As shown in Figure 7.37 and discussed in Section 7.12, the procedure for interpolation o...

As shown in Figure 7.37 and discussed in Section 7.12, the procedure for interpolation or upsampling by an integer factor N can be thought of as ,a cascade of two operations. For exact band-limited interpolation, the filter in Figure 7.37 is an ideal lowpass filter. In any specific application, it would be necessary to implement an approximate lowpass filter. In this problem, we explore some use-ful constraints that are often imposed on the design of these approximate lowpass filters.

(a) Suppose that is approximated by a zero-phase FIR filter. The filter is to be designed with the constraint that the original sequence values get reproduced exactly; that is,

This guarantees that, even though the interpolation between the original sequence values may not be exact, the original values are reproduced exactly in the interpolation. Determine the constraint on the impulse response h[n] of the lowpass filter which guarantees that eq. (P7.51-1) will hold exactly for any sequence .

(b) Now suppose that the interpolation is to be carried out with a linear-phase, causal, symmetric FIR filter of length N; that is

where is real. The filter is to be designed with the constraint that the original sequence values get reproduced exactly, but with an integer delay α, where α is the negative of the slope of the phase of ; that is,

Determine whether this imposes any constraint on whether the filter length N is odd or even.

(c) Again, suppose that the interpolation into be carried out with a lin causal, symmetric FM filter, so that

where is real. The filter is to be designed with the constraint original sequence values get reproduced exactly, but with a delay M that is not necessarily equal to the slope of the phase; that is,

Determine whether this impose any constraint on whether the filter lengthy N is odd or even.