Z-Transform
Convolution works, but algebra is faster. The Z-transform converts recursive difference equations into polynomial equations — making filter design and stability analysis clean and tractable.
Time-domain convolution is infinite for IIR filters. Stability requires checking every h[n] sample. Filter design is nearly impossible without the right tool.
Each sample x[n] is weighted by z raised to −n. The complex variable z = rejω — on the unit circle r=1, this becomes the DTFT.
- δ[n] ↔ 1 (all z)
- u[n] ↔ z/(z−1), |z| > 1
- aⁿu[n] ↔ z/(z−a), |z| > |a|
- cos(ω₀n)u[n] ↔ rational with poles at ±ω₀
Three properties power all filter analysis: linearity, time shift, and convolution → multiplication.
Apply Z-transform to both sides of the difference equation. Every delay z−k multiplies. The ratio H(z) = Y(z)/X(z) is the transfer function — a rational polynomial.
→ H(z) = z/(z−a)
- No infinite sums — just polynomial algebra
- Poles and zeros reveal filter behavior
The substitution z = esT maps the s-plane to the z-plane. Left half-plane → inside unit circle. Imaginary axis → unit circle.
- X(z) = Σ x[n]·z⁻ⁿ — generalizes DTFT to the full complex plane
- Key pairs: δ↔1, u↔z/(z−1), aⁿu↔z/(z−a)
- Time shift property: delay k → multiply by z⁻ᵏ
- Convolution in time = multiplication in z-domain
- LCCDE → rational transfer function H(z) = Y(z)/X(z)
- z = e^(sT) links Laplace and Z-transform
- Left half s-plane ↔ inside unit circle in z-plane