Inverse Z-Transform Calculator
Find the discrete sequence from a standard-form Z-transform expression.
About Inverse Z-Transform Calculator
Finds the inverse Z-transform x[n] = Z⁻¹{X(z)} for standard forms. Supported: • Z⁻¹{z/(z-a)} = aⁿ·u[n] • Z⁻¹{az/(z-a)²} = n·aⁿ·u[n] • Z⁻¹{1} = δ[n] • Z⁻¹{z/(z-1)} = u[n] • Z⁻¹{z/(z-1)²} = n·u[n]
Formula
Inverse Z-transform
x[n] = Z⁻¹{X(z)}
- x[n]
- Discrete sequence
- X(z)
- Z-domain transform
Worked example
With the default values below, Inverse Z-Transform Calculator returns x[n] value of 0.03125000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- X(z) formone of z/(z-a) → aⁿu[n], az/(z-a)² → n·aⁿu[n], 1 → δ[n], z/(z-1) → u[n], z/(z-1)² → n·u[n]
- z/(z-a) → aⁿu[n]
- a (base)
- 0.5
- Evaluate x[n] at n =minimum 0
- 5
Results
- x[n] value
- 0.03125000
- x[n] =
- x[n] = 0.5^n = 0.03125
Step by step
- 1Values used
- 2Inverse Z-transform
- 3x[n] value= 0.03125000
- 4x[n] == x[n] = 0.5^n = 0.03125
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency 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
How do I invert a complex X(z)?
Use partial fraction decomposition to break X(z)/z into simpler terms, then use the standard inverse transform pairs for each term.
What does u[n] mean?
u[n] is the unit step sequence: u[n] = 1 for n ≥ 0, and 0 for n < 0. It indicates the sequence is causal (right-sided).