonic bal- ance analysis can, therefore, handle intermodulation distortion provided there are not too many excitation tones. In the harmonic balance technique an initial esti- mate is required for the final waveshape, and this is re- fined interactively during analysis. The harmonic balance method computes the response of a nonlinear circuit by iteration, and the final result is a list of numbers that do not indicate which nonlinearities in the circuit are mainly responsible for the observed nonlinear behavior. Hence such a method is suitable for verification of circuits that have already been designed. This method does not present information from which designers can derive which circuit parameters or circuit elements they have to modify in or- der to obtain the required specifications [31]. While Vol- terra series analysis can provide such information, it is applicable only to weak nonlinearities. While viewed as a universal solution, and has been widely used, the harmonic balance analysis may be un- necessarily slow, cumbersome, and prone to subtle errors [32], especially for weak nonlinearities or when a nonlin- ear device is excited by very small signals. Volterra series analysis is generally more accurate than harmonic bal- ance for these types of problems, and it is several orders of magnitude faster than a harmonic balance analysis [32]. Moreover, Volterra series analysis integrates well with linear analysis tools, supporting simultaneous optimiza- tion of several parameters of the nonlinear system. There- fore, Volterra theory appears to be an ideal tool for circuits and systems that are not strongly nonlinear but have as- pects of linear and nonlinear circuits [32]. However, Vol- terra series analysis becomes very cumbersome above third-order products, and for products above fifth order, it loses most of its advantages over the harmonic balance analysis. The major disadvantage of Volterra series is the occasional difficulty in deciding whether the limitations to weakly nonlinear operation have been exceeded. In fact, Volterra-series analysis and the harmonic bal- ance technique complement each other [32]. Thus, while the Volterra series analysis works well in those cases where harmonic balance works poorly, the harmonic bal- ance works well where the Volterra series works poorly. Volterra series analysis is, therefore, not appropriate for mixers, frequency multipliers, saturated power amplifiers, and similar strongly driven and/or hard nonlinearities. Volterra series analysis is suitable for small-signal ampli- fiers, phase shifters, attenuators, and similar small-signal and/or soft nonlinearities. Another technique for analyzing nonlinear systems is the describing function. This approach can yield closed- form expressions for a feedback system that contains an isolated static nonlinearity in the feedback loop [33]. Since it is not possible to map all nonlinear circuits and systems to such a feedback system, the describing function method has restricted applications. 6. PASSIVE INTERMODULATION (PIM) While the concept of intermodulation in active devices such as amplifiers, filters, and mixers is familiar and well documented, the effects of intermodulation in passive com- ponents such as directional couplers, cables, coaxial con- nectors, power splitters, antennas, and electromechanical and solid-state programmable attenuators are less famil- iar and less documented. More recently, evidence has emerged that PIM has an impact in other system equip- ment, such as amplifiers and extenders, fiber nodes, and interface units [34]. Poor mechanical contact, dissimilar metals in direct contact, ferrous content in the conductors, INTERMODULATION MEASUREMENT 2219 debris within the connector, poor surface finish, corrosion, vibration, and temperature variations are among the many possible causes of PIM. The sources of PIM have been studied extensively; see Refs. 35–43 and the refer- ences cited therein. Similar to the intermodulation prod- ucts in active devices, PIM is generated when two or more RF signals pass through RF passive devices having non- linear characteristics [41,42]. Generally the nonlinearities of RF passive devices consist of contact nonlinearity and material nonlinearity [43]. Contact nonlinearity refers to all metal contact nonlinearities causing nonlinear cur- rent–voltage behavior, such as the tunneling effect, micro- discharge, and contact resistance. Material nonlinearity refers to the bulk material itself. Magnetoresistivity of the transmission line, thermal resistivity, and nonlinear hystresis of ferromagnetic material are good examples [43]. PIM generation in RF passive devices is caused by the simultaneous appearance of one or more of these PIM sources, and the overall performance is often domi- nated by one principal PIM source [43]. In the case of antennas, PIM is generated not only by the same PIM sources as in general RF passive components but also by the external working environment, such as conducting metal materials. Over the years Eq. (1) was used to describe the nonlin- ear current/voltage conduction characteristics of passive components, (see, e.g., Refs. 37–39 and the references cited therein). While this approach results in simple expres- sions for the magnitudes of the harmonics and intermod- ulations products resulting from multisinusoidal excitations, it suffers from the following shortcomings. In order to predict high-order harmonic or intermodulation product magnitudes, it is necessary to determine coeffi- cients of terms of similar order in the polynomial. A pre- requisite to obtaining coefficients of high-order polynomial terms is measurement of output products of the same or- der. For example, to obtain the coefficients of a fifth-order polynomial, it is necessary to measure the output fifth-or- der components. With increasing use of narrowband com- ponents in multicouplers used in base stations of mobile radio systems, it becomes difficult to determine high-order coefficients in the nonlinear characteristic because the measured high-order product amplitudes from which they are computed are influenced to an unknown extent by the system selectivity [44]. To overcome these prob- lems, an exponential method has been used to predict the intermodulation arising from corrosion [45]. 7. INTERMODULATION CHARACTERIZATION Although it is important to understand the origin of in- termodulation and the engineering techniques for avoid- ing it, it is equally important to be able to characterize it objectively, preferably in a way that correlates well with the subjective perception of the intermodulation. The abil- ity to characterize an imperfection in this way is an important step toward eliminating it as a system perfor- mance degradation. Several techniques for characterizing intermodula- tion distortion have been proposed. While some of these techniques measure the total intermodulation distortion, others distinguish between the various intermodulation products. The latter are preferred, for subjective percep- tion of intermodulation shows that equal amounts of total intermodulation disortion differ widely in their effect ac- cording to how the total is made up. Depending on the signal characteristics, techniques for characterization of intermodulation distortion can be clas- sified into two categories: (1) steady-state techniques, where characterization is performed on the assumption that the input to the system under consideration is a mul- tisinusoidal signal, and (2) dynamic techniques, where characterization is performed on the assumption that the input to the system under consideration is formed of a sinusoidal signal superimposed on another signal char- acterized by rapid changes of state, for example, a square wave or a sawtooth wave. While steady-state techniques can be used to characterize both RF and audio systems, dynamic techniques are generally used for characterizing only audio systems. 7.1. Steady-State Techniques 7.1.1. The Intercept Point. Increasing the signal level at the input to a weakly nonlinear device will cause the IMPs to increase at the output. In fact, the increase in IMP am- plitudes is faster than the increase in the output version of the input signal. For increasing fundamental input power, the fundamental output power increases in a linear man- ner, according to the gain or loss of the device. At some point, gain compression occurs and the fundamental out- put power no longer increases with input power. The out- put power of the second-order intermodulation products also increases with fundamental input power, but at a faster rate. Recall that, according to the simple intermod- ulation theory, the second-order intermodulation changes by 2dB per 1dB of change in the fundamental. Similarly, the third-order intermodulation changes by 3 dB per 1 dB of change in the fundamental. Thus, on a logarithmic scale, as shown in Fig. 2, the lines representing the second- and third-order intermodulation products have twice and three times, respectively, the slope of the fundamental line. If there were no gain compression, the fundamental input power could be increased until the second-order in- termodulation eventually caught up with it, and the two output power levels would be equal. This point is referred to as the second-order intercept point (IP2). The third-or- der intermodulation product also increases faster than the fundamental, and those two lines will intersect at the third-order intercept point (IP3). Rarely can either of these two points be measured directly, due to the gain compression of the fundamental. Instead, the intercept points are extrapolated from measurements of the funda- mental and intermodulation products at power levels below the point where gain compression occurs. The intercept points are usually specified in dBm and may refer to either the output or the input; the two points will differ by the gain of the system under consideration. The second-order and third-order intercept points are fig- ures of merit that are independent of the signal level. 2220 INTERMODULATION MEASUREMENT Therefore, the intermodulation performance of two differ- ent systems can be compared quite easily if their intercept points are known [46]. Using the intercept point it is easy to calculate the rel- ative intermodulation level corresponding to a given input signal level. In fact, the difference between the level of the second-order intermodulation and the fundamental signal level is the same as the difference between the fundamen- tal signal level and the intercept point. Thus, if the second- order intercept point is þ15 dBm and the fundamental signal level is À10 dBm (both referred to the output of the device), the difference between these two values is 25 dB. Therefore, the second-order intermodulation products will be 25 dB below the fundamental, or À35 dBm. So the in- tercept point allows easy conversion between fundamental signal level and the intermodulation level. The difference between the level of the third-order in- termodulation products and the fundamental signal level is twice the difference between the fundamental signal level and the third-order intercept point. (Note that the second-order intercept point is not the same as the third- order intercept point.) Suppose that the third-order inter- cept point is þ5 dBm and the fundamental signal is À25 dBm, both referred to the output of the device. The difference between the intercept point and the fundamen- tal is 30 dB, so the third-order intermodulation products will be 2 times 30 dB down from the fundamental. The relative distortion level is À60 dB, and the absolute power of the intermodulation products is À85 dBm. It is important, however, to note that the preceding analyses assume that the second-order and third-order intermodulation curves have slopes of 2 and 3dB/dB, re- spectively. Thus, theoretically, the intercept points are not functions of the input power level. If a power sweep is performed, it is expected that the intercept points will remain constant. The intercept points can, therefore, be calculated from measurements at only one power level. However, if the input signal exceeds a certain limit, the amplitudes of the output fundamentals and the resulting intermodulation products will start to saturate, and the intercept points will usually drop off, indicating an invalid measurement. It is essential to know this limit. It is par- ticularly useful for high-dynamic-range circuits and sys- tems with relatively low output powers where the intermodulation is low, but only for signals that are low enough. Expanding the model of Eq. (2) to include fourth- and fifth-order terms [47] can achieve this. Moreover, at low power levels, the intercept points will start to change as the noise floor of the measuring instru- ment, usually a spectrum analyzer, is approached, thus indicating an invalid measurement. It is important, there- fore, to look at the variation of the intercept points as functions of power as this provides a good way of checking the valid measurement range. 7.1.2. Two-Tone Test. The two-tone test is extensively used in characterizing a wide range of devices. Magnetic tapes [48]; microwave and millimeter-wave diode detec- tors [49]; analog-to-digital converters [50,51]; gamma cor- rectors [52]; and electrical components such as resistors, capacitors, inductors, as well as contacts of switches, con- nectors, and relays [53] are a few examples. The two-tone test is also used to characterize the performance of the basilar membrane of the cochlea [54]. The two-tone test can also be used to determine the transfer characteristic of a nonlinear device modeled by the polynomial approximation of Eq. (2). With the input formed of two properly selected frequencies o 1 and o 2 , and if the second-order and third-order intermodulation prod- ucts are measured separately, it is possible to find, from the measured data, the coefficients of the quadratic and cubic terms k 2 and k 3 , respectively, in the polynomial ap- proximation of Eq. (2). If in addition, the IMPs are mea- sured at two sets of values of o 1 and o 2 , it is possible to identify the dominant physical nonlinear process from the variation of IMPs with test frequencies [13]. The two-tone test can also be used to determine the complex transfer characteristic of a nonlinear device exhibiting AM/AM nonlinearity only with fixed phase shift between the output and the input. In this case a complete set of measurement for all the two-tone inter- modulation products produced by the nonlinearity at two different power levels is necessary [55]. If the device under consideration exhibits both AM/AM and AM/PM non- linearities, then determination of a unique set of polyno- mial coefficients requires a complete set of intermodula- tion measurements at three different power levels [55]. The set obtained at the highest power level will decide the amplitude range within which the characterization will be valid. • • IP2 IP3 Input power (dBm/tone) O u t p u t p o w e r ( d B m ) −100 −60 −20 −80 0 −40 0 40 80 a b c Figure 2. Third-order and second-order intercept points are de- termined by extending the fundamental, the second- and the third-order intermodulation transfer function lines: (a) Funda- mental transfer function, slope ¼1; (b) second-order intermodu- lation, slope ¼2; (c) third-order intermodulation, slope ¼3. IP3— third-order intercept point; IP2—second-order intercept point. INTERMODULATION MEASUREMENT 2221 According to the basic assumption that the nonlinear- ities are represented by polynomials, high-accuracy rep- resentation of the device characteristics will require difficult accurate measurements of higher-order intermod- ulation products, in addition to increased complications and considerable efforts involved in the analysis [55]. Another difficulty from which this method suffers arises from the necessity of measuring complete sets of two-tone intermodulation products spread over a relatively wide frequency range, which consequently may impose strin- gent specifications on the measuring instruments and techniques if accurate measurements are to be achieved. In the two-tone test the inband IMPs are used to de- scribe a device, a circuit or a system nonlinearity. Mea- surements are made in or near the frequency range of interest. In this test, the input signal consists of two fre- quencies, o 1 and o 2 of equal amplitude and a fixed amount of frequency spacing. At the output of the circuit or the system under test the amplitudes of the third-order inter- modulation products 2o 1 Ào 2 and 2o 2 Ào 1 are measured. The intermodulation distortion is defined as the ratio be- tween the root sum square of the intermodulation prod- ucts and the root sum square of the twin-tone amplitudes. Unless a wave analyzer or a spectrum analyzer is avail- able, the implementation of the two-tone test invariably require amplification of the whole output spectrum to ob- tain components o 1 and o 2 on a normalized value (100%). Then, o 1 and o 2 are suppressed, and the remaining com- ponents 2o 1 Ào 2 and 2o 2 Ào 1 are measured with an AC voltmeter or oscilloscope. Especially at audiofrequencies, this approach requires steep filters, one set of filters for each set of o 1 and o 2 . For the same reason o 2 Ào 1 cannot be too low, so it will never be a really narrowband system. This narrowband aspect is particularly important for higher frequencies, where equalizers, in the reproduction audio channel, may give unequal amplification of the com- ponents in the spectrum [56]. In the audiofrequency range several versions of the two-tone test are available [56–59]. 7.1.3. Three-Tone Test. In this test, again, specific in- band IMPs are selected to characterize the overall system nonlinearities [60]. The more even spectral distribution and flexibility, while still allowing discrete frequency eval- uation, make this an attractive test for multifrequency systems such as communication and cable television sys- tems. In this test three equal-amplitude tones are applied to the input of the nonlinear system under consideration. Thus V i ¼Vðcos o 1 t þ cos o 2 t þ cos o 3 tÞ ð6Þ Combining Eqs. (2) and (6), and using simple trigonomet- ric identities, it is easy to show that the third-order term, k 3 V i 3 will contribute, to the output spectrum, the following: 1. Three components at frequencies o 1 , o 2 and o 3 each with amplitude given by A 1 ¼ 15 4 k 3 V 3 ð7Þ 2. Three components at frequencies 3o 1 , 3o 2 , 3o 3 each with amplitude given by A 3 ¼ 1 4 k 3 V 3 ð8Þ 3. Twelve components at frequencies 2o m Æo n ; m; n¼ 1–3, each with amplitude given by A 21 ¼ 3 4 k 3 V 3 ð9Þ 4. Four components at frequencies o m Æo n Æo p ; m; n; p¼1–3, each with amplitude given by A 111 ¼ 3 2 k 3 V 3 ð10Þ Equations (9) and (10) show that an intermodulation prod- uct of frequency o m Æo n Æo p is 6 dB higher in level than an intermodulation product of frequency 2o m Æo n : Inter- modulation distortion is defined as the ratio between the amplitude of one of the intermodulation products of fre- quency o m Æo n Æo p and the amplitude of one of the three output tones. In this test the choice of frequencies o 1 ,o 2 ,o 3 used to perform the measurement is important. This is because a system’s intermodulation performance may not be constant over its operating frequency range. The three-tone test is widely used to characterize the performance of RF amplifiers used in television broadcast transposers, where the vision carrier, color subcarrier, and sound carrier frequency components interact in the pres- ence of amplifier nonlinearities. If the three frequency components are represented as single frequencies (o the vision carrier, o sc the color subcarrier, and o s the sound carrier with amplitudes V v , V sc , and V s , respectively), then the input signal can be expressed as V i ¼V v cos o v t þV sc cos o sc t þV s cos o s t ð11Þ Combining Eqs. (2) and (11), and using simple trigono- metric identities, it is easy to show that the third-order term of Eq. (2) produces, among others, two in-band in- termodulation components given by V ip ¼ 3 2 k 3 V v V sc V s cosðo v þo s Ào sc Þt þ 3 4 k 3 V s V 2 sc cosð2o sc Ào s Þt ð12Þ Intermodulation performance of the transposer is mea- sured by taking the transposer out of service and using the three-tone simulation of a composite video and sound sig- nal, given by Eq. (11), as its input. The three levels and frequencies vary from system to system. Typical levels, below the peak synchronous pulse level, are V v ¼ À6 dB, V sc ¼17 dB, and V s ¼ À10 dB. Under these conditions, the first term of Eq. (12) is the most visible, and the second term will be much lower in amplitude, typically 17 dB less. Using a spectrum analyzer, the relative amplitude of the major in-band intermodulation is measured and 2222 INTERMODULATION MEASUREMENT referenced to the level of peak synchronous pulse. Usually, the permissible level of the major in-band intermodulation component is À53 dB below the reference level. This three-tone test method is slow and requires spectrum an- alyzers with relatively wide dynamic ranges. Moreover, it measures the system performance at one luminance level and one chrominance level. Thus, it does not test the sys- tem over its full operating range [61]. The inadequacy of the internationally accepted three- tone test method can be overcome by using a modified col- orbar test signal [61]. The colorbars are applied to the transposer via a test transmitter. The colorbars and sound carrier therefore apply the three tones to the transposer, changing levels in rapid succession. With suitable pro- cessing, based on sampling the demodulated colorbar sig- nal for short intervals corresponding to a selected color, intermodulation levels can be measured simultaneously at seven different luminance levels and can be shown in his- togram form [61]. 7.1.4. Noise Power Ratio (NPR) Test. In the NPR test, the input to the device under test is obtained from a white- noise source that is bandlimited to the instantaneous fre- quency range of interest. This emulates a situation with many simultaneous input signals. Provided that none of the signals dominate, according to the central-limit theo- rem, the resulting voltage obtained when many uncorre- lated signals are added will approach a Gaussian distribution. True white noise covers a frequency range of interest continuously, unlike discrete signals. The NPR test measures the amount of intermodulation products power between two frequency ranges of white Gaussian noise. A white-noise generator is used with its output frequency range limited by a bandpass filter ac- cording to the bandwidth of the device under test. A quiet channel is formed by a switchable band-reject filter, as shown in Fig. 3. Then, the resulting white-noise signal is applied to the input of the device under test. At the output of the device under test is a receiver which is switch-tuned to the frequency of the band-reject filter used to produce the quiet channel. The NPR test is widely used for eval- uating the intermodulation performance of systems whose input signal spectrum distribution can be approximated by that of white noise. However, the NPR may be degraded by the noise floor of the system under test, especially under very low loading conditions. It may also be degraded by the distortion products produced under high loading conditions [62]. 7.1.5. Cross-Modulation. Cross-modulation occurs when modulation from a single unwanted modulated signal transfers itself across and modulates the wanted signal. Cross-modulation is troublesome primarily if the desired signal is weak and is adjacent to a strong unwanted signal. Even when the carrier of the strong unwanted signal is not passed through the system, the modulation on the unde- sired carrier will be transferred to the desired carrier. Cross-modulation is, therefore, a special case of intermod- ulation. Recall that when the input to a non- linear system is formed of a two-tone signal of the form of Eq. (3), then the amplitudes of the output components at frequencies o 1 and o 2 will be given by b 1 ¼k 1 V 1 þ 3 4 k 3 V 3 1 þ 3 2 k 3 V 1 V 2 2 ð13Þ and c 1 ¼k 1 V 2 þ 3 4 k 3 V 3 2 þ 3 2 k 3 V 2 1 V 2 ð14Þ respectively. Thus, the output obtained at each frequency o 1 and o 2 , is dependent on the amplitude of the signal component of the other frequency. If the amplitude of the wanted unmodulated carrier is V 1 and the instanta- neous amplitude of the unwanted amplitude-modulated carrier is V 2 ðtÞ ¼V 2 ð1 þmcos o m tÞ ð15Þ then, using Eq. (13), the amplitude of the wanted carrier will be b 1 ¼k 1 V 1 þ 3 4 k 3 V 3 1 þ 3 2 k 3 V 1 V 2 2 ð1 þmcos o m tÞ 2 ð16Þ For small values of m and with k 3 5k 1 , Eq. (16) can be ap- proximated by b 1 ffik 1 V 1 þ3k 3 V 1 V 2 2 mcos o m t ð17Þ Thus the wanted carrier will be modulated by a modulation index p¼3 k 3 k 1 V 2 2 m ð18Þ The cross-modulation factor is then defined as K ¼ p m ð19Þ Thus, one frequency will be modulated by the modulation of the other frequency. Similar results can be obtained if the unwanted carrier is FM-modulated. Frequency P o w e r ( d B ) ∆ δω o ω b a a A B Figure 3. The output spectrum of a noise–power ratio measure- ment. (a) injected noise; (b) noise and intermodulation generated in the measurement bandwidth do by the DUT. NPR¼AÀB. INTERMODULATION MEASUREMENT 2223 Cross-modulation can be measured as the change in the amplitude of the wanted unmodulated carrier as a func- tion of the amplitude of the unwanted unmodulated car- rier. This is the procedure recommended by the NCTA (National Cable Television Association) standard cross- modulation measurement [63]. Alternatively, cross-modu- lation can be measured using the definition of Eq. (19): measuring percentage modulation that appears on an un- modulated desired carrier due to the presence of an un- desired modulated carrier, divided by the percentage modulation on the undesired carrier [64]. Cross-modulation can also be measured using two equal-amplitude carriers. The wanted carrier, o 2 , is un- modulated while the unwanted carrier, o 1 , is FM-modu- lated. The output spectrum clearly shows the frequency deviation of the wanted carrier. Moreover, it can be shown that the frequency deviation of the intermodulation com- ponents, of the output spectrum, is larger than that of the original FM-modulated unwanted carrier. For the inter- modulation product of frequency ao 1 7bo 2 , the deviation will be multiplied by a. Thus, it may be easier to measure the cross-modulation by measuring the deviation of an in- termodulation product rather than the deviation of the wanted unmodulated carrier [65]. 7.1.6. Differential Gain. Differential gain (DG), a pa- rameter of special interest in color-TVengineering, is con- ventionally defined as the difference in gain encountered by a low-level high-frequency sinusoid at two stated in- stantaneous amplitudes of a superimposed slowly varying sweep signal. In video signal transmission, the high-fre- quency sinusoid represents the chromatic signal and the low-frequency sinusoid represents the luminance signal. Corresponding to the theoretical conditions of the differ- ential measurement, DG measurement is performed by a signal of the form of Eq. (3) with o 2 bo 1 and V 2 !0:0 at V 1 ¼0.0 and X [66]. Therefore, recalling that when the in- put to a nonlinear system is formed of a two-tone signal of the form of Eq. (3), the amplitude of the output component at frequency o 2 will be given by c 1 ¼k 1 V 2 þ 3 4 k 3 V 3 2 þ 3 2 k 3 V 2 1 V 2 ð20Þ Thus, DG can be expressed as DG¼1 À k 1 þ 3 4 k 3 V 2 2 k 1 þ 3 4 k 3 V 2 2 þ 3 2 k 3 X 2 ð21Þ DG can, therefore, be considered to some extent as a mea- sure of the intermodulation performance of a system under test. 7.1.7. Dynamic Range. Dynamic range can be defined as the amplitude range over which a circuit or a system can operate without performance degradation. The mini- mum amplitude is dictated by the input thermal noise and the noise contributed by the system. The maximum am- plitude is dictated by the distortion mechanisms of the system under consideration. In general, the amount of tolerable distortion will depend on the type of signal and the system under test. However, for the purpose of an ob- jective definition the maximum amplitude will be consid- ered the input signal level at which the intermodulation distortion is equal to the minimum amplitude [67]. The dynamic range can, therefore, be considered to some ex- tent as a measure of the intermodulation performance of a system under test. A useful working definition of the dynamic range is that it is (1) two-thirds of the difference in level between the noise floor and the intercept point in a 3 kHz band- width [68] or (2) the difference between the fundamental response input level and the third-order response input as measured along the noise floor (sometimes defined as 3 dB bandwidth above the noise floor) in a 3 kHz bandwidth, as shown in Fig. 4. Reducing the bandwidth improves dy- namic range because of the effect on noise. Because the power level at which distortion becomes intolerable varies with signal type and application, a ge- neric definition has evolved. The upper limit of a net- work’s power span is the level at which the power of one IM product of a specified order is equal to the network’s noise floor. The ratio of the noise floor power to the upper- limit signal power is referred to as the network’s dynamic range (DR). Thus the DR can be determined from [69] DR n ¼ n À1 n ½IP n;in ÀMDS ð22Þ where DR n is the dynamic range in decibels, n is the order, IP in is the input intercept power in dBm, and MDS is the minimum detectable signal power in dBm. Alternatively, in receiver circuits the spurious-free dynamic range (SFDR) and the intermodulation-free • IP3 (a) (b) (c) Dynamic range −40 0.0 40 80 120 160 −40 0.0 40 80 120 160 O u t p u t l e v e l ( d B a b o v e i n p u t ) Input level (dBuV) Figure 4. The dynamic range is the difference between the fun- damental response input level and the third-order response input as measured along the noise floor: (a) fundamental response; (b) third-order intermodulation response; (c) noise floor. 2224 INTERMODULATION MEASUREMENT dynamic range (IFDR) are widely used to quantify the capability of the receiver to listen to a weak station, without disturbance from an intermodulation product generated by strong stations on other frequencies. The SFDR and the IFDR are in fact measures of how strong two signals can be before the level of their intermodulation products can reach the noise floor of the receiver. The SFDR, or the IFDR, is defined as the difference in decibels between the power levels of the third-order intermodula- tion IM3 (assuming that there is only a third-order non- linearity) and the carrier when the IM3 power level equals the noise floor at a given noise bandwidth. It can be expressed as [70] SFDR¼ 2 3 ½IIP3 ÀEINÀ10 log 10 ðNBWÞ ð23Þ where IIP3 is the third-order input intercept point, EIN in (dB/Hz) is the equivalent input noise, and NBW (in Hz) is the noise bandwidth. 7.1.8. Adjacent- and Cochannel Power Ratio Tests. In modern telecommunication circuits, signals constituting one or more modulated carriers are handled. Character- ization of the intermodulation performance of such cir- cuits cannot, therefore, be performed using two-tone and three-tone input signals; a combination of equally spaced tones—in practice, more than B10 sinusoids [71], with constant power and correlated or uncorrelated phases— would be more appropriate [72]. Because of the nonlinearity of the device under test, intermodulation products will be generated. These inter- modulation products can be classified as adjacent-channel distortion when their frequencies are located to the right or to the left of the original spectrum, or cochannel dis- tortion when their frequencies are located exactly over the original spectrum. The adjacent-channel power ratio (ACPR) is defined as the ratio between the total linear output power and the total output power collected in the upper and lower adjacent channels [73]. The cochannel power ratio (CCPR) is defined as the ratio between total linear output power and total distortion power collected in the input bandwidth [73]. The intermodulation distor- tion ratio (IMR) is the ratio between the linear output power per tone and the output power of adjacent-channel tones [73]. In fact, the ACPR, CCPR, and IMR distortion measure- ments are simple extensions of the two-tone intermodula- tion measurement [74]. However, it is important to first generate a very clean multitone signal. This can be easily achieved using the technique described in Ref. 75. 8. INTERMODULATION MEASUREMENT 8.1. Measurement Equipment 8.1.1. Multitone Tests. A block diagram of the system used for multitone intermodulation measurement is shown in Fig. 5. The multiple-frequency source can be implemented from two or three synthesized sine/square/ triangular-wave generators. Amplifier/attenuator pairs can be added at the output of each generator. Bandpass filters can also be added to suppress the harmonic con- tents at the output of each generator. For RF measure- ments, harmonic suppression and isolation between different generators is achieved by using amplifier/circu- lator combinations and cavity resonators [76]. The syn- thesized sources are combined using hybrids or combiners of adequate isolation. Spectral purity at this point is cru- cial to the accuracy of the measurement. The multitone output is fed to the device under test (DUT). The output of the DUT is fed to the spectrum analyzer. For RF measure- ments, the output of the DUT can be fed to directional couplers. The outputs of the directional couplers are fed to a television oscilloscope and/or a spectrum analyzer. Alternatively, for audiofrequency measurements, the intermodulation components of interest can be filtered out, using bandpass filters, and fed to AC voltmeters. For audiofrequency measurements, resistive combiners are widely used for combining the outputs of two or more signal generators. 8.1.2. Measurement Using a Microcomputer. Intermod- ulation can also be measured using a microcomputer [77]. A block diagram of this technique is shown in Fig. 6. This technique is based on measuring the single-tone input– output characteristic of the DUT using a vector voltmeter. The output of the vector voltmeter is fed to a microcom- puter that converts it into three digital data lines repre- senting the input amplitude, the output amplitude, and the phase lag between the input and output signals. After storing the data, the microcomputer increments the am- plitude of the input signal. After storing all the necessary data, the microcomputer, using a stochastic method, cal- culates the amplitudes of the intermodulation components of the DUT. Although the procedure reported in Ref. 77 uses a stochastic method for calculating the amplitudes of the intermodulation components resulting from a two- tone input signal, the same procedure can be applied to any number of input tones using different analytic tech- niques for modeling the nonlinear characteristics of the DUT. Alternatively, microcomputers can be added to the measurement setup of Fig. 5 to 1. Control the frequencies of the signal sources, espe- cially in the millimeter-wave range, where the A A BPF BPF BPF C DUT SG1 SG2 SA C Figure 5. Block diagram of the two-tone test setup; multitone tests require additional signal generators, combiners, amplifiers, and bandpass filters (SG—signal generator; A—amplifier; BPF— bandpass filter; C—combiner; DUT—device under test; SA—spec- trum analyzer). INTERMODULATION MEASUREMENT 2225 difference in frequencies between the signal sources may be less than 0.001 of the base signal frequency [78]. 2. Scan the base signal frequency over the measure- ment range of interest in predefined steps [79]. 3. Correct the power from each source so that power delivery to the DUT will be the same across the whole frequency range scanned. 4. Read and calculate the parameters of interest dur- ing the measurements [80,81]. 8.1.3. Noise Power Ratio Test. Figure 7 shows a block diagram of a noise power ratio test setup [62]. The setup consists of a white-noise generator that applies an accu- rate level of white Gaussian noise power with known bandwidth (equaling Do and centered around o 0 ) to the DUT. The output of the DUT is measured with the band- reject filter out. When the band-reject filter, with band- width¼do and centered around o 0 , is switched in, a nar- row band of frequencies is attenuated by about 70 dB, and a quiet channel, of width do and centered around o 0 , is formed as shown in Fig. 3. At the output of the DUT, the noise power is measured in the quiet channel, using a bandpass filter with bandwidth do and centered around o 0 . This noise power is due to the thermal noise and the intermodulation introduced by the DUT. The NPR is the ratio between the noise power measured without the band-reject filter inserted before the DUT to that mea- sured with the band-reject filter inserted. The white-noise generator corrects the loading power level for the inser- tion loss of the band-reject filter. 8.1.4. Noise Floor and SFDR Test. Figure 8 shows a test setup for measurement of noise floor and the SFDR of a communication link [70]. To measure the noise floor of the communication link, the transmitter is switched off. Then the noises of the low-noise amplifier and the spectrum an- alyzer are measured. Switching the transmitter on in- creases the noise floor by the transmitter noise and therefore the difference between the two noise measure- ments is the noise generated by the transmitter. To measure the SFDR, the input power is decreased until the IM3 level equals the noise floor. Recall that de- creasing the input power by 1dB decreases the IM3 level by 3 dB. However, this is true only if the third-order non- linearity is dominant. Higher-order nonlinearities will contribute to the third-order intermodulation (IM3), and DC DC DUT VV MC SG Figure 6. Block diagram of a microcomputer-based intermodu- lation measurement setup (SG—signal generator; DC—direction- al coupler; DUT—device under test; VV—vector voltmeter; MC— microcomputer). • • • • • • • • • • • • WNG BPF1 BRF DUT BPF2 PM Figure 7. Block diagram of the noise power ratio test setup (WNG—white-noise genera- tor; BPF1—bandpass filter with bandwidth do centered around o 0 ; BRF—band-reject fil- ter with bandwidth do centered around o 0 ; DUT—device under test; BPF2—bandpass fil- ter with bandwidth do centered around o 0 ; PM—power meter). Communication Link R LNA SA SG1 SG2 CIR2 CIR1 C T Figure 8. Setup for noise floor and SFDR measurement (SG— signal generator; CIR—circulator; C—combiner; T—transmitter; R—receiver; LNA—low-noise amplifier; SA—spectrum analyzer). 2226 INTERMODULATION MEASUREMENT in such cases the measured SFDR will be different from calculations obtained using Eq. (23). 8.1.5. Externally Induced Intermodulation Test. This is a two-tone test with one signal applied to the input and the other signal applied to the output [9]. A test setup is shown in Fig. 9. Two directional couplers are used to gauge both the forward-carrier power and the intermod- ulation product levels. Two more directional couplers are added to inject the interfering signal and to measure the actual injected value using the spectrum analyzer. 8.2. Measurement Accuracy 8.2.1. Multitone Tests. For accurate measurements of the intermodulation products using multitone tests, it is essential to reduce, or remove, the nonlinear distortion originating in the signal sources and/or the measurement equipment. Measurement accuracy may, therefore, be affected by the purity of the signal sources, the linearity of the combiners, and the performance of the spectrum analyzer. 8.2.2. Signal Sources. Measurement of the amplitudes of the intermodulation components requires the use of two or more signals. The frequencies of these signals must be noncommensurate. Otherwise, harmonics in one source might interfere with the fundamental(s) of other signal(s) and thus interfere with the desired intermodulation com- ponents. Ideally the signal generators would produce perfect si- nusoids, but in reality all signals have imperfections. Of particular interest here is the spectral purity, which is a measure of the inherent frequency stability of the signal. Perhaps the most common method used to quantify the spectral purity of a signal generator is its phase noise [82]. In the time domain, the phase noise manifests itself as a jitter in the zero crossings of a sine wave. In the frequency domain, the phase noise appears as sidebands surround- ing the original frequency. Thus, mixing with other fre- quencies, due to the nonlinearities of the DUT, would result in additional intermodulation products. It is, there- fore, important to consider the intermodulation due to phase noise when calculating the intermodulation perfor- mance of the DUT [83]. Signal generators with automatic level control (ALC) may produce signals with unwanted modulation. The ALC is implemented by rectifying the output signal of the gen- erator and feeding back the resulting DC voltage to drive an amplitude modulator. If a second signal is applied to the output of the signal generator, the detector will pro- duce a signal at the point of difference between the two frequencies. This signal will modulate the generator’s out- put. The frequency of the modulation sidebands will share the same spectral lines as the intermodulation products of interest. Isolating the signal generators and the combin- ers can minimize such an effect. This can be achieved by ensuring that there is as much attenuation as possible between them. 8.2.3. Combiners. Measurement of intermodulation products is performed by applying to the input of the cir- cuit, or the system, under test a signal consisting of two or more different frequencies obtained from different signal generators. The outputs of the signal generators are, therefore, combined by a combiner. The combiner must provide sufficient isolation between the signal sources to reduce the possibility of producing intermodulation prod- ucts before the combined input signal is applied to the circuit or the system under test. While resistive combiners are adequate for input signal levels up to a few millivolts, for larger voltage levels the use of power combiners may be inevitable [84]. Insertion of an attenuator in each arm of the combiner helps minimize the distortion components resulting from the interaction between the two signal sources. Such components, if generated, should be at least 80 dB below the fundamental components. A simple test to determine whether adequate isolation has been achieved can be effected by introducing a vari- able attenuator between the signal source combiner and the DUT in Fig. 6. This is set to a low value during mea- surements, but at setup, when IMPs have been located on the spectrum analyzer, increasing the attenuation by 3 dB will result in a reduction in the observed IMP level. If this reduction is only 3 dB, it has to be assumed that the IMP observed has originated in the signal sources, not in the DUT. If, however, the reduction is 6 dB for a second-order IMP or 9dB for a third-order IMP [see Eq. (4)], then it is safe to assume that the IMP has originated in the DUT or the spectrum analyzer. Alternatively, a technique that attenuates the parasitic intermodulation products that result from the interaction between the generators of the fundamental components, before the input of the spectrum analyzer, was described in Ref. 85. A block diagram of the technique is shown in Fig. 10. The input to the system under test is formed by SG2 A BPF DC SA DUT SG1 DC • PM Figure 9. Measurement of externally induced intermodulation can be performed by using two tones: one injected at the input and one injected at the output of the DUT (SG—signal generator; DC—directional coupler; PM—power meter; SA—spectrum analyzer; BPF—bandpass filter; A—amplifier). INTERMODULATION MEASUREMENT 2227 combining the outputs of two signal generators at fre- quencies o 1 and o 2 in the combiner. The first hybrid com- biner/splitter (HCS1) splits the combined signal into two branches with voltage transfer ratio a¼a and b¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Àa 2 p at the first and second outputs. Using Eq. (1), and assum- ing that the system under test and the compensator have identical nonlinear characteristics, the inputs of the sec- ond hybrid combiner/splitter (HCS2) can be expressed as V a ¼ N n¼0 k n ðaV i Þ n ð24Þ and V b ¼ N n¼0 k n ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Àa 2 p V n i _ _ ð25Þ Using Eqs. (24) and (25), the output of the second hybrid combiner/splitter (HCS2), with voltage transfer ratio op- posite in sign and equal to the reciprocal of that of HCS1, can be expressed as V out ¼ N n¼0 Àk n ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Àa 2 p ðaV i Þ n Àa ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Àa 2 p V n i _ _ _ _ ð26Þ According to Eq. (26), broadband compensation occurs for the linear components of the combined signal, with n¼1. Thus, all the linearly transformed spectral components are eliminated. This is also true for the intermodulation components that may result from the nonlinear interac- tion between the two signal generators. The output of HCS2 can, therefore, be applied directly to the spectrum analyzer. This technique does not require complicated high-order selective filters and can attenuate the parasitic intermod- ulation components and the fundamental frequency com- ponents by about 50 dB over a wide range of frequencies differing by 7–10 octaves. However, it requires a compen- sator with a nonlinear characteristic similar to that of the system under test. 8.2.4. Spectrum Analyzers. Spectrum analyzers are widely used in measuring the intermodulation perfor- mance of electronic circuits and systems. Internal circuits of the spectrum analyzers are, themselves, imperfect and will also produce distortion products [46]. The distortion performance of the analyzers is usually specified by the manufacturers, either directly or lumped into a dynamic range specification. The performance of the analyzer can be stretched, however, if the nature of these distortion products is understood. Amplitudes of the distortion products, resulting from the internal circuits of the analyzer, can be reduced by reducing the signal levels at the analyzer’s input. Thus, using internal and/or external attenuators can reduce the input signal levels to the analyzer and hence reduce its distortion products and improve the intermodulation mea- surement range of the spectrum analyzer. However, re- duced input levels to the analyzer mean reduced signal-to- noise ratio, and the distortion component to be measured may be buried in the noise. While reducing the resolution bandwidth of the analyzer can reduce noise, this may lead to slower sweep rate. Thus, achieving an optimum dynam- ic range involves tradeoffs between input signal levels and analyzer distortion. Usually, datasheets of analyzers will contain information about noise level in each resolution bandwidth and distortion products generated by the ana- lyzer for each input level. Using this information, one can determine the dynamic range of the analyzer for various input levels [86]. Whenever good selectivity, as well as sensitivity and dynamic range, are of prime importance, test receivers may be used in preference to spectrum analyzers [6]. Al- ternatively, if the frequencies of the intermodulation com- ponents of interest are sufficiently lower (or higher) than the fundamental frequencies, then lowpass (or highpass) filters can be used to remove the fundamental components that would give rise to other nonlinear distortion compo- nents in the spectrum analyzer. Attenuation factors of 80 dB or more, at frequencies outside the band of interest, are recommended. The insertion loss of the lowpass (or the highpass) filter should be as small as possible; 0.4dB or less is recommended. If the frequency of the intermodulation component of interest is not sufficiently higher (or lower) than the fun- damental frequencies, then it would be necessary to have complicated multiple-section high-order filters with am- plitude–frequency characteristics that are nearly rectan- gular. Such filters will change, to some extent, the amplitude of the intermodulation components, and this will complicate calculation of the intermodulation perfor- mance of the system under test. A method for compensat- ing for a large fundamental component, thus allowing the measurement of small intermodulation components in its presence, was described in Ref. 87. A block diagram of the compensation method is shown in Fig. 11. The input to the system under test is formed of one large amplitude signal at frequency o 1 and one small amplitude signal at frequency o 2 with o 1 5o 2 . The output of the system under test contains fundamental compo- nents at frequencies o 1 and o 2 , and intermodulation com- ponents at frequencies o 2 7no 1 , n¼1, 2, y, N. In order to measure the small amplitude intermodulation compo- nents, it is necessary to avoid applying to the analyzer the fundamental component at frequency o 2 . This can be achieved as follows. The output of the system under test is SG1 SG2 C HSC1 DUT CO HSC2 SA a b V i • Figure 10. A technique for attenuating the intermodulation products resulting from interaction between the signal genera- tors of the fundamental components (SG—signal generator; C— combiner; DUT—device under test; BRF—band-reject filter; PS— phase shifter; DA—differential amplifier). 2228 INTERMODULATION MEASUREMENT fed to the second band-reject filter BRF2 to suppress the fundamental component at o 1 . The output of the signal generator of frequency o 2 is fed to the first band-reject fil- ter BRF1 to suppress any component at frequency o 1 be- fore reaching the phase shifter through the combiner. The phase shifter compensates, at the frequency o 2 , the phase shift through the system under test. Ideally, the voltages of frequency o 2 at the inputs of the differential amplifier are equal. Thus, the output of the differential amplifier at frequency o 2 is ideally zero. In practice, the output voltage at o 2 will be attenuated by 50–60 dB [6]. The output of the differential amplifier, with suppressed fundamental component at frequency o 2 , can be applied to the spectrum analyzer. This compen- sation technique, which entails additional filters and matching units, can be used only for broadband measure- ments with o 1 5o 2 . Although spectrum analyzers using digital IF sections may not suffer from the internally generated distortion, discussed above, they may suffer from the relatively low-level distortion products resulting from the analog- to-digital conversion. The amplitudes of these products is usually less sensitive to the amplitude of the signal components. 8.2.5. Noise Power Ratio Test. The accuracy of the noise power ratio (NPR) test is affected mainly by two factors: (1) the noise floor of the amplifier that will dominate under very low loading conditions and (2) the distortion products produced under very high loading conditions. It is, there- fore, recommended to sweep the loading between two pre- specified start and stop levels. The NPR is measured at different levels, and the largest measured value of NPR is considered as the worst case. 8.2.6. Microcomputer-Based Tests. 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In many such problems, an exact solution does not exist and a direct numerical solu- tion may not be feasible because of the very large number of degrees of freedom. Typical applications include the so- lution of large systems of differential and integral equa- tions that may involve thousands or millions of unknown variables. For these problems it may not be possible to generate and store a full system matrix, and then solve it directly (e.g., by inversion, factorization, or Gauss elimi- nation). Iterative methods only need to apply an operator (or system matrix) to the solution at each iteration. They are particularly well suited for the solution of sparse ma- trix systems because a large percentage of the operations involved are negligible. Mathematically, an iterative algorithm starts with an initial approximate solution, and repeatedly applies an operator to the solution to improve its accuracy at each iteration. Eventually the solution should converge to a given level of accuracy. Convergence is the primary issue associated with any iterative method. The solution may converge very slowly if the iterative operator is not well conditioned, or it may even diverge. Figure 1 illustrates the basic iterative loop. Initial approximate solution End Yes No Is solution sufficiently accurate? Apply iterative operator to obtain improved solution from previous solution Figure 1. Schematic diagram of an iterative solution. ITERATIVE METHODS 2231 There are two broad categories of iterative methods: stationary and nonstationary. Stationary methods are characterized by an operator that does not change with each iteration. Classical iterative methods are included in this category, such as Jacobi and Gauss–Seidel. Conjugate- gradient methods are included in the class of nonstation- ary methods, wherein some parameters in the operator change with each iteration [1]. In general, conjugate-gradient iterative methods have better convergence properties than do classical iterative methods when compared over a wide range of problems. The convergence of classical methods tends to be very problem-dependent. In fact, classical methods are often based on the underlying physics of a particular scenario. For example, a classical iterative algorithm may be de- signed to model the multiple electromagnetic (EM) wave scattering between two or more objects. Such an algorithm could be very rapidly convergent for that problem, but slowly convergent or even divergent for a different prob- lem. On the other hand, conjugate-gradient methods have theoretically guaranteed convergence if the system matrix is nonsingular, although in practice the limited numerical precision of a computer may cause the algorithm to stall. The convergence of any iterative method may be im- proved by altering the formulation so that it is better con- ditioned. This is referred to as preconditioning the operator or system of equations. The accuracy of the solu- tion at each iteration may be gauged in terms of the re- sidual error, which is a measure of how well the solution satisfies the original system of equations. 2. HISTORICAL REVIEW OF ITERATIVE METHODS IN ELECTROMAGNETICS Iterative methods in EM did not become popular until ad- vances in computer technology made it possible to solve large systems of equations. Classical iterative methods were developed to model physical EM interactions be- tween different parts of a geometry. Thiele et al. first de- veloped a hybrid technique to combine physical optics and the method of moments in 1982 [2–4]. The solution iter- ates between the optically lit region and the shadow re- gion of an arbitrary scattering geometry. This method was extended further and made more general by Hodges and Rahmat-Samii [5], including the interactions between an- tennas and their supporting platform. Domain decompo- sition was used by Sullivan and Carin to break up a method-of-moments (MoM) problem into multiple, sim- pler, solution regions [6]. Iterative method of moments and iterative physical optics have been used to solve multi- bounce problems such as the EM scattering from large open-ended cavities [7,8]. Classical iterative methods have also been applied extensively to compute the scattering from rough surfaces. The forward-backward method de- veloped by Holliday et al. [9], and the method of ordered multiple interactions of Kapp and Brown [10], take ad- vantage of the dominant forward and backward propaga- tion of EM waves over a rough surface. The generalized forward–backward method extended this work to include an obstacle on the rough surface by modifying the matrix splitting used in the forward–backward method [11]. Com- parisons of stationary with nonstationary iterative meth- ods are presented in Refs. 12–14. The conjugate gradient (CG) method was developed in 1952 by Hestenes and Stiefel [15]. However, like the clas- sical iterative methods, it was not used in the area of elec- tromagnetics until advances in computers made it possible to solve large linear systems. Sarkar and Rao used the CG method to solve method of moments problems in 1984 [16], and Sarkar and Arvas presented a more general CG de- velopment for eletromagnetics problems in 1985 [17]. The CG–fast Fourier transform method (CG-FFT) became pop- ular for solving quasiplanar geometries in the late 1980s [18,19]. The development of fast integral equation meth- ods, such as the CG-FFT, the fast multipole method [20], the adaptive integral method [21], and the precorrected FFT method [22] gave CG methods a boost. These methods greatly reduce the computational cost of applying the in- tegral equation operator, thereby allowing very large sys- tems of equations to be solved. 3. MATRIX NOTATION FOR ITERATIVE METHODS The solution of a system of equations with N degrees of freedom, or unknown variables, may be expressed in ma- trix format as Ax ¼b ð1Þ where A is an NÂN system matrix, x is a column vector containing the N unknown coefficients, and b is a known excitation-dependent column vector. The individual ele- ments of this equation may be expressed as b m ¼ N n¼1 A mn x n ð2Þ This matrix equation is obtained by discretizing the EM operator governing the problem of interest, whether it is from a differential equation or integral equation formula- tion. The unknown quantity, such as the EM fields or equivalent currents, are expanded into a set of N known basis functions with unknown coefficients compris- ing the column vector x. The N basis functions are tested (or sampled) with N test functions to yield a system of N equations. The preceding equation for the unknown coefficients may be solved using direct matrix inversion or factoriza- tion. However, the operational complexity for the direct approach is O(N 3 ), that is, of order N-cubed. This means that the number of computations necessary to solve the system is proportional to N 3 , which may be too costly when there are thousands or millions of unknowns. Iterative methods have an operational complexity of no more than O(N 2 ) per iteration, which is the cost of doing one matrix– vector multiplication. So as long as the solution converges quickly, the iterative method is much more efficient. Iterative methods seek to solve Eq. (1) by succes- sively improving an initial solution to a desired degree of 2232 ITERATIVE METHODS accuracy. The residual error vector is a measure of the accuracy of the solution after k iterations and is defined by r ðkÞ ¼b ÀAx ðkÞ ð3Þ The residual error norm, or simply the residual error, is the length of this vector normalized to the length of the excitation vector, r ðkÞ _ _ _ _ = b k k, where jjr ðkÞ jj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ðkÞ r ðkÞ p . Here, the inner product (or vector product) of two column vectors is Hermitian, defined by ab ¼ N n¼1 a à n b n ð4Þ where the * superscript denotes the complex conjugate. (Note: The CG algorithm described later does not use the complex conjugate in the inner product, as will be made apparent.) The residual error tells us how well the solu- tion satisfies the system of equations, and is most often used as the criterion for halting the iterations. The abso- lute error vector is defined by e ðkÞ ¼x ðkÞ Àx ð5Þ where x is the exact solution to (1). The spectral radius of a matrix is defined as the magnitude of its largest eigenval- ue. This quantity is important for determining conver- gence of classical iterative methods, whereas the eigenvalue spectrum of a matrix determines the conver- gence of CG methods. 4. CLASSICAL ITERATIVE METHODS As is apparent from the historical review presented ear- lier, classical iterative methods are often used to solve problems via a physical decomposition of the geometry, sometimes even using a different solution technique for each region. All of these methods can be cast in the form of matrix splittings, where the original system matrix is de- composed in some manner that makes the problem easier to solve. Figure 2 shows some common matrix splittings. We will focus on the lower–upper (LU) triangular split- ting. The block-diagonal and banded matrix splittings are extensions of the LU splitting, where the diagonal D is replaced by the block-diagonal or banded portion of the matrices. Likewise, the hybrid decomposition is a special case of the block-diagonal splitting with only two blocks on the diagonal. All the matrix splittings of Fig. 2 have the general form A¼MÀN. We may then write an iterative equation from Eq. (1) as Mx ðkÞ ¼Nx ðkÀ1Þ þb ð6Þ starting with some initial solution candidate x (0) and solv- ing repeatedly. It is easy to show that if x (k) ¼x (k À1) , then Eq. (1) is satisfied and x (k) ¼x. To solve (6) for x (k) , we need Mto be easily invertible or factorizable. Diagonal matrices are trivial to invert, and block-diagonal matrices are easily inverted by inverting each block independently of the other blocks. Lower triangular and upper triangular matrices are also easy to invert via backward and forward substitutions, respectively [1]. All of these types of inver- sions are computed much more efficiently than inverting the entire system matrix A. The absolute error vector at the kth iteration may be shown to be e ðkÞ ¼ðM À1 NÞ k ðx ð0Þ ÀxÞ ð7Þ Therefore, the spectral radius of the matrix ðM À1 NÞ must be less than unity to guarantee convergence [1]. This en- sures that the absolute error approaches zero as k goes to infinity. The residual error vector for the kth iteration may be shown to be r ðkÞ ¼Nðx ðkÞ Àx ðkÀ1Þ Þ which is easily computed by saving the matrix–vector product Nx ðkÀ1Þ from the previous iteration. Some com- mon iterative algorithms based on the matrix splittings of Fig. 2 are discussed next. 4.1. Jacobi Iteration This is the simplest classical iteration algorithm. We choose M¼D and N¼ À(LþU), so the iterative equation becomes Dx ðkÞ ¼b ÀðLþUÞx ðkÀ1Þ ð8Þ The magnetic field integral equation (MFIE) has this form, which is also used in the iterative physical optics L U D A 11 A 21 A 22 A 12 (a) (b) (c) (d) Figure 2. Some common matrix splittings. (a) hybrid decomposi- tion; (b) lower–upper triangular; (c) block-diagonal; (d) banded. ITERATIVE METHODS 2233 technique [8]. The operational cost is O(N 2 ), which is the cost of computing the matrix–vector product ðLþUÞx ðkÀ1Þ on the right-hand side (RHS) of (8). 4.2. Gauss–Seidel Method This is an improvement over simple Jacobi iteration [1]. Here we choose M¼DþL and N¼ ÀU, resulting in ðDþLÞx ðkÞ ¼b ÀUx ðkÀ1Þ ð9Þ This equation is solved using forward substitution. This is easy to see by writing the expression for the individual elements as x ðkÞ m ¼ b m À mÀ1 n¼1 x ðkÞ n À N n¼mþ1 x ðkÀ1Þ n D m ð10Þ The elements x ðkÞ m are updated sequentially for m¼ 1,2,y,N, so the updated values can be used on the RHS of (10). The convergence of Gauss–Seidel is expected to be somewhat better than Jacobi, and with the same opera- tional cost. 4.3. Symmetric Gauss–Seidel Method The Gauss–Seidel method can be formulated using for- ward or backward substitution. A symmetric form of Gauss–Seidel iteration is obtained using both forward and backward substitution in the following two-step algo- rithm: ðDþLÞx ðkÀ1=2Þ ¼b ÀUx ðkÀ1Þ ðDþUÞx ðkÞ ¼b ÀLx ðkÀ1=2Þ ð11Þ This is the form of the forward–backward method [9], or the method of multiple ordered interactions [10]. This two-step algorithm has the same operational cost of a one- step algorithm because the half-matrix–vector product Lx ðkÀ1=2Þ is reused in step 2 of each iteration, and Ux ðkÀ1Þ may be saved from the previous iteration and reused. 4.4. Relaxation Unless the problem geometry is very well ordered, or the system matrix is strongly diagonally dominant, the clas- sical iterative algorithms above will probably have poor convergence properties. To improve convergence, a relax- ation parameter [1] (or damping coefficient), o may be in- troduced. This is a constant usually in the range 0ooo2 such that the relaxed iterative equations reduce to the basic equations above for o¼1. The relaxed Jacobi itera- tive equation is given by Dx ðkÞ ¼ob þ ð1 ÀoÞDÀoðLþUÞ ½ x ðkÀ1Þ ð12Þ It is easy to show that (12) reduces to (8) for o¼1, and if x ðkÞ ¼x ðkÀ1Þ then (1) is satisfied and x (k) ¼x for any non- zero o. Likewise, the relaxed Gauss–Seidel method, also known as successive overrelaxation (SOR) [1], is given by ðDþoLÞx ðkÞ ¼ob þ ð1 ÀoÞDÀoU ½ x ðkÀ1Þ ð13Þ The relaxed form of symmetric Gauss–Seidel is known as symmetric successive overrelaxation (SSOR) [1], and is given by ðDþoLÞx ðkÀ1=2Þ ¼ob þ ð1 ÀoÞDÀoU ½ x ðkÀ1Þ ðDþoUÞx ðkÞ ¼ob þ ð1 ÀoÞDÀoL ½ x ðkÀ1=2Þ ð14Þ Figure 3 shows a plot of the convergence of relaxed Jacobi, SOR, SSOR, and the biconjugate gradient stabilized (BCGS) algorithms for the problem of radar scattering from a perfect electrically conducting cylinder computed by the method of moments [14]. For this relatively simple problem, the SSOR method has the best convergence and the BCGS, the worst. However, for more arbitrary geom- etries the classical iterations may fail to converge, and may eventually diverge. 5. CONJUGATE-GRADIENT (CG) METHODS CG methods are superior to classical iterative methods in the sense that they are theoretically guaranteed to converge in no more than N iterations if the matrix is 0 10 20 30 40 50 60 70 80 90 100 0.001 0.002 0.003 0.005 0.01 0.02 0.03 0.04 0.05 0.1 0.2 0.3 0.4 0.5 1 R e s i d u a l e r r o r n o r m Iteration units SSOR SOR FB Jacobi Bi-CGStab 2m 2m x z y Frequency 300 MHz Vertical polarization Elevation angle 30 deg E −−i Figure 3. Convergence of classical iterative methods compared with BCGS. 2234 ITERATIVE METHODS nonsingular. (In practice, the numerical precision of the computer may limit this theoretical convergence.) They are not limited to specific types of problems or physics, and can be used as general matrix solvers. This is achieved by generating a sequence of search vectors that will eventu- ally span the entire N-space. The only difference between the various CG versions is how these search vectors are generated. In general, the search vectors are chosen such that each new vector is linearly independent of all previ- ous vectors, and the residual error is minimized. The basic CG method is applicable only to symmetric systems and is presented here first, followed by two popular methods for nonsymmetric systems, the modified biconjugate gradient (BCG) and the generalized minimum residual (GMRES) methods. An excellent resource for these and other CG algorithms is the Templates book [23], for which associat- ed computer subroutines are readily available. 5.1. CG Algorithm for Complex Symmetric Matrices The basic CG algorithm for solving the complex symmetric matrix equation A~ xx ¼ ~ bb is listed below. A complete deriva- tion is included later in this article. In the following algo- rithm the vector products are not Hermitian, that is, there is no complex conjugation as in (4). Conjugate Gradient Algorithm 1 Initialization: ~ vv ¼ ~ bb ffiffiffiffiffiffiffiffiffiffi b ~ A ~ bb _ ; ~ xx ¼0; ~ rr ¼ ~ bb Iteration: 1. ~ xx ¼ ~ xx þa~ vv; a ¼ v ~ ~ bb 2. ~ uu¼A~ vv; ~ rr ¼ ~ rr Àa ~ uu 3. Check jj ~ rrjj=jj ~ bbjj e; if yes, then stop, and ~ xx is a good approximation; if no, then continue. 4. ~ pp¼ ~ rr Àb~ vv; b¼ u ~ ~ rr 5. ~ vv ¼ ~ pp= ffiffiffiffiffiffiffiffiffiffi p ~ A~ pp _ 6. Go to step 1. Each iteration of the CG algorithm involves one matrix– vector multiplication, which is at most an O(N 2 ) operation. The residual error of the solution is checked in step 3. If it is less than some threshold error e, the iterations are halt- ed. This threshold level determines the accuracy of the solution. For most engineering applications a threshold level in the range 0.0001–0.01 yields sufficient accuracy. Of course, greater accuracy requires more iterations, and it is possible for the algorithm to stall before reaching a given threshold. The convergence properties of the CG method are discussed later in this article. 5.2. Modified Biconjugate-Gradient Method for Nonsymmetric Matrices In solving the magnetic field integral equation or combined field integral equations or many other RF engineering applications (such as hybrid finite-element/integral equa- tion formulation), we quite often end up with a nonsym- metric matrix equation A~ xx ¼ ~ bb, where A is an NÂN complex nonsymmetric matrix. There are many variants of Krylov-based methods for solving this equation; here we shall list the modified BCG method, followed by the GMRES method. In the following algorithm the vector products are not Hermitian; that is, there is no complex conjugation as in (4). Modified BCG Algorithm for Solving Complex Nonsymmetric Matrix Equations Initialization: ~ xx ¼ ~ xx T ¼0 ~ rr ¼ ~ bb ~ rr T ¼ ~ bb ~ pp¼ ~ rr ~ pp T ¼ ~ rr T ~ vv ¼ ~ pp ffiffiffiffiffiffiffiffiffiffiffiffiffi p ~ T A~ pp _ ~ vv T ¼ ~ pp T ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ~ A T ~ pp T _ Iteration: 1. ~ xx ¼ ~ xx þa~ vv ¼ ~ xx þ v ~ T ~ bb _ _ ~ vv ~ xx T ¼ ~ xx T þa T ~ vv T ¼ ~ xx T þ v ~ ~ bb _ _ ~ vv T 2. Compute ~ uu¼A~ vv ~ uu T ¼A T ~ vv T 3. ~ rr ¼ ~ rr Àa ~ uu ~ rr T ¼ ~ rr T Àa T ~ uu T 4. Check convergence. If jj ~ rrjj=jj ~ bbjj e, then stop and ~ xx is a good approximation; if not, continue. 5. ~ pp¼ ~ rr Àb ~ pp b¼ u ~ T ~ rr u ~ T ~ pp ~ pp T ¼ ~ rr T Àb T ~ pp T b T ¼ u ~ ~ rr T u ~ ~ pp T 6. ~ vv ¼ ~ pp ffiffiffiffiffiffiffiffiffiffiffiffi p ~ T A~ pp _ ~ vv T ¼ ~ pp T ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ~ A T ~ pp T _ 7. Go to step 1. In this BCG algorithm it is noted that two matrix–vector products need to be computed for each iteration, one with the original matrix A and the other with its transpose A T . This makes the BCG methods roughly twice as computa- tionally expensive as the CG method for symmetric sys- tems. The difference between the basic BCG and the modified BCG is that the former uses the Hermitian ma- trix A H (i.e., complex conjugate–transpose) instead of the transpose matrix A T . ITERATIVE METHODS 2235 5.3. GMRES Method for Nonsymmetric Matrix Equations Another useful Krylov space iterative method for nonsym- metric systems is the generalized minimum residual (GMRES) method [24]. Like the CG method, a sequence of linearly independent search vectors is generated. How- ever, unlike the CG method, the entire set of search vec- tors is saved in memory. Coefficients are found that give the minimum residual error over the complete set of search vectors. In essence, it is a ‘‘brute force’’ CG meth- od. The advantages are that only one matrix–vector prod- uct is computed per iteration and the transpose of the matrix is not needed. Furthermore, the GMRES method truly minimizes the residual at each iteration, so its con- vergence is monotonic. The disadvantage is that all the previous search vectors must be stored in memory. There- fore, the memory requirement grows with the number of iterations. This may not be a problem for dense system matrices for which the matrix storage is generally much larger than the storage of a set of search vectors (depend- ing, of course, on how many search vectors are stored). To alleviate the memory requirement, the GMRES al- gorithm may be restarted after a certain number of iter- ations. The solution vector after one set of iterations is used as the initial solution for the next set of iterations. However, the restarted version of the GMRES algorithm is not guaranteed to converge because the reduced set of expansion vectors may not span the entire solution space. The GMRES algorithm is listed in the third-edition book by Golub and Van Loan [1] and in Templates [23]. A simplified algorithm that is conceptually equivalent to GMRES, the generalized conjugate residual (GCR) method [25], is listed below. In the following algorithm the vector products are Hermitian, using complex conju- gation as in (4). Generalized Conjugate Residual Algorithm Initialization: x ¼0, r ¼b, p 1 ¼b, u 1 ¼Ap 1 Iteration: k¼1; 2; . . .: 1. x ¼x þap k , a ¼u k r= u k k k 2 . 2. r ¼r Àau k . 3. Check jjrjj=jjbjj e. If yes, then stop, and x is a good approximation; if no, then continue. 4. b i ¼ u i Ar u i k k 2 , for i ¼1; 2; . . . ; k. 5. p k þ1 ¼r À k i ¼1 b i p i . 6. u k þ1 ¼Ar À k i ¼1 b i u i . 7. Go to step 1. This algorithm is very similar to the basic conjugate- gradient method. Note that only one matrix–vector prod- uct is used per iteration (in step 4) if we store all the vectors p i and u i for i ¼1; 2; . . . ; k: It is also helpful to store u i k k 2 to avoid repeated computation in step 4. If storage becomes excessive, the algorithm may be restarted after the mth iteration starting with p 1 ¼p m and u 1 ¼u m . 6. PRECONDITIONERS FOR ITERATIVE METHODS The convergence rate of iterative methods, both classical and conjugate-gradient, can be very slow if the system matrix is not well-conditioned. As mentioned in the sec- tion on classical iterative methods, the convergence of these methods depends on the spectral radius of the iter- ation matrix ðM À1 NÞ. Similarly, the convergence rate of CG methods depends on the spectral properties of the sys- tem matrix (see Section 7 for a discussion). Certain for- mulations in electromagnetics give rise to poorly conditioned systems, such as the electric field integral equation (EFIE). Sometimes the formulation may be altered to give a better conditioned system, such as by converting the EFIE to the combined field integral equa- tion. The choice of basis functions may also affect the con- ditioning. Alternatively, one may apply a preconditioner matrix M to the original system as M À1 Ax ¼M À1 b Clearly, if the inverse of M approximates the inverse of A, then the solution of this system should be easier, or, math- ematically speaking, the matrix M À1 A should have better spectral properties than the original matrix. The precon- ditioner may be implemented in any iterative algorithm by replacing matrix–vector products of the form Ap with M À1 Ap, and the excitation vector b with M À1 b. There are cleverer ways to do this as described in Section 7. The preconditioner should improve convergence, while its inverse M À1 (or factorization) must be computed effi- ciently. It is not a coincidence that the preconditioner ma- trix M uses the same symbol as the classical iterative matrix splitting M. In fact, the M matrix of all of the clas- sical iteration matrix splittings discussed here and shown in Fig. 2 may be used as a preconditioner, namely, diago- nal, block-diagonal, lower or upper triangular, and banded. Classical splittings often mimic wave interactions, which make them useful as preconditioners. From the matrix- splitting point of view, we want the matrix Mto contain the ‘‘dominant’’ portion of the system matrix A. Then the in- verse of M will approximate the inverse of A, and the it- erative algorithm should therefore converge rapidly. A very effective preconditioner for the EFIE with sub- sectional basis functions is described in Ref. 26. The pre- conditioner M is a sparse version of A, which contains the matrix entries corresponding to basis interactions within a specified distance. Incomplete factorization is used to compute a sparse factorization of M. In fact, there is a large class of preconditioners that use incomplete factor- ization. Some common preconditioning approaches for iterative algorithms are discussed in Refs. 1 and 23. 7. THEORY OF THE BASIC CG METHOD The basic CG method is applicable only to symmetric sys- tems. Consider the following complex symmetric matrix equation Ax ¼b ð15Þ 2236 ITERATIVE METHODS where A is an NÂN complex symmetric matrix, x is the solution column vector, and b is the RHS excitation col- umn vector. Before we derive the CG method, let us try to answer a few related questions first. A-Conjugate Condition. Given a set of basis column vectors v 0 ; v 1 ; . . . ; v nÀ1 f g with noN, how to determine the best ap- proximate solution ~ xx app ¼ nÀ1 i ¼0 c i ~ vv i ð16Þ that solves (15). The answer is the Galerkin method, or weighted residual. For each column vector, we shall form a residual column vector ~ RR¼ ~ bb ÀA~ xx app ð17Þ Note also that equation (16) can be written in matrix form as ~ xx app ¼V~ cc V¼ ~ vv 0 ~ vv 1 Á Á Á ~ vv nÀ1 _ ¸ ; ~ cc ¼ c 0 c 1 . . . c nÀ1 _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ ð18Þ Subsequently, by requiring that the residual vector ~ RR be orthogonal to all the basis vectors is equivalent to solving the coefficient vector ~ cc through the following reduced ma- trix equation V t AV _ _ ~ cc ¼V t ~ bb ð19Þ Let us take a closer look at Eq. (19). If the reduced matrix V t AV turns out to be an identity matrix, then the coeffi- cients can be simply computed by c i ¼v ~ i ~ bb; v ~ ¼ð ~ vvÞ t ð20Þ What is more is that, as will be seen later in this section, there is no need to store all these basis vectors in order to find the approximate matrix solution ~ xx app . Requiring V t AV I implies that the basis vectors need to satisfy v ~ i A~ vv j ¼d ij ð21Þ which is called the A-conjugate condition. Before moving on to derive the CG methods, lets take a few moments to restate what we have discussed in a more fundamental way. You see, as in many applications, to solve equations, whether infinite dimensional problems (integral equation formulations), or finite-dimensional problems (like matrix equations), the Galerkin method is a very good method of choice. Once again, in applying the Galerkin method, we shall need to establish what the trial and test function spaces are. When the operators are sym- metric, some would argue that they need to be positive definite as well, we can simply have both the trial and test functions be the same. The next logical question will be how to generate these basis vectors that span the trial and test function spaces. As basic linear algebra taught us, these basis vectors at least need to be linearly indepen- dent, preferably orthonormal. This is where the A-conju- gate condition comes in. When the operator is symmetric, we can, with some violation when the operator is not pos- itive definite, define the vector inner product as ~ vv i ; ~ vv j ¸ _ ¼v ~ i A~ vv j ð22Þ As you shall see, different definitions of the ‘‘inner prod- uct’’ lead to different variants of CG methods. For the matrix equation A~ xx ¼ ~ bb with a nonzero initial solution ~ xx i , it is always possible to solve for the correction equation A~ xx 0 ¼ ~ bb 0 ¼ ~ bb ÀA~ xx i _ _ , and with ~ xx ¼ ~ xx 0 þ ~ xx i . There- fore, without loss of generality, we shall assume that we will solve A~ xx ¼ ~ bb with initial guess zero. We shall derive the CG method by induction. k ¼0: With the initial solution ~ xx ð0Þ ¼0, the residual vector is simply ~ rr ð0Þ ¼ ~ bb. The trial space for solving the matrix equation can now be established as V 0 ¼ ~ vv 0 ½ ¼MGS A ~ rr 0 f g ð23Þ The notation MGS A ~ aa 0 ~ aa 1 Á Á Á ~ aa nÀ1 _ _ means making a orthonormal basis from the n column vectors, ~ aa 0 ~ aa 1 Á Á Á ~ aa nÀ1 , through the modified Gram–Schmidt (MGS) process and the inner product is defined by the A-conjugate condition: ~ vv 0 ¼ ~ rr 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi r ~ 0 A~ rr 0 _ ð24Þ Notice that, in equation (24), we have an expression r ~ 0 A~ rr 0 . If the matrix A is positive definite, the expression r ~ 0 A~ rr 0 will always be a positive nonzero number, thus Eq. (24) will always be valid. Since in our case A is a complex sym- metric matrix, it is possible that r ~ 0 A~ rr 0 0 even though ~ rr 0 O0. This is referred to as ‘‘breakdown’’ in the CG meth- od. Although, in practical computation, it rarely occurs, but when the matrix A is poorly conditioned, it is possible that r ~ 0 A~ rr 0 % 0 and thus causes slow and even failure to converge in the CG process. It should be emphasized here that many researchers object to the use of CG method to non-positive-definite matrix equations; in reality, with good preconditioners (a topic which is of paramount im- portance) the CG method may be used to solve complex symmetric matrix equations. k ¼1: The best solution in the trial space V 0 ¼span ~ vv 0 f g, from the Galerkin method, for the matrix equation ITERATIVE METHODS 2237 A~ xx ¼ ~ bb is ~ xx ð1Þ ¼c 0 ~ vv 0 ; c 0 ¼v ~ 0 ~ bb ð25Þ Subsequently, the residual vector ~ rr ð1Þ can be obtained as ~ rr ð1Þ ¼ ~ bb ÀA~ xx ð1Þ ¼ ~ rr ð0Þ Àc 0 A~ vv 0 ð26Þ Since the solution is solved through Galerkin method, and v ~ 0 ~ rr ð1Þ ¼0, it is certain that ~ rr ð1Þ is linearly independent with vectors from V 0 ¼span ~ vv 0 f g; therefore it would be a good idea to have V 1 ¼V 0 [ span ~ rr ð1Þ _ _ ¼MGS A ~ vv 0 ~ rr ð1Þ _ _ ð27Þ Consequently, our new basis vector is determined through the modified Gram–Schmidt (MGS) process: ~ ww¼ ~ rr ð1Þ Àb~ vv 0 ; b¼v ~ 0 A~ rr ð1Þ ¼r ~ ð1Þ A~ vv 0 ~ vv 1 ¼ ~ ww ffiffiffiffiffiffiffiffiffiffiffiffi w ~ A ~ ww _ ð28Þ It is easy to verify that v ~ 0 A~ vv 1 ¼v ~ 1 A~ vv 0 ¼0 and v ~ 0 A~ vv 0 ¼ v ~ 1 A~ vv 1 ¼1. Moreover, we see from equation (26), A~ vv 0 2 V 1 . To summarize, at k ¼1, we have the following condi- tions: 1. V 1 ¼span ~ vv 0 ~ vv 1 _ _ ¼MGS A ~ vv 0 ~ rr ð1Þ _ _ 2. V t 1 AV 1 ¼I 3. A~ vv 0 2 V 1 kth iteration: At this moment, we have the trial space V kÀ1 ¼span ~ vv 0 ~ vv 1 Á Á Á ~ vv kÀ1 _ _ and it satisfies 1. v ~ i A~ vv j ¼d ij ; i; j ¼0; 1; . . . ðk À1Þ 2. A~ vv i 2 V kÀ1 ; i ¼0; 1; . . . k À2 ð Þ The best matrix solution in the trial space V kÀ1 ¼ span ~ vv 0 ~ vv 1 Á Á Á ~ vv kÀ1 _ _ is then ~ xx ðkÞ ¼ kÀ1 i ¼0 c i ~ vv i ¼ kÀ1 i ¼0 v ~ i ~ bb _ _ ~ vv i ¼ ~ xx ðkÀ1Þ þ v ~ kÀ1 ~ bb _ _ ~ vv kÀ1 ¼ ~ xx ðkÀ1Þ þa~ vv kÀ1 ð29Þ and of course, the residual vector is computed through ~ rr ðkÞ ¼ ~ bb ÀA~ xx ðkÞ ¼ ~ rr ðkÀ1Þ ÀaA~ vv kÀ1 ð30Þ From the Galerkin method it follows that v ~ i ~ rr ðkÞ ¼0; i ¼0; 1; . . . k À1 ð Þ ð31Þ Since A~ vv i 2 V kÀ1 ; i ¼0; 1; . . . k À2 ð Þ, we also have v ~ i A~ rr ðkÞ ¼0; i ¼0; 1; . . . k À2 ð Þ ð32Þ Subsequently, the next basis vector will be computed by ~ pp¼ ~ rr ðkÞ À kÀ1 i ¼0 b i ~ vv i ¼ ~ rr ðkÞ À kÀ1 i ¼0 v ~ i A~ rr ðkÞ _ _ ~ vv i ¼ ~ rr ðkÞ À v ~ kÀ1 A~ rr ðkÞ _ _ ~ vv kÀ1 ¼ ~ rr ðkÞ Àb~ vv kÀ1 ð33Þ and ~ vv k ¼ ~ pp ffiffiffiffiffiffiffiffiffiffi p ~ A~ pp _ ð34Þ Consequently, ~ vv 0 ~ vv 1 Á Á Á ~ vv k _ _ is an A-conjugate basis for the trial space V k . This process continues until ~ rr ðkÞ _ _ _ _ is very small at a certain iteration k; it then implies, for all practical purposes, that ~ xx ðkÞ is the solution to the matrix equation A~ xx ¼ ~ bb. Note that the process is extremely simple and the recursive nature of the process makes it possible not to store all the basis vectors. The detailed induction argument above leads directly to the basic CG algorithm 1 listed earlier in this article. 7.1. Convergence Rate of Conjugate-Gradient Methods There are two features that can make CG converge fast: (1) eigenvalue clusters and (2), a good condition number of the matrix. To see why eigenvalue clusters are good for CG method, let’s look at the following theorem. Theorem 1. Assume that matrix A, which is a diagonal- izable NÂN symmetric matrix, has only k distinctive eigenvalues, namely lðAÞ ¼ l 0 l 0 . . . l 0 ¸fflfflfflfflfflfflfflfflfflfflfflfflfflffl..fflfflfflfflfflfflfflfflfflfflfflfflfflffl¸ n 0 l 1 l 1 . . . l 1 ¸fflfflfflfflfflfflfflfflfflfflfflfflfflffl..fflfflfflfflfflfflfflfflfflfflfflfflfflffl¸ n 1 . . . l kÀ1 l kÀ1 . . . l kÀ1 ¸fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl..fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl¸ n kÀ1 _ _ _ _ _ _ n 0 þn 1 þ . . . þn kÀ1 ¼N ð35Þ then the dimension of the Krylov space K m ~ vv 0 ; A ð Þ ¼ ~ vv 0 A~ vv 0 Á Á Á A mÀ1 ~ vv 0 _ _ ð36Þ will always be bounded by k regardless of the initial vector ~ vv 0 and m: dim K m ~ vv 0 ; A ð Þ ½ k ð37Þ Proof: Let ~ ee p i be the ith eigenvector corresponding to eigenvalue l p of the matrix A: A~ ee p i ¼l p ~ ee p i ; i ¼0; 1; . . . n p À1 _ _ ð38Þ 2238 ITERATIVE METHODS Since these eigenvectors form a complete set of basis vec- tors, any column vector ~ vv 0 can be written as a linear com- bination of these eigenvectors: ~ vv 0 ¼ kÀ1 p¼0 n p À1 i ¼0 c p i ~ ee p i ¼ kÀ1 p¼0 ~ vv ðpÞ 0 ~ vv ðpÞ 0 ¼ npÀ1 i ¼0 c p i ~ ee p i ð39Þ It then follows that A~ vv 0 ¼ kÀ1 p¼0 n p À1 i ¼0 l p c p i ~ ee p i ¼ kÀ1 p¼0 l p n p À1 i ¼0 c p i ~ ee p i ¼ kÀ1 p¼0 l p ~ vv ðpÞ 0 ð40Þ Moreover, we have A n ~ vv 0 ¼ kÀ1 p¼0 l n p ~ vv ðpÞ 0 ð41Þ This means that for any Krylov vector A n ~ vv 0 , it can always be written as a linear combination of k independent vec- tors ~ vv ð0Þ 0 ; ~ vv ð1Þ 0 Á Á Á ~ vv ðkÀ1Þ 0 . Thus, we conclude that dim½K m ð ~ vv 0 ; AÞ ¼ dimf ~ vv 0 A~ vv 0 . . . A mÀ1 ~ vv 0 g k ð42Þ regardless of the initial vector and the iteration number m. Consequently, in applying the CG method, or any Krylov-based methods, to solve a matrix equation with k distinctive eigenvalues, CG converges in at most k iterations. Next, let’s examine the effect of condition number on the convergence rate of the CG methods. To gain more in- sight, let us assume further that matrix A is an NÂN symmetric positive definite (SPD) matrix. With this as- sumption, we can state a fact that at the mth iteration, the CG method produces the same solution as the following minimization problem. Minimization: Seek ~ xx ðmÞ 2 K m ~ bb; A _ _ ¼ ~ bb A ~ bb Á Á Á A mÀ1 ~ bb _ _ such that the quadratic form x ~ Àx ~ ðmÞ _ _ A ~ xx À ~ xx ðmÞ _ _ ð43Þ is minimized. Since A is SPD, and its eigenvectors form a complete set of basis vectors, we can express the RHS vector ~ bb as follows: ~ bb¼b 0 ~ ee 0 þb 1 ~ ee 1 þ Á Á Á þb NÀ1 ~ ee NÀ1 ¼ NÀ1 i ¼0 b i ~ ee i ð44Þ It is easy to show then the exact solution ~ xx is ~ xx ¼ b 0 l 0 ~ ee 0 þ b 1 l 1 ~ ee 1 þ Á Á Á þ b NÀ1 l NÀ1 ~ ee NÀ1 ¼ NÀ1 i ¼0 b i l i ~ ee i ð45Þ Furthermore, a general trial vector in the Krylov space in the mth iteration is of the form ~ vv ¼ mÀ1 i ¼0 c i A i ~ bb _ _ ¼ NÀ1 i ¼0 c 0 þc 1 l i þ Á Á Á þc mÀ1 l mÀ1 i _ _ b i ~ ee i ð46Þ Subsequently ~ xx À ~ vv ¼ NÀ1 i ¼0 1 l i 1 Àc 0 l i Àc 1 l 2 i ÀÁ Á Á Àc mÀ1 l m i _ _ b i ~ ee i ð47Þ and a quadratic functional F ~ vv ð Þ can be defined as F ~ vv ð Þ ¼ðx ~ Àv ~ ÞA ~ xx À ~ vv ð Þ ð48Þ Substituting (46) and (47) into Eq. (48), we have Fð ~ vvÞ ¼ NÀ1 i ¼0 ½1 Àc 0 l i Àc 1 l 2 i ÀÁ Á Á Àc mÀ1 l m i 2 1 l i b 2 i max 0iNÀ1 l i ½1 Àc 0 l i Àc 1 l 2 i ÀÁ Á Á Àc mÀ1 l m i 2 NÀ1 i ¼0 1 l i b 2 i ¼ max 0iNÀ1 l i ½1 Àc 0 l i Àc 1 l 2 i ÀÁ Á Á Àc mÀ1 l m i 2 x ~ A~ xx ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ð49Þ Since the CG solution is the same as the one that mini- mizes the quadratic functional, we have F ~ xx ðmÞ _ _ ¼ min ~ vv2K m ð ~ bb;AÞ Fð ~ vvÞ x $ A~ xx ¸ ¸ ¸ ¸ ¸ ¸ min f c 0 c 1 Á Á Á c mÀ1 g max 0iNÀ1 l i ½1 Àc 0 l i Àc 1 l 2 i ÀÁ Á Á Àc mÀ1 l m i 2 ¼ x $ A~ xx ¸ ¸ ¸ ¸ ¸ ¸ min P m ð0Þ ¼1 max 0iNÀ1 P m ðl i Þ ¸ ¸ ¸ ¸ 2 ð50Þ where P m l ð Þ is the mth polynomial in l. If we arrange the eigenvalues of A in ascending manner, namely, l 0 l 1 Á Á Á l NÀ1 , then we can replace the best appro- ximation problem on the discrete set with the best ITERATIVE METHODS 2239 approximation problem on the interval l 0 l NÀ1 _ ¸ . Note that we have min Pmð0Þ ¼1 max 0iNÀ1 l i P m ðl i Þ ¸ ¸ ¸ ¸ min Pmð0Þ ¼1 max l 0 ll NÀ1 P m ðlÞ ¸ ¸ ¸ ¸ ð51Þ The solution to the minmax problem on an interval is known; namely min P m ð0Þ ¼1 max l 0 ll NÀ1 P m ðlÞ ¸ ¸ ¸ ¸ ¼ 1 T m l NÀ1 þl 0 l NÀ1 Àl 0 _ _ max l 0 ll NÀ1 T m l NÀ1 þl 0 À2l l NÀ1 Àl 0 _ _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ð52Þ where T m x ð Þ ¼ 1 2 x þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 À1 p _ _ m þ x À ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 À1 p _ _ m _ _ is the Chebyshev polynomial. Also, since max À1x1 T m x ð Þ ¸ ¸ ¸ ¸ ¼1 and À1 ½ðl NÀ1 þl 0 À2lÞ=ðl NÀ1 Àl 0 Þ 1, we then find min P m ð0Þ ¼1 max l 0 ll NÀ1 P m ðlÞ ¸ ¸ ¸ ¸ ¼ 1 T m l NÀ1 þl 0 l NÀ1 Àl 0 _ _ ¼2 s m 1þs 2m ; s ¼ 1 À ffiffiffiffiffiffiffiffiffiffiffi l 0 l NÀ1 _ 1 þ ffiffiffiffiffiffiffiffiffiffiffi l 0 l NÀ1 _ ð53Þ In conclusion, the convergence rate of CG method, mea- sured in A norm is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ~ Àx ~ ðmÞ _ _ A ~ xx À ~ xx ðmÞ ð Þ ¸ 2 s m 1þs 2m ffiffiffiffiffiffiffiffiffi x ~ A~ xx _ ð54Þ 7.2. Preconditioned CG method We conclude the previous section by observing that CG method works well on matrices that are either well-con- ditioned or have just a few distinct eigenvalues. For many RF engineering applications (such as the electric field in- tegral equation), the system matrix equations are usually not suitable directly for CG method. However, if a proper preconditioning matrix, M¼C t C, can be found, then the system matrix can be transformed into A~ xx ¼ ~ bb )A 0 ~ zz ¼ ~ bb 0 A 0 ¼ C t _ _ À1 AC À1 _ _ ; ~ xx ¼C~ zz; ~ bb 0 ¼ C t _ _ À1 ~ bb ð55Þ Applying the CG algorithm 1 to the transformed matrix equation results in the following algorithm. CG Algorithm 2 Initialization: ~ vv 0 ¼ ~ bb 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi b 0 ~ A 0 ~ bb 0 _ ; ~ zz ¼0; ~ rr 0 ¼ ~ bb 0 Iteration: 1. ~ zz ¼ ~ zz þa~ vv 0 ; a ¼v ~ 0 ~ bb 0 . 2. ~ uu 0 ¼A 0 ~ vv 0 ; ~ rr 0 ¼ ~ rr 0 Àa ~ uu 0 . 3. Check jj ~ rr 0 jj=jj ~ bb 0 jj e. If yes, then stop, and ~ zz is a good approximation; if no, then continue. 4. ~ pp 0 ¼ ~ rr 0 Àb~ vv 0 ; b¼u ~ 0 ~ rr 0 . 5. ~ vv 0 ¼ ~ pp 0 = ffiffiffiffiffiffiffiffiffiffiffiffiffi p ~ 0 A 0 ~ pp 0 _ . 6. Go to step 1. Of course, once we have ~ zz, then we can obtain ~ xx via ~ xx ¼C À1 ~ zz. However, it is possible to avoid explicit refer- ence to the matrix C À1 by defining ~ pp 0 ¼C~ pp; ~ zz ¼C~ xx and ~ rr 0 ¼ðC t Þ À1 ~ rr in every CG iteration. Indeed, if we substi- tute these definitions into CG algorithm 2 and recall ~ bb 0 ¼ðC t Þ À1 ~ bb, then we obtain CG Algorithm 3 Initialization: C~ vv ¼ C t _ _ À1 ~ bb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ~ C À1 C t _ _ À1 _ _ A C À1 C t _ _ À1 _ _ ~ bb ¸ ; ~ zz ¼0; ~ rr ¼ ~ bb Iteration: 1. C~ xx ¼C~ xx þaC~ vv; a ¼v ~ ~ bb. 2. C t _ _ À1 ~ uu¼ C t _ _ À1 A~ vv; C t _ _ À1 ~ rr ¼ C t _ _ À1 ~ rr Àa C t _ _ À1 ~ uu. 3. Check jj ~ rr 0 jj=jj ~ bb 0 jj e, If yes, then stop, and ~ xx is a good approximation; if no, then continue. 4. C~ pp¼ C t _ _ À1 ~ rr ÀbC~ vv; b ¼ u ~ C À1 C t _ _ À1 ~ rr. 5. C~ vv ¼C~ pp= ffiffiffiffiffiffiffiffiffiffi ~ ppA~ pp _ . 6. Go to step 1. Finally, the entire algorithm can be simplified by using the preconditioner M¼C t C directly instead of referring to C or C t . This is then the preconditioned CG algorithm. Preconditioned CG Algorithm Initialization: ~ zz ¼0; ~ rr ¼ ~ bb; ~ pp¼M À1 ~ rr; ~ vv ¼ ~ pp= ffiffiffiffiffiffiffiffiffiffi p ~ A~ p p _ Iteration: 1. ~ xx ¼ ~ xx þa~ vv; a ¼v ~ ~ bb. 2. ~ uu¼A~ vv; ~ rr ¼ ~ rr Àa ~ uu. 3. Check jj ~ rrjj=jj ~ bbjj e. If yes, then stop, and ~ xx is a good approximation; if no, then continue. 4. ~ pp¼M À1 ~ rr Àb~ vv; b ¼u ~ M À1 ~ rr. 5. ~ vv ¼ ~ pp= ffiffiffiffiffiffiffiffiffiffi p ~ A~ pp _ . 6. Go to step 1. 2240 ITERATIVE METHODS BIBLIOGRAPHY 1. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins Univ. Press, Baltimore, 1996. 2. T. J. Kim and G. A. Thiele, A Hybrid diffraction technique— general theory and applications, IEEE Trans. Anten. Propag. 30:888–897 (1982). 3. M. Kaye, P. K. Murthy, and G. A. Thiele, An iterative method for solving scattering problems, IEEE Trans. Anten. Propag. 33:1272–1279 (1985). 4. P. K. Murthy, K. C. Hill, and G. A. Thiele, A hybrid-iterative method for scattering problems, IEEE Trans. Anten. Propag. 34:1173–1180 (1986). 5. R. E. Hodges and Y. Rahmat-Samii, An iterative current- based hybrid method for complex structures, IEEE Trans. Anten. Propag. 45:265–276 (1997). 6. A. Sullivan and L. Carin, Scattering from complex bodies us- ing a combined direct and iterative technique, IEEE Trans. Anten. Propag., 47:33–39 (1999). 7. D. D. Reuster and G. A. Thiele, A field iterative method for computing the scattered electric fields at the apertures of large perfectly conducting cavities, IEEE Trans. Anten. Prop- ag. 43:286–290 (1995). 8. F. Obelleiro, J. L. Rodriguez, and R. J. Burkholder, An itera- tive physical optics approach for analyzing the electromag- netic scattering by large open-ended cavities, IEEE Trans. Anten. Propag. 43:356–361 (1995). 9. D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, Forward- backward: A new method for computing low-grazing angle scattering, IEEE Trans. Anten. Propag. 44:722–729 (1996). 10. D. A. Kapp and G. S. Brown, A new numerical method for rough surface scattering calculations, IEEE Trans. Anten. Propag. 44:711–721 (1996). 11. M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces, IEEE Trans. Anten. Propag. 47:961–969 (1999). 12. J. C. West and J. M. Sturm, On iterative approaches for elec- tromagnetic rough-surface scattering problems, IEEE Trans. Anten. Propag. 47:1281–1288 (1999). 13. A. R. Clark, A. P. C. Fourie, and D. C. Nitch, Stationary, non- stationary, and hybrid iterative method of moments solution schemes, IEEE Trans. Anten. Propag. 49:1462–1469 (2001). 14. R. J. Burkholder, On the use of classical iterative methods for electromagnetic scattering problems, Proc. 4th Conf. Electro- magnetic and Light Scattering by Nonspherical Particles: Theory and Applications, Vigo, Spain, Sept. 20–21, 1999, pp. 65–72. 15. M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Stand. 49:409– 436 (1952). 16. T. K. Sarkar and S. M. Rao, The application of the conjugate gradient method for the solution of electromagnetic scattering from arbitrary oriented wire antennas, Trans. Anten. Propag. 32:398–403 (1984). 17. T. K. Sarkar and E. Arvas, On a class of finite step iterative methods (conjugate directions) for the solution of an operator equation arising in electromagnetics, Trans. Anten. Propag. 33:1058–1066 (1985). 18. T. K. Sarkar, E. Arvas, and S. M. Rao, Application of FFT and the conjugate gradient method for the solution of electromag- netic radiation from electrically large and small conducting bodies, Trans. Anten. Propag. 34:635–640 (1986). 19. J. D. Collins, J. M. Jin, and J. L. Volakis, A combined finite element-boundary element formulation for solution of 2-di- mensional problems via CG-FFT, Electromagnetics, 10:423– 437 (1990). 20. R. Coifman, V. Rokhlin, and S. Wandzura, The fast multipole method for the wave equation: A pedestrian prescription, IEEE Anten. Propag. Mag. 35:7–12 (1993). 21. E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, A fast in- tegral equation solver for electromagnetic scattering prob- lems, Radio Sci. 31:1225–1251 (1996). 22. J. R. Phillips and J. White, A precorrected-FFT method for electrostatic analysis of complicated 3D structures, IEEE Trans. Comput. Aid. Design Integr. Circ. Syst. 16:1059–1072 (1997). 23. S. Barnett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM Publications, Philadel- phia, 1993. 24. Y. Saad and M. Schultz, GMRES: A generalized minimal re- sidual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7:856–869 (1986). 25. S. C. Eisenstat, H. C. Elman, and M. H. Schultz, Variational iteration methods for nonsymmetric systems of linear equa- tions, SIAM J. Num. Anal. 20:345–357 (1983). 26. J.-F. Lee and R. J. Burkholder, Loop star basis functions and a robust preconditioner for EFIE scattering problems, IEEE Trans. Anten. Propag. 51:1855–1863 (2003). FURTHER READING A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, Wiley, New York, 1997 (a good general resource for computational electromagnetics, including the fi- nite-element and finite-difference methods, the method of mo- ments, basis expansions, and solution methods). O. Axelsson, Iterative Solution Methods, Cambridge Univ. Press, 1996 (a good source for iterative methods, in general). Matrix Computations by Golub and Van Loan [1] is an excellent ref- erence on matrix theory, solution of matrix systems, and iter- ative algorithms. The Templates book [23] presents many iterative algorithms and their underlying theories, along with a discussion of precondi- tioners and parallelization; it is available online at http:// www.netlib.org/linalg/html_templates/Templates.html, and the associated software may be downloaded from http:// www.netlib.org/templates/. ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9GHz RAMEZ L. GERGES IEEE-ITSC Standards Committee Goleta, California 1. INTRODUCTION This article describes ongoing activities to create a new family of standards that supports the emerging Intelligent ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz 2241 Transportation Systems (ITS) and telematics wireless markets. ITS-WAVE is a radiocommunication system in- tended to provide seamless, interoperable services to sur- face transportation systems. After an initial overview of the ITS-WAVE family of standards, more emphasis will be given to the radio (lower layers) part of the system, and the use of orthogonal frequency-division multiplexing (OFDM) for the physical layer [1]. 1.1. ITS, Telematics, and Wireless Interoperability The Intelligent Transportation Systems (ITS) initiative was created by Congress in the Intermodal Surface Trans- portation Efficiency Act of 1991 (ISTEA) to improve the mobility and safety of the surface transportation system. ITS is defined as those systems utilizing synergistic tech- nologies and systems engineering concepts to develop and improve transportation systems of all kinds. Communica- tion and information technologies are at the core of road- side infrastructure and in-vehicle systems. These technologies promise to enhance mobility by improving the way we monitor and manage traffic flow, clear inci- dents, reduce congestion, and provide alternate routes to travelers. The telematics industry is focused on driver comfort and safety, and while ‘‘telematics’’ in general has meant the blending of computers and telecommuni- cations, it is used within the ITS community with the connotation of ‘‘automotive telematics’’ or the in-vehicle subsystem of ITS. In 1999, the Federal Communications Commission (FCC) allocated the 5.850–5.925-GHz band for use by the ITS radio service for both public safety and commercial ITS applications. Many standards development organiza- tions (e.g., IEEE, IETF, ISO) are engaged in the process of achieving an end-to-end ITS wireless interoperability. This article addresses Wireless Access in Vehicular Envi- ronments (WAVE), which is currently being developed un- der the IEEE WG 802.11, WAVE Study Group. 1.2. ITS Radio Services The proposed ITS-WAVE standard addresses broadband wireless communications that operate in a long range (r1000m) and at a high data rate [27Mbps (megabits per second)] for all ITS applications. The proposed lower- layer standard currently addresses communications between roadside units and mostly high-speed, but occa- sionally stopped and slow-moving, vehicles or between high-speed vehicles. The ITS new spectrum will be used to support multiple applications to enhance public safety and transportation mobility and can be categorized as follows: 1. Public Safety: The primary use of this band is to offer services such as emergency vehicle signal preemp- tion and announcements for work zones. While the FCC has allocated the 4.9GHz for communications between first responders, the 5.9-GHz band is ex- pected to allow first responders to communicate with the general driving public on roads and freeways. 2. Mobility: Services such as electronic toll, vehicle probes, traveler information, and public transporta- tion integration are expected to enhance the trans- portation system performance. 3. Driver Safety: New features such as support of col- lision avoidance and warnings for excessive speed and railroad crossings are expected to improve sys- tem performance. More recently, vehicle manufac- turers and telematics providers have shown interest in the ITS-WAVE standards. There is no other ra- diocommunication technology that can support the real-time requirements for vehicle-to-vehicle com- munications. 1.3. ITS-WAVE Development History Attempts to develop standards for the wireless ITS envi- ronment date back to the early 1990s, when California adopted the Title 21 regulation to achieve a common stan- dard for ‘‘toll collections.’’ The dedicated short-range com- munications (DSRC) standard at 900 MHz, and Title 21 [2], predated the ITS initiative, and addressed only the electronic toll collection; it was not intended to support a national interoperable wireless ITS standard. The Intermodal Surface Transportation Efficiency Act of 1991 (ISTEA) funded many research ITS programs. In the mid-1990s, the author (then with the New Technology program at Caltrans) initiated some of the first technical studies to develop an integrated wireless communications system for all ITS applications [3]. In 1996, the U.S. Na- tional System Architecture identified wireless communi- cations as one of the critical enabling technologies needed to support many of the ITS services. Later, the USDoT funded more studies, and the California Department of Transportation (Caltrans) established the Testbed Center for Interoperability (TCFI) to study and test end-to-end wireless interoperability. In May 1997, the Intelligent Transportation Society of America (ITSA) filed a Petition for Rulemaking, requesting that the FCC allocate 75 MHz of spectrum in the 5.850–5.925-GHz band on a coprimary basis for DSRC-based ITS services. In 1998 at the IEEE– Vehicular Technology Conference, the author suggested to leverage the economical feasibility of the IEEE 802.11 to achieve wireless ITS interoperability [4]. In 1999, the FCC amended Parts 2 and 90 of the Commission’s Rules to al- locate the 5.850–5.925-GHz band to the Mobile Service for Dedicated Short Range Communications of Intelligent Transportation Services. 1 The USDoT funded the Ameri- can Society for Testing and Materials (ASTM) to initiate the standard writing group for the DSRC at 5.9 GHz. In 2000, TCFI tested the first video relay to a moving vehicle at highway speed using OFDM technology. 2 The success- ful test paved the way to use broadband technologies for wireless ITS. Later, the ASTM selected the OFDM Forum proposal to use the IEEE 802.11a [5,6] as the basis for the new standard. 3 The new DSRC standard is now being 1 ET Docket 98-95, 14 FCC Record 18221. 2 Wireless LAN provided the OFDM equipment at 2.4 GHz; Fur- ther infromation is available at http://www.wi-lan.com. 3 The OFDM-Forum proposal (802.11 RA) suggested changing the physical layer of the IEEE 802.11 to match the requirements of the ‘‘road access’’ environment. (http://www.ofdm-forum.com). 2242 ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz completed within the IEEE WG 802.11. The Study Group (SG) decided to use the ASTM standard [7] as the basis for the ITS-WAVE proposal. 4 As the proposed standard is not limited to short-range applications, the SG has named it the Wireless Access in Vehicular Environments (WAVE) instead of DSRC. This will also avoid any confusion with the single carrier technology in use in the United States (900-MHz band) or in Japan and Europe (at 5.8 GHz but different standards). On December 17, 2003 the FCC adopted the rules for the ITS band. It is expected that the new standard (possibly 802.11p) will be completed by the end of 2005. 2. ITS RADIO SERVICES SYSTEM-LEVEL DESCRIPTION 2.1. Spectrum Allocation for ITS, Telematics, and Public Safety The Broadband ITS Radio Service (ITS-RS) establishes a common framework for providing wireless services in the 5.850–5.925-GHz band. This band is allocated for ITS-RS applications by the FCC. 5 Figure 1 shows the spectrum allocation in the 4.9–5.9-GHz band. The differences be- tween the ITS-WAVE and the IEEE 802.11 WLAN sys- tems stem from the fact that the ITS-WAVE operates in a licensed band, and it establishes reliable communications between units operating at full vehicle mobility, a different environment than the indoor WLAN. These communications may occur with other units that are (1) fixed along the roadside or above the roadway, (2) mounted in other high-speed moving vehicles, (3) mounted in stationary vehicles, (4) mounted on mobile platforms (e.g., watercraft, buoy, or a robotic platform), or (5) porta- ble or handheld. In-vehicle communications units are called onboard units (OBUs). Communication units fixed along the freeways, over the road on gantries or poles, or off the road in private or public areas, are called roadside units (RSUs). The WAVE RSUs may function as stations or as access points (APs) and the WAVE OBUs only have functions consistent with those of stations (STAs). The common function between all RSUs is that these station- ary units control access to the radiofrequency medium for OBUs in their communication zone or relinquish control to broadcast data only. The vehicular mobility environment requires that we design a system that can survive both the time-dispersive (frequency-selective) multipath fading and the frequency- dispersive (time-selective) fading environment. Tests con- ducted at the Testbed Center For Interoperability (TCFI) at UCSB show that we may encounter up to 400ns of delay spread and up to 2200 Hz of Doppler spread as explained later. Single-carrier transmission, with a time- domain equalizer, has an inherent limitation due to con- vergence and tracking problems which arises as the number of taps increase. A coded OFDM (COFDM) approach similar to the IEEE 802.11a/g standard offered a more robust, as well as economically feasible solution ITS-RS China Japan USA Europe 4.900 5.000 5.150 5.250 5.030 5.091 5.150 5.150 5.350 5.350 5.470 Indoor 200 mW / Outdoor 1 W EIRP DFS & TPC DFS & TPC 4.900 5.000 5.100 5.200 5.300 5.400 5.500 5.600 5.700 5.800 5.900 6.000 DFS: Dynamic Frequency Selection TPC: Transmit Power Control Freq./GHz Homeland Security Indoor 200 mW EIRP Outdoor 1W EIRP ITS-RS Max mean Tx power Max peak Tx power 5.725 5.725 5.85 5.825 5.850 5.725 5.925 Outdoor 4W EIRP Frequency allocations 4.9-5.9 GHz: Figure 1. Spectrum allocation at the 5-GHz Band. (This figure is available in full color at http:// www.mrw.interscience.wiley.com/erfme.) 4 The ASTM E2213-02 was approved but not published because of copyright issues with the IEEE. 5 Title 47, Code of Federal Regulations (CFR), Part 90, Subpart M. ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz 2243 based on the success of the WLAN industry. This economic feasibility also made the COFDM approach a better can- didate than the single-carrier transmission with a fre- quency-domain equalizer. Although, the latter has the same complexity and may have some good features [e.g., avoids the PAPR (peak-to-average power ratio) issues]. The 802.11a scheme would not be able to tolerate the delay spread expected in the WAVE environment. Figure 2 shows the impact of delay spread on a 16-QAM signal constellation for a 64-subcarrier OFDM system [1]. The channel has a two-ray multipath; the second ray is 6dB lower than the first one: (1) delay spread less than guard time (Fig. 2a), (2) delay spread greater than guard time by 3% of the FFT interval (Fig. 2b), and (3) delay spread greater than guard time by 9% of the FFT inter- val (Fig. 2c). We proposed to double the guard interval (GI) to be more multipath-tolerant; in principle, using half the mas- ter clock should double the GI and scale down the channel bandwidth to 10 MHz, a desired outcome to increase the number of channels within the allocated spectrum. Of course the maximum data rate will be reduced to 27 Mbps, which is still adequate for demanding ITS applications (e.g., video relay). WLAN chips manufacturers (e.g., In- tersil and Atheros) confirmed the feasibility of the ap- proach using their current 802.11a implementations. 6 It is expected that products with the correct front end operat- ing at 5.9GHz (10 MHz bandwidth) will be available as the market develops. In order to accommodate the more dynamic vehicle en- vironment with essentially the same radio technology, and provide priority to public safety communications, the ITS community is proposing a complementing set of standards under the IEEE SCC32. These standards address the upper layers including a different channel structure and access mechanism than that of the IEEE 802.11 as ex- plained later. 2.2. System Architecture and SDO Coordination The International Standards Organization (ISO) and the IEEE are coordinating their standards development efforts to achieve ITS wireless interoperability. To this end a common CALM/WAVE architecture has been developed as shown in Figs. 3 and 4. The current scope of the IEEE-WAVE proposed project is to create an amendment of IEEE 802.11 to support com- munication between vehicles and the roadside and be- tween vehicles while operating at speeds up to a minimum of 200km/h for communication ranges up to 1000 m. The amendment will support communications in the 5-GHz bands; specifically, the 5.850–5.925-GHz band within North America with the aim to enhance the mobility and safety of all forms of surface transportation, including rail and maritime transportation. Amendments to the PHY and MAC will be limited to those required to support com- munications under these operating environments within the 5-GHz bands. The IEEE SCC32 sponsors the IEEE P1556, DSRC Security and Privacy and the ITS-WAVE (upper layers). The WG P1556 is proposing a dual-certificate system for public safety and vehicle safety to balance security and anonymity requirements. The IEEE WG P1609 architec- ture adopted IPV6 as the method of handling upper-layer applications. It consists of a series of four standards: 1. P1609.4 defines the channelization approach and considers integration issue with the IEEE 802.11e and IEEE 802.11 h. 2. P1609.3 is based on the IPv6 specification and may include a broad range of supporting standards de- fined by the Internet Engineering Task Force (IETF). It defines IPv6 addressing and configura- tion issues, network services (e.g., WAVE router ad- vertisement), and all the WAVE management entities needed for registration and ‘‘service table’’ exchanges. 3. P1609.2 defines applications services. 4. P1609.1 defines a resource manager for onboard units (OBUs). The ISO-Transport Information & Control (TC204) Working Group 16 is developing standards for wide-area wireless communications for transport information and control. ISO-TC204-WG16 is developing the communica- tion air interface for long and medium range (or short media) (CALM) architecture. The CALM scope includes 0 0 2 4 6 −2 −4 −6 0 2 4 6 −2 −4 −6 0 2 4 6 −2 −4 −6 −2 −4 −6 2 4 6 (a) 0 −2 −4 −6 2 4 6 0 −2 −4 −6 2 4 6 (b) (c) Figure 2. Impact of delay spread. (This figure is available in full color at http://www.mrw. interscience.wiley.com/erfme.) 6 WLAN products from Intersil are now part of Conexant (http:// www.conexant.com), Atheros ( http://www.atheros.com). 2244 ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz communications between fixed sites and switching be- tween communication media (e.g., 3G cellular and WAVE), as well as issues such as handover and mobility management. CALM mandates end-to-end system inter- operability at all levels. CALM-M5 is adopting the IEEE- WAVE proposal for the lower layers at 5 GHz. 2.3. Basic Concept of Operation The ITS-WAVE typically consists of two types of radio devices. The first type is always used while stationary, usually permanently mounted along the roadside, and is referred to as the roadside unit (RSU). The second is SNMP agent (RFC 1157) App 1 App Data Sockets App 2 App Data Sockets OBU IVN App 3 App Data Sockets UDP Networking Services IVN L2/L1 UDP (RFC 768) Networking Services (IPv6 – RFC 2460) SNMP MIB SME (1609.3) Logical Link Control (802.2) Channelization (1609.4) MAC (802.11p) PHY (802.11p) WME (1609.3) MLME (802.11p) PLME (802.11p) IVN L2/L1 IVN: In- Vehicle Network Figure 3. Wave architecture. CME (Commun ication Managem ent Entity) ISO LME (Link Managem ent Entity) Common Station, PHY, MAC, LLC Managers Service Access Point – Management Service Access Point – Data Transfer CALM M5 CALM M5 3G cellular std CALM 3G NETWORK INTERFACE Routing and Media Switching based on IPv6 ISO 21210-2 Directory Services Convergence Layer Convergence Layer Layer 5-7 INTERNET Non-CALM- aware Point-to-point Non-CALM- aware IP (Internet) CALM-Aware APPLICATIONS Figure 4. CALM architecture. (This figure is available in full color at http://www.mrw. interscience.wiley.com/erfme.) ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz 2245 mobile, mounted on board vehicles, and is referred to as the onboard unit (OBU). Three types of communication are supported: a command/response type between a ser- vice provider and a service user, a broadcast to listener, and a peer-to-peer type that does not identify either device as controlling the actions of the other. OBUs and RSUs can initiate both types of communication. The command/ response type includes various forms of transactions be- tween a service provider and a user of that service. To en- sure scalable interoperability between ITS-WAVE units, the proposed standards define two levels of implementa- tions. A minimal implementation only supports the lower layers, those below the network layer, and will be referred to as a ‘‘WAVE radio.’’ An implementation that has the full ITS-WAVE protocol stack is referred to as a ‘‘WAVE de- vice.’’ Multiple devices interact with each other through the Networking Services (IEEE P1609). Figure 5 represents the current ITS-WAVE band plan. The ITS-WAVE uses a ‘‘control channel’’ and any of six ‘‘ser- vice channels’’. Licensing of both roadside RSUs and OBUs are necessary to prevent unauthorized use of the control channel. OBUs should be licensed by rule, since these devices are mobile and can operate nationwide, communi- cating with any other ITS-WAVE devices within range. The onboard units (OBUs) are required to listen on the control channel every few hundred milliseconds, in order to check for public safety messages. The messages on the control channel are of variable length, but are generally kept short, to permit maximum access to the channel. Control channel access will be performed via a standard IEEE 802.11a, Multiple Access with Collision Avoidance (CSMA/CA). By default, all devices when turned on are tuned to the control channel. If an ITS-WAVE device desires to transmit, but detects another message being broadcast on the control channel, it must wait before at- tempting to transmit. A request to send (RTS) is initiated, and time is granted first to high priority (public safety) broadcasts, then to lower-priority transmissions. The same control channel is used for roadside-to-vehicle, ve- hicle-to-roadside, and vehicle-to-vehicle communications. ‘‘Control channel interval’’ and ‘‘service channel interval’’ are controlled by RSU beacon frames. Since the control channel will be fixed throughout the nation, all ITS-WAVE devices will be able to access those services in an interop- erable matter. A registration process must occur before a WAVE device can be considered ‘‘ready for operation’’; the RSU broad- casts beacon frames that include the ‘‘provider service ta- ble’’ (PST) and the ‘‘WAVE router advertisement’’ (WRA) on the control channel. Application initialization proce- dures are based on SNMP, and the designated service channel, priority, and power level are indicated in the PST. At the end of the application initialization state, the RSU commands the OBU to switch to the designated service channel. The RSU, now on the service channel, receives UDP datagrams sent by the OBU. The RSU routes datagrams to and from the applications indicated by the global IPv6. The description above is included to give an idea about the basic concept of operations, with the understanding that the proposed standards are now under development. The P1609.3, 1609.4, and the P1556 are currently the most critical part of the WAVE family of standards as they require integration and coordination with many other standards such as IEEE 802.11e/h/i and many of the IETF recommendations. Shared public safety/private Control Med Rng Service Short Rng Service Dedicated public safety High avail Intersections 40 dBm Not currently implemented Power limit Power limit Power limit Not currently implemented 44.8 dBm 33 dBm 23 dBm Uplink Downlink Public safety Veh-Veh Ch 172 Public safety/ private Ch 174 Public safety/ private Ch 176 Control channel Ch 178 Public safety/ private Ch 180 Public safety/ private Ch 182 Ch 184 Public safety intersections Canadian special license zones* Frequency (GHz) 5 . 8 2 5 5 . 8 3 0 5 . 8 3 5 5 . 8 4 0 5 . 8 4 5 5 . 8 5 0 5 . 8 6 0 5 . 8 6 5 5 . 8 7 0 5 . 8 7 5 5 . 8 8 0 5 . 8 8 5 5 . 8 9 0 5 . 8 9 5 5 . 9 0 0 5 . 9 0 5 5 . 9 1 0 5 . 9 1 5 5 . 9 2 0 5 . 9 2 5 5 . 8 5 5 Figure 5. ITS-wave band plan. (This figure is available in full color at http://www.mrw. interscience.wiley.com/erfme.) 2246 ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz It is expected that WAVE radios that implement only the lower layers will develop first, as they leverage the existing IEEE 802.11a standard and chip technology. 3. DESIGNING FOR WIRELESS VEHICULAR ENVIRONMENTS 3.1. Channel Impairments There is extensive literature on different statistical mod- els of the communication channels at different frequency bands [8]. However, limited data of actual field measure- ments for vehicle–vehicle and vehicle–roadside communi- cation are available at the ITS-WAVE frequency band. Statistical models include large-scale path loss models and small-scale fading models: 1. Large-scale propagation models characterize the mean received power over large transmitter–receiv- er separation distances. It is used to estimate radio coverage area of a transmitter. 2. Small-scale (fading) models characterize the rapid fluctuations of the received signal strength and phase over a very short distance. Multipath struc- ture (power delay profile) is used to measure and describe the fading effects. Both large- and small-scale fading models are needed for packet error rate characterization. 3.1.1. Time-Dispersive (Frequency-Selective) Multipath Fading Channel. A time-dispersive channel is defined as a channel for which the delay spread is much wider than the signal duration. The classification of a channel as time dispersive is therefore dependent on the data rate of the system. For a single carrier, high-data-rate systems, time- dispersive channels are commonly encountered. This type of fading is often referred to as frequency-selective because the signal may be simultaneously faded at one frequency and not at another. OFDMis robust against delay spread by design because of the longer symbol time and the fact that each subcarrier experiences a flat-fading channel. Similar to the IEEE 802.11a, the insertion of guard interval, and the use of forward error correction (FEC) are essential de- sign elements to the coded OFDM scheme employed in the WAVE physical layer. This multipath rejection capability was one of the main reasons for selecting COFDM instead of a single-carrier system, especially for ITS applications that operate at longer ranges and at high data rates. Short-range systems typically experience significantly smaller delay spreads than does a longer-range system. Previous studies show that 90% RMS (root-mean-square) delay spread is less than 100ns for typical short-range applications (e.g., toll collection) in urban environments. 7 RMS delay spread could be up to 300 ns [9,10] in a non- line-of-sight (NLoS) heavy-multipath environment, as may be expected in a freeway urban environment. In order for subcarriers to perceive a flat-fading chan- nel, the bandwidth (subcarrier spacing) must be less than the ‘‘coherence bandwidth’’ (B c ) of the channel. 8 B c is the bandwidth of the channel variation in frequency and is defined [8] as B c ¼1=5s where s is the RMS delay spread of the channel. The ITS- WAVE has a subcarrier spacing (bandwidth) of 156kHz, and each subcarrier will encounter flat fading as long as so100 ns (for rangeo300m). For long ranges (large delay spread), the pilot channels are available to estimate the channel in the frequency domain if they are well struc- tured. In order to use the pilots for channel estimation, the pilot spacing in frequency has to be less than B c (B2 MHz for s ¼500ns). This may not be the case using the current pilot structure of the IEEE 802.11a (pilot spacing ¼14  156 kHz ¼2.18 MHz4B c ). Interpolation of pilot subcar- rier in the current structure may not be sufficient to track the frequency selective fades. It is expected that the first generation WAVE radios, those using modified 802.11a chips, will be limited in range and may not be suitable for long-range public safety applications. 3.1.2. Frequency-Dispersive (Time-Selective) Fading Channel. Frequency-dispersive channels are classified as channels that have a Doppler spread larger than the channel bandwidth. Doppler spread is a direct result of multiple Doppler shifts which are caused by motion of the transmit and/or receive antenna. Doppler shifts can also result from reflections off of moving objects. 9 Distortion of the power spectrum of the received signal results from Doppler spread, which can be approximated by the Dopp- ler spread B d B d¼ f m . cos a where f m¼ v . f c /c, where v is the vehicle speed in m/s, f c is the carrier frequency in Hz, c is the speed of light in m/s, and a is the angle between the direction of vehicle travel and the ray of the communication path. In the case of the ITS-WAVE where vehicle speeds of r120 mph (193 km/h) must be supported (public safety), the maximum Doppler shift for a vehicle traveling directly toward the roadside antenna would be about 1100Hz at 5.9 GHz, and much less for vehicle–vehicle communication (two vehicles head- ing in the same directions). Time-selective fading caused by Doppler spread is described by the coherence time (T c ) of the channel. T c represents the duration over which the channel character- istics do not change significantly, and is defined [8] as T c ¼0:423=f m 7 For transmit–receive (Tx/Rx) separation of 30–300m, both LoS and NLoS (Xiongwen, etc.; IEEE JSAC 2002). 8 B c is defined as the bandwidth over which the frequency corre- lation function is above 0.5. 9 If a sinusoidal signal is transmitted over a fading channel (com- monly referred to as a constant wave), the Doppler spread B d is defined as the range of frequencies over which the received Dopp- ler spectrum is essentially nonzero. ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz 2247 At a vehicle speed of 120 mph and a frequency of 5.9GHz, T c ¼400ms. When using pilot symbols at the start of a packet, the assumption is that channel variations during the rest of the packet are negligible. This limits the packet duration to less than T c , and places an upper limit on the packet size. At a data rate of 3Mbps, 1 2 -code-rate BPSK- modulated signal, the maximum packet size 10 is 135 bytes. Although this suggests that higher-order modulation would give better performance, as their transmission time is shorter, these modulation schemes degrade more in the presence of channel impairment. 3.2. The ITS-WAVE Physical Layer The ITS-WAVE physical layer is based on using the robustness of the coded orthogonal frequency-division multiplexing (OFDM) signal to achieve the required per- formance in the wireless vehicular environments. OFDM is a special case of multicarrier modulation (MCM), which is the principle of transmitting data by dividing the data- stream into several parallel bitstreams and modulating each bitstream onto individual subcarriers. Each subcar- rier is a narrowband signal, resulting in long bit intervals. High data rates are achieved by using multiple orthogonal subcarriers for a single data transmission. The OFDM system differs from traditional MCM in that the spectra of the subcarriers were allowed to overlap under the restric- tion that they were all mutually orthogonal. An orthogo- nal relationship between subcarriers is achieved if there are integer numbers of subcarrier frequency cycles over the symbol interval. This orthogonality guarantees that each subcarrier has a null at the center frequency of all other subcarriers as shown in Fig. 6. Orthogonality is achieved with precision by modulating the subcarriers with a discrete Fourier transform (DFT), which is implemented in hardware with the fast Fourier transform (FFT). By transmitting several symbols in par- allel, the symbol duration is increased proportionately, which reduces the effects of intersymbol interference (ISI) caused by the dispersive fading environment. Additional multipath rejection and resistance to intercarrier inter- ference (ICI) is realized by cyclically extending each sym- bol on each subcarrier. Rather than using an empty guard space, a cyclic extension of the OFDM symbol is used to ensure that delayed replicas of the OFDM symbol will always have an integer number of cycles within the FFT interval. This effectively converts the linear convolution of the channel to a circular one, as long as the cyclic prefix (CP) is longer than the impulse response of the channel. The penalty of using a CP is loss of signal energy propor- tional to the length of the CP. In order to avoid excessive bit errors on individual subcarriers that are in a deep fade, forward error control (FEC) is typically applied. The ITS-WAVE physical layer organizes the spectrum into operating channels. Each 10-MHz channel is com- posed of 52 subcarriers. Four of the subcarriers are used as pilot carriers for monitoring path shifts and ICI, while the other 48 subcarriers are used to transmit data symbols. Subcarriers are spaced 156.25 kHz apart, giving a total bandwidth of 8.8MHz. The composite waveform, consist- ing of all 52 subcarriers, is upconverted to one of the seven channels between 5.850 and 5.925 GHz. As shown in Fig. 7, channels are numbered from À26 to 26. Sub- carrier 0 is not used for signal processing reasons, and pilot subcarriers are assigned to subcarriers À21, À7, 7, and 21. To avoid strong spectral lines in the Fourier trans- form, the pilot subcarriers transmit a fixed bit sequence as specified in the IEEE 802.11a using a conservative mod- ulation technique. Table 1 compares the ITS-WAVE and the IEEE 802.11a parameters. Table 2 lists ITS-WAVE baseband modulation values. 3.2.1. Structure of the WAVE Physical Layer. The phy- sical layer is structured as two sublayers: the physical- layer convergence procedure (PLCP) sublayer and the physical-medium-dependent (PMD) sublayer. The PLCP communicates to MAC via primitives through the physi- cal-layer service access point (SAP); it prepares the PLCP protocol data unit (PPDU) shown in Fig. 8. The PPDU provides for asynchronous transfer of the MAC protocol data unit (MPDU) between stations. The PMD provides Table 1. Comparison of 802.11a and ITS-WAVE Parameters Parameter 802.11a ITS-WAVE Channel bandwidth (MHz) 20 10 Subcarrier spacing (kHz) 312.5 156.25 T FFT (ms) 3.2 6.4 T GI (ns) 800 1600 T SYM (ms) 4 8 Channel symbol rate (Msps) 12 6 Minimum data rate (BPSK) (Mbps) 6 3.0 Maximum data rate (64-QAM) (Mbps) 54 27 A B C D E Tone Figure 6. Subcarrier Orthogonality in OFDM systems. (This figure is available in full color at http://www.mrw.interscience. wiley.com/erfme.) Center frequency Carrier number −25 −20 −15 −10 −5 5 15 20 25 10 0 Figure 7. Structure of an Operating Channel. 10 Packet duration¼[10(16þ2)(80þ1)(80þ135)(8)(2/48)(80)]  100ns ¼400ms. 2248 ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz actual transmission and reception of the physical layer entities via the wireless medium, interfaces directly to the medium, and provides modulation and demodulation of the transmission frame. 3.2.2. Roles of Preamble, Training Sequences, and Pi- lots. The ITS-WAVE specifies a preamble at the start of every packet as shown in Fig. 9. The PLCP preamble consists of 10 short training sym- bols, each of which is 1.6 ms, followed by two long training symbols, each of which is 6.4 ms including a 3.2-ms prefix that precedes the long training symbol. The long training sequence contains a guard interval, T GI2 , and two long training symbols, each 6.4ms in duration. The short sym- bols are used by the receiver for synchronization [signal detection, AGC (automatic gain control), diversity selec- tion, frequency offset estimation, and timing synchroniza- tion]. The long symbols are used to fine-tune the frequency offset and channel estimates. This training sequence is transmitted over all 52 subcarriers and is QPSK-modu- lated. In terms of algorithmic complexity, carrier frequen- cy offset and timing recovery are by far the most difficult to determine. The phase-locked-loop (PLL) on the radio sub- system is responsible only for maintaining the 5-ppm volt- age-controlled oscillator (VCO) requirement. Digital signal processing is used, independent of the VCO, to remove the carrier frequency offset. It is important to note that once the carrier frequency offset is deter- mined by the digital baseband hardware, there is no time to provide a feedback signal to the WAVE radio’s VCO since a PLL network will take too long to eliminate the offset. The training sequences are followed by the SIGNAL symbol, which is a single BPSK-modulated OFDM data symbol containing information about the packet such as data rate. After preamble transmission, any common frequency offset is tracked via the four pilot subcarriers as shown in Fig. 10. It is not necessary to use pilots to estimate the channel as long as the channel remains fairly stationary over the duration of a single packet. The four pilot signals facilitate coherent detection throughout the duration of the packet. The remaining subcarriers carry the data body of the packet. The pilot spacing is selected to be less than the coherent bandwidth of the channel, as explained later. 3.2.3. ITS-WAVE Performance Issues. The performance of an OFDM receiver is affected by several factors, most of which fall into the categories of hardware limita- tions and channel impairments. Hardware limitations, Table 2. ITS-WAVE Baseband Modulation Data Rate (Mbps) Code Rate Modulation N_CBPS N_DBPS 3 1 2 BPSK 48 24 4.5 3 4 BPSK 48 36 6 1 2 QPSK 96 48 9 3 4 QPSK 96 72 24 1 2 16-QAM 192 96 18 3 4 16-QAM 192 144 24 2 3 64-QAM 288 192 27 3 4 64-QAM 288 216 PLCP - Header Rate 4-bits Reserved 1-bit Length 12-bits Parity 1-bit Tall 6-bits Service 16-bits PSDU Tall 6-bits Pad bits Coded - OFDM BPSK rate = 1/2 Coded - OFDM Rate Indicated by signal symbol PPDU Data Variable number of OFDM symbols Signal (1) OFDM symbol PLCP preamble 12 - symbols Figure 8. PPDU frame. Signal detect, AGC, diversity selection Coarse freq. offset estimation timing synchronize Channel and fine frequency offset estimation RATE LENGTH SERVICE + DATA DATA 16 + 16 = 32 s 10 x 1.6 = 16 s 2 x 1.6 + 2 x 6.4 = 16.0 s 1.6 + 6.4 = 8.0 s 1.6 + 6.4 = 8.0 s 1.6 + 6.4 = 8.0 s t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 G12 T 1 T 2 GI GI GI SIGNAL Data 1 Data 2 Figure 9. ITS-WAVE PLCP structure. ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz 2249 particularly clock accuracy and oscillator stability, affect the synchronization accuracy of the receiver. The channel impairments discussed in Section 3.1 include ‘‘delay spread’’ and ‘‘Doppler spread,’’ which result in frequency- selective fading and time-selective fading, respectively. OFDM is extremely sensitive to receiver synchroniza- tion imperfections, which can cause degradation of system throughput and performance. The overlap between sub- carriers leads to a system that is extremely sensitive to imperfections in carrier frequency synchronization. Also, multiplexing symbols onto multiple subcarriers results in a system that is extremely sensitive to imperfections in timing synchronization. This requires that the receiver architecture be structured to correct for frequency, timing, and sampling. Figure 11 is a simplified block diagram [1] depicting the major processing modules associated with the ITS-WAVE physical layer. 3.2.3.1. Synchronization. Synchronization is a big hur- dle in OFDM systems. The ITS-WAVE physical layer uses the same synchronization scheme as in the IEEE 802.11a; it usually consists of three processes: 1. Frame detection 2. Carrier frequency offset estimation and correction 3. Sampling error correction Frame detection is used to determine the symbol boundary so that correct samples of the symbol frame can be taken. The first 10 short symbols are identical and are used for frame detection. The received signal is corre- lated with the known short-symbol waveform that pro- duces correlation peaks. The received signal is also correlated with itself with a delay of one short symbol, which creates a plateau for the length of 10 short symbols. If the correlation peaks are within the plateau, the last peak is used as the position from where the start of the next symbol is determined. Frequency offset estimation uses the long training symbols, which are two FFT symbols back-to-back. The Frame detection: 10 short symbol Frequency offset estimation: Two FFT symbol back-to-back Data Pilot 52 sub-carriers Frequency S y m b o l Figure 10. Pilot structure. (This figure is available in full color at http://www.mrw. interscience.wiley.com/erfme.) Coding Interleaving Binary input data Binary output data QAM mapping Pilot insertion Serial to parallel RF TX DAC Parallel to serial Add cyclic extension and windowing IFFT (TX) FFT (RX) Deinterleaving Decoding QAM demapping Channel correction Parallel to serial RF RX ADC Remove cyclic extension Timing and frequency synchronization Serial to parallel Symbol timing Figure 11. Basic OFDM block diagram. 2250 ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz corresponding chips of the two FFT symbols are then correlated to estimate the frequency offset. Channel esti- mation uses the same two OFDM symbols as the frequen- cy offset estimation. Once the frame start is detected, frequency offset is estimated and signal samples are com- pensated, the two long symbols are transformed into fre- quency domain by FFT. After performing FFT on the preambles, the frequency-domain values are compared with the known preamble values to determine the chan- nel response. 3.2.3.2. Carrier Frequency Offset. The ITS-WAVE (like the 802.11a) specifies that the carrier frequency and sym- bol clock be derived off the same oscillator. This allows the receiver to compute symbol clock drift directly from the carrier frequency offset (e.g., ppm error). Frequency syn- chronization must be applied before the FFT. Without a carrier frequency offset, the peak of any subcarrier corre- sponds to the zero crossings of all other subcarriers. When there is a random frequency offset, there is no longer an integer number of cycles over T FFT , resulting in ICI. The degradation in SNR that occurs due to random frequency offset is approximated by D [1] in decibels D % 10 3 ln 10 ðpDFT FFT Þ 2 E s N 0 DF is the frequency offset and W ( ¼1/T FFT ) is the band- width of the composite OFDM waveform (subcarrier spac- ing). In essence, any carrier frequency offset results in a shift of the received signal in the frequency domain. This frequency error results in energy spillover between sub- carriers, resulting in loss of their mutual orthogonality. The approximation states that the degradation increases with the square of normalized frequency offset. The major tradeoffs encountered when selecting an appropriate car- rier frequency offset correction algorithm include speed, accuracy, and performance under noisy conditions. Short training symbols can recognize offsets as high as 312.5 kHz [ 1 2 Â(1/1.6 ms)]. However, their short duration results in reduced accuracy since they produce only 16- point FFT samples per symbol. Although there are 10 short training symbols, 5 or 6 are consumed during RSSI, AGC, and timing recovery. Long training symbols provide a much more accurate estimate of the frequency offset since they produce 4 times as many FFT points compared to the short training symbol. However, their long extent limits the discernable frequency offset to 78 kHz [ 1 2  (1/6.4 ms)] as shown in Fig. 12. Noise imparts variance on the final offset estimate, thereby mitigating its accuracy. 3.2.3.3. Symbol Timing. Errors in symbol timing syn- chronization manifest as ISI and nonuniform phase shift to the constellation points. Both of these effects naturally lead to degradation of bit error rate (BER). The fast fourier transform (FFT) demodulation process accumulates over exactly one 6.4-ms OFDM interval. If the start of the sym- bol time is not accurately established, the FFT demodu- lation process will operate on two adjacent symbols leading to ISI as shown in Fig. 13. Coarse synchroniza- tion can resolve to within half the sampling period and remove ISI. However, the residual sampling time offsets must be identified, or a nonuniform phase shift will be imparted to the constellation points. The 6.4-ms FFT window is divided up into 64 time in- stants separated by 100ns. Each point of the FFT is com- puted at a rate of 10 Msps (megasamples, i.e., 10,000 samples, per second), which corresponds to 64 discrete frequency-domain samples of the composite 6.4-ms symbol. Since these 64 samples are 100ns apart [T s ¼100ns], the Long sync and Data symbol spectrum: 52 sub-carriers 0 freq Coarse frequency estimate must place 52 sub-carriers To within ½ frequency-bin of their true location: +/− 78.125 KHz 0.15625 MHz freq 0 Short sync spectrum: 12 sub-carriers +1 +1 +1 −1 −1 −1 −1 −1 +1 +1 +1 +1 0.625 MHz Figure 12. Carrier frequency offset. T Multi-path components T g Sampling start max T x T Figure 13. ISI and sampling-time offset. (This figure is available in full color at http://www.mrw.interscience.wiley.com/erfme.) ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz 2251 range of the maximum detectable sampling offset ranges from À50 to þ50 ns. This sampling time offset manifests itself in the frequency domain as a phase shift, which is proportional to the subcarrier frequency. Subcarriers at the high end of the frequency range are affected dispro- portionately relative to those subcarriers at the low end. The effect of this phase shift on BER can be devastating, as symbols that map to subcarriers at the edges will experi- ence a phase shift that rotates the constellation point out of its reliable detection region. Sampling frequency offset does not negatively impact performance on a ‘‘symbol per symbol’’ basis. However, it can have harmful effects over large numbers of symbols. The ITS-WAVE proposal calls out a 5ppm static center frequency offset from the VCO for analog-to-digital/digi- tal-to-analog clocks and carrier VCOs. At 10 MHz, a 5ppm figure corresponds to a 50 Hz offset, which means that one of the clocks is toggling 50 Hz faster than the other. In the period of one 10-MHz clock (100 ns), one clock will advance past the other by 0.5 psc. If we take into account the num- ber of samples per symbol and the number of symbols in a large packet, we find that over a time span of 50 symbols the sampling instants for symbols will have shifted by 2 ns. 11 This timeshift will manifest itself in the frequency domain as a phase shift proportional to the subcarrier fre- quency. This is clearly a receiver steady-state issue, and can’t be detected during training. During receiver track- ing, this offset is taken care of by processing the pilots and feeding back corrections to an interpolator. 3.2.4. ITS-WAVE Adjacent-channel and Cochannel Inter- ference. Effects of adjacent-channel and cochannel inter- ference has been studied using simulation [11], a Simulink model developed to evaluate these types of interference as shown in Fig. 14. In the model shown in Fig. 14 we consider the type of the device and apply the corresponding spectrum mask as Bernoulli random binary generator Bernoulli binary Bernoulli random binary generator1 Bernoulli binary Transmitter In1 In1 Out1 Out1 Interferer Int_gain Tx_gain K− [80x1] [96×1] [96×1] [80x1] [80x1] K− K− [80x1] K− [80x1] [80x1] [80x1] Tx_path_loss [80x1] Receiver sensitivity [80x1] In 1 Out 1 [96x1] [96x1] Tx Rx Error rate calculation In1 Out1 Receiver [96x1] In_path_loss [80x1] AWGN AWGN Channel OFDM demo: Initial settings Figure 14. Enhanced simulation model. (This figure is available in full color at http:// www.mrw.interscience.wiley.com/erfme.) Table 3. ITS-WAVE Device Class Spectral Mask Device Class 7 4.5 MHz Offset 75MHz Offset 75.5MHz Offset 710MHz Offset 715MHz Offset A 0 À10 À20 À28 À40 B 0 À16 À20 À28 À40 C 0 À26 À32 À40 À50 D 0 À35 À45 À55 À65 Table 4. ITS-WAVE Classes and Transmit Power Levels Device Class Maximum Device Output Power (dBm) A 0 B 10 C 20 D 28.8 Table 5. ITS-WAVE Receiver Sensitivity Data Rate (Mbps) Minimum Sensitivity (dBm) 3 À85 4.5 À84 6 À82 9 À80 12 À77 18 À70 24 À69 27 À67 11 0.5psc/sample Â80 samples/symbol Â50 symbols ¼2 ns. 2252 ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz given in Table 3. The model also considers the fact that the devices operate at the maximum power output according to Table 4, which reflects the increase of the out-of-band attenuation for higher power devices. The model takes into account the minimum receiver sensitivity as per Table 5. The channel path loss is modeled according to the two-segment model with a breakpoint of 164m as given by LðdÞ dB ¼20 log ðdÞ þ43:05 ðdo164mÞ LðdÞ dB ¼40 log ðdÞ À1:263 ðd 3 64 mÞ This is typical for models that use ray tracing [12], where the path loss is generally proportional to 1/d 2 before the breakpoint and 1/d 4 after the breakpoint. The breakpoint represents the point at which the first Fresnel zone touch- es the ground, wherein the reflected ray off the surface of the ground cancels some of the power of the direct ray. The breakpoint is approximated by d bp D(4h t h r )/l, where h t is the transmit antennal height and h r is the receive anten- na height. 3.3. The ITS-WAVE MAC Layer Generally, for reliable system operation, the MAC must be properly designed to match the physical layer so that its impairments do not cause undue degradation at higher layers. The IEEE 802.11 MAC is a very complex protocol; it took over 10 years of development with the support of dozens of corporations developing products for the WLAN market. The ITS-WAVE Study Group intends to use the IEEE 802.11a MAC without modification, except for changes to the management information base (MIB). The management information specific to each layer is rep- resented as a MIB for that layer. The generic model of MIB-related management primitives exchanged across the management SAPs is to allow the SAP user entity to either GET the value of a MIB attribute, or to SET the value of a MIB attribute. The invocation of a SET:request primitive may require that the layer entity perform certain defined actions. Figure 15 depicts these generic primitives. 4. CHALLENGES AND FUTURE DEVELOPMENTS 4.1. Validation, Verification, and Testing Developing the ITS-WAVE family of standards is a com- plex task, and the fact that these standards will be sup- porting safety applications makes validation, verification, testing, and system integration critical steps for develop- ing this market. The USDoT has begun this process through funding the Vehicle Safety Communications Con- sortium (VSCC) and other industry participants. At the Caltrans Testbed Center For Interoperability (TCFI), we have developed lab and field infrastructure [11] in order to support these activities as the standards mature. Figure 16 shows data collected over the air using Agilent’s equipment (VSG, VXI, and PSA) at TCFI. In addition, we have demonstrated passing data between the test equip- ment and the simulation tool (Simulink). Validation of the MAC layer will be a special challenge; currently we are experimenting with Telelogic’s TAU G2, which supports both the Specifications and Description Language (SDL) and the Unified Modeling Language (UML). 12 SDL is an ITU formal language that was used to describe the orig- inal IEEE 802.11 MAC specifications. 4.2. System Integration: The Santa Barbara Radio Access Network Beyond the development of the ITS-WAVE family of stan- dards and the availability of telematics products, the de- velopment of the ITS wireless market will require reliable roadside infrastructure [13]. This infrastructure requires feeder and backhaul networks that may use both fixed wireless and landline networks. Figure 17 shows one such infrastructure that has passed the planning stage: Santa Barbara’s Radio Access Network (RAN) [14]. The RF plan- ning of 28 sites has been completed, and some sites are Data link Physical layer MAC MAC SAP MAC Management Entity PHY Management Entity PLCP PMD SAP MIB MIB PLME GET/SET PLME GET/SET Station Management Entity MLME GET/SET PHY SAP PMD Figure 15. GET and SET operations. 12 Further information is available at http://www.telelogic.com. ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz 2253 being installed through the collaboration of Caltrans, local governments, and the university (UCSB). SB-RAN is currently part of a new proposal to develop a public safety testbed, which addresses wireless infrastructure interoperability (WII) issues for both the ‘‘first responder’’ (4.9 GHz) and the ITS (5.9-GHz) bands. 4.3. Observations and Future Developments While WAVE standards efforts are progressing within the IEEE, there are remaining questions yet to be answered regarding many issues such as the following: * Security architecture (P1556 and 802.11i) * Multiple-channel devices and the current ‘‘concept of operations’’ * Interference mitigation in a real environment * MAC extension and its relation to IEEE 802.11e/h * ‘‘Pilot structure’’ and its impact on dedicated public safety channels * Fast handover * IP-based internetworking * Wireless infrastructure interoperability As we move from the descriptive phase of the standard development to the performance and testing phases, the need for better protocol development tools will continue to be a challenge. Roadside system integration issues are rarely ad- dressed in the wireless ITS community. Issues such as cost-effective feeder and backhaul networks to support the wireless infrastructure are considered implementation issues and are outside the scope of the national efforts. 4 2 * 6 9 7 8 10 11 13 12 14 15 16 18 17 22 20 19 23 25 27 21 26 24 28 1 3 5 * TV Hill * Monopole Figure 17. Santa Barbara’s RAN site locations. (This figure is available in full color at http:// www.mrw.interscience.wiley.com/erfme.) Figure 16. Over-the-air ITS-WAVE signal—18 Mbps. (This figure is available in full color at http://www.mrw.interscience.wiley.com/erfme.) 2254 ITS RADIO SERVICE STANDARDS AND WIRELESS ACCESS IN VEHICULAR ENVIRONMENTS (ITS-WAVE) AT 5.9 GHz Presently, the ITS industry is addressing market-enabling applications such as vehicle safety and toll 0applications. The more general ITS ‘‘public safety’’ applications such as work-zone safety (WZS), public protection disaster relief (PPDR), and homeland security, are assumed to be the role of public agencies. The potential gains from considering advanced tech- nologies such as software-defined radios (SDR), and mul- tiple-input/multiple-output (MIMO) systems have not yet been investigated for wireless ITS applications. 5. SUMMARY AND CONCLUSION This article examined the emerging ITS-WAVE family of standards, with emphasis on the mobile vehicle environ- ment and the lower-layer standard. Our findings con- firmed the validity of adopting the IEEE 802.11a as the basis for the broadband wireless ITS standard. The OFDM Forum proposal, originally submitted to the ASTM for wireless road access, is now well understood and it has been accepted by the ITS industry. The physical-layer, proposal is ready for standardiza- tion with the exception of the new pilot structure issue. Long-range dedicated public safety cannot be realized without resolving this issue. The data-link layer proves to be more challenging, as we integrate other IEEE stan- dards (e.g., 802.11e/h/i). As new devices and systems are introduced, there will be a need for demonstration projects, testing standards, compliance certification, and performance benchmarks. Acronyms AGC ASTM BER BPSK CALM Caltrans CFR CP CSMA/ CA DFT DoT DSRC FCC FFT ICI IEEE IETF ISI ISO ISTEA ITS ITSA ITS-RS Automatic gain control American Society for Testing and Materials Bit error rate Binary phase shift keying Communications air interface for long and medium range California Department of Transportation Code of Federal Regulations Cyclic prefix Multiple access with collision avoidance Discrete Fourier transform Department of Transportation Dedicated short-range communications Federal Communications Commission Fast Fourier transform Intercarrier interference Institute of Electrical and Electronics Engineers Internet Engineering Task Force Intersymbol interference International Standards Organization Intermodel Surface Transportation Efficiency Act of 1991 Intelligent Transportation Systems Intelligent Transportation Society of America ITS Radio Services ITU MAC MCM MIB MME MPDU Msps OBUs OFDM OMG PAPR PER PHY PLCP PMD PPDR PPDU PSA QAM QPSK RAN RSSI RSUs SAP SDL SDO SNMP SNR TCP/IP UCSB UML UNII VSA VSCC WAVE WG 802.11 WII WLAN WZS International Telecommunication Union Medium access control Multicarrier modulation Management information base MAC management entity MAC protocol data unit Megasymbol per second Onboard units Orthogonal frequency-division multiplexing Object Management Group Peak-to-average power ratio Packet error rate Physical layer (OSI) Physical-layer convergence procedure Physical-medium-dependent Public protection disaster relief PLCP protocol data unit Power spectral analyzer Quadrature amplitude modulation Quadrature phase shift keying Radio access network Received signal strength indicator Roadside units Service access point Specification(s) and Description Language Standards development organization Simple Network Management Protocol Signal-to-noise ratio Transmissions Control Protocol/Internet Protocol University of California, Santa Barbara Unified Modeling Language Unlicensed national information infrastruc- ture Vector spectrum analyzer Vehicle Safety Communications Consortium Wireless Access in Vehicular Environments Work Group 802.11—WLAN standards Wireless infrastructure interoperability Wireless local-area network Work-zone safety BIBLIOGRAPHY 1. 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