Question

In Nth order differential equation, the characteristics of bilinear transformation, let z=re^jw,s=o+jΩ Then for s = \(\frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}})\), the values of Ω, ℴ are

(a) ℴ = \(\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})\), Ω = \(\frac{2}{T}(\frac{2rsinω}{1+r^2+2rcosω})\)

(b) Ω = \(\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})\), ℴ = \(\frac{2}{T}(\frac{2rsinω}{1+r^2+2rcosω})\)

(c) Ω=0, ℴ=0

(d) None

I had been asked this question during an internship interview.

This interesting question is from IIR Filter Design by the Bilinear Transformation topic in portion Discrete Time Systems Implementation of Digital Signal Processing

Answer 1

Correct choice is (a) ℴ = \(\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})\), Ω = \(\frac{2}{T}(\frac{2rsinω}{1+r^2+2rcosω})\)

The explanation is: s = \(\frac{2}{T}(\frac{z-1}{z+1}) \)

= \(\frac{2}{T}(\frac{re^jw-1}{re^jw+1})\)

= \(\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω}+j \frac{2rsinω}{1+r^2+2rcosω})(s = ℴ+jΩ)\)