Methods
Three systematic techniques for recovering a time-domain sequence from its Z-domain representation: partial fractions, long division, and the formal contour integral.
back to Time
Design happens in the Z-domain. Implementation happens in time. The inverse Z-transform bridges the two — recovering h[n] from a pole-zero specification or solving difference equations.
Decompose X(z)/z into first-order terms. Each residue A_k = (z − p_k)X(z)/z at z = p_k. ROC determines causality of each term.
The same pole p gives different sequences depending on the region of convergence chosen.
Power Series
Divide numerator by denominator in ascending powers of z⁻¹. Each quotient coefficient is a direct time-domain sample: the coefficient of z⁻ⁿ is x[n].
Formal definition: x[n] = (1/2πj) ∮ X(z) zn−1 dz. By Cauchy's residue theorem this equals the sum of residues of X(z)zn−1 at poles inside the contour C — identical to partial fractions.
H(z) = z²/(z² − 0.9z + 0.81) has poles at 0.9e±jπ/3. The impulse response is a damped sinusoid: h[n] = (0.9)ncos(nπ/3) u[n].
- ROC is essential — same X(z) → different x[n] for different ROCs
- Partial fractions: decompose into first-order terms, read from table
- Long division: coefficients of power series = time-domain samples
- Contour integral: formal definition, equals residue sum = partial fractions
- Conjugate poles at (r, ±θ) → damped sinusoid rⁿcos(nθ + φ)u[n]
- |z| > |p| → causal; |z| < |p| → anti-causal