If we are given a basic filter module (a hardware or computer subroutine), it is sometim...

If we are given a basic filter module (a hardware or computer subroutine), it is sometimes possible to use it repetitively to implement a new filter with sharper frequency-response characteristics. One approach is to cascade the filter with itself two or more times, but it can easily be shown that, while stopband errors are squared (thereby reducing them if they are less than 1), this approach will increase the passband approximation error. Another approach, suggested by Tukey (1977), is shown in the block diagram of Figure P7.52-1. Tukey called this approach “twicing.”

(a) Assume that the basic system has a symmetric finite-duration impulse response; i.e.,

Determine whether the overall impulse response g[n] is (i) FIR and (ii) symmetric.

(b) Suppose that H(e^jω) satisfies the following approximation error specifications:

It can be shown that if the basic system has these specifications, the overall frequency response G(e^jω) (from x[n] to y[n]) satisfies specifications of the form

Determine A, B, C, and D in terms of δ₁ and δ₂. If δ₁ << 1 and δ₂ << 1, what are the approximate maximum passband and stopband approximation errors for G(e^jω)?

(c) As determined in part (b), Tukey’s twicing method improves the passband approximation error, but increases the stopband error. Kaiser and Hamming (1977) generalized the twicing method so as to improve both the passband and the stopband. They called their approach “sharpening.”The simplest sharpening system that improves both passband and stopband is shown in Figure P7.52-2. Assume again that the impulse response of the basic system is as given in part (a). Repeat part (b) for the systemof Figure P7.52-2.

(d) The basic system was assumed to be noncausal. If the impulse response of the basic system is a causal linear-phase FIR system such that

how should the systems of Figures P7.52-1 and P7.52-2 be modified? What type(s) (I, II, III, or IV) of causal linear-phase FIR system(s) can be used? What are the lengths of the impulse responses g[n] for the systems in Figures P7.52-1 and P7.52-2 (in terms of L)?