Copyright IEEE, 13th Stat. Signal Proc. Workshop (SSP), July 2005, Bordeaux (F)

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1 Copyrigt IEEE, 13t Stat. Signal Proc. Worksop SSP), 17-0 July 005, Bordeaux F) ROBUSTNESS ANALYSIS OF A GRADIENT IDENTIFICATION METHOD FOR A NONLINEAR WIENER SYSTEM Ernst Ascbacer, Markus Rupp Institute of Communications and Radio-Frequency Engineering Vienna University of Tecnology Gussausstrasse 5/389, A-1040 Vienna, Austria {ernst.ascbacer, markus.rupp}@tuwien.ac.at web: prototyping ABSTRACT Te gradient identification of te linear filter part of a nonlinear Wiener system wit unknown output non-linearity is investigated wit respect to robustness in a deterministic sense. In order to estimate accurately te linear filter coefficients, an adaptive nonlinear filter is placed at te output of te Wiener system wic compensates te output nonlinearity. Terefore, two adaptive algoritms work simultaneously. Local and global passivity relations are derived from wic information on te robustness of te algoritm can be extracted. 1. INTRODUCTION Nonlinear systems occur frequently in applications, e.g., in wireless and satellite communications te power amplifier is driven near saturation for efficiency reasons. A simple and low-complex model for suc nonlinear power amplifiers wit memory is a Wiener system [1]. A Wiener system is a series connection of a linear filter and a static memoryless) non-linearity in tat order. Suc a model can be used to model te AM/AM conversion [] of power amplifiers, describing te nonlinear) dependence of te amplitude of te output signal from te amplitude of te input signal including memory effects. Te problem is tat most often in real applications neiter te linear part nor te nonlinear part is known and must be identified, e.g., wit a low-complex gradient algoritm. A statistical analysis of te adaptation algoritm is ardly possible, often only information about convergence-in-te Researc reported ere was performed in te context of te network TARGET Top Amplifier Researc Groups in a European Team) and supported by te Information Society Tecnologies Programme of te EU under contract IST NOE, Te work was also supported by te Cristian Doppler Pilot Laboratory for Design Metodology of Signal Processing Algoritms, Gussausstr. 5/389, 1040 Vienna, Austria. mean can be extracted [3], assuming a specific output nonlinearity. A furter problem is tat te Wiener system is nonlinear-in-parameters, tus, witout a good initial estimate of te parameters te gradient algoritm locates possibly only a local minimum, resulting in a very inaccurate parameter estimation. In tis paper a metod for te identification of te linear filter parameters wit a gradient procedure is presented wic avoids te problem of local minima. Te convergence properties of te algoritm are analysed in a deterministic sense [4].. IDENTIFICATION METHOD A grapical representation of te Wiener system is sown in Fig. 1. u[n] v[n] H q 1 x[n] ) y[n] f ) + Fig. 1. A nonlinear Wiener system d[n] Bot, te linear filter and te static output non-linearity are assumed to be unknown. Te signals x[n] and y[n] are not observable, wic complicates te system identification. Te noise v[n] is te measurement noise. Since f ) is a nonlinear function, te input/output relation is nonlinear in te parameters,m of te linear filter H q 1 ) = M1 m=0,mq m, d[n] = f H u[n]) ) + v[n]. 1) Te minimisation of a quadratic cost-function for te estimation of te parameters of te linear filter o = arg min Ed[n] un ) ) )

2 Copyrigt IEEE, 13t Stat. Signal Proc. Worksop SSP), 17-0 July 005, Bordeaux F) wit = [ 0,..., M1 ],u n = [ u[n],...,u[nm +1] ] T and E ) denoting te expectation operator, can lead to poor estimation results, depending on te output non-linearity, see also Section 3 furter aead. Following an idea in [5] te adaptive sceme in Fig. is proposed. Here, te non-linearity g ) tries to compen- u[n] Hq 1 ) x L [n] + x N [n] g ) d[n] ẽ[n] Adaptive algoritm Fig.. Proposed adaptive identification sceme sate te negative effects of te output non-linearity f ) on te estimation of te linear filter parameters. Te objective function wic is minimised by te adaptive algoritm is J[n] = E x L [n] x N [n] ). 3) It is assumed tat f ) is invertible. Te inverse of te nonlinear map f 1 ) is approximated by g ), wic is represented using a series f 1 ) g ) = K w k ψ k ), 4) k=1 {ψ k } K k=1 being a set of basis-functions. Te objective function is terefore J[n] = Eu n wψ n ), 5) wit te signal-vector ψ n = [ ψ 1 d[n] ),...,ψk d[n] )] T and te parameter-vector w = [w 1,...,w K ]. Te trivial solution = w = 0 as to be excluded. Here, te coefficient 0 is fixed to one, 0 = 1. Tis is no restriction since Wiener systems are invariant to scaling [6], meaning tat te linear filter can be assumed w.l.o.g. to be monic. Now, a reduced parameter vector as to be estimated, = [ 1,..., M1 ], giving te error ẽ[n] = u[n] + u n wψ n, 6) wit u n = [ u[n 1],...,u[n M + 1] ] T..1. Gradient updates Derivation of te objective function wit respect to te parameter vectors and w and simplification of te expectation operator leads to te update equations n = n1 µ [n]ẽ a [n]u T n, n 0, 0 given 7) w n = w n1 + µ w [n]ẽ a [n]ψ T n, n 0, w 0 given. 8) Here, ẽ a [n] = u[n] + n1u n w n1 ψ n 9) is te disturbed a-priori error. In te error-vector form te update equations read n = n1 + µ [n]ẽ a [n]u T n 10) w n = w n1 µ w [n]ẽ a [n]ψ T n, 11) wit te parameter error-vectors n = n, w n = w w n, were te denotes te optimal values, n,w n are te estimated parameter-vectors at time n. Te disturbance v[n] is indirectly included in te above definition of te a-priori error, namely via te signal ψ n. Te task is to analyse te stability of te algoritm in 10) and 11) and possibly device bounds for te step-sizes µ [n] and µ w [n] wic guarantee stable operation of te algoritm. Since a stocastic analysis witout a-priori knowledge of te output non-linearity is not feasible, te analysis is carried out deterministically... Local passivity relations Decomposition of te disturbed a-priori error into ẽ a [n] = e a,w [n] e a, [n] + v e [n] 1) wit e a,w [n] = w n1 ψ n, e a, [n] = n1u n, and v e [n] = v g [n] + v v [n], wereby v g [n] = f 1 d[n] ) w ψ n 13) v v [n] = f 1 y[n] ) f 1 d[n] ) 14) = y f 1 y[n] ) v[n] + O v[n] ) are reflecting te errors due to te approximation of f 1 ) using g ), see 13), and te influence of te measurement noise, see 14), wic primarily depends on te first derivative of te function f 1 ). Te error ẽ a [n] can be decomposed into two different expressions, ẽ a [n] = e a,w [n] + v w [n] = e a, [w] v [n] 15) wit te noise terms v w [n] = e a, [n] + v e [n] 16) v [n] = e a,w [n] v e [n]. 17) Assuming no noise v[n] te noise v v [n], see 14), vanises. Te remaining disturbance is v g [n], due to te approximation of te inverse nonlinear function wit a truncated series, cf. 4). At te oter and, if v[n] does not vanis but f 1 ) = g ), te noise v g [n] = 0 but not v v [n], resulting in v e [n] = v v [n]. Furter, additional disturbance, i.e., te

3 Copyrigt IEEE, 13t Stat. Signal Proc. Worksop SSP), 17-0 July 005, Bordeaux F) undisturbed a-priori errors e a, [n] and e a,w [n] appear and couple te two adaptive systems. Te update equations in te error-vector form read now n = n1 µ [n]e a, [n] + v [n])u T n 18) w n = w n1 µ w [n]e a,w [n] + v w [n])ψ T n. 19) Te equations are coupled via te noise terms v [n],v w [n] wic depend on te undisturbed a-priori errors of te respectively oter system, see 17) and 16). Te local passivity relations can now be stated as follows [4]: If ten µ [n] < 1 u n = [n] 0) µ w [n] < 1 ψ n = µ w [n] 1) n + µ [n] e a, [n] ) n1 + µ [n] v [n] ) < 1 ) w n + µ w[n] e a,w [n] ) w n1 + µ w[n] v w [n] ) < 1. 3) Te two relations a coupled via te noise terms te adaptation processes are not independent. Te relations ) and 3) wit te conditions 0) and 1) fulfilled, define two contractive forward maps: as long as µ [n] < [n] and µ w [n] < [n] local stability of te individual parts is guaranteed. For one iteration step te error energy, te l -norm of te parameter error-vector and te squared magnitude of te undisturbed a-priori error, are guaranteed to remain smaller tan te disturbance energy, te l -norm of te parameter error-vector at te previous iteration step wit te squared magnitude of te noise terms v [n] and v w [n]. From 16) and 17) can be seen tat tese noise terms contain te undisturbed a-priori errors of te respectively oter system. Tus, no matter ow large tese errors are, te error energies nominators in te passivity relations) are smaller tan te energy of te disturbance...1. Feed-back structure Straigtforward manipulation leads to te update equations n = n1 [n] e a, [n] + v [n] ) u T n 4) w n = w n1 µ w [n] e a,w [n] + v w [n] ) ψ T n 5) wit time-index n omitted) v = 1 µ ) e a, }{{} feedback v w = 1 µ ) w µ w e a,w } {{ } feedback µ e a,w + v e ) }{{} disturbance 6) µ w e a, v e ). 7) µ } w {{} disturbance Since in 4) and 5) te step-sizes are equal to [n] and µ w [n], te energy relations are now n + µ [n] e a, [n] ) n1 + µ [n] v [n] ) = 1 8) w n + µ w[n] e a,w [n] ) w n1 + µ w[n] v w [n] ) = 1, 9) and define two lossless forward maps, T and T w. Te feedback is defined via 6) and 7). Using te small-gain teorem [7] te conditions for stability of te system 4) and 5) can be devised. Since te forward pat is lossless, te stability depends entirely on te gain of te feedback pats. Terefore, if 1 µ [n] [n] < 1 30) 1 µ w[n] µ w [n] < 1, 31) te system is locally stable. Tis yields te following limits for te step-sizes: 0 < µ [n] < [n] 3) 0 < µ w [n] < µ w [n]. 33) Te upper bounds are now twice as large as te bounds obtained in 0) and 1). In Fig. 3 te feedback structure corresponding to te update equation for te linear-filter parameters is sown. Te a-priori error e a,w [n] constitutes an additional disturbance, wic adds to te disturbance v e [n]. Te condition for applying te small-gain teorem is tat te energy of te complete disturbance-signal time n omitted) v = µ µ v e µ µ µ w µw e a,w 34) is bounded. Tis can be guaranteed if te energy of bot components, e a,w [n] and v e [n] are also bounded. Te energy of te a-priori error e a,w [n] depends on te stability of te second update-equation, wic can be assured if te step-size is in te derived limit, cf. 33). Te energy of te noise v e [n] depends on te approximation of te inverse f 1 ), cf. 13), and on te noise v[n], cf. 14), and is assumed to be bounded.

4 Copyrigt IEEE, 13t Stat. Signal Proc. Worksop SSP), 17-0 July 005, Bordeaux F) n1 v + µ v + µ µ v e µ µ µ µ w µw e a,w q 1 T 1 µ µ e a, Fig. 3. Feedback structure only te branc corresponding to te update-equation for te linear system is sown. Te oter branc, corresponding to te update-equation for te parameters of te nonlinear filter, is similar.3. Global passivity relations Rewriting te local relations 8) and 9) n + [n] e a, [n] ) = n1 + [n] v [n] ) 35) w n + µ w[n] e a,w [n] ) = w n1 + µ w[n] v w [n] ) n 36) and summation of bot sides over te te finite time-orizon n = 1,...,N gives [n] e a, [n] ) 1 + N µ w [n] e a,w [n] ) w 1 + N [n] v [n] ) 37) µ w [n] v w [n] ), 38) wic, after insertion of te noise terms 6) and 7) and simple manipulations gives omitting te time-index n) e a, 1 1 γ 1 39) + δ e a,w + v e ) ) µ w e 1 a,w w 1 γw 1 40) + δw µ w e a, + v e ) ), wereby γ = max,...,n 1 µ [n] [n] 41) γ w = max,...,n 1 µ w[n] µ w [n] 4) µ [n] δ = max,...,n [n] 43) µ w [n] δ w = max,...,n µ w [n]. 44) Terefore, for global stability te following conditions must old: γ < 1 0 < µ [n] < [n], n = 1,...,N 45) γ w < 1 0 < µ w [n] < µ w [n], n = 1,...,N, 46) wic are te same conditions as for local stability, except tat te conditions must old for te wole time-orizon. Te error-energies on te left-and side of 39) and 40) depend on te size of te feedback-gains γ and γ w, as well as on te size of δ and δ w. E.g., coosing µ [n] = [n] minimises te gain 1/1 γ ), since γ = 0, but δ = 1, wic can result in a relatively ig error-energy, since te disturbance-energy is not attenuated. 3. SIMULATION RESULTS For illustration purposes an example is analysed and simulations are carried out. A reference Wiener system wit a linear FIR filter wit 17 taps defining a bandpass filter and a static non-linearity given by fx) = tan1.5x) is feeded wit a wite gaussian sequence wit variance equal to one. For tis example no measurement noise is added. Te filter g ) is a Taylor series wit only uneven terms up to te sevent order, te basis-functions being {ψd)} 7 k=1 = {d,d 3,d 5,d 7 }. Te algoritm is run 50 times and averages are taken to approximate te relative misadjustment ) m [n] = E n. 47) 0 Four different cases are investigated, in eac case te initial values for te filter parameter are w 0 = [1,0,0,0] and 0 = [1,0,...,0]. Te learning curves are sown in Fig. 4: Case 1 Here, te step-sizes are µ [n] = [n],µ w [n] = 0, tus only te linear part is identified. A relative misadjustment of minimally 1.5 db is acieved. Te system is stable. Since no noise at te output is added, te remaining misadjustment is entirely due to te influence of te nonlinear output filter in te Wiener

5 Copyrigt IEEE, 13t Stat. Signal Proc. Worksop SSP), 17-0 July 005, Bordeaux F) system. By a reduction of te amplitude of te input signal, te error can be reduced, since te output non-linearity looses influence, i.e., te Wiener system beaves more linear. Case In tis case, te step-sizes are µ [n] = [n] and µ w [n] = Bot step-sizes are smaller tan te respective limits, as can be assured by observing µ w. A significant reduction of te relative misadjustment of approx. 5 db compared to case 1 can be acieved if te output of te Wiener system is nonlinearly filtered by g ). Case 3 If µ [n] = µ w [n] = µ[n] wit µ[n] = 1/ u n + ψ n ), bot step-sizes are witin teir limits. Te system is guaranteed to be stable. Te remaining error is in tis case relatively large. Case 4 A furter increase of te step-sizes to µ w [n] = 0.55 and µ [n] = 1.5 [n] brings te system to instability in te time-orizon simulated. Te bound for te step-size µ w [n], namely µ w [n], is exceeded approx. in only 0. % of te time averaged over te realisations). From tis it can be seen tat te obtained bounds are relatively tigt. m / [db] Case 4 Case 3 Case 1 15 Case # of iterations Fig. 4. Learning curves for te first simulation example. No noise is added at te output, v[n] = 0. In te second example, see Fig. 5, wite, zero-mean gaussian noise v[n] wit a standard-deviation of σ v = 0.05 is added at te output. Te resulting SNR, defined as ) y[n] SNR [db] = 10log v[n] 48) is approx. 1 db. Case 1 and Case are simulated again. Altoug significant noise is added only a sligtly larger misadjustment compared to te first example of approximately db results, empasising tat te principal cause of te misadjustment is due to te influence of te nonlinear function at te output of te linear filter of te Wiener system. m / [db] Case, wit noise Case 1, wit noise # of iterations Fig. 5. Learning curves for te second simulation example. Noise is added at te output, resulting in an SNR 1 db. 4. CONCLUSIONS A metod for te identification of te linear filter of a Wiener system as been analysed wit respect to robustness in te l -sense. Te influence of te unknown output nonlinearity is reduced by an adaptive static nonlinear filter in order to improve te estimation quality for te linear-filter parameters. Terefore, two adaptive algoritms run simultaneously, wic complicates te stability analysis. Bounds for te step-sizes of bot update-equations for a robust beaviour could be derived and simulation results confirmed te teoretical predictions. 5. REFERENCES [1] E. Ascbacer and M. Rupp, Modelling and identification of a nonlinear power-amplifier wit memory for nonlinear digital adaptive pre-distortion, in Proc. IEEE International Worksop on Signal Processing Advances in Wireless Communications SPAWC 03), pp , June 003. [] A. A. M. Sale, Frequency-independent and frequencydependent nonlinear models of TWT amplifiers, IEEE Trans. Commun., vol. COM-9, no. 11, pp , Nov [3] N. J. Bersad, S. Boucired, and F. Castanie, Stocastic analysis of adaptive gradient identification of Wiener- Hammerstein systems for gaussian inputs, IEEE Trans. Signal Processing, vol. 48, no., pp , Feb [4] A. H. Sayed and M. Rupp, Error-energy bounds for adaptive gradient algoritms, IEEE Trans. Signal Processing, vol. 44, no. 8, pp , Aug [5] A. D. Kalafatis, L. Wang, and W.R. Cluett, Identification of Wiener-type nonlinear systems in a noisy environment, International Journal of Control, vol. 66, no. 6, pp , April [6] S. Boyd and L. O. Cua, Uniquiness of a basic nonlinear structure, IEEE Trans. Circuits Syst., vol. CAS-30, no. 9, pp , Sep [7] H. K. Kalil, Nonlinear Systems, Macmillan Pub., 199.