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.<>