Inverse Z-transform and Residues

There are 3 popular methods for solving the z-transform $X(z)\to x[n]$:

• Contour integration
• Power series expansion
• Partial fractions and table lookup (for rational functions) The last method is the easiest

Partial Fractions and Table Lookup (for rational functions)

We wish to have it on the following form:

$$X(z)=\frac{R_1}{1-p_1{z}^{-1}}+\frac{R_2}{1-p_2{z}^{-2}}+\dots$$

Where $R_k$ is the residue for pole $k$ and we inverse by ${\mathcal{Z}}^{-1}[X(z)]=x[n]$