Question

Which of the following condition is to be satisfied for the Fourier transform of a sequence to be equal as the Z-transform of the same sequence?

(a) |z|=1

(b) |z|<1

(c) |z|>1

(d) Can never be equal

The question was asked during an interview.

This interesting question is from Frequency Analysis of Discrete Time Signal in division Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Correct answer is (a) |z|=1

For explanation: Let us consider the signal to be x(n)

Z{x(n)}=\(\sum_{n=-∞}^∞ x(n)z^{-n} and X(ω)=\sum_{n=-∞}^∞ x(n)e^{-jωn}\)

Now, represent the ‘z’ in the polar form

=> z=r.e^jω

=>Z{x(n)}=\(\sum_{n=-∞}^∞ x(n)r^{-n} e^{-jωn}\)

Now Z{x(n)}= X(ω) only when r=1=>|z|=1.