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Z-Transform Calculator (with ROC)

Both directions: a sequence such as 0.5^n, n·0.8^n, cos(πn/4) or a finite sample list into X(z) with poles, zeros and the causal ROC, and a rational X(z) — q accepted as shorthand for z⁻¹ — back into causal samples and a closed form.

Related tools

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Z-Transform Calculator (with ROC) is free to use as often as you like, directly from this page. It sits under Calculus in our catalogue, alongside Fast Fourier Transform (FFT) Calculator and Inverse Laplace Transform Calculator.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is Z-Transform Calculator (with ROC)?

Both directions: a sequence such as 0.5^n, n·0.8^n, cos(πn/4) or a finite sample list into X(z) with poles, zeros and the causal ROC, and a rational X(z) — q accepted as shorthand for z⁻¹ — back into causal samples and a closed form.

When would I actually use this?

Checking a derivative or an integral you worked out by hand, finding where a function turns, and getting a numeric answer when no closed form exists.

What is the most common mistake?

Integrating across a discontinuity as if it were not there. A numeric method will happily return a finite value for an integral that diverges — check the domain before trusting the number.

How is Z-Transform Calculator (with ROC) different from Fast Fourier Transform (FFT) Calculator?

They sit next to each other but answer different questions. Fast Fourier Transform (FFT) Calculator: radix-2 Cooley–Tukey FFT of real or complex samples (3+4i, −2i): magnitude, phase, frequency bins and a bar plot, with Hann, Hamming and Blackman windows corrected for coherent gain and one-sided bins correctly doubled except DC and Nyquist. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Inverse Laplace Transform Calculator is the closest one after this: f(t) from a rational F(s) by partial fractions, handling repeated roots and irreducible quadratics — 3/((s+2)²+9) comes back as a damped sine — with every residue listed and the expansion checked against F(s) numerically.

What else is worth having open alongside it?

Laplace Transform Calculator and Antilog Calculator (inverse logarithm) — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

Symbolic results are exact; numeric ones come from adaptive quadrature or a standard step method, and the tool reports which. Where both are available, compare them — a large gap means the problem is ill-conditioned.

Further reading

All guides →
ExplainerThe Pigeonhole Principle Proves a Thing Exists Without Finding ItTen items in three containers force four into one of them. 367 people force a shared birthday. Neither statement tells you which container or which pair.ExplainerA Cone Is Exactly One Third of Its CylinderRadius 5, height 10: the cone holds 261.8 and the cylinder around it holds 785.4. The ratio is exactly a third, and it holds for any base shape at all.ExplainerOne Formula That Is Also a Rectangle and Also a TriangleParallel sides of 8 and 5 with a height of 4 give an area of 26 — the average width times the height. Make the two sides equal and it is a rectangle; shrink one to nothing and it is a triangle.ExplainerTwice as Wide Holds Four Times as Much; Twice as Tall Holds TwiceA cylinder of radius 5 and height 10 holds 785.4 units of volume. Double the radius and it holds 3,141.6; double the height instead and it holds 1,570.8.ExplainerForty Out and Sixty Back Averages Forty-EightThe harmonic mean of 40 and 60 is 48, not 50, because the slower leg takes longer and therefore counts for more. For 1, 2 and 4 it is 1.714 against an arithmetic mean of 2.33.ExplainerIf the Variance Is Not the Mean, It Is Not PoissonWith an average of five events per interval, exactly three occur 14.0 % of the time, none at all 0.7 %, and ten or more 3.2 % — about once every 31 intervals. The distribution has one parameter, and it sets both the mean and the variance.