Question

What is the expression for system function in z-domain?

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

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

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

(d) \(\frac{T}{2}[\frac{1-z^{-1}}{1+z^{-1}}]\)

I got this question in a job interview.

My enquiry is from Bilinear Transformations topic in division Digital Filters Design of Digital Signal Processing

Answer 1

The correct answer is (c) \(\frac{T}{2}[\frac{1+z^{-1}}{1-z^1}]\)

Explanation: We know that

y(n)-y(n-1)= \([\frac{x(n)+x(n-1)}{2}]T\)

Taking z-transform of the above equation gives

=>Y(z)[1-z^-1]=([1+z^-1]/2).TX(z)

=>H(z)=Y(z)/X(z)=\(\frac{T}{2}[\frac{1+z^{-1}}{1-z^1}]\).