The formal expression of the inverse z-transform requires the use of contour integrals in the complex plane: $ x(n) = \frac{1}{2 \pi j} \oint_{C} X(z) z^{n-1} \, dz $ For the analysis of [[discrete-time LTI system]] the [[transfer function]] is a rational function that can be inverted with the inspection method. The inspection method makes use of common z-transform pairs such as: $ a^{n}u(n) \xrightarrow[\cal ZT]{} \frac{1}{1-az^{-1}}, \, |z|>|a| $ A rational transfer function can be decomposed in a sum of terms similar to the z-transform above using the partial fraction expansion: $ \begin{align} X(z) &= \frac{\sum_{k=0}^{M}b_{k}z^{-k}}{\sum_{k=0}^{N}a_{k}z^{-k}} \\ &= \sum_{k=1}^{N} \frac{A_{k}}{1-d_{k}z^{-1}} \end{align} $