Question

Which of the following in z-domain is equal to s-domain of second order derivate?

(a) \((\frac{1-z^{-1}}{T})^2\)

(b) \((\frac{1+z^{-1}}{T})^2\)

(c) \((\frac{1+z^{-1}}{T})^{-2}\)

(d) None of the mentioned

I had been asked this question during a job interview.

My question is based upon IIR Filter Design by Approximation of Derivatives topic in portion Digital Filters Design of Digital Signal Processing

Answer 1

Correct option is (a) \((\frac{1-z^{-1}}{T})^2\)

To explain I would say: We know that for a second order derivative

d^2y(t)/dt^2=[y(n)-2y(n-1)+y(n-2)]/T^2

=>s^2 = \(\frac{1-2z^{-1}+z^{-2}}{T^2} = (\frac{1-z^{-1}}{T})^2\)