Question

If s=jΩ and if Ω varies from -∞ to ∞, then what is the corresponding locus of points in z-plane?

(a) Circle of radius 1 with centre at z=0

(b) Circle of radius 1 with centre at z=1

(c) Circle of radius 1/2 with centre at z=1/2

(d) Circle of radius 1 with centre at z=1/2

I had been asked this question at a job interview.

Origin of the question is IIR Filter Design by Approximation of Derivatives in section Digital Filters Design of Digital Signal Processing

Answer 1

Right option is (c) Circle of radius 1/2 with centre at z=1/2

To explain I would say: We know that

s=(1-z^-1)/T

=> z=1/(1-sT)

Given s= jΩ => z = 1/(1- jΩT)

Thus from the above equation if Ω varies from -∞ to ∞, then the corresponding locus of points in z-plane is a circle of radius 1/2 with centre at z=1/2.