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Analysis of the Filtered-X LMS Algorithm - Speech and Audio Processing, IEEE Transactions on

Analysis of the Filtered-X LMS Algorithm - Speech and Audio Processing, IEEE Transactions on
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  504 zyxwvutsrqpo EEE zyxwvuts RANSACTIONS ON SPEECH zyxwv ND AUDIO PROCESSING, VOL. 3, NO 6, NOVEMBER 1995 alysis of the Filtere -X LMS Algorit Elias Bjamason, zyxwvu ssociate Member, zyxwv EEE Abstract-The presence of a transfer function zyxwvu n the auxiliary- path following the adaptive filter and/or in the error-path, as in the case of active noise control, has been shown to generally degrade the performance of the LMS algorithm. Thus, he con- vergence rate is lowered, the residual power is increased, and the algorithm can even become unstable. To ensure convergene of the allgorithm, the input to the error correlator has to be filtered by a copy of the auxiliary-error-path transfer fundon. This paper presents an analysis of the Filtered-X LMS zyxwvut lgorithm using stochastic methods. The influence of off-line and on-line estimation of the error-path filter on the algorithm is also nves- tigated. Some derived bounds and predicted dynamic behavior are found to correspond very well to simulation results. I. INTRODUCTION HE active control of sound and vibration involves the zyxwv n- troduction of a number of controlled “secondary” sources driven such that the field generated by these sources interferes destructively with the field caused by the srcinal “primary” source. The extent to which such destructive interference is possible depends on the geometric arrangement zyxwvut f the primay and secondary sources and their environment, and on the spectrum of the field produced by the primary source [SI. Due to the time-variance of the acoustic path the system has to be adaptive. Fig. 1 shows a simple active noise cancellation system without feedback, using a loudspeaker as a secondary source. Here, zyxwvutsrq   describes the adaptive controller and I the interpolation filter used to reconstruct the estimated signd y(t). In Fig. 2(a) a model of the active cancellation system is shown. Here, h, describes the system in the auxiliary-path, modeling anti-aliasing filter, loudspeaker, and acoustic path from loudspeaker to microphone. The system in the error-path he models the path common to desired and estimated signd, including error microphone and anti-aliasing filter. Because of the system in the auxiliary-path, w has to include the inverse system of h, Wi z)/H, z) s the system to be identified). Measurements show that h, can be a nonminimum phase system. This results in making w noncausal, and therefore, degrading the cancellation effect. The drawback of the noncausal parts is decreased due to the inherent delay in acoustic system. The filter he does not affect the required model wo, but it does affect the behavior of the adaptive algorithm as does h,. primary source (noise) secondary source Fig. 1. Active noise cancellahon system without feedback (b) Fig. 2. (a) h, can be moved through the summing junction to form an equivalent system. (b) h, and he can be combined to form a new error-path system €unction h. Ln she same way, wb and the inverse system of the auxiliary path hgl can be combined to wo which is the system to be identified. the LMS algorithm. Thus, the convergence rate is lowered, the residual power is increased, and the algorithm can even become unstable. In order to stabilize the algorithm a filter identical to the filter h in the auxiliary-error-path (the com- bined auxiliary- and error-path filter will hereafter simply be referred to as error-path filter) is used to filter the reference input a (n) [24], [16] (see Fig. 1) to the algorithm. The recursive relation for updating the tap-weight vector in this case is (1) n + 1) = pu n) + PEf(++) where gf and .ru are vectors of dimension N, and uf(n) = ~(n) h, he = u(n) * h, 11. THE FILTERED-X LMS &GO“ y(n) = zoT(n) &(n), The existence of a filter in the auxiliary- and/or the error- Yf(n) = Y(n) * hz * he = Y(n) *h, path has been shown to generally degrade the performance of df n) = zyx ‘ n) *he = d(n) * h, Manuscnpt received July 18, 1993; revised February 21, 1995. = ddn) - Yf(4 + 4n). The author was with the hstitut fur Netzwerk- und Signdtheorie, Technis- che Hochschule Darmstadt, Merckstr. 25, 64283 Darmstadt, Germany. He is now with Rockwell, Vesturbrun 16, 1s-107 Reykjavik, Iceland. Normalization of LMS-type algorithms has been shown to greatly improve their convergence properties for colored IEEE Log Number 9414956. inputs, as well as making the algorithms independent of 1063-6676/95 04.00 0 1995 IEEE  BJARNASON ANALYSIS OF THE FILTERED-X LMS ALGORITHM zyxwvutsr 05 zy the energy of the input signal. In the case of the FXLMS algorithm, a normalization rule can be given (2) zyxwv   -f zyxwvutsr n)gb(n) p= zyxwvutsrqpo T Stochastic Analysis In the early 1960 s Widrow and Hoff [23] introduced the LMS algorithm. They showed that with a stationary stochas- tical input the expected value of the weight vector converged to the optimal Wiener solution. Starting with the first-order moments by Gersho, in 1969 [13], and with the second-order moments by Ungerbock, in 1974 [22], the analysis of the LMS algorithm for a stochastic input has been a long and slow- moving process. The real breakthrough was the introduction of the independence theory for an independent, identically distributed (IID) white-input process by Gardner, in 1984 [12]. In 1985, Feuer and Weinstein [l 11 carried out the calculation of the covariance matrix of the weight-error vector for a Gaussian input. Nitzberg [17] and [18], and later, Bershad [2], extended the analysis on the NLMS algorithm assuming a Gaussian input. Further extension of the analysis to a large group of input processes called spherically invariant random processes (SIRP) was done by Rupp [19], in 1993. In the following, the classical stochastical analysis methods will be introduced and applied to the FXLMS algorithm. In 1981, Widrow et al. [24] introduced the FXLMS algo- rithm, and showed that with a stationary stochastical input the expected value of the weight vector converges to the Wiener solution. This can be shown by minimizing the expected value of the squared filtered error ~[e;(n)] --) min. (3) leading to the Wiener solution RUfllfWO = T(df+e?,f)Uf' (4) Here, zyxwvutsrqp  df eqf)q is the expected value of the product of the filtered input vector gf(n) and the filtered output h (d(n) + e,(n)) = df n) + e,f(n), with d(n) being that part of the error that is predictable while the disturbance term e,(n) is the part of the error e(.) that is not predictable by the adaptive filter. The first-order moment of e,(n) is assumed zero and the second-order moment is called the minimum mean-squared error of the Wiener solution E[e,(n)l = 0, (5) E[e: zyxwvuts = Jmin. (6) This can be done by minimizing the covariance matrix E [g(n - )gT(n )] of the weight-error vector. It will be assumed that the input signals belong to a certain class of processes called SIRP's. For an introduction to SIRP's [7] and [21] are a good reading. The first-order moments of the processes are all assumed to be zero. Furthermore, it is assumed that the correlated random process ~(n) s a linear transformation A of an uncorrelated process ~(n) f the same order N The autocorrelation matrix (ACF) matrix & can then be obtained Let Q be an unitary orthogonal matrix that diagonalizes &. Then the unitary similarity transformation is given by The elements of the diagonal matrix A are the eigenvalues of the ACF matrix. Furthermore, QT can also diagonalize z   - It can be shown for SIRP's that the elements of the vector ~(n) re statistically independent with respect to each other, with the usage of spherical coordinates. A further implication of this, due to the symmetry of the joint density function, is that all moments containing at least one uneven order equal zero. A further important element of the analysis is the independence theory. This allows one to assume that the vectors g(i) and g j) are independent of each other for zy   # j. This assumption was introduced by Gersho, in 1969 [13] and later shown to be true for certain conditions by Mazo, in 1979 [15]. Clearly, there are many practical problems for which the independence theory is not satisfied. Nevertheless, experience with the LMS algorithm has shown that results gained by the application of the independence theory retain sufficient information about the structure of the adaptive process to serve as reliable design guidelines, even for some problems with highly dependent data samples [l]. For convenience, the algorithm for updating the weights of Furthermore, it is assumed that the disturbance term e,(n) is uncorrelated with the input process u(n). Also, it is assumed the FXLMS adaptive filter is given again that e,(n) is a white process. Modeling the error-path with a transversal filter h of order Mh, the mean-square of the filtered zyxwvutsr rror can be calculated ~[e;(n)] =E f(tr(R,,E[z(n - )gT(n - ] + Jmin). (7) This relation is derived in Appendix Al. In order to minimize E[e?(n)], he trace term has to be minimized. Mh-1 a O (14) (15)  506 zyxwvutsrqponmlkj EEE TRANSACTIONS ON SPEECH zyxwv ND zyxwv UDIO PROCESSING, VOL 3, NO 6, NOVEMBER 1995 First-OrderMoments: Building a weight-error vector for zyxwvu   zyxwv > zyxw ) and a simple delay D with a weight hD in the z (n) = ~(n) zoo at time n and modeling the error-pa& error path the stability bounds for the stepsize are given by with a transversal filter h of order hfh the following vector equation results Mh-1 Mh-1 c(n + 1) = c(n) - P hZh& -j) z=o j=o T(, - )c(n - zy   Mh-1 Mh-1 +P hah,E(n )eo n - ). z=o 3=0 (14) Using the independence theory [13] and 1151 and the ex- pectation operator leads to The term E[gf n)eof TI)] is neglected since its expectation is zero. Diagonalizing Ruu with the unitary matrix using the abbreviation y(n) = - 'E[g(n>], he following equation is obtained Normalizing results in exactly the same bounds that are calculated in161 for a sinusoidal input at R = 0 and T 0 < Q: < 2sin (2(2;+ 1)). There, it is further shown that these bounds are the minimum stability bounds for a sinusoidal input. The stability of the first-order moments is, therefore, a sufficient requirement for a sinusoidal input. In general, this is not the case. Thus, convergence of the second-order moments of the weight-error vector or the error signal is desired. Second-Order Moments: When investigating the behavior of the second-order moments, the system mismatch S(n) = E[gT(n)g(n)] ives a good insight into the properties of the algorithm. For the study of S(n) the trace of the covariance matrix &,(n) = E[4n)gT(n)] as to be evaluated. The mean squared error E[e2(n)] an also be derived from the covariance matrix by using the identity E[e2(n)] tr(KOO(n)&,,) + Jmin. For the covariance matrix the following expression is obtained in (22) at the bottom of this page. Using the abbreviation (n) = E[g( where Go n) epresents the covariance m error vector, leads to For each element of the vector g(n) a characteristic equation exists Mh-1 h'h-1Mh-1Mh-1Mh-1 2 - 1 + pxk h:ZFz = 0 ;for k = 1..N. (19) -P2 hahjhkhl 2=0 i=O j=o k=O kO To ensure stability of the first-order moments, the roots Therefore, attention is paid to the roots due to pAmax, where . E[g(n - )gT(n )gzl(n)g(n E)gT(n - k)] of the characteristic equation have to remain inside the zyxwv nit Mh-1 Mh-1 circle. Increasing PAk moves the roots out of the unit circle. + P JA, h?h;BUU. (23) z=O j=o A is the largest of the eigenvalues A, of the ACF matrix & For a sinusoidal input of an amplitude A Amax = A2/2 Again, let A be the matrix that transforms the uncorrelated process ~(n) o a correlated process g(n) and be the orthog-  BJAFNASON: ANALYSIS zyxwvutsrqponml F THE FILTERED-X zyxwvutsrq MS ALGORITHM zyxwvutsrq 07 onal matrix that diagonalizes zyxwvutsr Then E = zyxwv ;UT and - TBUuQ A. By using these identities and the abbreviation - TKij(n)Q = zyxwvut tJ(n) he following presentation of (23) is obtained in(24) and (25) at the bottom of this page. Considering the term with z(n) = h1/2Q21 n)h1/2 ith respect to spherically invariant processes [ 191, defining as the joint fourth order moment of the input process results in E[g(n )gT(n - )Z(n)g(n - Z)gT(n - zyx )lrS = (26) mi2 2) zyxwvutsrqpo   n) I j = i # = k, mi2’2)ZsT(n); j = 1 fa = k, mi2’2 E mm(n); 0; for r = s, for r # s, j = k # i = zyxw   else. ; Because all of the main diagonal elements of cij(n 1) depend only on the main diagonal elements of Ckz(n) nd due to the invariance of the trace to diagonalization, only the elements of the main diagonal of Coo(n) re of interest in calculating the system mismatch S n). By using the result of (28) and the abbreviation i2’2) mk2’2)/a: he following simpler representation for the diagonal elements of C,,(n + 1) and Cok(n 1) is obtained Mh-1 coo(n + 1) = Goo 4 - 2P h?hcOi(4 2=0 MI. -1 i=O j>i ML 1 Mh -1 i=o j=o i=O The vector is filled by the eigenvalues from the diagonal matrix A. Delayed LMS Algorithm: For the special case of a simple delay D in the error path with weight hD, (29) and (30) can be reduced to Equation (33) can be rewritten to (see Appendix A2) The function fDLMs depends on the delay D and the characteristics of the input signal such as eigenvalue ratio z   etc. For a stable algorithm the maximum possible value of the sum in (34) is Dc,,(n - D + 1). Therefore, the following inequality holds The higher the eigenvalue ratio the slower the convergence. Correspondingly the value of fDLMS increases. Substituting the results of (34) into (31) and using the z-transformation and the matrix inversion lemma results in the following equation for the second-order moments (see Appendix A3)  508 zyxwvutsrqpon EE zyxwvutsrqpon RANSACTIONS ON SPEECH zyxwvu ND AUDIO PROCESSWG, VOL. 3, NO. 6, NOVEMBER 1995 Fig. 3. Active system without feedback, using the F2tered-X zyxwvuts XS algorithm. where The entries of B(z) are where The elements of the matrix D(z) are defined with (40) at The characteristic equation of the discrete-time system the bottom of this page. described with (37) can be split into two polynomials N-1 1 - p2h$Gpi2)z-D X:dZz = 0, (41) dil = 0. (42) zyxwv =0 For a white process with identical eigenvalues, i.e.,& = X (for zyxwvutsrq   = 1..N), the characteristic equation can be reduced to zD+l - D + 2ph3 - p2 h4,X2(6p(N + 2) + ~~DLMS) 0, 143) + 2 2p 4 DX zyxwvutsrq   (6p’2) fDLMS) = 0. (44) zDtl - Dph&X 1- 0.5 0- -0.5 - -I - I -1. ii 5 -i zy 015 zyxw   0’5 i S Re(z.1 Fig. 4. Root loci of the Characteristic equatton for a whtte-input process wiW = 10,X = 1,a: = m?”) = 1 and D = 8. The convergence behavior of the DLMS algorithm depends on the roots of the characteristic equation. For N 2 2 (43) is the dominant one. Fig. 4 depicts the root locus of the characteristic equation for D = 8. It can be seen that D roots start in &e srcin. With increasing p they move radially away from the srcin under an angle of 27rilD; i = 0..D - 1. The initial root at the point 1,O) moves inwards along the positive real axis in the direction of the srcin. This root dominates the convergence and the stability of the algorithm. As p increases further, the negative square terms in p become dominant. The movement of the roots is reversed and they move back on the sane locus as before. When the negative and positive parts in ,U of the chaxacteristic equation cancel each other, the dominant root retums to the point (1, 0) and the other D roots return to the srcin. Increasing p further would migrate the dominant root outside the unit circle making the algorithm unstable. The stability condition for the algorithm can then be given by the following expression: From Fig. 5 it can be observed that the magnitude of the dominant root has a minimum in the middle of the stable interval. This indicates that the convergence will be fastest if the stepsize equals half of the upper bound. This can also be shown because this is exactly the point were the Oth order term in the characteristic equation reaches its highest positive value.
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