Z-Transform Calculator

Compute one-sided Z-transforms and causal inverse Z-transforms with MiniWebtool’s Z-Transform Calculator. 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.

Need help using this tool? Ask AI

Gemini (copy & open)

For Gemini, paste the copied question into the chat. You can also copy it below.

View the question

Help me understand MiniWebtool's Z-Transform Calculator: https://miniwebtool.com/z-transform-calculator/. What can I use it for, what inputs or settings do I need, and how do I use it and understand the output? Walk me through one simple example. Use this page as your source and answer in English.

3 free uses dailyNo sign-up requiredInstant Results

Z-Transform CalculatorTry it now — free ▼

Z-transform: Σ x[n] z⁻ⁿ turns 0.5ⁿ into z/(z − 0.5), for |z| > 0.5

Enter a sequence formula, a list of samples or a rational X(z) below to get the transform, region of convergence, poles and zeros, with the steps.

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.

At a Glance

Z-Transform Calculator converts a discrete-time sequence into its one-sided Z-transform X(z) and expands a rational X(z) back into causal samples. It also returns the region of convergence, poles and zeros.

Z-Transform Calculator: features at a glance
Feature	What you get	Good for
Forward transform	Accepts constants, a^n, na^n, n^2a^n, sine and cosine terms, damped sinusoids, u[n], delta[n-k] or a finite sample list such as 1, 2, 3.	Textbook signals and short FIR filter coefficient lists
Inverse transform	Enter a rational X(z) in z, z^-1 or q = z^-1, such as z/(z-0.8). The coefficient of q^n is shown as x[n].	Recovering samples from a transfer function
ROC, poles and zeros	Returns the causal region of convergence, outside the outermost pole, along with the poles and zeros.	Checking stability and convergence for causal systems
z-plane plot	Marks zeros with circles, poles with crosses and the unit circle with a dashed reference curve.	Seeing pole and zero positions at a glance
Sample count	Choose 4 to 40 samples to show in the sample table and bar chart.	Looking at the first terms of a sequence
Example presets	Buttons load examples such as geometric decay, weighted ramp, cosine sequence, single pole and two-pole response.	Learning the input syntax from a working example

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

Z-Transform Calculator: 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

Chat with your favorite AI

Gemini (copy & open)

For Gemini, paste the copied question into the chat. You can also copy it below.

View the question

Help me understand MiniWebtool's Z-Transform Calculator: https://miniwebtool.com/z-transform-calculator/. What can I use it for, what inputs or settings do I need, and how do I use it and understand the output? Walk me through one simple example. Use this page as your source and answer in English.

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

Z-Transform Calculator

About Z-Transform Calculator

At a Glance

Z-Transform Formula

Supported Inputs

How to Use

Common Z-Transform Pairs

FAQ

What is the Z-transform?

What is the region of convergence?

Which inverse Z-transform does this calculator return?

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

What sequence formulas are supported for forward transforms?

References

Chat with your favorite AI

Calculus:

Math Solver

Numbers & factors

Fractions & division

Powers, roots & logs

Algebra & equations

Calculus

Geometry & trigonometry

Top & Updated:

Download everything as one ZIP with Pro