Details

Linear time-invariant (LTI) discrete-time systems are commonly described from different perspectives, including a difference equation and a transfer function , where is a complex variable. The frequency response of the system can be obtained by evaluating on the unit circle. To visualize this, a 3D plot of is shown, with the unit circle highlighted. You can see how the locations of the poles and zeros affect the behavior of the system.

The transfer function is given by

.

The difference equation is given by

.

This Demonstration focuses on a specific scenario in which the poles are within the unit circle. This configuration ensures a bounded-input, bounded-output (BIBO) stable system. Showcasing different descriptions and representations leads to a better understanding of the performance and design of a first-order discrete-time system. You can vary the locations of the poles and zeros to show how a second-order system can be used as a lowpass, bandpass or highpass filter.

Reference

[1] F. T. Ulaby and A. E. Yagel, Signals and Systems: Theory and Applications, Michigan Publishing, 2018. (Jul 21, 2023) ss2.eecs.umich.edu.