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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.

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
Inverse unilateral Z of a causal elementary pair, evaluated at integer n≥0, with an engine-linked stem figure of x[n].
Scope
Mode locked to inverse
Verification
Engine tested · Specification checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.special-transform-cvp
Verification revision
2026-09-22.special-transform-cvp · 1/1 property · digest 06304bdbc847
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 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (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
9/9 golden · 2/2 CVP boundary · 4/4 invalid · 1/1 property · 3/3 cross-interface · 5/5 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 9/9 oracle-backed golden · 4/4 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 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.

GeometricZ^{-1}{z/(z−a)} = a^n
SineZ^{-1}{z sin(ω)/(z²−2z cos(ω)+1)} = sin(ω n)
iSame engine as /z-transform. Laplace and DFT are other capabilities. Forward Z lives on the hub.

How to use

1

Choose a pair and n≥0

geom uses a; sin/cos use ω. Default (1/2)^n at n=0 is 1.

2

Read x[n]

The matching X(z) formula is shown. Poles and ROC belong to the forward transform. The figure is a stem of x[n].

Example calculations

Common configurations with formula and result.

ϟ

(1/2)^n at n=0

f=geom, a=0.5, n=0

a^n
1
ϟ

Ramp at n=3

f=ramp, n=3

n
3

Inverse Z calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Inverse of the unilateral Z table of causal elementary pairs, evaluated at integer n≥0. This page is mode=inverse on math.transform.z. Not a second engine. Not residue contour inversion. Not Laplace. Not CAS.
What it calculates
Inverse unilateral Z of a causal elementary pair, evaluated at integer n≥0, with an engine-linked stem figure of x[n].
Inputs
  • f
  • n
  • a?
  • omega?
Outputs
  • value
  • f
  • roc
  • algorithm
Formula
table inverse of X(z)
Assumptions
  • Mode locked to inverse
  • Causal integer n≥0
  • The Result Card includes an engine-linked stem figure of x[n] (at most 16 indices, marker at the published n). Same table as REST/MCP.
  • Table of named pairs, not residue, not CAS
Units
  • dimensionless
Boundary conditions
  • n not an integer 0–64 → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
f=geom a=0.5 n=0 → 1
Validation cases

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

  • f=geom a=0.5 n=0 → 1
  • mode=laplace → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: DFT Calculator, Discrete Fourier Transform Calculator, FFT Calculator, Inverse DFT Calculator, Inverse FFT Calculator, Inverse Laplace Calculator, Laplace Transform Calculator, Z 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. › → 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. › → 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. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second Z engine?

No. It is mode=inverse on math.transform.z, the same engine as /calc/math/z-transform.

Is this residue inversion?

No. Table inverse of the same causal pairs. Not a contour integral. Not a CAS.

Where does this run?

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