Compute Z-transforms and inverses for discrete-time sequences. Plot pole-zero diagrams and regions of convergence.
The Z-transform is to discrete-time signals what the Laplace transform is to continuous ones: it converts a sequence x[n] into a function X(z) of a complex variable, turning difference equations into algebra and convolution into multiplication. This tool computes X(z) for a pasted sequence, computes x[n] back from a rational X(z), and plots the pole-zero diagram with its region of convergence in the z-plane. The thing most users get wrong: the transform is not complete without the ROC, and two different sequences can produce the exact same rational X(z).
A concrete input and expected output from the current implementation.
Input
x[n] = (0.5)^n u[n]
Expected output
X(z) = z/(z - 0.5) = 1/(1 - 0.5 z^(-1)), valid for |z| > 0.5. Pole at z = 0.5, zero at z = 0. Pole-zero plot: ROC is the exterior of the circle of radius 0.5 in the z-plane.
Each sample contributes the term (0.5 z^(-1))^n, so the sum from n = 0 is a geometric series closing to 1/(1 - 0.5 z^(-1)) = z/(z - 0.5), which converges exactly when |0.5 z^(-1)| < 1, that is |z| > 0.5. The single pole sits at the ratio 0.5, inside the unit circle, matching a stable causal sequence.
The tool evaluates X(z) = sum over all n of x[n] z^(-n). Finite sequences expand term by term into a polynomial in z^(-1): a length-4 pulse 1, 1, 1, 1 becomes 1 + z^(-1) + z^(-2) + z^(-3). Exponential terms like a^n u[n] form a geometric series closing to z/(z - a), valid only where the series converges, and that boundary becomes the ROC circle on the plot.
For the inverse, the tool takes a rational X(z), factors numerator and denominator into zeros and poles, and expands X(z)/z into partial fractions whose terms map back to standard pairs such as a^n u[n] or n a^n u[n]. The ROC picks which side of each pole the sequence occupies: |z| > |a| gives a right-sided causal term, |z| < |a| an anti-causal one.
Almost always an ROC mismatch. The rational function alone is ambiguous: z/(z - 0.5) inverts to (0.5)^n u[n] for |z| > 0.5 and to -(0.5)^n u[-n-1] for |z| < 0.5, with rings between poles giving two-sided sequences. The tool can return only one, so it follows the ROC you give, or a causal default if you give none. State the ROC and the result will match.
The FFT samples a signal at N frequencies on the unit circle; the Z-transform is a function of the whole complex variable z. Evaluated on the unit circle it becomes the discrete-time Fourier transform, and the DFT is that function sampled at N equally spaced angles. The pole-zero diagram adds what frequency samples hide: where the transform blows up, which side the sequence extends on, and where the sum converges.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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