Inverse Z Calculator

What does this inverse z calculator compute?

Inverse of the unilateral Z table of causal elementary pairs, evaluated at integer n≥0.

The published formula or method is table inverse of X(z). 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, n=0 with a^n. 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.