In this problem, we consider the effect of mapping continuous-time filters to discrete-t...

In this problem, we consider the effect of mapping continuous-time filters to discrete-time filters by replacing derivatives in the differential equation for a continuous-time filter by central differences to obtain a difference equation. The first central difference of a sequence x[n] is defined as

and the kth central difference is defined recursively as

For consistency, the zeroth central difference is defined as

(a) If X (z) is the z-transform of x[n], determine the z-transform of (k){x[n]}. The mapping of an LTI continuous-time filter to an LTI discrete-time filter is as follows: Let the continuous-time filter with input x(t) and output y(t) be specified by a

differential equation of the form

Then the corresponding discrete-time filter with input x[n] and output y[n] is specified by the difference equation

(b) If H_c(s) is a rational continuous-time system function and H_d (z) is the discrete-time system function obtained by mapping the differential equation to a difference equation as indicated in part (a), then

Determine m(z).

(c) Assume that Hc(s) approximates a continuous-time lowpass filter with a cutoff frequency of Ω = 1; i.e.,

This filter is mapped to a discrete-time filter using central differences as discussed in part (a). Sketch the approximate frequency response that you would expect for the discrete-time filter, assuming that it is stable.