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 

∆⁽¹⁾{x[n]} = x[n + 1] – x[n – 1],

and the kth central difference is defined recursively as 

∆^(k){x[n]} = ∆⁽¹⁾{∆ ^(k–1){x[n]}}.

For consistency, the zeroth central difference is defined as 

∆⁽⁰⁾{x[n]} = x[n].

(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) output y(t) be specified by a difference 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 

H_d(z) = H_e(s)|_{s = m}(z)’ 

Determine m (z).

(c) Assume that H_c(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.

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