Common Z-Transform Table
For discrete-time signal processing and system analysis
What is the Z-Transform?
The Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation.
Reference Table
Here are the most common Z-transform pairs used in engineering.
| Signal x[n] (n ≥ 0) | Z-Transform X(z) | ROC (Region of Convergence) |
|---|---|---|
δ[n] (Unit Impulse) |
1 |
All z |
u[n] (Unit Step) |
z / (z - 1) |
|z| > 1 |
a^n * u[n] (Exponential) |
z / (z - a) |
|z| > |a| |
n * u[n] (Unit Ramp) |
z / (z - 1)^2 |
|z| > 1 |
n * a^n * u[n] |
az / (z - a)^2 |
|z| > |a| |
sin(ωn) * u[n] |
(z * sin(ω)) / (z^2 - 2zcos(ω) + 1) |
|z| > 1 |
cos(ωn) * u[n] |
(z^2 - zcos(ω)) / (z^2 - 2zcos(ω) + 1) |
|z| > 1 |
Properties of ROC
- The ROC cannot contain any poles.
- For casual sequences (right-sided), the ROC is the exterior of a circle: |z| > r.
- For stable systems, the ROC must include the unit circle (|z| = 1).