Fair and Square Computation of Inverse ${cal Z}$ -Transforms of Rational Functions

All methods presented in textbooks for computing inverse Z-transforms of rational functions have some limitation: 1) the direct division method does not, in general, provide enough information to derive an analytical expression for the time-domain sequence x(k) whose Z-transform is X(z) ; 2) computation using the inversion integral method becomes labored when X(z)zk-1 has poles at the origin of the complex plane; 3) the partial-fraction expansion method, in spite of being acknowledged as the simplest and easiest one to compute the inverse Z-transform and being widely used in textbooks, lacks a standard procedure like its inverse Laplace transform counterpart. This paper addresses all the difficulties of the existing methods for computing inverse Z -transforms of rational functions, presents an easy and straightforward way to overcome the limitation of the inversion integral method when X(z)zk-1 has poles at the origin, and derives five expressions for the pairs of time-domain sequences and corresponding Z-transforms that are actually needed in the computation of inverse Z -transform using partial-fraction expansion.