Z-Transform Calculator

Step-by-Step Forward & Inverse Symbolic CAS Free · No Signup

Compute forward and inverse Z-transforms with detailed step-by-step solutions. Features a complete Z-transform pairs reference table, region of convergence, discrete stem plots, and a built-in Python compiler. Essential for digital signal processing, control systems, and difference equations.

Z-Transform
Enter a function of n, or use AI above
Type an expression above…
x[n]X(z)ROC
all z
|z| > 1
|z| > 1
|z| > |a|
|z| > |a|
|z| > 1
|z| > 1
|z| > 1
|z| > 1
n^2    z^3    (z-1)^2
(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))
Multiplication: Use * explicitly: n*(1/2)**n
Powers: n^2 or (1/2)^n
Fractions: 1/2, 3/4, Rational(1,3)

Result

Z

Enter an expression and click Compute

Compute forward or inverse Z-transforms with step-by-step solutions.

x[n] vs n (Stem Plot)

Compute a transform to see the discrete stem plot.

Python Compiler

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.

Applications

🔧

Digital Filters

Design and analyze FIR and IIR filters using transfer functions H(z) in the Z-domain.

⚙

Control Systems

Discrete-time control system design, stability analysis via pole-zero placement in the z-plane.

🎤

Audio & Speech

Audio codec design, speech synthesis, echo cancellation, and adaptive filtering algorithms.

📡

Communications

Channel equalization, error correction coding, and digital modulation/demodulation systems.

Frequently Asked Questions

The Z-transform converts a discrete-time sequence x[n] into a complex function X(z) using the sum Z{x[n]} = sum from n=0 to infinity of x[n]*z^(-n). It is the discrete-time counterpart of the Laplace transform and is fundamental in digital signal processing and control systems.
The most common method for computing the inverse Z-transform is the residue method: compute X(z)*z^(n-1), find the poles of the denominator, and sum the residues at each pole. Alternatively, use partial fraction decomposition on X(z)/z, then look up each term in a Z-transform table.
The region of convergence is the set of complex values of z for which the Z-transform sum converges. For causal sequences (n >= 0), the ROC is the exterior of a circle |z| > r. The ROC determines the uniqueness of the inverse Z-transform.
The Laplace transform is for continuous-time signals f(t) and uses integration, while the Z-transform is for discrete-time sequences x[n] and uses summation. The relationship z = e^(sT) connects them, where T is the sampling period.
A transfer function H(z) = Y(z)/X(z) describes the input-output relationship of a discrete-time linear system. It characterizes digital filters and is used to analyze stability (all poles inside the unit circle), frequency response, and system behavior.
Yes, this calculator is completely free with no signup required. You get symbolic computation via SymPy, step-by-step solutions, discrete stem plots, LaTeX export, and a built-in Python compiler.

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