Digital Signal Processing Basics
When the frequency response of a discrete-time LTI system is obtained from its transfer function H(z), along which of the following is z moved?
- AAlong the unit circle of the z-plane (z = e^(jω), |z| = 1)
- BAlong the positive real axis of the z-plane (z = r, where r is a real number greater than 0)
- CAlong the imaginary axis of the z-plane (z = jω)
Show answer and explanation
Answer: A. Along the unit circle of the z-plane (z = e^(jω), |z| = 1)
The frequency response is H(z) with z = e^(jω) substituted; as ω goes from −π to π, z travels once around the unit circle. z = 1 (ω = 0) corresponds to DC and z = −1 (ω = π) to the Nyquist frequency. For the continuous-time Laplace transform the frequency response is found on the imaginary axis s = jω, and in discrete time the unit circle plays that role.
- A. Correct. It is evaluated on the unit circle with z = e^(jω).
- B. On the positive real axis, only the point z = 1 (DC), where it meets the unit circle, is part of the frequency response.
- C. Evaluating on the imaginary axis applies to s of the continuous-time Laplace transform.