Let h n0 (n) be the FIR least squares inverse filter of length N with delay n 0 for a sequence...

Let h_{n0} (n) be the FIR least squares inverse filter of length N with delay n_{0} for a sequence g(n), i.e.,

The coefficients h_{n0} (n) are the solution to the Toeplitz equations (seep. 174 in Chapter 4)

which may be solved efficiently using the Levinson recursion. Since the value for the delay no that produces the smallest least squares error is typically unknown, to find the optimum value for n_{0} these equations must be solved for each value of n_{0}, beginning with n_{0} = 0. Instead of using the Levinson recursion to solve these equations repeatedly, it is possible to take advantage of the relationship between g_{n0} and g_{n0+ 1},

to derive a recursion for h_{n0} (n). In this problem we derive this recursion which is known as the Simpson sideways recursion [26].

(a) The solution for no = 0 may be found using the Levinson-Durbin recursion. Show how to generate the solution for n_{0} = 1 from the solution for no = 0 in less than 4N multiplications and divisions where N is the length of the inverse filter h_{n0}· Note that any information generated in the Levinson-Durbin recursion (for no = 0) can be used to construct the new solution.

(b) Generalize the result of part (a) to obtain a recursion that will successively construct the solution for all n_{0} > 0. Again your method should have less than 4N multiplications and divisions at each step.

(c) Write an expression for the error E_{n0} at the n0th step of the recursion in terms of the coefficients g(n) and the coefficients of the least squares inverse filter h_{n0} (n).

(d) Write a MATLAB program that implements the Simpson sideways recursion.

(e) How can this recursion be used to find the inverse of a Toeplitz matrix?