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Z-transform for discrete-time signal analysis. Z-transform and inverse z-transform for discrete-time signal analysis.
Table-based implementation covering common discrete-time sequences. The z-transform of a sequence x[n] is X(z) = Σ x[n] z⁻ⁿ.
§Supported transforms (forward)
| Time domain | Z domain |
|---|---|
| c (constant) | c·z/(z−1) |
| aⁿ | z/(z−a) |
| n·aⁿ | a·z/(z−a)² |
| sin(ωn) | z·sin(ω) / (z²−2z·cos(ω)+1) |
| cos(ωn) | z·(z−cos(ω)) / (z²−2z·cos(ω)+1) |
| aⁿ·sin(ωn) | a·z·sin(ω) / (z²−2a·z·cos(ω)+a²) |
| aⁿ·cos(ωn) | z·(z−a·cos(ω)) / (z²−2a·z·cos(ω)+a²) |
| nᵏ·x[n] | (−z d/dz)ᵏ X(z) (so n²aⁿ, n³, …) |
| δ[n−k] | z⁻ᵏ |
| H(n−k) | z⁻ᵏ·z/(z−1) |
| C(n, k) | z/(z−1)^(k+1) |
| 1/n! | e^(1/z) |
| aⁿ·x[n] | X(z/a) |
| x[n−k]·H(n−k) | z⁻ᵏ·X(z) |
Plus linearity (sum of terms) and constant factor extraction.
The inverse handles rational X(z) through partial fractions (terms
z/(z−a)ᵐ → C(n, m−1) a^(n−m+1), 1/(z−a)ᵐ via the delay rule),
constants (δ\[n\]), z⁻ᵏ (δ\[n−k\]), z⁻ᵏ X(z) (delay) and the
trigonometric forms.
Unit samples in inverse results are KroneckerDelta(n, k); on input both
KroneckerDelta(n, k) and DiracDelta(n − k) are accepted.
Discrete unit steps in inverse results are written H(n − k + 1/2):
for integer n this is exactly u[n − k] (1 for n ≥ k, else 0)
under every convention for H(0). Both H(n − k) and H(n − k + 1/2)
are accepted on input, with H(0) read as 1.