A HAM M.ERSTEIN
`
`PREDISTOR TIO N LINEARIZATION DESIGN BASED ON
`
`THE IND IRECT LEARNING ARCHITECTURE
`
`Lei Ding
`Raviv Raich
`G. Tong Zhou
`School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250, USA
`
`ABSTRACT
`
`Power amplifiers (PAs) are inherently nonlinear devices and
`are used in virtually all communications systems. Digital
`baseband predistortion is a highly cost effective way to lin­
`earize the PAs, but most existing architectures assume that
`the PA has a memoryless nonlinearity. For wider bandwidth
`applications such as WCDMA, PA memory effects can no
`longer be ignored, and memoryless predistortion has limited
`effectiveness. In tHis paper, we model the PA as a Wiener
`system and construct a Hammerstein predistorter, obtained
`using an indirect learning architecture. Linearization per­
`formance is demonstrated on a 3-carrier UMTS signal.
`
`1.. INTRODUCTION
`
`Power amplifiers �rAs) are indispensable components in a
`communication system and are inherently nonlinear. It is.
`well known that there is an approximate inverse relationship
`between the PA efficiency and its linearity. Hence, nonlinear
`PAs are desirable from an efficiency point of view. The price
`paid for higher efficiency is that nonlinearity causes spec­
`tral regrowth (broadening) which leads to adjacent channel
`interference. It also causes in-band distortion which de­
`grades the bit error rate (BER) performance. Newer trans­
`mission formats such as CDMA and OFDM are especially
`vulnerable to PA nonlinearities, due to their high peak to
`average power ratio; i.e. large fluctuations in their signal
`envelopes. In order to comply with spectral masks imposed
`by regulatory bodies and to reduce BER, PA linearization
`is necessary.
`Of all linearization techniques, digital baseband predis­
`tortion is among tlie most cost effective. A predistorter is a
`functional block that precedes the PA. It generally creates
`an expending nonlinearity since the PA has a compressing
`characteristic. Ideally, we would like the PA output to be
`a scalar multiple of the input to the predistorter-PA chain.
`For a memoryless PA, (i.e. ; the current output depends only
`on the current input) , memoryless predistortion is sufficient.
`There has been intensive research on memoryless predistor­
`tion during the past decade [3].
`For wider bandwidth applications such as WCDMA, PA
`memory effects can no longer be ignored. Moreover, higher
`power amplifiers such as those used in wireless basestations
`exhibit memory effects. The cause of memory effects can
`be electrical or electro-thermal as suggested in [7]. Memo­
`ryless predistortion for a PA with memory often results in
`
`poor linearization performance. Although Volterra series is
`a general nonlinear model with memory, its predistortion
`is complex and its real-time implementation difficult. In
`[2], Clark et.aI. used a Wiener model; i.e., a linear time­
`invariant (LTI) system followed by a memoryless nonlin­
`earity, to capture the nonlinear memory effects in the PA
`associated with wideband signals. In this paper, we also
`adopt the Wiener PA model, which has the advantage that
`its predistortion can be easily carried out. A Hammerstein
`system is a memoryless nonlinearity followed by a LTI sys­
`tem, and can therefore linearize a Wiener PA.
`In the current literature, predistorters with memory
`mainly fall into the data predistorter category [5, 6), in the
`sense that predistortion is applied before the pulse shap­
`ing filter. T he main drawback of data predistortion is its
`dependence on the signal constellation and the pulse shap­
`ing filter. Both Volterra model based [5) and Hammerstein
`model based [6] data predistorters have been proposed. In
`[5), Volterra data predistorter is constructed using the in­
`direct learning architecture. In [6), the Hammerstein data
`predistorter is obtained using a stochastic gradient method.
`As opposed to data predistortion, we shall pursue sig­
`nal predistortion in this paper; i.e., predistortion occurs
`after the pulse shaping filter. To construct a Hammerstein
`predistorter, one approach is to first identify the Wiener
`PA and then find the Hammerstein predistorter as its in­
`verse. Since Wiener system identification is generally more
`difficult to carry out than Hammerstein system identifica­
`tion, we pursue an alternative approach which generates
`the Hammerstein predistorter without first identifying the
`Wiener PA. Unlike [6], our Hammerstein predistorter will
`be constructed using an indirect learning architecture sim­
`ilar to the one used in [5]. In this setup, finding the predis­
`torter is essentially equivalent to identifying a Hammerstein
`system. the PA can be modeled as a Wiener system,
`
`2.
`
`INDIRECT LEARNING ARCHITECTURE
`
`Fig. 1 shows the indirect learning structure that is used
`for Hammerstein predistorter identification. The PA has a
`Wiener structure (LTI followed by memoryless nonlinear­
`ity). The feedback path labeled "Predistorter Training"
`(block A) has a Hammerstein structure if we view y(n)/ K
`as its input and zen) as its output. The actual predistorter
`is an exact copy of the feedback path (copy of A); it has
`x(n) as its input and zen) as its output. Ideally, we would
`like yen) = K:z;(n), which renders zen) = zen) and the er-
`
`0-7803-7402-9/02/$17.00 ©2002 IEEE
`
`III - 2689
`
`Authorized licensed use limited to: Eric Berg. Downloaded on July 08,2024 at 18:51:21 UTC from IEEE Xplore. Restrictions apply.
`
`PETITIONERS EXHIBIT 1008
`Page 1 of 4
`
`

`

`is a classical Hammerstein system identification problem. If
`no additional assumptions are made on the system's input
`signal yen), iterative Newton and Narendra-Gallman algo­
`rithms are �he two most popular iterative estimation meth­
`ods [4). The two algorithms exhibit similar performance as
`shown in 14). The main drawback of these algorithms is
`
`that they are sensitive to the initial guesses and may con­
`
`verge to a local minimum. A recent method proposed by
`Bai 11] uses an optimal two stage identification algorithm,
`which can lead to a global optimum. The model structure
`introduced in [1) is a Hammerstein system followed by a
`memoryless nonlinearity. However, we can easily modify
`the results of [1) to suit our model. Note that for a given
`set of {y(n),z(n)} values, the bq's and the C2k+l'S are not
`C2A:+1 by the same constant yields the same model). To
`unique (i.e.; multiplying bq with a constant and dividing
`avoid this problem, we assume,that E�o Ibql2 = 1 and the
`real part of bo is positive as suggested in [1).
`Next, we will review the Narendra-Gallman (NG) and the
`optimal two stage identification (tSISVD) algorithms.
`
`3.1. Narendra-Gallman algorithm
`The NG algorithm starts with initial guesses for the ap and
`bq coefficients, denoted by a�O) and b�O), respectively. At
`the ith iteration eq. (3) can be rewrit�en as
`(K-l)/2
`L C2A:+lU21c+l(n)
`"=0
`Q
`E b�i)y(n - q)ly(n - q)121o•
`q=O
`
`p
`
`z(n) - Ea�i)z(n - p) =
`p=1
`
`(4).
`
`At this stage out objective is to solve for C21c+l' Using
`matrix notations we can reformulate eq. (4) as
`Zo - Za(i) = Uc,
`
`where Z = [ZI' . . • ,zp) , Zl = [Or, z(l), . .. , z(N _l»)T, where
`01 is a 1 x l all zero vector, a(i) =
`[a�i), ... ,al)jT, U =
`. The least-squares solution for eq. (5) is
`[Ul, .. ·,UK� ·U21o+1 = [U2k+1(1)'''·,'II.21o+1(N») ,and c =
`[Cl,··· ,CK)
`
`(5)
`
`(6)
`
`where H denotes Hermitian transpose. In the second step,
`based on the ��+':l 's obtained, we rewrite eq. (3) as,
`zo=Za+Vb =[ZVj [:],
`where V = JVI,"" vQ), VI = [Or, v(I),. . . , v(N _l)]T, b =
`[bo, ... ,bQ) ,and v(n) is given in eq. (1). The least-squares
`
`(7)
`
`solution for eq. (7) is,
`
`to the first step and continue until the algorithm converges.
`With the new aci+1) and f,(i+l) estimates, we can go back
`
`(8)
`
`Figure 1. The indirect learning architecture for the
`Hammerstein predistorter.
`ror term e(n) = O. Given yen) and zen), our task is to find
`the parameters of block A, which yields the predistorter.
`The algorithm converges when the error energy Ile(n)W is
`minimized.
`Here we consider that the PA characteristics do not
`change rapidly with time - changes in PA characteristics
`are often due to temperature drift, aging etc which have
`long time constants. After gathering a block of yen) and
`zen) data samples, the training branch (block A) can pro­
`cess the data off-line, which lowers the requirement of the
`processing power of the predistortion system. Once the pre­
`distorter identification algorithm has converged, the new set
`of parameters are plugged into the high speed predistorter,
`which can be readily implemented by Application-Specific
`Int�grated Circuits (ASIC) or Field Programmable Gate
`Arrays (FPGA) . When the predistorter coefficients have
`been found and it is believed that the PA characteristics
`are hardly changing, the setup in Fig. 1 can be run in open
`100Pi i.e., we temporarily shutdown the training branch, un­
`til changes in PA characteristics require a new predistorter.
`
`IDENTIFICATION OF THE
`3.
`HAMMERSTEIN PREDISTORTER
`The predistorter training branch can be described by:
`(K-l)/2
`v(n) = E C21c+l y(n)ly(n)12Ic,
`1c=0
`Q
`P
`z(n) = E apz(n - p) + E bqv(n),
`p=1
`q=O
`which implies that for the predistorter, we model the memo­
`ryless nonlinearity as an odd-order polynomial and the tTl
`system as a general pole/zero system. Combining the two
`equations above, we obtain
`
`(1)
`
`(2)
`
`P
`
`zen) = L apz(n - p) +
`)
`(CK-I)/2
`� bq � C21c+ly(n - q)ly(n _ q)12A:
`Q
`Given y(n) and zen), our objective is to estimate the ap, bq
`and C2lc+l coefficients. Parameter estimation of this model
`
`•
`
`(3)
`
`rn - 2690
`
`Authorized licensed use limited to: Eric Berg. Downloaded on July 08,2024 at 18:51:21 UTC from IEEE Xplore. Restrictions apply.
`
`PETITIONERS EXHIBIT 1008
`Page 2 of 4
`
`

`

`they appear together as the coefficient on the r.h.s. of eq.
`(3), if we define
`
`eq. (3), we obtain
`p
`
`p=1
`
`9=0
`
`.=0
`
`matrix form, we obtain
`
`where G =
`
`(9)
`
`(10)
`
`(11)
`
`Eq. (9) can be alternatively expressed as
`
`D=
`
`diS
`
`dQ3
`
`.
`dQl
`
`=bcT,
`
`(13)
`
`trix D has rank one, a natural way to estimate b and c from
`D is to perform a singular value decomposition (SVD) on D
`and then find the eigenvectors corresponding to the largest
`singular value. Let the SVD of b be given by.
`
`3.2. Optimal two stage identification algorithm
`Since the difficulty in estimating the bq's and C2lc+1'S is that
`we can first estimate dq,2k+ l using least-squares and then
`find bq and C2k+1 from dq,21c+l. Substituting eq. (9) into
`zen) = E Qpz(n - p)
`Q (K-l)/�
`+ E E dq,2lc+l gq,2Ic+l(n),
`where gq,2k+l(n) = yen -q)ly(n - q)121c. Rewriting in a
`Zo = Za+ Gd = [Z G] [ : ] •
`[gOI.···,gOKT .. ·,gQl, .. ·,gQK], gq,21c+1 =
`[gq,2Ic+l(I),···,gq,21c+l(N)] , and d =
`[dOl,···,doK,···,
`dQ1,'" ,dQK]T. The least-squares solution for eq. (11) is
`[ : ] = (lZ G]H[Z GJr1 [Z G]HZo.
`(12)
`[dol
`dos
`dOK 1
`dll
`dlK
`dQK
`where b = [bo •. • •• bQ]T. c = [Cl, . . . • CKJT. Since the ma­
`min{(C2+11.(K+1I/2}
`i=1
`
`D =
`
`aiPw[i .
`
`(14)
`
`E
`where I-'i's and lIi'S are Q + 1 and (K + 1)/2 dimensional
`orthonormal vectors, respectively. Then b and C can be
`estimated as
`
`b = S,.l-'l, C = S,.all1;,
`(15)
`where· denotes conjugate and s,. is the first non-zero ele­
`ment of 1-'1. These estimates can be shown to be the closest
`band c to 0 in the least-squares sense [1].
`In summary, the NG algorithm is a simple and. robust
`algorithm. Although it may have convergence problems. it
`can perform well in many cases as will be shown in the next
`section. The LS/SVD algorithm avoids the potential local
`minimum problem of the NG algorithm. However. using
`SVD to find the bq's and C2Ic+l'S may not result in the best
`bq's and C2.+l·S that minimize the squared error criterion.
`Our exanlples in the next section will show that both work
`well for identifying the Hammerstein predistorter although
`one may outperform the other in a particular scenario.
`
`-Ie!!.
`
`0
`0.5
`-0.5
`Normalized Frequency
`
`(a) Output
`Figure 2. Comparison of the PSDs.
`without predistortlonj (b) Output with memory­
`less predistortionj (c) Output with Hammerstein
`predistortion, NG and LS/SVD algorithms (similar
`performance).
`
`4. SIMULATIONS
`In this section. we illustrate through computer simulations
`the performance of the Harnmerstein predistorter identified
`using the indirect learning architecture. In the first exanl­
`pie. the LTI portion of the Wiener PA model has a pole/zero
`form. whose system function is given by
`
`H( ) =
`z
`
`1 + 0.3z-2
`1- 0.2z-1·
`
`(16)
`
`For the memoryless nonlinear portion of the Wiener PA
`model, we use a 5th order nonlinearity with coefficients,
`
`C6
`
`21.3936 + 0.4305j,
`
`(17)
`
`Cl = 14.9740 + 0.0519j, ca = -23.0954 + 4.9680j,
`
`which were extracted from an actual Class AB PA.
`The baseband input signal is a 3-carrier Universal Mobile
`Telecommunications System (UMTS) signal. Hammerstein
`predistorter identification is carried out based on 8000 data
`samples. The predistorter parameters usually converge af­
`ter a few iterations. Next, we compare the spectra of the
`input and output signals to asses the effectiveness of the
`predistorter in reducing spectral regrowth. In this exam­
`ple. we assume that the LTI portion of the Hammerstein
`predistorter is a pole/zero system with two poles and one
`zero (correct model orders for the inverse of the H (z) of
`eq. (16)). In addition, we make the assumption that the
`nonlinearity of the predistorter is 5th order.
`Performance of predistorter identified with the LS/SVD
`and NG algorithms is demonstrated in Fig. 2. Both algo­
`rithms fully suppress the spectral regrowth exhibited by the
`PA output when no predistortion is applied. In contrast, we
`
`does not fully suppress the spectral regrowth.
`In the second exanlple, the LTI portion of the Wiener PA
`
`merstein predistorter is assumed to be FIR 85 well. Our
`objective here is to see whether the algorithm can correctly
`
`observe in Fig. 2 that 5th order memoryless predistortion
`is H(z) = 1+0.3z-2 (FIR), and the LTI portion of the Ham­
`
`III - 2691
`
`Authorized licensed use limited to: Eric Berg. Downloaded on July 08,2024 at 18:51:21 UTC from IEEE Xplore. Restrictions apply.
`
`PETITIONERS EXHIBIT 1008
`Page 3 of 4
`
`

`

`-.'O'----..---r----.-----,
`-:10 ....
`-30
`
`-101,..---.......... ---...-----.-----,
`
`-:10
`
`. . .
`
`. . .
`
`. .
`
`. . . .•. . . . . . . . .
`
`.. .
`
`-30 .
`-40
`-50 ' . .
`
`-60
`
`�
`Q Ul
`
`II.
`
`-o�,5---�0 ---O:>-:J-----.J
`-10!!1�1 ------:
`Normalized Frequency
`
`-IO!!I!-I ------:-o:!:,5 :----�0---::"0.5,.-------l
`Normalized Frequency
`
`Figure S. Comparison of the PSDs.
`(a) Output
`without predistortion; (b) Output with memory­
`less predistortion; (c) Output with Hammerstein
`predistortion (NG); (d) Output with Hammerstein
`predistortion (LS/SVD).
`identify an FIR filter that approximates the inverse of the
`FIR system in the PA. We assume that the FIR system in
`the predistorter has 15 taps. The 'results are shown in Fig.
`3. The two algorithms exhibit different behaviors in this
`time: the NG algorithm performs worse than the LS/SVD
`algorithm. When examining the concatenated response of
`the two LTI blocks (one from the Wiener PA and the other
`from the Hammerstein predistorter) , we observe that the
`predistorter's LTI system identified by the NG algorithm
`can only compensate for the PA's LTI system within the
`signal bandwidth. However, the LS/SVD algorithm is able
`to find a good FIR system for the predistorter, both within
`and outside of the signal bandwidth.
`In the third example, we perturb� the Wiener PA model
`coefficients so it is a full Volterra model (not W iener any
`more) . Our objective is to see whether the Hammerstein
`predistorter has any robustness. The result is shown in
`Fig. 4. We still observe significant reduction of spectral
`regrowth with the Hammerstein predistorter.
`In all cases, memoryless predistortion is not very effective
`in suppressing spectral regrowth, which underscores the no­
`tion that PA memory effects must be taken into account
`when designing the predistorter.
`
`5. CONCLUSIONS
`We employed the indirect learning structure to identify the
`Hammerstein predistorter for a PA modeled by a Wiener
`model. We compared the performance of two Hammerstein
`system identification algorithms; i.e., the NG and LS/SVD
`algorithms, in this context. For a Wiener model with a
`simple pole/zero LTI structure, both algorithms show sim­
`ilar performance. However, when the LTI portion of the
`Wiener PA as well as that of the Hammerstein predistorter
`are FIR, the LS/SVD algorithm outperforms the NG algo­
`rithm. Simulation results illustrate the effectiveness of the
`proposed predistorter design.
`
`Acknowledgment: This work was supported in part by
`
`Figure 4. Comparison of the PSDs.
`(a) Output
`without predistortion; (b) Output with memory­
`less predistortion; (c) Output with Hammerstein
`predistortion (NG); (d) Output with Hammerstein
`predistortion (LS/SVD); (e) Input signal.
`National Science Foundation grant MIP 9703312 and by
`the State of Georgia's Yamacraw Initiative. The authors
`would also like to thank Dr. Zhengxiang Ma and Dr. Dennis
`R. Morgan of Bell Laboratories, Lucent Technologies for
`inspiring this work.
`
`REFERENCES
`[lJ E. W. Bai, "An optimal two stage identification al­
`gorithm for Hammerstein-Wiener nonlinear systems,"
`Pmc. American Contr. Con/., pp. 2756-2760, Philade­
`phia, Pennsylvania, Jun. 1998.
`[2J C. J. Clark, G. Chrisikos, M. S. Muha, A. A.
`Moulthrop and C. P. Silva., "Time- domain envelope
`measurement technique with application to wideband
`power amplifier modeling," IEEE 7mns. Microwave
`Theory Tech., vol. 46, no. 12, pp. 2531- 2540, Dec.
`1998.
`[3J S. C. Cripps, RF Power Amplifiers for Wireless Com­
`munications, Artech House, Norwood, MA, 1999.
`[4) E. Eskinat, S. H. Johnson, and W. L. Luyben, "Use
`of Hammerstein models in Identification of nonlinear
`systems," AIekE J., vol. 37, no. 2, pp. 255-267, Feb.
`1991.
`[5) C. Eun and E. J. Powers, "A predistorter design for a
`memory-less nonlinearity proceded by a dynamic linear
`system," Proc. GLOBECOM, pp. 152-156, Singapore,
`Nov. 1995.
`[6J H. W. Kang, Y. S. Cho, and D. H. Youn, "On com­
`pensating nonlinear distortions of an OFDM system
`using an efficient adaptive predistorter," IEEE 7'rans.
`Commun., vol. 47, no. 4, Apr. 1999.
`[7J J. H. K. Vuolevi, T. Rahkonen, and J, P. A. Manninen,
`"Measurement technique for characterizing memory ef­
`fects in RF power amplifiers," IEEE 7'rans. Microwave
`Theory Tech., vol. 49, no. 8, pp. 1383-1388, Aug. 2001.
`
`I11-2692
`
`Authorized licensed use limited to: Eric Berg. Downloaded on July 08,2024 at 18:51:21 UTC from IEEE Xplore. Restrictions apply.
`
`PETITIONERS EXHIBIT 1008
`Page 4 of 4
`
`