Question

If s=σ+jΩ and z=re^jω, then what is the condition on σ if r>1?

(a) σ > 0

(b) σ < 0

(c) σ > 1

(d) σ < 1

This question was addressed to me in unit test.

I'd like to ask this question from Bilinear Transformations in portion Digital Filters Design of Digital Signal Processing

Answer 1

Correct option is (a) σ > 0

To explain I would say: We know that if = σ+jΩ and z=rejω, then by substituting the values in the below expression

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

=>σ = \(\frac{2}{T}[\frac{r^2-1}{r^2+1+2rcosω}]\)

When r>1 => σ > 0.