Z-Transform Calculator
Compute the Z-transform of standard discrete sequences from coefficient parameters.
About Z-Transform Calculator
Computes the Z-transform X(z) = Σ x[n]·z⁻ⁿ for standard discrete sequences. Supported transforms: • Z{aⁿu[n]} = z/(z-a), |z|>|a| • Z{n·aⁿu[n]} = az/(z-a)², |z|>|a| • Z{δ[n]} = 1, all z • Z{u[n]} = z/(z-1), |z|>1 • Z{n·u[n]} = z/(z-1)², |z|>1 • Z{sin(ωn)u[n]} = z·sin(ω)/(z²-2z·cos(ω)+1) • Z{cos(ωn)u[n]} = z(z-cos(ω))/(z²-2z·cos(ω)+1)
Formula
Z-transform definition
X(z) = Σ x[n]·z^(-n) for n = 0 to ∞
- X(z)
- Z-domain transform
- x[n]
- Discrete sequence
- z
- Complex variable
Worked example
With the default values below, Z-Transform Calculator returns X(z) value of 1.33333333. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Sequence typeone of aⁿ·u[n] (geometric), n·aⁿ·u[n], δ[n] (unit impulse), u[n] (unit step), n·u[n] (ramp), sin(ωn)·u[n], cos(ωn)·u[n]
- aⁿ·u[n] (geometric)
- a (base)
- 0.5
- ω (angular frequency)
- 0.5
- Evaluate X(z) at z =
- 2
Results
- X(z) value
- 1.33333333
- Z{x[n]} = X(z)
- Z{0.5^n · u[n]} = z/(z - 0.5)
- ROC
- |z| > 0.5
Step by step
- 1Values used
- 2Z-transform definition
- 3X(z) value= 1.33333333
- 4Z{x[n]} = X(z)= Z{0.5^n · u[n]} = z/(z - 0.5)
- 5ROC= |z| > 0.5
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax or fees or charges into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What is the Z-transform used for?
It's the discrete-time equivalent of the Laplace transform, used in digital signal processing and control systems to analyze discrete-time systems.
What is the region of convergence (ROC)?
The ROC defines the values of z for which the Z-transform sum converges. It determines whether the system is causal and stable.