Z-Transform Formula
The Z-transform converts discrete-time signals into the complex frequency domain.
The digital equivalent of the Laplace transform, used in DSP and digital control.
The Formula
z = ejωT on the unit circle (DTFT)
The Z-transform converts a discrete-time sequence x[n] into a function of the complex variable z. It is to digital signal processing (DSP) what the Laplace transform is to analog circuits. The Z-transform enables algebraic manipulation of difference equations, just as the Laplace transform turns differential equations into algebraic equations.
Common Z-Transform Pairs
| Sequence x[n] | Z-transform X(z) | Region of Convergence |
|---|---|---|
| Unit impulse δ[n] | 1 | All z |
| Unit step u[n] | z / (z − 1) | |z| > 1 |
| a^n u[n] | z / (z − a) | |z| > |a| |
| n a^n u[n] | az / (z − a)² | |z| > |a| |
| cos(ω&sub0;n) u[n] | z(z − cosω&sub0;) / (z² − 2z cosω&sub0; + 1) | |z| > 1 |
Example 1 — Difference Equation Solution
Solve the difference equation y[n] − 0.5y[n−1] = x[n] with x[n] = u[n] (unit step) and zero initial conditions.
Take Z-transform: Y(z) − 0.5z&sup-1;Y(z) = X(z) = z/(z−1)
Y(z)(1 − 0.5z&sup-1;) = z/(z−1) → Y(z) = z/(z−1) × z/(z−0.5)
Y(z)/z = z/((z−1)(z−0.5)) = A/(z−1) + B/(z−0.5) [partial fractions]
A = z/(z−0.5)|_{z=1} = 1/0.5 = 2; B = z/(z−1)|_{z=0.5} = 0.5/(−0.5) = −1
y[n] = 2(1)^n − 1(0.5)^n = 2 − (0.5)^n for n ≥ 0 — approaches steady state of 2 exponentially
Example 2 — Digital Filter Design
A first-order IIR low-pass filter has transfer function H(z) = (1−a)/(1 − az&sup-1;). For a = 0.9, find the −3 dB frequency (normalized).
On the unit circle: z = e^(jω), H(e^(jω)) = 0.1/(1 − 0.9e^(−jω))
At DC (ω = 0): |H| = 0.1/0.1 = 1
|H| = 1/√2 when |1 − 0.9e^(−jω)|² = 2 × 0.01 = 0.02
Solving: ω ≈ 0.211 radians/sample ≈ 6.7% of the sampling rate. This filter has a gentle 6.7% bandwidth.
When to Use It
Use the Z-transform when:
- Designing and analyzing digital filters (IIR, FIR)
- Solving linear difference equations analytically
- Implementing digital control systems (PID controllers in microcontrollers)
- Analyzing stability of digital systems (poles must be inside the unit circle)
- Converting analog filter designs to digital equivalents (bilinear transform)