Z-Transform in MATLAB

Notes on Z-Transform in MATLAB

The Z-transform is the discrete-time counterpart of the Laplace transform, used for analyzing digital filters, control systems, and signal processing. MATLAB's Signal Processing Toolbox provides powerful functions for symbolic and numerical Z-transform computation, pole-zero analysis, and time-domain responses.

Key Functions

• ztrans() (Symbolic Math Toolbox): Symbolic Z-transform.
• tf() + dimpulse(), dstep(): Numerical responses for transfer functions.
• zplane(z,p) or zplane(sys): Pole-zero plot in the z-plane (unit circle for stability).

MathWorks example of zero-pole plot using zplane.

Residue plot from partial fraction expansion.

• residuez(): Partial fraction expansion (residues, poles, direct term) for inverse Z-transform.

Simple pole-zero map example.

• freqz(): Frequency response of digital filter (from z-domain).
• impz(): Impulse response from filter coefficients.

Basic Steps

1. Define Discrete Transfer Function:
   MATLABCopy

   b = [1 0];
   a = [1 -0.5];
   % Example: H(z) = z / (z - 0.5)  sys = tf(b, a, -1);
   % Ts = -1 indicates discrete (z-domain)

2. Pole-Zero Plot:
   zplane(b, a); grid on;

   MATLABCopy

   title('Pole-Zero Plot');

3. Impulse Response:
   impz(b, a, 20); % First 20 samples

Example discrete-time impulse response plot.

4. **Partial Fraction Expansion**:  ```matlab  [r, p, k] = residuez(b, a);
% r: residues, p: poles, k: direct term  disp([r p k]);

Symbolic Example

syms z n  H = z / (z - 0.5);
h = iztrans(H);
% Inverse:  (0.5)^n * u(n)

• Stability: Poles inside unit circle → stable.
• Use fvtool(b,a) for interactive filter visualization (magnitude, phase, etc.).
• Combine with filter() to process signals.