Skip to content
Allin
Sign up · Sign in

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

All Calculus tools →
Fast Fourier Transform (FFT) CalculatorRadix-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.Inverse Laplace Transform Calculatorf(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.Laplace Transform CalculatorF(s) from f(t) using the standard table plus first shifting, second shifting, multiplication by a polynomial in t and product-to-sum — and an explicit refusal, with the reason, for anything the table does not cover.Antilog Calculator (inverse logarithm)The antilogarithm undoes the logarithm: antilog_b(y) = bʸ, so log_b(bʸ) = y. Pick base 10, e, 2 or your own, enter the exponent, and the result comes back with the inverse relationship spelled out and the round trip checked. Bases of zero or less, and base 1, are rejected with the reason.Bernoulli ODE SolverSolves y′ + P(x)y = Q(x)yⁿ by showing the substitution v = y^(1−n) that turns it into a linear equation, then integrating. The two degenerate cases are handled explicitly: n = 0 is already linear, n = 1 is separable.Convolution CalculatorLinear and circular discrete convolution with the sliding-and-summing steps shown term by term, plus a symbolic continuous convolution solved exactly through the Laplace convolution theorem.Derivative CalculatorCompute single, partial, implicit and directional derivatives symbolically.Direction Field / Slope Field PlotterDraws f(x, y) as a grid of tangent dashes, so the shape of every solution is visible before a single one is computed. Add initial conditions and each spawns an RK4 curve that stays tangent to the field.

Z-Transform Calculator (with ROC) is free to use as often as you like, directly from this page. It covers 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 — adjust any of them and the result follows immediately.

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.

What does it take into account?

It factors in 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. Change any of them and the output follows immediately.

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 is the one to open when you need it to 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 Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.ExplainerWhat Is a Z-Score? Standard Deviations Above the MeanA z-score is z = (x − μ) / σ — how many standard deviations a value sits from the mean. Learn how to compute it, read the standard normal curve, and turn it into a percentile.How-toHow to Find the Distance Between Two PointsUse the distance formula d = √((x₂−x₁)² + (y₂−y₁)²) to measure the straight-line distance between two points on a plane, with a worked example and the midpoint too.How-toHow to Calculate the Slope of a Line: Rise Over RunFind the slope of a line from two points with m = (y₂ − y₁) / (x₂ − x₁), and read what a positive, negative, zero, or undefined slope tells you.