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)=$\frac{T}{2}[\frac{1+z^{-1}}{1-z^1}]$.