Time Constants and Learning Curves of LMS Adaptive Filters.

Shensa, M.

doi:10.21236/ada063317

Cited by 12 publications

(7 citation statements)

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“…Assuming that the adaptive weights and the input vector are independent, Shensa [23] showed that the convergence of the weight vector can be expressed as…”

Section: Lms Convergencementioning

confidence: 99%

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A Two-Stage Approach for Improving the Convergence of Least-Mean-Square Adaptive Decision-Feedback Equalizers in the Presence of Severe Narrowband Interference

Batra

Zeidler

Beex

2007

EURASIP J. Adv. Signal Process.

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It has previously been shown that a least-mean-square (LMS) decision-feedback filter can mitigate the effect of narrowband interference (L.-M. Li and L. Milstein, 1983). An adaptive implementation of the filter was shown to converge relatively quickly for mild interference. It is shown here, however, that in the case of severe narrowband interference, the LMS decision-feedback equalizer (DFE) requires a very large number of training symbols for convergence, making it unsuitable for some types of communication systems. This paper investigates the introduction of an LMS prediction-error filter (PEF) as a prefilter to the equalizer and demonstrates that it reduces the convergence time of the two-stage system by as much as two orders of magnitude. It is also shown that the steady-state bit-error rate (BER) performance of the proposed system is still approximately equal to that attained in steady-state by the LMS DFE-only. Finally, it is shown that the two-stage system can be implemented without the use of training symbols. This two-stage structure lowers the complexity of the overall system by reducing the number of filter taps that need to be adapted, while incurring a slight loss in the steady-state BER.

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“…Assuming that the adaptive weights and the input vector are independent, Shensa [23] showed that the convergence of the weight vector can be expressed as…”

Section: Lms Convergencementioning

confidence: 99%

“…The eigenvalues for the correlation matrix given by (2) and (29) can be found [7,23,37] to be equal to…”

Section: Eigenvaluesmentioning

confidence: 99%

A Two-Stage Approach for Improving the Convergence of Least-Mean-Square Adaptive Decision-Feedback Equalizers in the Presence of Severe Narrowband Interference

Batra

Zeidler

Beex

2007

EURASIP J. Adv. Signal Process.

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“…It can be shown that [4] 2 cash 8 -cos(w -S() 00 cosh a -cos( - (l-2pa __)2k + ___2_ (141102)2k (17) Our two mode approximation has produced two time constants, r1 = a2/43.100282 and = 1 / 8pa. In the case of the filter, eq.…”

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confidence: 98%

“…As a consequence it is possible to obtaia assume i(o) = 0. It then follows from (3) and (4) a simple representation of the spectrum of the LMS that filter as a function of time (i.e., during convergence).…”

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The spectral dynamics of evolving LMS adaptive filters

Shensa¹

ICASSP '79. IEEE International Conference on Acoustics, Speech, and Signal Processing

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C!479-7/1c/ofl),_)4c,),rIo. 7cc © t97' 1EFfis the well known solution to the Wiener-Hopf equation ([1], [2]). It is well known that the time constants governing the learning curves of LMS adaptive filtersThe first term in (3), a transient, represents are determined by the eigenvalues of the input the decay of the old state; and the second term, correlation matrix. In this paper it is shown the growth of the new state w. Inasmuch as unthat for sufficiently long filters, a transforfavorable initial conditions may arbitrarily pronation to the frequency domain diagonalizes the long convergence, we focus on the growth state and matrix. As a consequence it is possible to obtaia assume i(o) = 0. It then follows from (3) and (4) a simple representation of the spectrum of the LMS that filter as a function of time (i.e., during convergence).The theoretical results are compared with w*_c(k) = 2ji 2 (I-2pR)3p = (I_2pRw*. (5) a computer simulation. j=k Let A be the eigenvalues and e' the corresponding I BACKGROUND eigenectors of R. Taking the vector norm of (5), we have Least mean square (U(S) adaptive filters are typically updated through the algorithm (6)where "" represents scalar multiplication. A sinc(k) = d(k) -2 w(k) x(k) ilar expression may be derived for the convergence of the mean square error (k)E(s2(k)) to its limiting value It is easily shown ([2], [4])where v denotes the L-dimensional vector of filter that in the absence of gradient noise (which is of weights, x the reference input vector, p the feedorder Lp*) back constant, and d the primary input or desired L response.In the case of the adaptive lime em-(k)* = 2 (l_2pAv)21\jev.w*j2 (7) hancer (ALE), the components of x are delayed samples of d. A detailed exposition of these algo-The curves (6) and (7) as a function of k may be termed the learning curves of the filter and Let = E(w) be the expected value of the output respectively. Each of the L terms relaxes weight vector; = E(dx); and R(k_i)RkCE(xkxf) geometrically with a logarithmic slope of be the correlation matrix of the input. Them, for log (l-2pAv)2 -4pXv, thus exhibiting a time constationary inputs, the recursion equation for the stant of mean of w is given approximately (R and w are as-(8) sumed independent, cf [i], [2])by i(k+1) = (I-2pR)(k) + 2ppWe remark that, generally, the maximum time constant, l/4PAmin ,ia a very conservative estimate It is easily seen by direct substitution that for the behavior of the learning curves.

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