On the probability density function of the LMS adaptive filter weights

The joint probability density function of the weight vector in least-mean-square (LMS) adaptation is studied for Gaussian data models. An exact expression is derived for the characteristic function of the weight vector at time n+1, conditioned on the weight vector at time n. The conditional characteristic function is expanded in a Taylor series and averaged over the unknown weight density to yield a first-order partial differential-difference equation in the unconditioned characteristic function of the weight vector. The equation is approximately solved for small values of the adaption parameter and the weights are shown to be jointly Gaussian with time-varying mean vector and covariance matrix given as the solution to well-known difference equations for the weight vector mean and covariance matrix. The theoretical results are applied to analyzing the use of the weights in detection and time delay estimation. Simulations that support the theoretical results are also presented. >

[1]  B. Widrow,et al.  Adaptive antenna systems , 1967 .

[2]  Neil J. Bershad,et al.  Error probabilities for DS spread-spectrum systems using an ALE for narrow-band interference rejection , 1988, IEEE Trans. Commun..

[3]  Neil J. Bershad.  Further results on the joint characteristic function of the complex scalar LMS adaptive weight , 1988 .

[4]  H. O. Posten.  Multidimensional Gaussian Distributions , 1964 .

[5]  Neil J. Bershad,et al.  Comments on "Time delay estimation using the LMS adaptive filter-static behavior" , 1985, IEEE Trans. Acoust. Speech Signal Process..

[6]  N. Bershad,et al.  Time delay estimation using the LMS adaptive filter--Dynamic behavior , 1981 .

[7]  William A. Gardner,et al.  A Unifying View of Second-Order Measures of Quality for Signal Classification , 1980, IEEE Trans. Commun..

[8]  N. Bershad,et al.  The complex LMS adaptive algorithm--Transient weight mean and covariance with applications to the ALE , 1983 .

[9]  W. Burdic.  Detection of Narrowband Signals Using Time-Domain Adaptive Filters , 1978, IEEE Transactions on Aerospace and Electronic Systems.

[10]  K. Senne,et al.  Performance advantage of complex LMS for controlling narrow-band adaptive arrays , 1981 .

[11]  J. Zeidler,et al.  A performance analysis of adaptive line enhancer-augmented spectral detectors , 1981 .

[12]  D. M. Chabries,et al.  Adaptive enhancement of multiple sinusoids in uncorrelated noise , 1978 .

[13]  T. Kailath,et al.  SOME USEFUL PROBABILITY DISTRIBUTIONS , 1965 .

[14]  B. Widrow,et al.  Adaptive noise cancelling: Principles and applications , 1975 .

[15]  J. Zeidler,et al.  Second-order output statistics of the adaptive line enhancer , 1979 .

[16]  N. Bershad,et al.  Tracking characteristics of the LMS adaptive line enhancer-response to a linear chirp signal in noise , 1980 .

[17]  J. Treichler,et al.  Transient and convergent behavior of the adaptive line enhancer , 1979 .

[18]  L. Griffiths.  Rapid measurement of digital instantaneous frequency , 1975 .

[19]  Neil J. Bershad.  On the optimum gain parameter in LMS adaptation , 1987, IEEE Trans. Acoust. Speech Signal Process..

[20]  L. Milstein,et al.  Performance Analysis of Narrow-Band Interference Rejection Techniques in DS Spread-Spectrum Systems , 1984, IEEE Trans. Commun..

[21]  Lian Zuo Qu,et al.  On the joint characteristic function of the complex scalar LMS adaptive weight , 1984 .

[22]  L. Jolley.  Summation of Series. , 1962 .

[23]  Laurence B. Milstein,et al.  An Approximate Statistical Analysis of the Widrow LMS Algorithm with Application to Narrow-Band Interference Rejection , 1985, IEEE Trans. Commun..

[24]  Ehud Weinstein,et al.  Convergence analysis of LMS filters with uncorrelated Gaussian data , 1985, IEEE Trans. Acoust. Speech Signal Process..

[25]  W. Gardner.  Learning characteristics of stochastic-gradient-descent algorithms: A general study, analysis, and critique , 1984 .