Journal ArticleAbstract-Convergence analysis of stochastic gradient adaptive filters using the sign algorithm is presented in this paper. The methods of analysis currently available in literature assume that the input signals to the filter are white. This restriction is removed for Gaussian signals in our analysis. Expressions for the second moment of the coefficient vector and the steady-state error power are also derived. Simulation results are presented, and the theoretical and empirical curves show a very good match

Cho, Sung Ho

Mathews, V. John

English

The University of Utah: J. Willard Marriott Digital Library

450 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-35, NO.4, APRIL 1987 Improved Convergence Analysis of Stochastic Gradient Adaptive Filters Using the Sign Algorithm V. JOHN MATHEWS, MEMBER, IEEE, AND SUNG HO CHO Abstract-Convergence analysis of stochastic gradient adaptive fil-ters using the sign algorithm is presented in this paper. The methods of analysis currently available in literature assume that the input sig-nals to the filter are white. This restriction is removed for Gaussian signals in our analysis. Expressions for the second moment of the coef-ficient vector and the steady-state error power are also derived. Sim-ulation results are presented, and the theoretical and empirical curves show a very good match. 1. INTRODUCTION STOCHASTIC gradient adaptive filters using nonlinear correlation multipliers have received a great deal of attention recently [1], [2], [4]-[6], [11], [14], [15]. In this paper we are concerned only with adaptation' using the sign algorithm, where the coefficient vector is updated using H(n + 1) = H(n) + p,X(n) sign (e(n)), (1) where H (n) is the vector of N coefficient values at time n, X (n) is the primary input vector to the filter, p, is a time-invariant convergence parameter, and e(n) is the er-ror in estimating the reference input d (n) using the pri-mary input vector X (n ), i. e. , e(n) = d(n) - HT(n) X(n), (2) where ( . ) T denotes the transpose of ( . ). . Earlier analyses of the system described by (1) and (2) assume that the input signals are zero mean and white [4], [6]. In patiicular, the input correlation matrix is assumed to be diagonal, with R)JX = E{ X(n) XT(n)} = (J~I. (3) In many practical situations, this assumption is grossly violated and the convergence curves using the above model tend to show faster than true convergence, espe-cially when the eigenvalue spread for the input autocor-relation matrix is large (see Fig. 1). In this paper we will undo the whiteness assumption for the input data signals. We will assume that the primary and reference input signals are jointly Gaussian, zero mean signals. As in many convergence analyses of this Manuscript received May 13, 1986; revised October 24, 1986. The authors are with the Department of Electrical Engineering, Univer-sity of Utah, Salt Lake City, UT 84112. IEEE Log Number 8612883. E{h1 (n)} 1.0 ,--r I , (a) -0.1 0 Time(n) 10000 E{h2 (n)) 0.2 (b) ..... - - - ---0.15 --3 0 Time(n) 10000 E{h3 (n)} 0.05 (c) '-3 2 -0.3 0 Time(n) 10000 Fig. 1. Comparison of the theoretical and simulation results for the mean behavior of the filter coefficients: (1) simulation results, (2) mean values predicted by (15), and (3) mean values predicted by white signal as-sumption. type [10], [12], we will also assume that the input pairs { X (n ), d (n ) } are mutually uncorrelated for different val-ues of n. (Note that we are not restricting the nature of the input autocorrelation matrix Rxx.) H(n) is then un-correlated with {X (n ), d (n ) }, since H (n) depends only on inputs at time n - 1 and before. This assumption is not true in general, but it produces results that are very close to the true behavior of the system if p, is chosen to be small [10]. The analyses that do not use this assump-tion can be found in [3] and [7] for least mean square (LMS) adaptive filters. 0096-3518/87/0400-0450$01.00 © 1987 IEEE MATHEWS AND CHO: CONVERGENCE ANALYSIS OF STOCHASTIC GRADIENT ADAPTIVE FILTERS 451 The rest ofthe paper is organized as follows. In Section II, we present the convergence analysis of the sign algo-rithm assuming jointly Gaussian input signals. In this sec-tion,. we also derive expressions for the second moment of the coefficient vector and also the steady-state error power. A simulation example that verifies the expressions derived is presented in Section III. Concluding remarks are made in Section IV. II. CONVERGENCE ANALYSIS We start by evaluating the statistical expectation of both sides of (1). This gives E { H (n + 1)} = E { H (n)} + JL E { X (n) sign (e (n ) ) } . (4) Since d (n) and X (n) are zero mean. and jointly Gaussian, the error sequence, conditioned on the coefficient vector H (n), is also zero mean and Gaussian [say, with variance (J; I H (n )]. We can then use the fact [9] that for an arbitrary Borel function G ( . ), and Gaussian X and e E{XG(e)} = E{Xe} E-I{e 2 } E{eG(e)}, (5) and express E { X (n) sign (e (n ) ) } as E { X ( n) sign (e ( n ) ) } = E { E [ X ( n) sign (e ( n ) ) I H ( n ) l} ~ E r ~ \ /{X(n l e(nlIH(nl }} l~; (JelH n (6) where we have made use of the fact that the mean absolute value of a Gaussian random variable with zero mean and variance (J2 is (.J2hr) (J. In this analysis we will approx-imate the standard deviation of the error sequence, con-ditioned on the coefficient vector by the unconditional standard deviation of the error sequences, i.e., (7) This approximation is valid for small values of JL. Now (6) can be rewritten as - E { X ( n) sign (e (n »)} = ~ _(1 ) E { X (n) e (n)} . ~; (Ie n Expanding for e (n) using (2), E{X(n) e(n)} = E{ X(n) d(n)} (8) -E{X(n)XT(n)H(n)}. (9) Under the assumption that X (n) is uncorrelated with H(n), equation (9) becomes E{X(n) e(n)} = RXd - RxxE{H(n)} , (10) where RXd is the cross-correlation vector of X (n) and d (n ). Substituting (10) and (8) in (4) we get E{H(n+ 1)} JL ~. = E{H(n)} + -(-) - (RXd - RxxE{H(n)}). (11) (Je n 7r Let us define the misalignment vector V (n) as V(n) = H(n) - Hopt , where (12) Hopt = RxiRxd (13) is the optimum coefficient vector. Also, let K(n) = E{VCn) VT(n)} (14) define a second moment of the misalignment vector. At this point it is convenient to rewrite (11) using the mis-alignment vector V(n). Using (12), (11) can be equiva-lently expressed as E{V(n + Il} ~ (1- a,(nl ~Rxx) E{V(nl}. (15) It is easy to show that the misalignment vector will con-verge to the zero vector if the convergence constant JL is chosen so that r,:;- (Je (n ) o < JL < 'V27r -A.-' I (16) where Ai (i = 1, 2, . . . , N) are the eigenvalues of the autocorrelation matrix Rxx. A more restrictive, but suffi-cient and simpler, condition forlhe convergence (in mean) of the system is r,:;- .;r:;;. ( ) o < Jl < 'V27r { } , 17 tr Rxx where ~min = E{d2(n)} - R3cdHopt (18) is the minimum mean-squared estimation error, and tr { . } denotes the trace of { . }. An inspection of (16) will im-mediately show that if the convergence does occur, the root mean-squared estimation error (J e (n ) at time n is such that () JLAmax (Je n > r,:;-' 'V27r (19) We now need an expression for (Je(n) to complete the analysis. For this, (J; (n ) = E{e 2 (n)} = E{ (d(n) - HT(n) X(n))(d(n) - HT(n) X(n»)T} = E{d 2 (n)} - 2E{HT(n)} R;d + E{HT(n) X(n) . XT(n) H(n)} = ~min + E{ VT(n) X(n) XT(n) V(n)} = ~min + tr {RxxK(n)}: (20) 452 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-35, NO, 4, APRIL 1987 where we have once again made use of the fact that H (n ) and {X(n), d(n)} are uncorrelated. To evaluate (Te(n), we also need to know K(n). An expression for K(n) is derived next. A. Second Moment Behavior o/the Weights From (1) E{(HoPI + V(n + l»)(Hopt + V(n + l»)T} = E{(HoPI + V(n»)(Hopt + V(n»)T} + p.2RXX + p.E{(HoP! + V(n»)XT(n) sign (e(n»)} + p.E{X(n)(Hopt + V(n)/ sign (e(n»)}. (21) Equation (21) can be simplified as K(n + 1) = K(n) + p.2RXX + p.E{ yen) XT(n) sign (e(n»)} + p.E{X(n) VT(n) sign (e(n»)}. (22) The expectations in the third and fourth terms of the right-hand side of (22) can be evaluated as follows: E { V( n) XT (n) sign (e (n)) } E{E[V(n) XT(n) sign (e(n»)! V(n)}} E {V(.) ~ \) EIXT(n) e(')1 V(.) ljl l ~; (TelH n ( ~ 1 [T T :~ = E iV(n) - () RXd -(HOPI + Yen») Rxx]. 'lr (TelH n l ) ~ -E [ V(n) VT(n)R", ~ \ )} (23) l ~; (TelH n ~ ~ _(1 ) K(n)Rxx. (24) ~; (Je n Similarly, E{X(n) VT(n) sign [e(n)l} ~ 1 - -(-) RxxK(n). 'If (Te n (25) Substituting (24) and (25) in (22) will yield K(. + 1) = K(n+ - P~ u,;.) R",] + R",[ p'l- P~ .,;.) K(.) 1 (26) Note that in going from (23) to (24), we have once again made use of the approximation in (7). Closed-form expressions for the limiting (steady-state) values of the second moment matrix and error power are derived next. Let (Te( 00) and K( 00) denote the limiting values of (T e (n) and K (n ), respectively. We will now show that K( 00) exists for sufficiently small values of p,. Using (20), since K(n) converges, u;(n) also does. Let Q be an orthonormal matrix that diagonalizes Rxx. Pre- and postmultiplying both sides of (26) by QT and Q, respectively, we get K'(n + I) where K'(n) = QTK(n)Q, (28) A QTRxxQ:::::: diag [AI' A2' .. ,. , AN]' (29) We are then able to decompose the matrix equation (27) into the scalar form as Kij(n + 1) = (1 - "~.,;n) [A, +Ail) where . Kij(n) + p. 2}../)(i - j), (30) [1, if i = j 5(i - j) = 0, otherwise, (31) and Kij is the (i, j )th element of K'. Note that K' (n) is a , symmetric matrix. Mean square convergence of the. weight vector will oc-cur if p. belongs to the range r;:;- uA n) o < p. < v21r A + A' I 1 (32) As before, a sufficient, but more stringent, condition for mean square convergence is (33) Once again, it is easy to see that if mean-squared conver-gence does occur, (34) for all n. To find the steady-state error, we can solve for (Te( 00) and K ( 00 ) from (20) and (26) after taking the limits as n MATHEWS AND CHO: CONVERGENCE ANALYSIS OF STOCHASTIC GRADIENT ADAPTIVE FILTERS 453 goes to 00. In the limit, (26) becomes K(oo) ~ K(oo) [I - ~$; U,/OO) RXX] + RXX[ ~ 'I - ~ $; U,r'OO) K( 00 ) l which gives K( 00) ~ ~~ U,~oo) I, Using (36) in (20) after taking the limits, we get a;( 00) = ~min + tr {RxxK( oo)} = ~min + aae ( 00), where a =,~ G (f Ai)' 2~2 i=1 (35) (36) (37) (38) Solving for ae ( 00) in (37) and retaining the positive root, we get a + ..Ja2 + 4~min ae ( 00) = 2 . (39) If J1- is very small, the steady-state error power can be approximated as u~(OO) ~ <m'" + ~~ (,~J~.) .J<m'"' (40) Equation (40) is obtained by squaring (39) and ignoring all terms containing J1- 2. Remarks: 1) For very small values of J1-, we can approximate E{HT(n)X(n)Xrcn)H(n)} in (20) as . E{HT(n) X(n) XT(n) H(n)} =:; E{ HT(n)} RxxE{H(n)}. (41) In (41), the basic assumption is that the variations in the weight vector are much smaller than the mean value of the weight vector itself. Under this assumption, the expression for the error power (20) can be approximated by' , a;(n) =:; ~min + E{ VT(n)} RxxE{ V(n)}. (42) Ifwe substitute (42) in (15) and also assume that the input signals to' the adaptive filter are white, the expression de-rived for the convergence of the weight vector (15) is the same as that given by [6]. . 2) For the LMS algorithm [12], E {V(n + I)} and a;( 00) are given by [13] E { V( n + 1)} = (I - J1-Rxx) E { V( n ) } , (43 ) and u;( 00) ~ <m'" + ~ <m'" (,~J~) (44) To have the same steady-state mean-squared error for both the LMS and sign algorithms, we must choose the con-vergence constant J1-s for the sign algorithm to be ~s ~ ~,$;'hm'"' (45) where J1-L is the convergence constant for the LMS algo-rithm. For J1- = J1-s = J1-L .J2!1r .J~min.., (15) becomes { } ( 2~ ) { } E V(n + 1) = 1-;: aAn) J1-LRXX E V(n) . . (46) Since..J~min:::;; ae(n),wehavethat(2!1r)(..J~min!(Je(n» < 1, implying that the sign algorithm will always con-verge slower than the LMS algorithm when the steady-state errors are the same. This result agrees with those in [4] and [6]. However, for the same value of J1-, it IS pos-sible that the sign algorithm can converge faster than the LMS algorithm [14], [15]. Also, if J1- is very small, the difference in steady-state errors between the two methods may not be very large. Ill. A SIMULATION EXAMPLE For verifying the expressions derived in Section II, we chose a third-order predictor for a third-order autoregres-sive signal described by x(n) = 0.9x(n - 1) - O.lx(n - 2) - 0.2x(n - 3) + Hn), (47) where ~ (n) is a white, zero mean Gaussian signal with variance such that the variance of x (n) is 1. Note that the eigenvalue ratio of the signal is 16.32. The results pre-sented are comparisons of the theoretical curves with en-semble averages of 900 independent simulations using 10000 data samples each. The convergence constant was chosen to be 0.005. Fig. 1 shows plots ofthe mean values of the weights. For comparison, we have also plotted the theoretical curves obtained using the white input signal assumption [6]. We can see that the results of Section II agree with those of the simulations fairly closely, while the white noise assumptiqn gives misleading results in this case. The diagonal elements of the K (n) matrix and their steady-state values obta~ned using our theoretical model [(36) and (39)] and also simulation experiments are com-pared in Fig. 2 and Table I, respectively. Here, the steady-state values for the simulation are obtained as the mean values of the last 5000 samples of the corresponding curves. Once again, the theoretical and empirical results show very close match. (Note that in Fig. 2 both curves are always positive. We used negative values for the ver-tical axis only for ease of displaying the results.) 454 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL ASSP-35, NO.4, APRIL 1987 (K(n)} 1 1 0.9 (a) 2 -0.1 0 Time(n) 10000 {K(n)) 2,2 0.07 (b) ~ -0.01 0 Time(n) 10000 {K(n)} 3,3 0.05 (c) 1 \/?---0.005 0 Time(n) 10000 Fig. 2. Comparison of the theoretical and simulation results for the mean-squared beha.vi9r of the filter coefficients: (1) simulation curve and (2) theoretical curve. TABLE I COMPAIUSON OF THE T!iEORETlCA~ AND SIMU~ATION RESU~TS FOR THE STEADy-STATE MEAN-SQUARED VALUES OF THE MISALIGNMENT VECTOR {K(OO)}I,I {K(oo) h2 {K(oo)h,3 Theoretical Result 0.002006 0.002006 0.002006 IV. CONCLUSIONS Simulation Result 0.002110 0.002023 0.001998 In this paper, we presented convergence analysis for the sign algorithm when the signals involved are Gaussian. Expressions for the expected value of the weight vector, its second moments, and also the steady-state error power were derived. These derivations were done without mak-ing any assumptions on the nature of the autocorrelation matrix ofthe input vector and, because of this, the expres-sions developed show a much better match with simula-tion results than previously published results [4], [6] which assumed that the input signals were white. REFERENCES [1] N. 1. Bershad, "Comments on 'Comparison of convergence of two algorithms for adaptive FIR digital filters,'" IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 1604-1606, Dec. 1985. [2] -, "On the optimum data nonlinearity in LMS adaptation," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 69-76, Feb. 1986. [3] R. R. Bitmead and B. D. 0, Anderson, "Performance of adaptive estimation algorithms in dependent random environment," IEEE Trans. Automat. Contr., vol. AC-25, pp. 788-794, Aug. 1980. [4] T. A. C. M. Claasen and W. F. G. Mecklenbrauker, "Comparison of the convergence of two algorithms for adaptive FIR digital filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. 670-678, June 1981. [5] -, "Authors' reply to 'Comments on "Comparison of the conver-gence of two algorithms for adaptive FIR digital filters," , ", IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 202-203, Feb. 1986. [6] D. L. Duttweiler, "Adaptive filter performance with nonlinearities in the correlation multiplier," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-30, pp. 578-586, Aug. 1982. [7] D. C. Farden, "Stochastic approximation with correlated data," IEEE Trans. Inform. Theory, vol. IT-27, pp. 105-113, Jan. 1981. [8] Y. Iiguni, H. Sakai, and H. Tokumaru, "Convergence properties of simplified gradient adaptive lattice algorithms," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 1427-1434, Dec. 1985. [9] T. Koh and E. 1. Powers, "Efficient methods to estimate correlation functions of Gaussian processes and their performance analysis," submitted for publication, 1985. [10] J. E. Mazo, "On the independence theory of equalizer convergence," Bell Syst. Tech, 1., vol. 58, no. 5, pp. 963-993, May-June, 1979. [11] N. A. M. Verhoeckx and T, A. C. M. Claasen, "Some considerations on the design of adaptive digital filters equipped with the sign algo-rithm," IEEE Trans. Commun., vol. COM-32, pp. 258-266, Mar. 1984, [12] B. Wid row et al., "Adaptive noise cancelling: Principles and appli-cations," Proc. IEEE, vol. 63, pp. 1692-1716, Dec. 1975. [13] -, "Stationary and nonstationary learning characteristics of the LMS adaptive filter," Proc. IEEE, vol. 64, pp. 1151-1162, Aug. 1976. [14] D. H. Youn, V. 1. Mathews, and S. H. Cho, "An efficient lattice predictor algorithm for instantaneous frequency estimation," Signal Processing, vol. 10, pp. 75-81, 1986. [15] -, "An efficient algorithm for lattice filter/predictor," in Proc. Int. Con! Acoust., Speech. Signal Processing. Tampa, FL, Mar. 1985, pp. 1181-1184. V. John Mathews (S'82-M'85), for a photograph and biography, see p. 349 of the March 1987 issue of this TRANSACTIONS. Sung Ho Cho was born in Seoul, Korea, on Feb-ruary 21, 1960. He received the B.E. degree in electronic engineering from Hanyang University, Seoul, Korea, in 1982, and the M,S. degree in electrical and computer engineering from the Uni-versity of Iowa, Iowa City, in 1984. He is currently a Teaching Assistant at the Uni-versity of Utah, Salt Lake City, working toward the Ph.D. degree. His research interests include adaptive filtering, spectrum estimation, and sto-chastic processes. 

Improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm

https://collections.lib.utah.edu/dl_files/52/7c/527cc8888fe8f373c0fd7e92cd01c4c813e397c7.pdf

Improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm

Abstract

Similar works

Full text

Available Versions

The University of Utah: J. Willard Marriott Digital Library