Adaptive Filters

Problems and Computer Projects

PROBLEMS

Problem IV.1 (Auto-regressive process) A unit-variance white-random process s(i) is fed into a first-order auto-regressive model with transfer function , where a is real. The output process is denoted by u(i); it is referred to as an auto-regressive process of order 1, written as AR(1). Assume |a| < 1. Show that the auto-correlation sequence of u(i) is given by r(k) = Eu(i)u(i – k) = a^|k|, for all integer values k. If u(i) is fed into an adaptive filter of order M, what is the covariance matrix of the resulting regressor u_i?

Problem IV.2 (Finite alphabets) Consider an adaptive filter with a regressor vector u_i that possesses shift-structure, namely, u_i = [u(i) u(i - 1) … u(i - M + 1)]. The entries of u(i) are realizations of a binary random variable, i.e., they are ±1 with probability 1/2. Assume initially that all variables are real-valued.

1. (a) Show that the EMSE that would result when the filter is trained using LMS is given by . Show that this result is exact, i.e., it holds irrespective of any approximations.
2. (b) Assuming << M, show that a similar conclusion holds when the filter is trained using -NLMS with .
3. (c) Likewise, assuming e ...