Z-Transform
The Z-transform extends the DFT to analyze system behavior — transfer functions, stability, and frequency response.
Definition
X(z) = Σₙ₌₋∞^∞ x[n] · z⁻ⁿ where z ∈ ℂ
The DFT is the Z-transform evaluated on the unit circle: z = e^(j2πk/N).
Region of Convergence (ROC)
The set of z values where X(z) converges. Determines uniqueness and causality.
Causal signal (x[n] = 0 for n < 0): ROC is outside a circle |z| > r. Anti-causal signal: ROC is inside a circle |z| < r. Stable system: ROC includes the unit circle |z| = 1.
Common Z-Transform Pairs
| x[n] | X(z) | ROC | |---|---|---| | δ[n] | 1 | All z | | u[n] (unit step) | z/(z-1) | |z| > 1 | | aⁿu[n] | z/(z-a) | |z| > |a| | | nu[n] | z/(z-1)² | |z| > 1 |
Transfer Function
For an LTI system y[n] = x[n] * h[n]:
H(z) = Y(z) / X(z)
Difference equation: y[n] = Σ bₖx[n-k] - Σ aₖy[n-k]
H(z) = (b₀ + b₁z⁻¹ + ... + bₘz⁻ᴹ) / (1 + a₁z⁻¹ + ... + aₙz⁻ᴺ)
Poles and Zeros
Zeros: Values of z where H(z) = 0 (numerator roots). Output is zero at these frequencies.
Poles: Values of z where H(z) = ∞ (denominator roots). Resonance at these frequencies.
Stability: All poles must be inside the unit circle (|pole| < 1 for causal systems).
Pole-zero plot: Visualize system behavior. Zeros (○) and poles (×) in the z-plane. Frequency response is determined by the distance to poles and zeros from the unit circle.
Frequency Response from Z-Transform
Evaluate H(z) on the unit circle: z = e^(jω)
H(e^(jω)) = |H(e^(jω))| · e^(j∠H(e^(jω)))
Magnitude response: |H(e^(jω))| — gain at frequency ω. Phase response: ∠H(e^(jω)) — phase shift at frequency ω.
Near a pole: Magnitude increases (resonance). Near a zero: Magnitude decreases (notch).
Relationship to Laplace Transform
The Z-transform is the discrete-time equivalent of the Laplace transform.
z = e^(sT) where T = sampling period, s = Laplace variable
The unit circle in the z-plane maps to the imaginary axis in the s-plane.
Applications in CS
- Filter design: Specify desired pole/zero locations → determine filter coefficients.
- Control systems: Discrete-time controller design and stability analysis.
- Audio processing: Analyze and design audio effects (reverb, EQ, compressor).
- Communication systems: Channel modeling, equalization, modulation analysis.