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.