Z-Transform Calculator
Compute the Z-transform of standard discrete sequences from coefficient parameters.
Inputs
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
Values used
Sequence type = aⁿ·u[n] (geometric); a (base) = 0.5000; ω (angular frequency) = 0.5000; Evaluate X(z) at z = = 2
Z-transform definition
X(z) = Σ x[n]·z^(-n) for n = 0 to ∞
X(z) value
= 1.33333333
Z{x[n]} = X(z)
= Z{0.5^n · u[n]} = z/(z - 0.5)
ROC
= |z| > 0.5
How it works
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
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.