The z-Transform and System Analysis
Stub. The z-transform and its inverse, poles and zeros, and using the z-transform to analyze an LTI system's transfer function and stability.
Turns convolution into multiplication: the z-transform maps a sequence to a function of a complex variable, under which convolution becomes ordinary multiplication and a difference equation becomes an algebraic one. Poles and zeros of the resulting rational function encode the system’s behavior directly — the pole locations determine stability and the character of the impulse response (decaying, growing, oscillatory), and the transfer function (the z-transform of the impulse response) is the single object that summarizes an LTI system for both convolution and difference-equation descriptions.
Outline (TODO — flesh out each)
- The z-transform — definition, region of convergence
- Poles and zeros
- The inverse z-transform (partial fractions, power series)
- System analysis via the z-transform
- The system transfer function
- Stability in terms of pole locations