June 2, 2021
The z-transform is defined as:
Unit Circle in Z-domain
The z-transform is a generalization of the discrete-time Fourier transform.
Therefore, the DTFT is the unit circle portion of the z-transform.
The N-point DFT or FFT is N samples around the unit circle. The Fourier transform, which is a special case of the s-domain Laplace transform, can be roughly mapped into the z-plane by use of the bilinear transform.
The roots of the numerator are called zeros, and the roots of the denominator are called poles.
Determine Frequency Response
The frequency response of a filter is determined by the interaction of a unit vector rotating around the unit circle with the poles and zeros of the filter.
- The unit vector at rotation ω=0 corresponds to DC (0Hz).
- The unit vector at rotation ω=π (180°) corresponds to Fs/2 or the Nyquist frequency.
When the tip of the unit vector gets close to a zero, the filter magnitude response is pushed downwards because zero is a root of the numerator polynomial. When the tip of the unit vector gets close to a pole, the filter magnitude response is pushed upwards because a pole is a root of the denominator polynomial.
Pole-zero locations are important for:
- Cadzow, James A., Foundations of Digital Signal Processing and Data Analysis. New York: Macmillan Publishing Company, 1987.
- Greenhoe, Daniel J., A Book Concerning Digital Signal Processing. Self-published, 2019.