Performance Comparison of Two Implementations of the Leaky. LMS Adaptive Filter. Scott C. Douglas. University of Utah. Salt Lake City, Utah 84112
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1 Performance Comparison of Two Implementations of the Leaky LMS Adaptive Filter Scott C. Douglas Department of Electrical Engineering University of Utah Salt Lake City, Utah 8411 Abstract{ The leaky LMS adaptive lter can be implemented either directly or by adding random white noise to the input signal of the LMS adaptive lter. In this paper, we analyze and compare the mean-square performances of these two adaptive lter implementations for system identication tasks with zero mean i.i.d. input signals. Our results indicate that the performance of the direct implementation is superior to that of the random noise implementation in all respects. However, for small leakage factors, these performance dierences are negligible. Simulations verify the results of the analysis and the conclusions drawn. submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING EDICS Category No. SP.6.4 Please address correspondence to: Scott C. Douglas, Department of Electrical Engineering, University of Utah, Salt Lake City, UT (801) FAX: (801) Electronic mail address: World Wide Web: 0 Permission of the IEEE to publish this abstract separately is granted. 0 This research was supported in part by NSF Grant No. MIP
2 1 Introduction The leaky least-mean-square (LMS) adaptive lter is a useful variant of the LMS adaptive lter for several communications and signal processing tasks [1,, 3, 4, 5]. The leaky LMS coecient update is given by W k+1 = (1? )W k + e k X k (1) e k = d k? W T k X k ; () where W k = [w 0;k w 1;k w L?1;k] T is the L-dimensional coecient vector, X k = [x k x k?l+1] T is the input signal vector, d k is the desired response signal, e k is the error signal, is the step size, and is the leakage parameter. For = 0, equations (1){() describe the coecient updates for the standard LMS adaptive lter. An approximate analysis of the updates in (1){() for > 0 shows that the coecients slowly decay towards zero values if the desired response signal is uncorrelated with the input signal vector []. Recently, a more accurate analysis of the leaky LMS algorithm for jointly Gaussian input and desired response signals has enabled useful comparisons of the behaviors of the leaky LMS and LMS adaptive lters to be made [6]. It is well-known that a stochastic variant of the leaky LMS adaptive lter can be implemented by adding zero-mean random white noise to the input signal prior to the application of the LMS adaptive lter [7]. The resulting coecient updates are W k+1 = W k + e k X k (3) e k = d k? W T k X k ; (4) where the noisy input signal vector X k is dened as X k = X k + M k ; (5) and M k = [m k m k?l+1] T is a noise vector with zero-mean uncorrelated elements. If the noise power m = E[m ] is chosen as k m = ; (6) then it can be shown that the mean behaviors of the algorithms in (1){() and (3){(4) are approximately the same. If the noise signal m k can be easily generated (using a maximum-length shift register in VLSI hardware, for example), then this stochastic implementation of the leaky LMS adaptive lter avoids the L multiplies used to compute the scaled coecient vector (1? )W k in (1). However, to obtain an error signal that is free of the noise introduced within the adaptation process, the error e(n) in () must be computed, and thus the two algorithms in (1){() and 1
3 (){(4) are of similar complexity. Note that the algorithm in (){(4) is also obtained in situations where the input signal is dithered prior to sampling, and thus this alternative implementation is of practical interest in its own right. Even though the mean behaviors of the two algorithms are similar, it is not clear how the mean-square performances of the two algorithms dier in any particular situation. The mean-square behavior of an adaptive lter is a more accurate measure of its overall performance and stability characteristics than its mean behavior; thus, it is not clear which algorithm is to be preferred in any particular situation. In this paper, we compare the mean-square performances of the two algorithms in (1){() and (){(4), respectively, assuming a system identication desired response signal model of the form d k = WoptX T k + n k ; (7) where W opt is the optimal coecient vector, n k is an uncorrelated noise signal, and x k and m k are assumed to be independent, identically-distributed (i.i.d.) signals with even-symmetric probability density functions p X (x) and p M (m), respectively. In addition, we shall also assume that vectors within the sequences fx k g and fm k g are independent of each other and of the noise sequence n k. Such assumptions are similar to the independence assumptions often used in analyses of this sort [7, 8]. While never true for an FIR lter conguration, these assumptions lead to reasonably accurate descriptions of the adaptation behaviors of the algorithms, and they allow meaningful comparisons of dierent algorithms to be made. Through our analyses, we show the following: For any particular value of, the range of stable step sizes for the LMS adaptive lter with noisy inputs is smaller than that for the leaky LMS adaptive lter. For any stable values of and, the LMS adaptive lter with noisy inputs converges no faster than the leaky LMS adaptive lter. For any stable values of and, the LMS adaptive lter with noisy inputs has a higher steady-state excess mean-square error (MSE) than that for the leaky LMS adaptive lter. Thus, from a performance standpoint, the leaky LMS adaptive lter is to be preferred in this situation. Simulations verify the analytical results and the above conclusions. Analysis.1 Leaky LMS Adaptive Filter For our analyses, we dene the coecient error vector as V k = W k? W opt : (8)
4 With this denition, we can write the leaky LMS adaptive lter update in (1){() as V k+1 = ((1? )I? X k X T k )V k + n k X k? W opt : (9) Taking expectations of both sides of (9) and employing our assumptions, we nd that where = x E[x ] for our input signal model. k E[V k+1 ] = (1? ( x + ))E[V k]? W opt ; (10) To determine a description for the mean-square behavior of (1){(), we post-multiply both sides of (9) by their respective transposes and take expectations of both sides of the resulting equation. This operation results in E[V k+1 V T k+1 ] = E[((1? )I? X kx T )V k kvk T ((1? )I? X kx T? k )] + n x I + W opt Wopt T?? E[((1? )I? X k X T )V k k]wopt T + W opte[vk T ((1? )I? X kx T k )] ; (11) where n = E[n ]. Employing the assumptions described above and relying on results already k available for the LMS adaptive lter with i.i.d. inputs [8], we can take the trace of both sides of (11). After simplifying the result, we nd the update given by tre[v k+1 V T ] = k+1 1? ( + ) + x ((L? 1 + x ) 4 + x + x ) tre[v k V T ] k + L x n + jjw opt jj? (1? ( x + ))W T opt E[V k]; (1) where x = E[x 4 k ]=4 x. Note that this update depends on the mean coecient error vector E[V k ]. We can dene the (L + 1)-dimensional vectors Y k and B and matrix A as " tre[vk V Y k = T ] # " # k ; B = ; (13) E[V k ] (L x n + jjw opt jj )?W opt " 1? ( x + ) + A = ((L? 1 + x ) 4 x + + x )?(1? ( x + ))W T opt 0 (1? ( x + ))I # ; (14) respectively. With these denitions, we can represent the updates in (10) and (1) as Y k+1 = AY k + B: (15) Note that the excess MSE at time k is given by M SE;k = E[(V T k X k ) ] = xtre[v k V T k ]; (16) where tre[v k V T ] is the rst entry of Y k k. Thus, the form of (15) allows us to determine both the transient and steady-state mean-square behaviors of the leaky LMS adaptive lter for a known initial state vector Y 0. 3
5 For stability, we can determine the values of and that guarantee that all of the eigenvalues of A are less than one in magnitude. Because of the triangular form of A in (14), this transition matrix has L eigenvalues equal to 1? ( x + ) and one eigenvalue equal to the rst entry in the matrix. It can be easily shown that a necessary and sucient condition to guarantee all of the eigenvalues of A to be less than one is j1? ( x + ) + ((L? 1 + x ) 4 x + + x )j < 1; (17) from which we determine the mean-square stability conditions on to be 0 < < ( + ) x (L? 1 + x ) 4 + : (18) x + x For step sizes that satisfy (18), we can solve for the steady-state excess MSE by rst calculating lim Y k = (I? A)?1 B; (19) k!1 from which the excess MSE is simply the rst entry of the steady-state value of Y k scaled by x. Using the form of A and B in (13){(14), we nd after some algebra that lim Y k = k!1 where we have dened 6 4 +? L x n + (L? + x ) 4 x ( x + )? ((L? 1 + x ) 4 x + x + )? x + W opt ; (0) = ( x + ) jjw optjj : (1) From this result, we nd that the steady-state excess MSE for the leaky LMS adaptive lter is? M SE;ss = 4 x L n + (L? + x ) x x + ( x + )? ((L? 1 + x ) 4 x + + x ) : () The rst term on the right-hand-side of () is the excess MSE due to the bias in the lter coecients in steady-state. The second term is the excess MSE due to coecient uctuations caused by a nonzero adaptation speed.. LMS Adaptive Filter with Noisy Input Signals We now analyze the behavior of the LMS adaptive lter with noisy input signals. Similar to the leaky LMS adaptive lter, we write the coecient updates in (3){(4) in terms of the coecient error vector as V k+1 = (I? X k X T k )V k + n k X k? X k M T k W opt ; (3) 4
6 where X k is as dened in (5). Taking expectations of both sides and using our assumptions, we determine that the mean coecient error vector evolves according to Thus, if m m = in what follows. E[V k+1 ] = (1? ( x + m ))E[V k]? mw opt : (4) =, the evolution equation in (4) is the same as that in (10). Therefore, we set To determine a mean-square description of the adaptation behavior of this algorithm, we postmultiply both sides of (3) by their respective transposes and take expectations of boths sides. The resulting relation is E[V k+1 Vk+1] T = E[(I? X k X T k )V k V T k (I? X k X T k )] + ( x + ) I n E[(I? X k X Tk )V k W ToptM k X Tk ] + E[X k M Tk W opt V Tk (I? X k X Tk )]? + E[X k M T W k opt WoptM T k X T ]: (5) k Similar to the rst analysis, we take the trace of both sides of (5) and employ our analysis assumptions to simplify the resulting equation, as given by tre[v k+1 V? T k+1 ] = 1? (x + ) + ((L? 1 + x )x 4 + (L + ) x + (L? 1 + m) ) tre[v k Vk T? ]? 1? ((L + )x + (L? 1 + m )) Wopt T E[V k] where we have dened m = E[m 4 k ]=. + L( x + ) n + (L x + (L? 1 + m))jjw opt jj ; (6) As in the previous case, we can describe the mean-square evolution equation for the LMS adaptive lter with noisy input signals as Y k+1 = AY k + B; (7) where A and B are given by " # (L + 1) A = A + x + (L? + m ) ((L + 1) x + (L? + m ))W T opt (8) 0 0 " B = B + n L + # (L x + (L? + m ))jjw opt jj ; (9) 0 respectively. As in the case of the rst analysis, the stability of the LMS adaptive lter with noisy input signals is guaranteed if the rst element of A is less than one in magnitude. This criterion results in step size bounds of 0 < < ( x + ) (L? 1 + x ) 4 x + (L + ) + (L? 1 + x m): (30) 5
7 In addition, we can solve for the stationary point of (7) for stable step sizes in a similar manner as before; the resulting excess MSE for the LMS adaptive lter with noisy inputs is M SE;ss = x + x(l( x + ) n + (L 4 x?1 + (L? 6 + x + m ) x + L) x ) ( x + )? ((L? 1 + x) 4 x + (L + ) x + (L? 1 + m) ) : (31) Using these results, we can compare the mean-square behaviors of the two adaptive lters. 3 Performance Comparison Examining the upper step size bounds in (18) and (30), we note that the denominator of the upper bound in (30) is always greater than that of (18). Thus, the range of stable step sizes for the LMS adaptive lter with noisy inputs is in general smaller than that for the leaky LMS adaptive lter, and the dierence between these two ranges increases if either L,, or m increases. As for the convergence rates of the two systems, we note that the transition matrices A and A share L eigenvalues, and thus we can compare the eigenvalues represented by the rst entries of both matrices directly. From (8), we see that the rst entry of A is always larger than the corresponding entry of A for positive leakage factors. Since both of these entries are always positive, the convergence speed of the LMS adaptive lter is always slower than the convergence speed of the leaky LMS adaptive lter with the same step size and leakage factor. The dierence in convergence speeds becomes negligible as the step size and leakage factors are decreased, however, and thus we can conclude that the two adaptive systems converge at nearly the same rate if both and are suitably small-valued. We now compare the steady-state excess MSE of the two adaptive systems. We can express M SE;ss in () as M SE;ss = x + c 1 c ; (3) where c 1 and c are the numerator and denominator, respectively, of the second term on the righthand-side of (). Comparing this equation with (31), we see that M SE;ss? x = c? 1 + x L n + (L 4 x?1 + (L? 4 + m ) x + L) x : (33) c? ((L + 1) x + (L? + m )) For any value of L and any distribution of m k, the numerator and denominator of the right-handside of (33) will be larger than c 1 and smaller than c, respectively. Thus, the steady-state excess MSE of the LMS adaptive lter with noisy inputs is always greater than that of the leaky LMS adaptive lter with the same values of and. Combining the above results, we see that the noisy LMS adaptive lter always performs more poorly than the leaky LMS adaptive lter for i.i.d. input signals. The dierence in performance will in general increase if either, m, and/or L are increased. 6
8 4 Simulation Results We now verify the above conclusions via simulation. An L = 50 coecient system identication task was chosen for these simulations, where W opt = [1 1 1] T. The input and observation noise signals were both chosen to be zero-mean white Gaussian with x = 1 and n = 0:00001, respectively. For the noisy LMS adaptive lter, the noise signal m k was chosen to be zero-mean white Gaussian-distributed. One hundred simulation runs were averaged in each case. Figure 1 shows the convergence of the total coecient error power tre[v k V T ] for the leaky k LMS, noisy LMS, and LMS adaptive lters in this situation, where = 0:01 and = 10?5. As can be seen, the leaky LMS adaptive lter's convergence rate is nearly the same as that for the LMS adaptive lter with noisy inputs; however, the steady-state coecient error power is about 50 times larger for the noisy LMS adaptive lter. Also plotted are the convergence behaviors of the systems as obtained from the analyses, showing that the theory accurately predicts convergence behavior. Figure shows the total coecient error powers in steady state for the three algorithms as a function of for = 10?5. As can be seen, the LMS adaptive lter with noisy inputs has a larger error in steady-state as compared to that of the leaky LMS and LMS adaptive lters. Simulated behavior closely matches the theoretical predictions of performance. Since both of the leaky LMS adaptive lters converge at nearly the same rate for small step sizes, a useful quantity for comparison purposes is the fraction of the additional steady-state excess MSE due to coecient uctuations for the noisy LMS adaptive lter with respect to that of the leaky LMS adaptive lter, given by = M SE;ss? M SE;ss : (34) M SE;ss? x Figure 3 plots as a function of for the case described above, with = 0:01. Clearly, the behavior of follows two dierent trends for small and large leakage factors, respectively. It can be shown for large signal-to-noise ratios xjjw opt jj = n 1 and for x that xjjw opt jj x n + jjw opt jj : (35) Moreover, the maximum value of occurs at = x n =jjw opt jj and is approximately max 1 p SNR d ; (36) where SNR d = xjjw opt jj = n is the signal-to-noise ratio of d k. We can see that the analysis is accurate in predicting simulated behavior, as indicated by the simulation results. It should be noted from (35) that the value of tends to zero as the leakage factor is reduced. If one considers = 0:1 to be an acceptable fraction of excess MSE that can be tolerated in the 7
9 adaptation process, then for values of satisfying x > 10 SNR d ; (37) both the leaky LMS and noisy LMS adaptive lters have similar behaviors. In other words, if the input-signal-power-to-leakage ratio x = is large relative to the signal-to-noise ratio of d k, either implementation gives satisfactory performance. 5 Conclusions In this paper, we have provided an analysis of two competing implementations of the leaky LMS adaptive lter. We have shown through both theory and simulation that by adding noise to the input signal of the the LMS adaptive lter, one obtains a system whose mean behavior is similar to that of the leaky LMS adaptive lter. However, for every mean-square performance criterion studied, the mean-square behavior of this adaptive lter is worse than that of the leaky LMS adaptive lter, and this performance dierence is particularly large for large signal-to-noise ratios and moderate values of the leakage factor. We have also identied the range of leakage factors for which both implementations perform satisfactorily. Simulations verify the analyses and indicate the magnitude of the performance dierences. References [1] D.L. Cohn and J.L. Melson, \The residual encoder: An improved ADPCM system for speech digitization," IEEE Trans. Comm., vol. COMM-3, no. 9, pp , September [] R.D. Gitlin, H.C. Meadors, and S.B. Weinstein, \The tap-leakage algorithm: An algorithm for the stable operation of a digitally-implemented fractionally-spaced adaptive equalizer," Bell Sys. Tech. J., vol. 61, no. 10, pp , October 198. [3] J.R Treichler, C.R. Johnson, Jr., and M. G. Larimore, Theory and Design of Adaptive Filters (New York: Wiley-Interscience, 1987). [4] J.R. Gonzal, R.R. Bitmead, and C.R. Johnson, Jr., \The dynamics of bursting in simple adaptive feedback systems with leakage," IEEE Trans. Circuits Systems, vol. CAS-38, no. 5, pp , May [5] J. Cio and Y.-S. Byun, \Adaptive lters," in Handbook of Digital Signal Processing, S.K. Mitra and J.F. Kaiser, eds. (New York: Wiley, 1993), pp [6] K.A. Mayyas and T. Aboulnasr, \The leaky LMS algorithm: MSE analysis for Gaussian data," accepted for publication in IEEE Trans. Signal Processing; to appear. [7] S.S. Haykin, Adaptive Filter Theory, 3rd. Ed., (Englewood Clis, NJ: Prentice-Hall, 1995). [8] W.A. Gardner, \Learning characteristics of stochastic-gradient-descent algorithms: A general study, analysis, and critique," Signal Processing, vol. 6, no., pp , April
10 List of Figures Figure 1: Convergence of total coecient error power, theory and simulation: leaky LMS, noisy LMS, and LMS adaptive lters, white Gaussian input signals, = 0:01, = 0: Figure : Steady-state total coecient error power as a function of, theory and simulation: leaky LMS, noisy LMS, and LMS adaptive lters, white Gaussian input signals, = 0: Figure 3: Penalty factor as a function of the leakage factor, leaky and noisy LMS adaptive lters, = 0:01. 9
11 10 Total Coefficient Error Power Theory Leaky LMS (Sim.) Noisy LMS (Sim.) LMS (Sim.) number of iterations Figure 1: Convergence of total coecient error power, theory and simulation: leaky LMS, noisy LMS, and LMS adaptive lters, white Gaussian input signals, = 0:01, = 0:
12 10 - Steady-State Coefficient Error Power Theory Leaky LMS (Sim.) Noisy LMS (Sim.) LMS (Sim.) mu Figure : Steady-state total coecient error power as a function of, theory and simulation: leaky LMS, noisy LMS, and LMS adaptive lters, white Gaussian input signals, = 0:
13 Theory Simulation Eq. (35) alpha gamma Figure 3: Penalty factor as a function of the leakage factor, leaky and noisy LMS adaptive lters, = 0:01. 1
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