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Z Transform Calculator

Unilateral Z table of causal elementary pairs: a^n, step, ramp, n a^n, sin, cos, damped sin/cos. Evaluate X(z) or x[n]. Discovery /inverse-z. Not DFT. Not Laplace. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 3/3 declared partitions (z, inverse, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Unilateral Z or inverse of a causal elementary pair, evaluated at real z or integer n, with an engine-linked figure of X(z) in the ROC or a stem of x[n].
Scope
Unilateral, causal n≥0
Verification
Engine tested · Source checked · v1.0.4 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.4 · CVP protocol 1.0.0-proposed · Evidence 2026-10-12.transform-runtime-review
Verification revision
2026-10-12.transform-runtime-review · 1/1 property · digest ba53df66ade3
Legacy regression
10/10 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 5/5 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
656/656 golden · 2/2 CVP boundary · 12/12 invalid · 1/1 property · 18/18 cross-interface · 5/5 CVP contract · Manifest
Sources
Sources
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 656/656 oracle-backed golden · 12/12 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.4 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

DefinitionX(z) = Σ_{n=0}^∞ x[n] z^{−n}
GeometricZ{a^n} = z/(z−a), |z|>|a|
iTable pairs only. Z{(1/2)^n} at z=1 is 2. Z{1} at z=2 is 2. Inverse at n=0 of (1/2)^n is 1. DFT and Laplace are other canonicals.

How to use

1

Choose Z or inverse, and a pair

Forward needs |z| in the ROC. Inverse needs integer n≥0. geom uses a; sin/cos use ω.

2

Read X(z) or x[n]

The formula and ROC are part of the result. Poles sit on the ROC boundary. The figure is X(z) on the real ROC, or a stem of x[n].

Example calculations

Common configurations with formula and result.

ϟ

Z{(1/2)^n} at z=1

f=geom, a=0.5, z=1

z/(z−a)
2
ϟ

Z{1} at z=2

f=step, z=2

z/(z−1)
2
ϟ

Inverse (1/2)^n at n=0

mode=inverse, f=geom, a=0.5, n=0

a^n
1

Z Transform calculator specification

Version 1.0.4 · Engine tested

Calculation status

Review policy · Evidence

Definition
The unilateral Z-transform is X(z)=Σ_{n=0}^∞ x[n] z^{−n}. This seed evaluates a table of causal elementary pairs at a real z (forward) or integer n (inverse). Discovery /inverse-z. Not a CAS, not residue inversion, not two-sided, not Laplace, not DFT.
What it calculates
Unilateral Z or inverse of a causal elementary pair, evaluated at real z or integer n, with an engine-linked figure of X(z) in the ROC or a stem of x[n].
Inputs
  • mode
  • f
  • z?
  • n?
  • a?
  • omega?
Outputs
  • value
  • f
  • roc
  • algorithm
Formula
X(z)=Σ x[n] z^{−n} (table)
Assumptions
  • Unilateral, causal n≥0
  • Real z in the ROC
  • Table of named pairs, not a CAS
  • The Result Card includes an engine-linked figure of X(z) on the real ROC (pole at |z|=|a|) or a stem of x[n] (at most 16 indices). Same table as REST/MCP.
  • Not DFT / FFT
  • Not Laplace
  • Not residue / two-sided
  • Not a pole-zero Argand of complex z
Units
  • dimensionless
Boundary conditions
  • z not in ROC → INVALID_INPUT
  • n not an integer 0–64 → INVALID_INPUT
  • unknown mode (including dft, laplace, cas) → INVALID_MODE
Example
mode=z f=geom a=0.5 z=1 → 2
Validation cases

3 published on this page · 10/10 tests · Production surface contract 5/5 · View evidence

  • mode=z f=geom a=0.5 z=1 → 2
  • mode=z f=step z=2 → 2
  • mode=inverse f=geom a=0.5 n=0 → 1
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
  • Oppenheim & Schafer, Discrete-Time Signal Processing — unilateral Z table
    Supports: Z{a^n}=z/(z−a); Z{1}=z/(z−1); ROC |z|>pole
  • MIT Signals and Systems — Lecture 22: The z-Transform
    Supports: The z-transform, its relation to the discrete-time Fourier transform, inverse transform and region of convergence. A rational expression alone does not identify the time sequence without its ROC. Mathematical reference only; not validation or endorsement of this implementation.
Calculation version
1.0.4

Background

Interpretation and common distinctions.

What does this z transform calculator compute?

The unilateral Z-transform is X(z)=Σ_{n=0}^∞ x[n] z^{−n}. This calculator evaluates a table of causal elementary pairs at a real z (forward) or integer n (inverse).

The published formula or method is X(z)=Σ x[n] z^{−n} (table). The following references support the stated mathematics or method:

  • MIT Signals and Systems — Lecture 22: The z-Transform: The z-transform, its relation to the discrete-time Fourier transform, inverse transform and region of convergence. A rational expression alone does not identify the time sequence without its ROC. Mathematical reference only; not validation or endorsement of this implementation.

Which inputs and limits should you check?

The table is unilateral and causal. Check the region of convergence and distinguish forward evaluation at z from an inverse sequence value at integer n.

The worked example uses f=geom, a=0.5, z=1 with z/(z−a). Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.transform.z v1.0.4 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Other calculators in this family: DFT Calculator, Discrete Fourier Transform Calculator, FFT Calculator, Inverse DFT Calculator, Inverse FFT Calculator, Inverse Laplace Calculator, Inverse Z Calculator, Laplace Transform Calculator . Explore all Transforms.

→ DFT Calculator Forward DFT X[k]=Σ x[n] e^{−2πi kn/N}, N≤64, unnormalized. Same engine as Discrete Fourier Transform. Radix-2 FFT is on /fft. Not inverse, not Laplace. Runs locally. › → Discrete Fourier Transform Calculator Direct DFT / IDFT and radix-2 FFT / IFFT of a length-N sequence, N≤64. FFT requires N=2^k. Numpy backward: unnormalized forward, 1/N inverse. Discovery /dft, /inverse-dft, /fft, /ifft. Not Laplace. Not CAS. Runs locally. › → FFT Calculator Radix-2 Cooley–Tukey FFT, N=2^k ≤64, unnormalized. Same engine as Discrete Fourier Transform. Numpy backward. Not a second DFT engine. Not CAS. Runs locally. › → Inverse DFT Calculator Inverse DFT x[n]=(1/N) Σ X[k] e^{2πi kn/N}, N≤64, Numpy backward. Same engine as Discrete Fourier Transform. Radix-2 FFT is on /fft. Not Laplace. Runs locally. › → Inverse FFT Calculator Radix-2 Cooley–Tukey inverse FFT, N=2^k ≤64, Numpy backward 1/N. Same engine as Discrete Fourier Transform. Not a second DFT engine. Not CAS. Runs locally. › → Inverse Laplace Calculator Inverse of the unilateral Laplace table: exp, sin, cos, step, ramp, t^n, damped sin/cos at t≥0. Same engine as Laplace Transform. Not Bromwich. Not Z. Not CAS. Runs locally. › → Inverse Z Calculator Inverse of the unilateral Z table: a^n, step, ramp, n a^n, sin, cos, damped sin/cos at integer n≥0. Same engine as Z Transform. Not residue. Not Laplace. Not CAS. Runs locally. › → Laplace Transform Calculator Unilateral Laplace table of causal elementary pairs: exp, sin, cos, step, ramp, t^n, damped sin/cos. Evaluate F(s) or f(t). Discovery /inverse-laplace. Not DFT. Not Z. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a CAS Z of an arbitrary sequence?

No. This seed is a unilateral table of named causal pairs, evaluated at a real z or integer n. Not residue inversion. Not symbolic.

Is this DFT or Laplace?

No. DFT is math.transform.dft. Laplace is math.transform.laplace. Not CAS.

Where does this run?

Locally in the browser by default. REST and MCP call the same z-transform engine.