In this problem, we consider a method of filter design that might be called autocorrelat...

In this problem, we consider a method of filter design that might be called autocorrelation invariance. Consider a stable continuous-time system with impulse response h_c(t) and system function H_c(s). The autocorrelation function of the system impulse response is defined as

and for a real impulse response, it is easily shown that the Laplace transform of Φc(τ ) is Φ(s) = H_c(s)H_c(−s). Similarly, consider a discrete-time system with impulse response h[n] and system function H(z). The autocorrelation function of a discrete-time system impulse response is defined as

and for a real impulse response,

Autocorrelation invariance implies that a discrete-time filter is defined by equating the autocorrelation function of the discrete-time system to the sampled autocorrelation function of a continuous-time system; i.e.,

The following design procedure is proposed for autocorrelation invariance when H_c(s) is a rational function having N 1^st-order poles at s_k, k = 1, 2, . . . , N, and M < N zeros:

1. Obtain a partial fraction expansion of Φ_c(s) in the form

2. Form the z-transform

3. Find the poles and zeros of Φ(z), and form a minimum-phase system function H(z) from the poles and zeros of Φ(z) that are inside the unit circle.

(a) Justify each step in the proposed design procedure; i.e., show that the autocorrelation function of the resulting discrete-time system is a sampled version of the autocorrelation function of the continuous-time system. To verify the procedure, it may be helpful to try it out on the 1^st-order system with impulse response.

and corresponding system function

(b) What is the relationship between |H(e^jω)|² and |H_c(jΩ)|²?What types of frequency response functions would be appropriate for autocorrelation invariance design?

(c) Is the system function obtained in Step 3 unique? If not, describe how to obtain additional autocorrelation-invariant discrete time systems.