Introduction:

In the analysis and design of digital control systems and digital signal processing, it is essential to understand the relationship between continuous-time (analog) and discrete-time (digital) systems. This relationship is commonly established through the mapping of the S-plane (Laplace domain) to the Z-plane (Z-transform domain).
The S-plane is used in continuous-time systems, where system behavior is analyzed using the Laplace transform. The Z-plane, on the other hand, is used for discrete-time systems using the Z-transform. The transformation from the S-plane to the Z-plane is typically achieved through the relation: $$ z = e^{sT} $$ where T is the sampling period. This exponential mapping helps in translating system characteristics such as stability, frequency response, and pole-zero locations from the continuous domain to the discrete domain. Understanding this mapping is crucial for tasks such as digital controller design, stability analysis, and the implementation of digital filters. It helps ensure that the behavior of the discrete system accurately reflects that of its continuous counterpart.
This experiment focuses on visualizing and analyzing the mapping of different regions and points in the S-plane to their corresponding locations in the Z-plane, thereby deepening the understanding of the interrelation between analog and digital systems.

In this experiment, we shall investigate how the locations of the poles and zeros in the s plane compare with the locations of the poles and zeros in the z plane.