Example 3.2.2: Digital derivator
The output of an analog derivator

can be approximated by:
y(t) = x'(t) x(t) - x(t-T)
T
Thus
y(n) = x'(n) x(n) - x(t-1) = (1/T) x(n) - (1/T) x(n-1) = h(k) x(n-k)
T

and hence the impulse response of this simple digital derivator is

=> h(0) = 1/T
h(1) = -1/T
h(n) = 0 elsewhere

This is an FIR system.

Transfer function:

H(z) = h(n) z-n = h(0) z0 + h(1) z-1 = 1/T - (1/T) z-1

=> H(z) = 1 (1 - z-1)
T


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