Z-Transform Table Calculator: Transforms and Inverse Transforms of Standard Sequences
The standard Z-transform pairs, evaluated at a point.
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How this calculator works
Choose the transform pair and whether you want the transform or the inverse.
Enter K, a and w for the pair.
Enter the value of z or the sample number n.
Formulas used
- F(z) = \sum_{n=0}^{\infty}f[n]z^{-n}
- \mathcal{Z}\{a^n\} = \frac{z}{z - a}
- \mathcal{Z}\{\sin\omega n\} = \frac{z\sin\omega}{z^2 - 2z\cos\omega + 1}
Worked example
With the default numbers in the calculator above:
- The sequence: Geometric: f[n] = K a^n, F(z) = K z / (z - a)
- Its Z-transform: See the formula (Z{f[n]} = F(z))
- F(z) at the value of z you entered: 1.33333
Change the numbers in the calculator to see the full working, every step, for your own case.
A second worked example
A different case, with these numbers: Transform pair Sine: f[n] = K sin(w n), F(z) = K z sin(w) / (z^2 - 2 z cos(w) + 1).
- The sequence: Sine: f[n] = K sin(w n), F(z) = K z sin(w) / (z^2 - 2 z cos(w) + 1)
- Its Z-transform: See the formula (Z{f[n]} = F(z))
- F(z) at the value of z you entered: 0.643667
Common questions
What is the Z-transform used for?
It does for discrete-time signals and digital filters what the Laplace transform does for continuous ones. It turns difference equations into algebra.
What is the region of convergence?
The values of z for which the sum converges. For a geometric sequence a to the n it is the size of z being bigger than the size of a.
How do I invert a complicated F(z)?
Use partial fractions to split it into standard terms, then invert each term from the table.
Related calculators
Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any signals and systems or digital control text (the Z-transform table).
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