HomeCalculatorsMathNumerical CalculusZ Transform Calculator
Math calculator

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 Engine tested · Specification checked · 10/10 tests · Production surface contract 3/3 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Unilateral Z or inverse of a causal elementary pair, evaluated at real z or integer n.
Scope
Unilateral, causal n≥0
Verification
Engine tested · 10/10 tests · Production surface contract 3/3 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 golden · 2 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

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.

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 1.0.0 · 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.
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
  • Not DFT / FFT
  • Not Laplace
  • Not residue / two-sided
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 3/3 · 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
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: Bessel I Calculator, Bessel J Calculator, Bessel K Calculator, Bessel Y Calculator, Beta Function Calculator, Brent Method Calculator, Definite Integral Calculator, DFT Calculator . Explore all Numerical Calculus.

→ Bessel I Calculator I_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel J Calculator J_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel K Calculator K_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel Y Calculator Y_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Beta Function Calculator Complete Euler B(x, y) = Γ(x)Γ(y)/Γ(x+y) for x>0, y>0. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally. › → Brent Method Calculator Brent’s method (golden section plus inverse parabolic interpolation) for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not root-finding Brent. Runs locally. › → Definite Integral Calculator Composite trapezoid, Simpson 1/3, or Gauss–Legendre of a polynomial, sin, or exp on [a, b]. Not indefinite symbolic integrate. Runs locally. › → 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. ›
}

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.