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

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.