Z-Transform & Digital Mathematics

Mathematics of discrete-time systems. Discrete-time mathematics: the Z-transform, regions of convergence and difference equations, the DFT and FFT, and digital filter design — the mathematical foundation of digital signal processing.

• Z-Transform — The Z-transform is the Laplace transform for discrete-time systems. Learn the definition and ROC, common pairs, the delay property z⁻¹, difference equations and transfer functions H(z), poles and zeros in the z-plane, stability (|p| < 1), frequency response on the unit circle, and mapping s to z (bilinear transform).
• DFT & FFT — The discrete Fourier transform computes the spectrum of sampled data, and the FFT computes it fast. Learn the DFT definition, frequency bins and resolution, the radix-2 FFT and its N log N cost, windowing and spectral leakage, zero padding, amplitude scaling, real-signal FFTs, and applications — with a Python spectrum analyser.
• Digital Filters & DSP — Digital filters process sampled signals in software or hardware. Learn FIR vs IIR, moving-average filters, windowed-sinc FIR design, linear phase and group delay, IIR biquads, filter order estimation, fixed-point implementation, DSP processors and MAC operations — with a Python FIR design.