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Z Transform Calculator

Evaluate eight causal Z-transform pairs or inverse sequences. Check real inputs, ROC limits, formulas, numerical warnings, graphs and shareable results.

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 · v1.0.5 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.5 · CVP protocol 1.0.0-proposed · Evidence 2026-10-28.metric-scope-v3
Verification revision
2026-10-28.metric-scope-v3 · 1/1 property · digest 8fac1edf05ef
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 8/8 (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. Raw HTTP REST/JSON-RPC transport rejection 5/5; raw-number source and conflicting pins are transport-only checks, not engine/SSR input-source parity.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
656/656 golden · 2/2 CVP boundary · 22/22 invalid · 1/1 property · 2/2 metamorphic · 26/26 cross-interface · 5/5 CVP contract · Manifest
Sources
Sources
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 656/656 oracle-backed golden · 22/22 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.5 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 1.0.5 · 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
1.0.5

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 v1.0.5 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.

Numerical precision and input policy

These named table identities use approximate binary64 evaluation, published to 12 significant decimal digits. The independent reference fixture tolerance is 2e-11 relative for each nonzero scalar; mathematical zero is checked exactly. This is a tested fixture policy, not a uniform error proof. Inputs a and z are dimensionless real values; omega is angular frequency in radians per sample. Forward z may be positive or negative, with |z| strictly greater than the stated conservative ROC radius, including zero-sine or zero-amplitude cases. Inverse n is an integer from 0 to 64. This table uses u[0]=1 and a^0=1 including a=0; negative a alternates with integer parity.

Finite decimal numbers and decimal strings are accepted. Omitted mode and pair default to z and geom; a defaults to 0.5 and omega to 1. Blank optional a/omega fields use those defaults. The active z or n is required; blank inactive coordinates are omitted during normalization. Supplied nonblank n must always be an integer0–64. Aliases method/mode, pair/sequence/f and w/omega must agree. Null, arrays, booleans, hexadecimal values and unknown fields are rejected. A nonzero decimal input that would become zero is rejected as NUMERIC_UNDERFLOW before conversion. The inverse entry fixes mode=inverse even if a different mode is in the URL; other explicit aliases still must agree with that mode.

Near an ROC boundary, the distance to the pole and the frequency affect sensitivity. Trigonometric zeros and geometric scales can amplify input or evaluation error. Stable normalized quotients avoid many intermediate scale failures; inverse phase-product rounding and coarse subnormal quantization have explicit NUMERICAL_RANGE_ERROR refusals. Unrepresentable nonzero values are not silently published as zero. Host transcendental functions remain approximate; cross-runtime fixtures do not certify every input. The inverse ROC describes the forward pair. The forward graph samples one connected real ROC branch; inverse stems show a labeled window of at most16 indices around n, not a complete infinite sequence. Missing or unresolved samples are reported, while the scalar value remains separately available.

Sharing and privacy

Interactive calculations run locally in the browser. Share URLs send the included inputs to the server and may appear in logs; REST and MCP receive their request inputs. Browser calculation history may store results on this device. Log retention, processing region and deletion guarantees are not established by this calculator. A shared version parameter identifies the source calculation version; reopening evaluates with the current engine and does not supply a historical execution environment.

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 Compute the direct DFT for real or complex sequences of 1–64 samples. Inspect every frequency bin and magnitude, share precise inputs and export full results. › → Discrete Fourier Transform Calculator Compute DFT, inverse DFT, FFT and IFFT for real or complex sequences up to 64 samples. Compare all bins, inspect magnitudes, share inputs and export results. › → FFT Calculator Compute radix-2 FFT for real or complex sequences with power-of-two lengths up to 64. Inspect all frequency bins, share inputs and export complete results. › → Inverse DFT Calculator Compute the inverse DFT with 1/N normalization for real or complex sequences up to 64 samples. Inspect all samples, share inputs and export complete results. › → Inverse FFT Calculator Compute radix-2 inverse FFT with 1/N normalization for sequences of 1–64 samples at power-of-two lengths. Inspect samples, share inputs and export all results. › → Inverse Laplace Calculator Evaluate inverse Laplace table pairs at nonnegative time. Check formulas, causal limits, numerical warnings, graphs and reproducible share links. › → Inverse Z Calculator Evaluate causal inverse Z-transform table pairs at integer sample indices. Check formulas, ROC, numerical warnings, stem plots and reproducible share links. › → Laplace Transform Calculator Evaluate eight causal Laplace pairs or their inverse with real inputs. See formulas, ROC limits, numerical warnings, graphs and shareable results. ›

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.