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TRANSACTIONSIEEE [IO] [Ill T. Aulin, “Errorprobability bounds for Viterbi detected continuous phase modulated signals” (abstract), in Int. Symp. Inform. Theory, Abstr. Papers, Santa Monica, CA. 1981, p. 135. J . B. Anderson, C:E. Sundberg, T. Aulin. and N. Rydbeck, “Power-bandwidthperformance of smoothed phasemodulation codes.” IEEE Trans. Commun., vol. COM-29. pp. 187-195. Mar. 1981. [ 121 T. Aulin and C.-E. Sundberg, “Continuous phase modulation: Part [I31 1715 O N COMMUNICATIONS, VOL. COM-29, NO. 11, NOVEMBER 1981 I-Full response signaling,” IEEE Trans. Commun.. vol.COM29. pp. 196-209,Mar.1981. T. Aulin. N.Rydbeck, and C.-E. Sundberg. “Continuous phase modulation: Part 11-Partial response signaling,” IEEE Trans. Commun., vol. COM-29. pp. 210-225, Mar. 1981. Frequency-Independent and Frequency-DependentNonlinear Models of TWT Amplifiers formula for each of the four aforementioned functions. For each of several cases examined, these formulas fit TWT measurements accurately-more so than previously reported formulas. In addition, the formulas permit a closed-form solution of the output signal for an input signal consisting of two phasemodulated carriers, and a solution containing a single integral when more than two such carriersare involved. The parameters of the models are obtainable via straightforward singletone measurement and computation procedures. Moreover, a simple interpretation of measurementsobtainedatdifferent frequencies permitsfrequency selectivity effectsto be included inaquadrature model. Thelatterfeature may be particularly useful in cavity-coupled TWT amplifiers and other components that are not as broad-band as helix-type TWT’s, or when broad-band input signals are involved. 11. PROPOSED FORMULAS FOR THE AMPLITUDE-PHASE MODEL Let the input signal be ADEL A. M. SALEH, SENIOR MEMBER, IEEE x ( t ) = r ( t ) cos Abstract-Simpletwo-parameterformulas arepresentedforthe functions involved in the amplitude-phase and the quadrature nonlinear models of a TWT amplifier, and are shown tofit measured dataverywell. Also, aclosed-formexpressionisderivedforthe output signal of a TWT amplifier excited by two phase-modulated carriers, and an expression containing a single integral is given when morethantwosuchcarriersareinvolved.Finally,afrequencydependentquadraturemodelisproposedwhoseparametersare obtainable from single-tone measurements. I. INTRODUCTION Traveling-wave tube (TWT) amplifiers, and power amplifiers in general, exhibitnonlineardistortionsinbothamplitude (AM-to-AM conversion)andphase (AM-to-PM conversion) [ 11-[ 131. Two equivalent frequency-independent bandpass nonlinear models of helix-type TWT amplifiers have been used intheliteraturetostudythe adverse effects of these nonlinearities on various communication systems.Theseare the amplitude-phasemodel [4] -[ 71 and the quadraturemodel [8] -[ 131, in which the portion of the output wave falling in the same spectral zone asthe band-limited input waveis described in terms of the envelope of the input wave, rather than its instantaneous value. To specify each model, one needs t o know two functions-the amplitude and phase functions for the former model, and the in-phase and quadrature functions for the latter. Several representations (to be discussed later) for these functions have been proposed in the literaturewhich are generally complexin form, or require the knowledge of many parameters. The purpose of this paper is t o present nonlinear models of TWT amplifiers that are based ona simple two-parameter Paperapprovedby theEditorforRadioCommunicationofthe IEEE Communications Society for publication without oral presentation. Manuscript received November 12, 1980; revised June 9, 1981. The author iswithBellLaboratories,Crawford Hill Laboratory, Holmdel, NJ 07733. [mot + $(t)] (1) where mo is the carrier frequency, and r ( t ) and $ ( t ) are the modulated envelopeandphase,respectively. [It is worth noting that r ( t ) may assume positive and negative values.] In the amplitude-phase model [4] -[ 71, the corresponding output is written as Y ( t ) = A [ r ( t ) J cos {mot + $ ( t ) + @ [ r ( t ) ]1 (2) where A ( r ) is an odd function of r, with a linear leading term representing AM-to-AM conversion, and @ ( I ) is an even function of r , with a quadratic leading term [ 11-[4] representing AM-to-PM conversion. Sunde [41 proposed the use of a soft-limiter characteristic t o represent the instantaneous amplitude response of a TWT which results in an envelope amplitude function A ( r ) that does not fall off beyond saturation as observed in practice. He also r2, which proposed t o represent @(r) by apolynomialin would require a large number of terms to fit realistic data, as is the case for polynomial representations in general. Berman and Mahle [SI suggested a three-parameter formula t o represent @(r), which will be used later forcomparison. Thomas, Weidner, and Durrani [7] proposed a four-parameter formula for A(?), which will also be used for comparison. Here, we propose t o represent A ( r ) and @(r) by the twoparameter formulas , +PUP) @(r) = oc@r /(I + P@r 1. A ( r ) = aar/(l (3 1 (4) Notethatfor very large r , A ( r ) is proportional to l/r, and @ ( r ) approaches a constant. Beforetesting theseformulas against experimentaldata and against theformulas of [ 5 ] and171,we introducethe quadrature model. 0090-6778/81/1100-1715$00.75 0 1981 IEEE 1716 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-29, NO. 1 1 , NOVEMBER 1981 x ( t ) = r cos[wot +JI] 1.0 - P(r) h D 0.8 ' .R *- d 0 E b 0.6. v a m U + 0 > 0.4' U Fig. 1. Quadrature nonlinear model of a power amplifier. e U 0 111. PROPOSED FORMULAS FOR THE QUADRATURE MODEL In the quadrature model [8]-[ 131, if the input is given by (l), the output is given by the sum of the in-phase and quadrature components (see Fig. 1) d t ) = p[r(t)l COS [mot+ W)] d t ) = -Q[r(t)l sin + $(t)l [ W O ~ (5a) 0.2 0.00.0 1 0.5 1.0 1.5 0 Input Voltoge ( normalized ) Fig. 2. TWT amplitude (*) and phase (0)data of BermanandMahle [5]. Thesolidlinesare plotted from (3) and (4), and the dashed lines from the Berman-Mahle [5] phaseformula,andfrom the Thomas-Weidner-Durrani [7] amplitude formula. (93) where P(r) and Q ( r ) are odd functions of r with linear and cubic leading terms, respectively. Actually, (5) can be deduced form ( 2 ) with P(r) = A (r) cos [@(r)] (6a) Q(r) = A ( r ) sin [@(r)]. (6b) Eric proposed t o represent P ( r ) and Q ( r ) as odd polynomials of r , which requires large numbers of terms t o fit realistic TWT data. Kaye, George, and Eric [ 9 ] , and Fuenzalida, Shimbo, and Cook [ 101 represented each function by a sum of Bessel functions of the first kind of order 1, which results in simplifying the calculation of the output spectrum. However, a large number of terms is still required to fit realistic data.HetrakulandTaylor [ 111-[ 131 used two-parameter formulas involving modified Bessel functions of the first kind, which will be used later for comparison. Here, we propose t o represent P ( r ) and Q ( r ) by the twoparameter formulas P(r) = apr/(l + &r2) + pqr212. ~ ( r =) a4r3/(1 (7) (8) Note that for very large r , both P ( r ) and Q ( r ) given above are proportional to l/r, while those given by Hetrakul and Taylor approach constantvalues. A useful property of.(7)and (8) is that I Q(r) = - aP(r) apP (9) "p-*"q,Pp+P4. Thus, if the spectrum of P(r) is calculated for a given r ( t ) , the correspondingspectrum of Q(r) is readily obtainedby differentiation. This property will be used in Section V. IV. FITTING THE FORMULAS TO EXPERIMENTAL TWT DATA To establish the accuracy of the formulas proposed in (3), (4), (7), and(8), experimental TWT amplitude-phaseand 0 1 2 3 4 5 6 Input Voltage ( millivolts 1 TWT in-phase (*) and quadrature ( 0 ) data of Hetrakul and Taylor [ 111 -1 131. The solid lines are plotted from (7) and (8), and the dashed lines from their own formulas [ 111-[ 131. Fig. 3. quadrature data were obtained from three different sourcesBerman and Mahle [SI, Hetrakul and Taylor [ 111-[ 131, and Kaye, George, and Eric [ 91. These data are plotted in Figs. 26 by asterisks for A ( r ) and P(r), and by circles for @(r) and Q(r). The units used on the coordinate axes are the same units used in the cited references. In Figs. 2 , 5, and 6, the inputand output voltages are normalized to their correspondingvalues at saturation. A minimum mean-square-error procedure is described in the Appendix for fitting the formulas of (3), (4), ( 7 ) , and (8) to theexperimentaldata.The resultsare plottedbythe solid lines in Figs. 2-6, which show an excellent fit in eachcase. The values of the (Y and p parameters, as well as the resulting rootmean-square (rms) errors, are given in Table I. The (Y parameters given in the table for @(r) give a dimension of radians when substituted in (4); however, the corresponding rms errors are given in degrees. The remaining quantities in the table have dimensions consistent with those used in the associated figures. The Thomas-Weidner-Durrani [ 71 amplitudeformula is plottedby adashed linein Fig. 2 forcomparison.The resulting rms error was 0.014 normalized volts, which, in spite TRANSACTIONS IEEE COMMUNICATIONS, ON VOL. COM-29, NO. 11, NOVEMBER 1981 1717 TABLE I OPTIMUM PARAMETERS ANP'RESULTING RMS ERRORS OBTAINED BY FITTING(3), (4), (7), AND (8) THROUGH VARIOUS EXPERIMENTAL TWT DATA (FIGS.'%) Data from Reference IS1 Input Voltoge ( millivolts ) Fig. 4. TWT Vplitude (*) and phase ( 0 ) data obtained from Hetrakul and Taylor [ 111-[ 133 through the use'of (6). The solid lines are plotted from (3) and (4). 1.0 - -u 0.8 N I M 0 6r 0.6 v H ;t: 0.4 > 4 a 0 2 0. i 0. eI 0.0 0.5 1.0 1.5 Input Voltoge ( normalized 1 Fig. 5. TWT in-phase (*) and quadrature ( 0 ) dataofKaye,George, and Eric 191. The solid lines areplotted from (7) and (8). of the large number of parameters involved, is slightly larger thanour'0.012vgue given inTable I. The Berman-Mahle [5] phase function is also plotted by a dashed line through their own phase data in Fig. 2. While that formula gives a good fituptosaturation, large errorsareencounteredbeyond .. . saturation. The Hetrakul-Taylor [ 11 ] -[ 131 in-phase and quadrature formulas are plotted through their own data in Fig. 3. The resulting rms errors for P(r) and Q(r) are 0.058 V and 0.040 V, respectively,whichare somewhat higher than our corresponding 0.057 and 0.023 values given in Table I. V. INTERMPDULATION ANALYSIS d P b 0.6 h " 3 0.4 w v 4 4 2 0.2 4 2. 2 0.0 0.0 0.5 1.0 Input Voltoge ( 1.5 normoltzed 2.0 One of the applications intended fora nonlinear model of a power' amplifier is t o be able ,tp ,find the output signals' associated with various input signals of practical interest. Here we"consider the case of multiple phase-modulated carriers. The quadrature m.odel o f (7) and (8) will be employed inthe anaiysis. A two-carrier signal will be considered first because it yie1d.s a closed-form expression for the output signal. Actually, the use of such a signal has been proposed for a satellite corn: munication system [ 141. Moreover, the results, with the phase modulatipn suppressed, can be used in conjunction with the twoitone nonl6earity test[ 31, [4],[ 151. A . Two Phase-Modulated Carriers Let the input signal be ) Fig. 6 . TWT amplitude (*) andphase (0)data of Kaye,George, and Eric [9].The solid lines'are plotted from (3) and (4). X(!) = v, cos [ W l f + J/l(t)] + v, cos [o,t + J / z ( t ) ] . (1 0) 1718 IEEE TRANSACTIONS COMMUNICATIONS, ON VOL. COM-29, NO. 1 1 , NOVEMBER 1981 with M(k1 1 k23 ***> kp) = im .[fi Jki(vis)l 4- r[P(r) + iC?(r)l ds J I (rs) d r (20) where J k is the Bessel function of the first kind of order k . Substituting (7) and (8) in (20), one can evaluate the integration over r throughthe use of (9) and the Hankel-type integral given in [ 16, (6.565.4), p. 6861 to obtain where K O and K 1 are the modified Bessel functions of the second kind of orders 0 and 1, respectively. For n = 2, (21) reduces to the closed-form expressions obtained in the previous section. However, numerical integration of (21) seems necessary if n ' > 2. A direct numerical integration of (20) would, of course, be more difficult since a double integral is involved. It can be shown that when n is very large, then the integral in (21) can be evaluated in terms of a finite sum involving the exponential integral. This important result will be given in a future paper [ 171 . VI. A FREQUENCY-DEPENDENT QUADRATURE MODEL Thus far, it has been implied that the characteristics of the TWT amplifier are independent of frequency over the band of interest, which is often the situation encountered in practice, especially whenhelix-type TWT's are employed. However, when broad-band input signals are involved,' or in cases where amplifiers and components are used that are not as inherently broad-band as helix-type TWT's, frequency-dependent a model is needed. Here we make a conjecture, to be explained later, that enables us t o infer such a model from single-tone measurements. 'Consider a single-tone test in which the input signal to the TWT has adjustable amplitude r and frequency f and,let the measured amplitude and phase of the output signal be A ( r , fl and @ ( r , f), respectively. Let be the measured small-signal phase response. Thus, @(r, f) @o(f) + 0 as r + 0. In fact, this phase difference is what was labeled @ ( I ) in Sections 11-IV. Using (6),oneobtainsthe in-phase and quadrature output signals 1714 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-29, NO. 11, NOVEMBER 1981 For each given f, one can fit the formulas of (7) and (8) t o these data as is described in the Appendix. This would result inthefrequency-dependentparameters a p ( f ) ,&,cf), aq(f), and &(f). One now can compute the functions Hp(f)= W) (25) Gp(f) =~ p ( f ) / W ) Hs(0 = (26) W) (27) Gg(f) = ag(f)/P43'2(f). Let us define the normalized envelope nonlinearities (28) frequency-independent +r2) = (30) r3/(1 + r212, Po(') = r/(1 (29) which are obtained from (7) and (8) by setting the a's and p's to unity. It is observed from. (7), (25), (26), (29), and Fig. 1 that the operation performed on a single-tone signal passing through the in-phase branch can be.divided into three steps: first, the input amplitude is scaled by H p c f ) ; next, the resulting signal passes through the frequency-independent envelope nonlinearity Po(r); and finally, the output amplitude is scaled perby GpCf). Three similar stepsapplyfortheoperation formed in the quadraturebranch. Now, performing each of the aforementioned frequency-dependent amplitude-scaling operations may beinterpreted as passing the signal through an appropriately locatedlinear filter having the corresponding real frequency response. This leads to the frequency-dependent quadrature model shown in Fig. 7. That model is, of course, valid for single-tone input signals by virtue of the procedure used to construct it. However, its validity for arbitrary input signals is, at this point, a conjecture that remains to be confirmed experimentally. The box in the output side of Fig. 7 is a linear, all-pass network having an amplitude response of unity, and a phase response of which is defined in (22).Fora single-tone input signal, that phase response can be absorbed totally or partially into the filters in each of the two branches of the model without affecting the output signal. However, for the modelto give anacceptablerepresentation of the power amplifier for arbitrary input signals, it may be necessary to absorb that phase response into the various filters in a particular manner. This point needs further investigation. Fig. 7. Proposedfrequency-dependentquadraturemodel of a TWT amplifier. Po(r) and Qo(r) are frequency-independent envelopenonlinearities given in (29) and(30); the H's and the G's,are real linear fiiters whosefrequency responses are defineditl(25)-(28), and is a linear all-pass network whose phase response is defined (22). in @o(n excited by two phase-modulated carriers. Moreover, when the input signal consists of:,Fore than two such carriers, it was shown that the output signal can be representedby a single integral. When the number of carriers is very large, it can be shown that t.his integral can be evaluated in terms of a finite sum involving the exponential integral.Thisresult,however, was not given here, and will be presentedin a future paper [ 171. A frequency-dependent quadrature model of a TWT amplifier has also been proposed (see Fig. 7). The various parameters of themodel can be obtained byapplyingminimum mean-square-error curve fittingtomeasurements involving a single tone of variable amplitude and frequency. APPENDIX CURVE-FITTING PROCEDURE The formulas(3), (4),(7), and (8) assume the general form where n = 1, 2, or 3, and u = 1 or 2. (Actually, noninteger values of u were considered in fitting the data in Section IV, and,remarkably, V = 1 or 2 was foundtobe very nearly optimum.) Given m measured pairs, (zi, ri), i = 1, 2, -., m , we need to find a and to fit (Al) to these data. Defining and employing standard minimummean-square-error curvefittingprocedure,oneobtainsthe required optimum values of a and 0 VII. CONCLUSIONS Simple two-parameterformulas have beenproposed for each of the functions of the amplitude-phase model of a TWT amplifier, as well as for each of the functions of the equivalent quadrature model. The formulas fit available TWT data very well. This implies that the nonlinear behavior of a given TWT can be accuratelyrepresented by only four parameters. The same is also true for the quadrature model formulas proposed by Hetrakul and Taylor [ 1 11 -[ 131. Those formulas, however, seem unnecessarily complicated. The simplicity of the proposed formulasresulted in a closed form expression fortheoutput signal of a TWT amplifier where all the summations are over i = 1-m. ACKNOWLEDGMENT The author thanks and suggestions. L. J. Greenstein for useful discussions i E i k TRANSACTIONS ON COMMUNICATIONS,VOL. cOM-29, No.1 1 , NOVEMBER 1720 REFERENCES J. .P. Laico, H. L. McDoweil, and C. R. Moster, “A mediumpower traveling-wave tube for 6000-Mc radio relay,” Bell Syst. Tech. J . , vol.35,pp:1285-1346,Nov.1956. R: C : Chapman,Jr.,and J . B. Millard,‘:Intelligiblecrosstalk between.frequencymodulated.carriersthrough AM:PM conversion,” IEEE Trans. Commun. S y s t . , vol. CS-12, pp. 160-166, June 1964. , R;G. MCdhurst and J. H. Roberts, “Distortion of SSB transmission due.to AM-PMconversion,”, IEEE Tians. Commun. S y s t . , vol. CS-12, pp. 166-176, June1964. E. D . ‘Sunde, :‘Intermodulationdistortion in multicarrier FM systems,”in IEEE I t i t . Conv.Rec., 1965,vol.13,pt.2,pp. , 1’30-146. A. L. Berrnan and C. H. Mahle,“Nonlinearphaseshift in trabeling-wave tubes as applied to multiple access communication satellitis.” IEEETrans. Commun. Technol., vol.COM-18,pp. 37-48,Feb:1970. 0 . Shimbo, “Effects of intermodulation. AM-PM conversion, and additive noise in multicarrier TWT systems,” Proc. IEEE, vol. 59, pp:230-238,Feb.1971. S . H. Durrani,“Digital C. M . Thomas, M. Y.Weidner,and IEEE Trans. amplitude-phasekeyingwithM-aryalphabets,” Cprnmuii., vol.COM,22.pp. 168-180. Feb.1974. M . . J . Eric.“Intermodulationanalysisofnonlineardevicesfor multiplecamerinputs,”Commun.Res.Centre,Ottawa,Ont., Canada, CRC Rep. 1234, Nov. 1972. 1981 A . R.Kaye,D. ‘ A . ,George,and. M. J., Eric,“Analysisand compensation, of bandpassnonlinearitiesforcommunications,” IEEE Trans. Commun. Technol., vol. COM-20, pp. 965-972, Oct. 1972. . ’ J: C. Fuenzalida; 0. Shimbo,,and W . . L..Cook,“Time-domain analysis of intermodulationeffectscaused by nonlinearamplifiers,” COMSAT Tech. Rev., vol. 3, pp. 89-143, Spring 1973. P. Hetrakul and D. P. Taylor, “Nonlinear quadrature model for a traveling-wave-tube-typeamplifier.” Electron: Lett., vol. 11, p. 50,Jan.23:1975. ,I , ”The effects of tradsponder nonlinearity on binary ‘CPSK signaltransmission,’’ IEEE Tiaris. Commun., vol. COM-24, pp. 5 4 6 5 5 3 , May 1976. “Compensators for bandpass nonlinearitieg in satellite 131 -, communications,” IEEE Trans. Aerosp. Electron. Sysr., vol. AES12,’pp. 509-514; July 1976. 141 D. H. Staelin and R: L: Harvey, “Archite,cture and economics for pervasive broadband.sate1lite networks.” in Proc. 1979 Int. ConJ Commun., June 1979, vol. 2;pp: 35.4.1-35.4.7. G . L . Heiter,“Characterization of nonlinearitiebinmicrowave dev,ices and systems,” IEEE Trans. Miciowave Theory Tech.,vol. MTT-21, pp. 797-805, Dec. 1973. I. S . Gradshteyn and I. M. Ryzhik,.Tables of Integrais, Series and Products. New York:Academic,1965. A.A. M. Saleh,“IntermodulationanalysisofFDMAsatellite systems employing compensated and uncompensatedTWT’s:’ to be published. 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