S. C. Douglas and W. Pan, "Exact expectation analysis of the LMS adaptive filter," IEEE Trans. Signal Processing, vol. 43, pp. 2863--2871, Dec. 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... are major problems that limit the performance of fast DSL, such as very high speed DSL (VDSL) modems [3] As a research on this problem, we present an adaptive linear transversal filter frequency domain equalizer for the audio band using the Widrow least mean square (LMS) algorithm (cf. 4][5]) We developed model for a typical local loop from transmission line characteristics and derived the adaptive equalizer that undo the effect of ISI and AWGN. The work is found to have laid the foundation for further endeavors on combating the effect of ISI, ICI, and AWGN in multi tone signaling ....

S. C. Douglas and W. Pan. Exact expectation analysis of the LMS adaptive filter. IEEE Transactions on Signal Processing, 43(12):2863--2871, Dec.1995.

....M = 6 the matrix has size 28; 181 Theta 28; 181) It is therefore computationally infeasible to work directly with Phi; the approach is feasible only for relatively small filter lengths. For this reason, reference [15] considered only the case M = 2 (i.e. filter with two taps) while reference [16] used the same method for orders up to M = 6 coupled with a numerical procedure (viz. the power method) for the evaluation of the eigenvalues of Phi. For larger filter lengths, we need to develop an alternative procedure for the estimation of max that does not work directly with the matrix ....

....will be fl 4 (k 1) oe 2 fl 1 (k) Gamma 2oe 2 fl 4 (k) oe 2 fl 5 (k) 2 oe 4 fl 6 (k) The state vector in this case will therefore be of dimension 7, Gamma k = fl 1 (k) fl 7 (k) T . This procedure can in principle be repeated for any filter order M (and, in fact, as shown in [16], similar state space models can be obtained even in situations where the a(k) form a correlated sequence) However, the number of state variables will grow exponentially fast with the filter order, which makes it infeasible to pursue this line of reasoning for larger filter lengths (larger than 7 ....

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S. C. Douglas and W. Pan. Exact expectation analysis of the LMS adaptive filter. IEEE Transactions on Signal Processing, 43(12):2863--2871, December 1995.

.... adaptive scheme for larger step sizes and without the independence assumptions 1 Will EALCs provide satisfactory information in these cases as well In the process of comparing results obtained from EALCs with results predicted by an exact theoretical analysis for such scenarios (as in [20] [21], 22] we noticed considerable differences between simulation and theory. These differences persisted no matter how many experiments we averaged. A first explanation of the discrepancies was to blame the simulation program and possible numerical errors. After careful study, however, we realized ....

....the findings by several simulation results. We may mention that there are already several works in the literature of adaptive filtering that relate to both kinds of analyses: mean square and almost sure analyses. As examples of mean square based studies, we may cite [7] 9] 15] 17] and [19] [21]; and for almost sure based studies [12] 14] 16] 18] In some of these works (e.g. 14] 16] it is actually shown that both methods of analyses provide the same estimates for the rates of convergence of an adaptive filter when the step size is vanishingly small, and that (at least for ....

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S. C. Douglas and W. Pan. Exact expectation analysis of the LMS adaptive filter. IEEE Trans. Signal Process., 43(12):2863-- 2871, December 1995.

....yet reached the wanted maturity. Current treatments contain concepts, like the independence assumption , which cannot be made plausible [2] and important issues, like stability, which are addressed on an ad hoc basis. On the other hand, modern (including computer aided) research contributions [4,5,6,7,21] require such a thorough mathematical understanding and contain such lengthy proofs that they cannot provide the basis for an introductory course on adaptive filtering in an engineering curriculum. To bridge this gap we study three basic aspects of LMS filtering. First, we will show that the ....

....and the first component renewed. Such a strong deterministic coherence between successive input vectors in a TDL structure obviously conflicts with the independence assumption. Recently, two ways have been proposed to avoid the illicit independence assumption. The first way owing to Douglas et al. [7,21] provides an exact computeraided mean and mean square performance analysis, 90 Proceedings of the ProRISC Workshop on Circuits, Systems and Signal Processing 1997 which, however, becomes rather laborious for multitap filters. The second method [20] yields analytic results, but is confined to the ....

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S.C. Douglas and Weimin Pan, "Exact expectation analysis of the LMS adaptive filter". IEEE Trans. Signal Processing, vol. SP-43 (1995), p. 2863-2871.

....moments and that the pair fd(m) X(m)g is independent of fd(n) X(n)g if m 6= n. These assumptions are similar to the independence assumptions commonly employed in statistical analyses of adaptive filters [Haykin 1996] For a discussion of the accuracy of such assumptions, see [Mazo 1979, Douglas and Pan 1995] We shall also assume that (e) is continuous and differentiable at all points, although this assumption can be relaxed if the joint probability distributions of d(k) and X(k) are suitably smooth. Using (55) we can express (53) as e(k) d(k) Gamma X(k) w e w(k) 56) k) Gamma X(k) e ....

S.C. Douglas and W. Pan, "Exact expectation analysis of the LMS adaptive filter," IEEE Trans. Signal Processing, vol. 43, pp. 2863-2871, Dec. 1995.

.... [15, 16] Such results could have potential benefits in selecting step sizes for these algorithms to guarantee their stable operation and to provide fast convergence, particularly as such results are difficult to obtain via statistical characterizations due to the assumptions used in such analyses [17, 18]. In this paper, we develop a general theory for understanding the behavior of adaptive filtering algorithms whose updates depend on the coefficient vectors at time k and k 1. Our technique attempts to characterize the stability of any such algorithm using the nonlinear relationship between ....

S.C. Douglas and W. Pan, "Exact expectation analysis of the LMS adaptive filter," IEEE Trans. Signal Processing, vol. 43, pp. 2863-2871, Dec. 1995.

....be compactly described, and thus general forms of the expectations of these terms cannot be expressed. Even so, the technique for determining these evolution equations given the input and noise statistics is straightforward. We have used the computer automated analysis technique described in [13] to derive the update equations for the mean coefficient error vector, given by E[V k N ] E[A k ]E[V k ] E[B k ] 16) as well as the coefficient error correlation matrix, given by E[V k N V T k N ] E[A k E[V k V T k ]A T k ] E[B k B T k ] 17) for input signals that is generated ....

....A T U k ; 18) where A = a 0 a 1 Delta Delta Delta aM Gamma1 ] T defines the correlation statistics of the input signal and U k = u k u k Gamma1 Delta Delta Delta u k GammaM 1 ] T , where u k is a zero mean i.i.d. signal. A description of the automated analysis technique appears in [13]. In this case, we enforce the assumptions previously described to simplify the forms of the equations produced by the analysis. 3.2.2 Approximate Analysis for Small Step Sizes We now present a second analysis of the sequential LMS algorithm for small step sizes. For this analysis, we note that ....

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S.C. Douglas and W. Pan, "Exact expectation analysis of the LMS adaptive filter," IEEE Trans. Signal Processing, vol. SP-43, no. 12, pp. 2863-2871, December 1995.

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S. C. Douglas and W. Pan, "Exact expectation analysis of the LMS adaptive filter," IEEE Trans. Signal Processing, vol. 43, pp. 2863--2871, Dec. 1995.

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S. C. Douglas and W. Pan, "Exact expectation analysis of the LMS adaptive filter," IEEE Trans. Signal Processing, vol. 43, pp. 2863--2871, Dec. 1995.

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