Discrete-Time Signals and Systems in the z-Domain

Trace the historical development of the z-transform from difference equations to its modern formulation.

Derive the discrete-time Fourier transform (DTFT) and establish its relationship to the continuous-time Fourier transform of a sampled signal.

Analyze the periodicity of the DTFT spectrum and its implications.

Derive the z-transform as a generalization of the DTFT and establish its region of convergence (ROC).

Analyze the relationship between the z-plane unit circle and the DTFT frequency response.

Apply z-transform properties to analyze discrete-time LTI systems.

Analyze the stability and causality of discrete-time systems using pole-zero locations in the z-plane.

Derive the relationship between the s-plane poles of an analog system and the z-plane poles of its discrete-time counterpart via impulse invariance.