Page 185 - Schaum's Outline of Theory and Problems of Signals and Systems

174 THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS [CHAP. 4

where X,(z ), . . . , Xn( z) are functions with known inverse transforms x,[n], . . . , xn[n].
From the linearity property (4.17) it follows that

C. Power Series Expansion:
The defining expression for the z-transform [Eq. (4.3)] is a power series where the
sequence values x[n] are the coefficients of z-". Thus, if X( z) is given as a power series in
the form

we can determine any particular value of the sequence by finding the coefficient of the
appropriate power of 2-'. This approach may not provide a closed-form solution but is
very useful for a finite-length sequence where X(z) may have no simpler form than a
polynomial in z - ' (see Prob. 4.15). For rational r-transforms, a power series expansion
can be obtained by long division as illustrated in Probs. 4.16 and 4.17.

D. Partial-Fraction Expansion:

As in the case of the inverse Laplace transform, the partial-fraction expansion method
provides the most generally useful inverse z-transform, especially when Xtz) is a rational
function of z. Let

Assuming n 2 m and all poles pk are simple, then

where

Hence, we obtain

Inferring the ROC for each term in Eq. (4.35) from the overall ROC of X(z) and using
Table 4-1, we can then invert each term, producing thereby the overall inverse z-transform
(see Probs. 4.19 to 4.23).
If rn > n in Eq. (4.321, then a polynomial of z must be added to the right-hand side of
Eq. (4.351, the order of which is (m - n). Thus for rn > n, the complete partial-fraction