Z-Transform Calculator

Compute one-sided Z-transforms and causal inverse Z-transforms. Enter a sequence formula, finite sample list, or rational X(z) to get the transform, ROC, poles and zeros, sample table, and visual z-plane analysis.

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x[0]

q²

X(z)

Sequence samples become powers of $z^{-1}$. Use q notation, direct $z^{-1}$, or common z-domain fractions. The calculator reports the causal ROC and draws the pole-zero picture.

Quick examples

Σ Forward Z-transform Sequence x[n] → X(z) ↺ Inverse Z-transform Rational X(z) → samples

Sequence or transform

Forward examples: 0.5^n, n*0.8^n, cos(pi/4*n), or 1, 2, 3.

Samples

Displayed x[n] terms.

Use q as shorthand for z^-1. All inverse results are causal one-sided expansions.

Embed Z-Transform Calculator Widget

About Z-Transform Calculator

The Z-Transform Calculator converts discrete-time sequences into $X(z)$ and expands rational transforms back into causal samples. It is designed for digital signal processing, control systems, recurrence relations, difference equations, and filter analysis where the pole-zero picture and region of convergence are just as important as the algebraic formula.

Z-Transform Formula

$$X(z)=\mathcal{Z}\{x[n]\}=\sum_{n=0}^{\infty}x[n]z^{-n}$$

This tool uses the one-sided convention by default. For inverse transforms, it rewrites the input as a rational function of $q=z^{-1}$, expands $X(q)=N(q)/D(q)$, and reads the coefficient of $q^n$ as $x[n]$.

Supported Inputs

Forward mode: finite sample lists such as 1, 2, 3, constants, a^n, n*a^n, n^2*a^n, sinusoids such as cos(pi/4*n), damped sinusoids, u[n], and delta[n-k].

Inverse mode: rational expressions in $z$, $z^{-1}$, or $q$. Examples include 1/(1 - 0.5*z^-1), z/(z - 0.8), and (1 + 2*q)/(1 - 0.75*q + 0.125*q^2).

Visual output: the z-plane plot marks zeros with ○, poles with ×, and the unit circle with a dashed reference curve so stability and frequency behavior are easier to inspect.

How to Use

Choose a transform direction. Select Forward Z-transform for a sequence x[n], or Inverse Z-transform for a rational X(z).
Enter the sequence or transform. For forward transforms, enter a formula such as 0.5^n, n*0.8^n, cos(pi/4*n), or a finite sample list. For inverse transforms, enter a rational expression such as 1/(1 - 0.5*z^-1).
Set the sample count. Choose how many x[n] samples to display in the table and bar chart.
Click Calculate. The calculator returns the Z-transform expression, causal ROC, poles, zeros, first samples, and computation notes.
Review the z-plane. Use the z-plane diagram to inspect pole locations, zero locations, and the unit circle reference.

Common Z-Transform Pairs

Sequence	Z-transform	Causal ROC
$a^n u[n]$	$\frac{1}{1-az^{-1}}$	$\|z\|>\|a\|$
$n a^n u[n]$	$\frac{az^{-1}}{(1-az^{-1})^2}$	$\|z\|>\|a\|$
$\cos(\omega n)u[n]$	$\frac{1-\cos(\omega)z^{-1}}{1-2\cos(\omega)z^{-1}+z^{-2}}$	$\|z\|>1$
$\sin(\omega n)u[n]$	$\frac{\sin(\omega)z^{-1}}{1-2\cos(\omega)z^{-1}+z^{-2}}$	$\|z\|>1$

FAQ

What is the Z-transform?

The Z-transform converts a discrete-time sequence x[n] into a complex-domain function X(z) = Σ x[n]z^-n. It is the discrete-time counterpart of the Laplace transform and is widely used for digital filters, signal analysis, control systems, and recurrence relations.

What is the region of convergence?

The region of convergence, or ROC, is the set of z values for which the infinite Z-transform sum converges. For right-sided causal sequences, the ROC is outside the outermost pole, so it has the form |z| greater than a pole radius.

Which inverse Z-transform does this calculator return?

This calculator returns the causal one-sided inverse Z-transform. It expands X(z) as a power series in q = z^-1, so the coefficient of q^n is the displayed x[n] sample.

Can I enter expressions like z/(z-a)?

Yes. The parser accepts z-domain expressions such as z/(z-0.5), as well as q notation where q = z^-1 and direct z^-1 notation such as 1/(1 - 0.5*z^-1).

What sequence formulas are supported for forward transforms?

The forward mode supports finite sample lists and common right-sided formulas including constants, a^n, n*a^n, n^2*a^n, sin(ωn), cos(ωn), scaled damped sinusoids, u[n], and delta[n-k].

References

Reference this content, page, or tool as:

"Z-Transform Calculator" at https://MiniWebtool.com/z-transform-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Apr 24, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Sequence	Z-transform	Causal ROC
\(a^n u[n]\)	\(\frac{1}{1-az^{-1}}\)	\(\|z\|>\|a\|\)
\(n a^n u[n]\)	\(\frac{az^{-1}}{(1-az^{-1})^2}\)	\(\|z\|>\|a\|\)
\(\cos(\omega n)u[n]\)	\(\frac{1-\cos(\omega)z^{-1}}{1-2\cos(\omega)z^{-1}+z^{-2}}\)	\(\|z\|>1\)
\(\sin(\omega n)u[n]\)	\(\frac{\sin(\omega)z^{-1}}{1-2\cos(\omega)z^{-1}+z^{-2}}\)	\(\|z\|>1\)