The system shown in Figure, approximately interpolates the sequence x[n] by a factor L. Suppose that the linear filter has impulse response h[n] such that h[n] = h[? n] and h[n] = 0 for |n| > (RL ? 1), where R and L are integers; i.e., the impulse response is symmetric and of length (2RL ? 1) samples.

(a) In answering the following, do not be concerned about the causality of the system; it can be made causal by including some delay. Specifically, how much delay must be inserted to make the system causal?

(b) What conditions must be satisfied by h[n] in order that y[n] = x[n / L] for n = 0, ? L, ? 2L, ? 3L?

(c) By exploiting the symmetry of the impulse response, show that each sample of y[n] can be computed with no more than RL multiplications.

(d) By taking advantage of the fact that multiplications by zero need not be done, show that only 2R multiplications per output sample are required.

Transcribed Image Text:

H(ej) x(n] y[n)