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Table lookup of the unilateral Z-transform for 11 standard pairs, with parameter substitution, region of convergence, derivation note, and optional numeric X(z₀) evaluation.
Execution
Fill in the form, run the tool, and review the result in one place.
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Tool usage guide
The Z-Transform Calculator performs fast unilateral Z-transform lookups across 11 standard discrete-time signal pairs. It substitutes sequence parameters into closed-form algebraic expressions, determines the Region of Convergence (ROC), provides concise derivation notes, and evaluates the transfer function at any valid complex or real point z₀.
Background
A student is analyzing the step response of a first-order digital low-pass filter with sequence x[n] = 0.5^n u[n].
Problem
Need to determine the closed-form Z-transform, ROC, and evaluate X(z) at z = 2.
How to use
Select the a^n sequence form, enter 0.5 for parameter a, specify 2 for evaluation point z₀, and set precision to 4 decimal places.
Outcome
Yields X(z) = z / (z - 0.5) with ROC |z| > 0.5 and computes X(2) = 1.3333.
Background
An engineer is modeling the oscillatory impulse response of a sampled second-order mechanical sensor.
Problem
Requires the algebraic transform and convergence boundary for sequence x[n] = 0.8^n · cos(0.5n).
How to use
Choose the a^n · cos(ω₀n) form, set parameter a to 0.8, set frequency ω₀ to 0.5 rad/sample, and enter z₀ = 2.
Outcome
Returns X(z) = z(z - 0.702066)/(z^2 - 1.40413z + 0.64) with ROC |z| > 0.8 and evaluates X(2) = 1.4172.
It is an algebraic transform defined as X(z) = Σ x[n]·z^(−n) summed from n = 0 to infinity, used to convert discrete-time domain signals into the complex frequency domain.
The calculator supports 11 common pairs: δ[n], u[n], a^n, n, n^2, n·a^n, sin(ω₀n), cos(ω₀n), a^n·sin(ω₀n), a^n·cos(ω₀n), and (−1)^n.
The ROC specifies the set of values in the complex z-plane for which the infinite summation converges, typically defined as |z| > |a| for causal unilateral signals.
No, the evaluation point z₀ must strictly satisfy the Region of Convergence condition to ensure the underlying power series converges.
Parameter a acts as the geometric base in power sequences and scales pole magnitudes in damped sinusoidal forms like a^n·cos(ω₀n).