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. The locations of the poles and zeros are essential in characterizing , providing insight into the behavior of the system.

The difference equation is given by

.

This Demonstration focuses on a specific scenario where the pole is positioned 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 pole and zero to show how a first-order system can be used as a lowpass 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.