Z-transform: X(z) = Σₙ x[n]z⁻ⁿ. Converts difference equations to algebraic equations. Poles determine stability (inside unit circle = stable). Common: Z{aⁿu[n]} = z/(z−a), ROC: |z|>|a|. Inverse via partial fractions or contour integration.
How to Use
Select a sequence:
- x[n]: Preset sequence
- X(z): Transform formula
- Analysis: Poles, zeros, ROC
Stability
Causal + stable ⟺ all poles inside unit circle |z|<1. Poles on unit circle = marginally stable (pure oscillation). Poles outside = unstable (growing). This is the discrete equivalent of Re(s)<0 for Laplace.
Properties
- Time shift: Z{x[n−k]} = z⁻ᵏX(z)
- Convolution: Z{x*y} = X(z)Y(z)
- Initial value: x[0] = lim z→∞ X(z)
Step-by-Step Instructions
- 1Select sequence.
- 2View X(z).
- 3Find poles/zeros.
- 4Determine ROC.
- 5Evaluate X(z₀).