How to use the Z-Transform Table and Calculator
- 1. Choose what you needA standard signal, a finite sequence, an inverse Z-transform or the table.
- 2. Enter X(z) or x[n]Use z⁻¹ coefficients like 1 −1.5 0.5, or an expression in z.
- 3. Get the answerX(z), ROC, poles, x[n] and the first samples.
Z-transform and its inverse
The Z-transform of a sequence is X(z) = Σ x[n]z⁻ⁿ, and the region of convergence (ROC) is the set of z where the sum converges. A right-sided signal has its ROC outside the largest pole, and a left-sided signal has it inside the smallest pole. To invert a rational X(z), divide by z, expand X(z)/z in partial fractions, multiply back by z and use the pairs z/(z − p) ↔ pⁿu[n] and z/(z − p)² ↔ n·pⁿ⁻¹u[n]. Repeated and complex poles are handled the same way.
Worked example: X(z) = 1/(1 − 1.5z⁻¹ + 0.5z⁻²) = z²/((z − 1)(z − 0.5)). Then X(z)/z = 2/(z − 1) − 1/(z − 0.5), so x[n] = (2 − 0.5ⁿ)u[n] = 1, 1.5, 1.75, 1.875 and so on. Long division gives the same samples. The ROC is |z| > 1.
Exam tips
- Always state the ROC. The same X(z) can belong to different sequences.
- A causal LTI system is stable when all of its poles are inside the unit circle.
- Use the initial value theorem, x[0] = lim X(z) as z → ∞, to check your answer.
- Complex pole pairs give a damped cosine: 2|c|·rⁿ·cos(θn + φ).
Frequently asked questions
What input formats does the inverse mode accept?
In z⁻¹ form, type coefficients from the constant term: 1 −1.5 0.5 means 1 − 1.5z⁻¹ + 0.5z⁻². In z form, type an expression such as z^2 or (z−1)(z−0.5).
Does it handle repeated poles?
Yes. Repeated poles give terms like n·pⁿ⁻¹, and the tool checks the closed-form answer against long division.
Which ROC does the inverse assume?
A causal (right-sided) sequence, with the ROC outside the largest pole. For other ROCs, the same partial fractions give different signals.
Is this tool free?
Yes. You fill in a short form once to see your result, and the other Uncodemy tools then open without it. All the maths runs in your browser.