Z-Transform
Reference for the Z-transform X(z) = Σ x[n] z^(-n).
Generalizes the discrete Fourier transform for digital filter design and analysis.
The Definition
The z-transform converts a discrete-time signal x[n] into a complex-valued function X(z) of the complex variable z. It generalizes the discrete-time Fourier transform: substituting z = e^(iω) recovers the DTFT.
Unilateral (Causal) Z-Transform
For causal signals (zero for n < 0), the sum starts at zero. This form is what most controls and filter applications use.
Variables
| Symbol | Meaning |
|---|---|
| x[n] | Discrete-time signal (sample at index n) |
| X(z) | Z-transform of x[n] |
| z | Complex variable, often written z = r × e^(iω) |
| r | Magnitude of z |
| ω | Normalized angular frequency (radians/sample) |
| ROC | Region of convergence — values of z for which the sum converges |
Example — Unit Step
Find the z-transform of the unit step x[n] = 1 for n ≥ 0.
X(z) = Σ_{n=0}^{∞} 1 × z^(−n) = Σ_{n=0}^{∞} z^(−n)
This is a geometric series with ratio z^(−1)
Converges when |z^(−1)| < 1, i.e. |z| > 1
X(z) = z / (z − 1), ROC: |z| > 1
Example — Exponential Decay
Find the z-transform of x[n] = a^n × u[n], where 0 < a < 1.
X(z) = Σ_{n=0}^{∞} a^n × z^(−n) = Σ_{n=0}^{∞} (a / z)^n
Geometric series with ratio a/z, converges when |a/z| < 1, i.e. |z| > a
X(z) = z / (z − a), ROC: |z| > a
Key Properties
| Property | x[n] | X(z) |
|---|---|---|
| Linearity | a × x[n] + b × y[n] | a × X(z) + b × Y(z) |
| Time shift | x[n − n₀] | z^(−n₀) × X(z) |
| Convolution | x[n] * h[n] | X(z) × H(z) |
| Multiplication by n | n × x[n] | −z × dX/dz |
| Modulation | a^n × x[n] | X(z / a) |
| Initial value | x[0] | lim_{z→∞} X(z) |
| Final value | lim_{n→∞} x[n] | lim_{z→1} (z − 1) × X(z) |
Transfer Functions
For a linear time-invariant discrete system with input x[n] and output y[n] related by a difference equation, the z-transform turns convolution into multiplication. The system's transfer function H(z) = Y(z) / X(z) captures the system completely.
Example — Simple Moving Average Filter
A 3-sample moving average filter: y[n] = (x[n] + x[n−1] + x[n−2]) / 3. Find H(z).
Take z-transform of both sides
Y(z) = (1/3) × (X(z) + z^(−1) X(z) + z^(−2) X(z))
Y(z) = (1/3) × X(z) × (1 + z^(−1) + z^(−2))
H(z) = (1 + z^(−1) + z^(−2)) / 3
Stability via Pole Locations
A discrete-time LTI system is causal and stable if and only if all poles of its transfer function lie strictly inside the unit circle in the z-plane (|z| < 1). This is the discrete-time analog of the left-half-plane stability rule for continuous systems.
| Pole location | Time response | Stability |
|---|---|---|
| Inside unit circle | Decays to zero | Stable |
| On unit circle | Oscillates with constant amplitude | Marginally stable |
| Outside unit circle | Grows without bound | Unstable |
When to Use It
- Analyzing digital filters (FIR and IIR)
- Solving linear difference equations with z-domain algebra
- Stability analysis of digital control systems
- Designing IIR filters (Butterworth, Chebyshev, elliptic) via bilinear transform
- Modeling sampled-data control systems and discretized continuous plants
- System identification from input-output measurements
Relationship to Other Transforms
The z-transform on the unit circle (z = e^(iω)) is the discrete-time Fourier transform. The z-transform is to discrete systems what the Laplace transform is to continuous systems — the mapping is approximately s = (z − 1) / T (forward Euler) or s = 2/T × (z − 1) / (z + 1) (bilinear / Tustin), where T is the sampling period.