Z-Transform Calculator
Forward & inverse Z{x[n]} · partial fractions · ROC · step-by-step
Quick examples
Common Z-transform pairs
| x[n] | X(z) | ROC |
|---|---|---|
| all z | ||
| |z| > 1 | ||
| |z| > 1 | ||
| |z| > |a| | ||
| |z| > |a| | ||
| |z| > 1 | ||
| |z| > 1 | ||
| |z| > 1 | ||
| |z| > 1 |
Syntax help
(1/2)^n (3/4)^n (-1)^n
n*(1/2)^n 2^n
sin(pi*n/3) cos(pi*n/4)
z/(z-1) z/(z-1)^2
z^2/((z-1)*(z-1/2))
Multiply:
n*(1/2)**n
Powers: n^2 or (1/2)^n
Enter an expression and click Compute
Compute forward or inverse Z-transforms with step-by-step solutions.
Compute a transform to see the discrete stem plot.
What is the Z-Transform?
The Z-transform converts a discrete-time sequence x[n] into a complex function X(z). It is defined as Z{x[n]} = Σn=0∞ x[n] · z−n, where z is a complex variable. It is the discrete-time counterpart of the Laplace transform.
The Z-transform is fundamental in digital signal processing, discrete-time control systems, and the analysis of difference equations and digital filters.
Key Properties
Linearity
Z{ax[n] + by[n]} = aX(z) + bY(z). The transform distributes over addition and scalar multiplication.
Time-Shifting
Z{x[n−k]} = z−kX(z). A delay of k samples corresponds to multiplication by z−k.
Z-Domain Differentiation
Z{n·x[n]} = −z·dX(z)/dz. Multiplication by n in the time domain becomes differentiation in the z-domain.
Convolution
Z{x[n]*y[n]} = X(z)·Y(z). Convolution of sequences becomes multiplication of their Z-transforms.