The ideal analog differentiator is described by

Y_a(t) = dx_a(t)/dt

Where x_a(t) is the input and y_a(t) the output signal.

(a) Determine its frequency response by exciting the system with the input x_a(t) = e^j2πF1.

(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

H(ω) = jω        |ω| ≤ π

Justify this definition by comparing the frequency response |H(ω)|, < H(ω) with that in part (b).

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

y(n) = x(n) – x(n – 1)

is good approximation of a differentiator at low frequencies.

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

x(n) = A cos(ω₀n + θ)