IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 1, JANUARY 1997
`
`223
`
`APPENDIX
`Proof of (12): We use (10), (11), the Schwarz inequality, and the
`unitarity of the spreading function
`
`A New Volterra Predistorter Based on
`the Indirect Learning Architecture
`
`Changsoo Eun and Edward J. Powers
`
`kLGH LGLH k2
`2
`2
`
`= kSGH SG SHk2
`
`=
`
`
`
`SG(
`
`0
`
`; 
`
`0)SH ( 
`
`0
`
`;  
`
`0)
`
`2
`
`2
`
`
`
`
`
`j2[  (1=2) (1=2+)+2  ] 1 d
` e
`
`0
`
`0
`
`d
`
`d d
`
`2
`
`2
`
`2
`
`2
`
`2
`
`
`
`
`
`jSG(1; 1)j2
`
`d1 d1
`
` 4
`
`jSH (2; 2)j2 sin2([ 2(1=2 )
`
`
`
`
`
`2(1=2 + ) + 222])d2d2 d d
`
`< 6400kGk2
`
`max
`
`j j< ;j j<
`
` sin2([12(1=2 ) 21(1=2 + ) + 222])
`
`
`
`jSH (; )j2
`
`d d
`
`
`
`
`
`
`
`= 6400 sin2(00(1 + 2jj))kGk2HSkHk2H S :
`
`
`
`REFERENCES
`
`[1] P. A. Bello, “Characterization of randomly time-variant linear channels,”
`IEEE Trans. Commun. Syst., vol. 11, pp. 360–393, 1963.
`[2] L. Zadeh, “Frequency analysis of variable networks,” in Proc. IRE, vol.
`38, Mar. 1950, pp. 291–299.
`[3] V. Filimon, W. Kozek, W. Kreuzer, and G. Kubin, “LMS and RLS
`tracking analysis for WSSUS channels,” in Proc. IEEE ICASSP–93,
`Minneapolis, MN, 1993, pp. III/348–351.
`[4] R. S. Kennedy, Fading Dispersive Communication Channels. New
`York: Wiley, 1969.
`[5] W. Kozek, “Matched Weyl-Heisenberg expansions of nonstationary
`environments,” Ph.D. dissertation, Vienna Univ. Technol., Austria, 1996.
`[6] J. D. Parsons, The Mobile Radio Propagation Channel. London, U.K.:
`Pentech, 1992.
`[7] A. W. Naylor and G. R. Sell, Linear Operator Theory in Engineering
`and Science. New York: Springer-Verlag, 1982.
`[8] K. A. Sostrand, “Mathematics of the time-varying channel,” in Proc.
`NATO Adv. Study Inst. Signal Processing Emph. Underwater Acoust.,
`vol. 2, 1968, pp. 25-1–25-20.
`[9] G. B. Folland, Harmonic Analysis in Phase Space. Annals Math. Studies.
`Princeton, NJ: Princeton Univ. Press, 1989.
`
`Abstract—In this correspondence, we present a new Volterra-based pre-
`distorter, which utilizes the indirect learning architecture to circumvent
`a classical problem associated with predistorters, namely that the desired
`output is not known in advance
`
`INTRODUCTION
`I.
`Nonlinear compensation techniques are becoming increasingly
`important to improve the performance of telecommunication channels
`by compensating for channel nonlinearities. Nonlinear compensators
`can be classified into two categories: equalizers and predistorters,
`located after and before the nonlinear channel, respectively.
`The pth-order inverse method [1] based on the Volterra series
`model was applied to predistorter design [2] for nonlinear telecom-
`munication channels. However, the design of the pth-order inverse
`system is very complicated and must be based on a known Volterra
`series model of the nonlinear channel. Moreover, as the order of
`nonlinearity grows higher, so does the design complexity of the
`pth-order inverse method.
`To the best of the authors’ knowledge, nonlinear predistorter design
`using a Volterra series model, other than the pth-order inverse method,
`has not been attempted. This is due to the fact that, beforehand, the
`desired output values of the predistorter are unknown. To overcome
`this difficulty, we utilize the indirect learning architecture and the
`recursive least square (RLS) algorithm. Specifically, we propose in
`this correspondence an indirect Volterra series model predistorter
`which is independent of a specific nonlinear model for the system to
`be compensated, as opposed to the pth-order inverse method which
`first requires a Volterra series model of the system or channel to be
`compensated. We use the phrase indirect Volterra to distinguish our
`new predistorter from the more common pth-order inverse approach,
`and in recognition that our approach rests upon utilizing the indirect
`learning architecture. Both 16–phase shift keying (PSK) and 16-
`quadrature amplitude modulation (QAM) will be used to demonstrate
`the efficacy of the new approach.
`
`II. VOLTERRA MODEL AND THE pTH-ORDER INVERSE METHOD
`In discrete time, a third-order Volterra series for a causal, finite-
`memory system becomes
`
`y[n] =
`
`N
`
`k=0
`
`+
`
`(1)
`k x[n k] +
`h
`
`(2)
`k; lx[n k]x[n l]
`h
`
`N
`
`N
`
`N
`
`N
`
`N
`
`k=0
`
`l=0
`
`(3)
`k; l; mx[n k]x[n l]x[n m]
`h
`
`l=0
`
`m=0
`
`k=0
`+ e[n]
`
`(1)
`
`Manuscript received December 6, 1995; revised August 21, 1996. This
`work is supported in part by the Joint Service Electronics Program AFOSR
`under Contracts F-49620-92-C-0027 and F-94620-95-C-0045.
`C. Eun was with the Electronics Research Center and the Department of
`Electrical and Computer Engineering, University of Texas, Austin, TX 78712-
`1084 USA. He is now with Daewoo Electronics Co., Ltd., Seoul, Korea.
`E. J. Powers is with the Electronics Research Center and the Department of
`Electrical and Computer Engineering, University of Texas, Austin, TX 78712-
`1084 USA (e-mail: ejpowers@mail.utexas.edu).
`Publisher Item Identifier S 1053-587X(97)00515-1.
`
`Authorized licensed use limited to: James Proctor. Downloaded on October 30,2024 at 14:17:09 UTC from IEEE Xplore. Restrictions apply.
`
`1053–587X/97$10.00 ª
`
`1997 IEEE
`
`PETITIONERS EXHIBIT 1015
`Page 1 of 5
`
`

`

`224
`
`IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 1, JANUARY 1997
`
`Fig. 1. The training architecture of the Volterra series model predistorter.
`
`Fig. 2. The satellite communication channel model reduced to base band.
`
`where N is the discrete system memory length; x[n] and y[n] the
`
`
`
`input and output, respectively; and h(1)k , h(2)k; l, and h(3)k; l; m are the
`discrete time domain Volterra kernels of order 1, 2, 3, respectively.
`The pth-order inverse method, based on the Volterra series model,
`was proposed by Schetzen in [1]. This method connects another
`Volterra system in series with the nonlinear system to be compensated
`such that the overall Volterra kernels of order greater than one and less
`than or equal to p are zero. However, the kernels of the overall system
`for orders higher than p are not zero, even though these higher order
`nonlinearities may not be present in the original system. They may
`have an undesirable effect on the nonlinear compensation depending
`on the input signal level.
`
`III. THE DIRECT VOLTERRA SERIES MODEL APPROACH
`To circumvent the problem of not knowing beforehand the desired
`output of a predistorter, we utilize the indirect learning architecture
`[3] indicated in Fig. 1. This algorithm uses two identical Volterra
`models for the predistorter and training. As the innovation [n]
`approaches zero, the overall output of the system y[n] approaches
`the system input x[n], i.e., the desired output of the overall system,
`since the inputs to the identical networks are equal. After the
`training session, the training network is removed. We will call this
`approach the indirect Volterra series model approach since we use
`an independent Volterra series model instead of the pth-order inverse
`system as a predistorter. Additional details follow.
`The third-order Volterra series model of (1) is expressed in a matrix
`form as
`
`d[n] = hxT [n]
`
`(2)
`
`002
`
`where d[n] is the output of the Volterra series model predistorter, the
`superscript T denotes the transpose of the matrix, h is the Volterra
`kernel vector, and x[n] is the input vector, which are defined by
`
`
`
`
`001; h(3)000; h(3)h = [h(1)0 ; h(1)1 ; ; h(1)N 1; h(3)
`
`
` ; h (3)klm; ; h(3)
`(N 1)(N 1)(N 1)]
`x[n] = [x[n]; x[n 1]; ; x[n N + 1]; jx[n]j2x[n]
`jx[n]j2x[n 1]; jx[n]j2x[n 2];
` ; jx[n N + 1]j2x[n N + 1]]:
`
`(3)
`
`(4)
`
`Here, n is the current sample time and N is the system memory length
`in the discrete time form. The overall system output y[n] is given by
`(5)
`
`y[n] = (d[n])
`
`(a)
`
`(b)
`
`Fig. 3. Constellations of the original (a) 16-PSK and (b) 16-QAM signals.
`A simple example of data assignment is shown for the 16-PSK constellation.
`
`where  is a nonlinear system function with memory and the vector
`d[n] is given by
`
`d[n] = [d[n]; d[n 1]; ; d[n M + 1]]:
`
`(6)
`
`Here, M is the memory duration of the nonlinear system.
`The output o[n] of the training Volterra series model is given by
`
`o[n] = hyT [n]
`
`(7)
`
`where y[n] is the vector of the sampled output of the overall system
`defined by
`
`y[n] = [y[n]; y[n 1]; ; y [n N + 1]; jy[n]j2y[n]
`jy[n]j2y[n 1]; jy[n]j2y[n 2]
` ; jy[n N + 1]j2y[n N + 1]]:
`
`(8)
`
`If the Volterra series models satisfy the following conditions
`
`if x[n] 6= y[n];
`
`then d[n] 6= o[n]
`
`and
`
`if x[n] = y[n];
`
`then d[n] = o[n]
`
`(9)
`
`then, as [n] = d[n] o[n] approaches zero, y[n] approaches x[n],
`thus so does y[n] to x[n].
`The kernel update of the training Volterra series system follows
`the modified RLS algorithm. The kernel vector at the sample time
`n is updated by
`
`^h(n) = ^h(n1) + k[n] [n]
`
`(10)
`
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`

`IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 1, JANUARY 1997
`
`225
`
`on board the satellite is limited. Amplifier saturation introduces
`a nonlinear dependence of the output amplitude and phase on
`input amplitude levels thereby resulting in AM/AM and AM/PM
`conversion, respectively. The amplitude A and the phase shift  of
`the output signal of the TWT are modeled by [4] as follows:
`
`A[r(t)] =
`
`[r(t)] =
`
`ar(t)
`1 +

ar2(t)
`pr2(t)
`1 +

pr2(t)
`
`(15)
`
`(16)
`
`where a,

a, p,

p are constants and r(t) is the normalized input.
`At the receiver a filter rejects noise and unwanted signals, and then
`decodes and detects. The demodulated signal d(t) is given by
`
`d(t) = A[r(t)]ejf (t)+[r(t)]g
`
`(17)
`
`where the symbol input to the channel is u(t) = r(t)ej[ (t)].
`Even though the TWT may be regarded as a zero-memory nonlin-
`ear system, the overall telecommunication channel shown in Fig. 2
`can be considered as a nonlinear system with memory.
`
`V. NUMERICAL EXPERIMENT RESULTS
`To compare the performance of the indirect and pth-order pre-
`distorters, we will use the normalized mean-squared error (NMSE)
`defined by
`
`K
`
`jxk ok j
`
`2
`
`NMSE =
`
`k=1
`
`K
`
`(18)
`
`2
`
`jxk j
`k=1
`
`(a)
`
`(b)
`
`Fig. 4. The constellation of (a) the 16-PSK signal and (b) the 16-QAM signal
`distorted by the linear filters and TWT of the satellite communication channel.
`The maximum amplitude of the input signal is one.
`
`where [n] is called the innovation that represents the information
`contained in the current desired output value which cannot be
`predicted by the previous kernel vector and is defined by
`[n] = d[n] ^h(n1)yT [n]
`= ^h(n1)(x[n] y[n])T
`
`(11)
`(12)
`
`and k[n] is the time-varying gain vector defined by
`
`k[n] =
`
`1P[n 1]yT [n]
`1 + 1y [n]P[n 1]yT [n]
`
`:
`
`(13)
`
`The matrix P[n] is updated as follows:
`
`P[n] = 1P[n 1] 1k[n]y [n]P[n 1]:
`
`(14)
`
`Note that the output y[n] of the overall system is used as the input
`to the training Volterra series model. The contents of the training
`Volterra series model are copied into those of the Volterra series
`model predistorter.
`
`IV. SATELLITE TELECOMMUNICATION CHANNEL
`In Fig. 2, we show a simplified base band satellite telecom-
`munication channel. In such channels, the data or symbols to be
`transmitted are passed through a pulse shaping filter for effective
`bandwidth use. At the satellite, the weak signal is amplified by a
`microwave power amplifier, usually a traveling wave tube (TWT)
`or a solid-state amplifier. The amplifier is normally operated near
`saturation to maximize power efficiency as the power available
`
`where K is the number of total test data, xk is the desired detector
`output of kth test data, and ok is the kth compensated output. Note
`that in the compensated system, the desired output is the input to the
`overall system.
`In this example, we use the base-band satellite communication
`channel model depicted in Fig. 2. We note that the linear filters, be-
`cause of their memory characteristics, cause intersymbol interference
`(ISI). The linear filter coefficients used in this example are given by
`
`h = [0:8; 0:1]
`
`g = [0:9; 0:2; 0:1]
`
`(19)
`(20)
`
`where h and g are the transmit and receiver filter coefficient vectors,
`respectively. These numbers are chosen to represent general inter-
`symbol interference in which the effect of previous symbols decreases
`with time.
`The nonlinearity caused by the TWT can be represented by
`(15), (16) with the typical values of constants a = 2:0,

a =
`1:0, p = =3,

p = 1:0 [5]. In Fig. 3 we show 16-PSK
`and 16-QAM constellations to be transmitted, and in Fig. 4 we
`present the distorted constellations after transmission through the
`nonlinear channel. Without compensation, high symbol error rate
`would seriously degrade system performance.
`For comparison purposes, we first modeled the system with a third-
`order Volterra series model in order to apply the pth-order inverse
`method. The memory span of the model was assumed to be three,
`since the memory of the ISI is three—one for the transmitter filter and
`two for the receiver filter. Then the pth-order inverse was determined.
`Unlike the pth-order inverse method, the indirect Volterra series
`model approach does not need a Volterra series model of the system
`to be compensated. Therefore, utilizing the linear filter coefficients
`
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`IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 1, JANUARY 1997
`
`(a)
`
`(c)
`
`(b)
`
`(d)
`
`Fig. 5. 16-PSK constellations of signals: (a) predistorted by the direct Volterra series model predistorter; (b) predistorted by the pth-order inverse system;
`(c) compensated by the direct Volterra series model predistorter (NMSE = 0.0007); and (d) compensated by the pth-order inverse system (NMSE =
`0.088) for the satellite communication channel. SNR = 1.
`
`and the parameters of the TWT, which were previously given, we
`obtain, using the indirect learning architecture, a Volterra series model
`predistorter to compensate the undesired nonlinearity. In both cases
`we consider third-order models.
`The channel reduces the amplitudes of the input signal as shown
`in Fig. 4. The predistorter should “amplify” the signal to compensate
`for the amplitude reduction. Therefore, if the level of the input to
`the predistorter is one, the level of the output from the predistorter
`becomes larger than one due to the amplification. If this amplified
`signal
`is fed to the communication channel,
`the input signal
`to
`the TWT may exceed the maximum input level that produces the
`maximum output power. Therefore, for the predistorter output to fit
`into the TWT input range for maximum output power, the maximum
`level of the input to the predistorter was reduced to 0.64. Therefore,
`the maximum level of the desired compensated output is 0.64. This
`can be seen in the compensated constellations in Figs. 5(c) and 6(c).
`This may be a drawback of the predistorter in this particular case,
`since the reduced signal amplitude means a lower output signal-
`to-noise ratio and shorter Euclidean distance between the symbols,
`which means a higher detection error probability.
`In the following, we compare the performance of the indirect
`Volterra and pth-order inverse distorters using both 16-PSK and
`16-QAM signals.
`In Fig. 5, we show results obtained from the two predistorters for
`500 16-PSK symbols. In Fig. 5(a) and (b), we show the constellations
`of 16-PSK signals predistorted by the indirect Volterra series model
`and the pth-order inverse predistorters, respectively. In Fig. 5(a),
`we see that TWT nonlinearity is compensated for by increasing
`
`level. Here, what
`the “gain” of the predistorter with the signal
`we mean by “gain” of a predistorter or a filter is the amplitude
`variation introduced by digital signal processing. The increasing
`gain of the predistorter results in nonuniform symbol distribution
`in which symbol density is densest at small amplitudes and becomes
`sparser with increasing amplitude. However, the pth-order inverse
`result of the predistorter shown in Fig. 5(b) does not show this
`compensation effect. In Fig. 5(c) and (d), we show the compensated
`output symbol constellation. The performance of the indirect Volterra
`series predistorter is quite good (NMSE = 0.0007). On the other
`hand, the output of the channel compensated by the pth-order inverse
`predistorter shows that the compensation is not as effective (NMSE =
`0.088). Although some of the effects of clustering have been reduced,
`we note significant amplitude and phase distortion remain.
`Next, we compare the two approaches using a 16-QAM signal. In
`Fig. 6, we show the predistorter output for the 16-QAM signal using
`both the indirect Volterra series model approach Fig. 6(a) and the pth-
`order inverse method Fig. 6(b). Examining Fig. 6(d) for the pth-order
`inverse we note that, although the clustering has been reduced, both
`significant amplitude and phase distortion remain, especially for the
`larger amplitudes. On the other hand, the performance of the indirect
`Volterra predistorter shown in Fig. 6(c) is quite good as manifested
`by its NMSE of 8:0  104, compared to 0.17 for the pth-order
`predistorter.
`The relatively poor performance of the pth-order predistorter, in
`both the 16-PSK and 16-QAM cases, may very well be due to
`the generation of higher order (>p) nonlinearities by the pth-order
`inverse predistorter.
`
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`227
`
`(a)
`
`(c)
`
`(b)
`
`(d)
`
`Fig. 6. The 16-QAM constellations of signals: (a) predistorted by the direct Volterra series model predistorter; (b) predistorted by the pth-order inverse
`system; (c) compensated by the direct Volterra series model predistorter (NMSE = 0.0008); and (d) compensated by the pth-order inverse system (NMSE
`= 0.17) for the satellite communication channel. SNR = 1.
`
`[3] D. Psaltis, A. Sideris, and A. A. Yamamura, “A multilayer neural
`network controller,” IEEE Contr. Syst. Mag., pp. 17–21, Apr. 1988.
`[4] A. A. M. Saleh, “Frequency-independent and frequency-dependent
`nonlinear models of TWT amplifiers,” IEEE Trans. Commun., vol.
`COM-29, no. 11, pp. 1715–1720, Nov. 1981.
`[5] S. Pupolin and L. J. Greenstein, “Performance analysis of digital
`radio links with nonlinear transmit amplifiers,” IEEE J. Select. Areas
`Commun., vol. SAC-5, no. 3, pp. 535–546, Apr. 1987.
`
`VI. DISCUSSION
`The results show that, as well as having a much simpler design
`procedure, the indirect Volterra series model approach has other
`advantages over the pth-order inverse method. The indirect Volterra
`series model approach does not need a specific nonlinear model, thus
`eliminating the need to first determine a pth-order Volterra model
`of the nonlinear channel. Furthermore, with the indirect Volterra
`approach, no pth-order inversion is required. The generation of higher
`order (greater than p) nonlinearities is reduced, since the design
`procedure is an optimization (in a least squares sense) instead of
`a system inversion.
`The indirect Volterra series model approach is preferable if the
`order of the nonlinearity is low (say, third or possibly fifth order).
`If the order of the nonlinearity is higher (greater than, say, third or
`fifth order), then both the indirect Volterra series model approach and
`the pth-order inverse method are usually not practical because of the
`increased complexity. For example, the number of unknown kernel
`coefficients increases rapidly. In such cases, neural-network-based
`predistorters, designed using the indirect learning architecture, appear
`to be an appropriate alternative, an approach we have investigated
`and will report upon elsewhere.
`
`REFERENCES
`
`[1] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems.
`New York: Wiley, 1980.
`[2] G. Lazzarin, S. Pupolin, and A. Sarti, “Nonlinearity compensation in
`digital radio systems,” IEEE Trans. Commun., vol. 42, pp. 988–999,
`Feb./Mar./Apr. 1994.
`
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