Title :
On the probability density function of the LMS adaptive filter weights
Author :
Bershad, Neil J. ; Qu, Lian Zuo
Author_Institution :
Dept. of Electr. Eng., California Univ., Irvine, CA, USA
Abstract :
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.<>
Keywords :
adaptive filters; filtering and prediction theory; least squares approximations; partial differential equations; probability; vectors; Gaussian data models; LMS adaptive filter weights; Taylor series; adaption parameter; covariance matrix; detection; first-order partial differential-difference equation; probability density function; time delay estimation; time-varying mean vector; weight vector; Adaptive filters; Covariance matrix; Data models; Delay effects; Difference equations; Differential equations; Least squares approximation; Partial differential equations; Probability density function; Taylor series;
Journal_Title :
Acoustics, Speech and Signal Processing, IEEE Transactions on
