Summary

This blog post walks through a clear, visual example of how points in the continuous-time s-plane map into the discrete-time z-plane and what that mapping means for stability and frequency behavior. Readers will gain intuition on how poles, zeros, and the jω-axis transform under sampling and common mapping methods used in digital filter design.

Key Takeaways

• Understand how the exponential mapping z = e^{sT} transfers s-plane locations into the z-plane
• Identify how left-half-plane (LHP) stability in s maps to the interior of the z-plane unit circle
• Recognize how the jω-axis maps to the unit circle and how frequency wrapping/aliasing can occur
• Apply mapping intuition to analog-to-digital conversion methods (e.g., bilinear transform vs. impulse invariance)
• Visualize how time-domain damping and oscillation in s reflect as radius and angle in the z-plane

Who Should Read This

Intermediate DSP or control engineers and graduate students who design or analyze digital filters and need practical intuition about s-to-z mappings for stability and frequency behavior.

TimelessIntermediate

Topics