The ideal analog differentiator is described by where is the input and the...

The ideal analog differentiator is described by

where is the input and the output signal.

(a) Determine its frequency response by exciting the system with the input

(b) Sketch the magnitude and phase response of an ideal analog differentiator band-limited to B hertz.

(c) The ideal digital differentiator is defined as

Justify this definition by comparing the frequency response with that in part (b).

(d) By computing the frequency response H(ω), show that the discrete-time system

is a good approximation of a differentiator at low frequencies.

(e) Compute the response of the system to the input