Signals and Systems - Electrical Engineering

542 C H A P T E R 9: The Z-Transform

9.5 One-Sided Z-Transform Inverse

Different from the inverse Laplace transform, which was done mostly by the partial fraction expan-

sion, the inverse Z-transform can be done in different ways. For instance, if the Z-transform is given

as a finite-order polynomial, the inverse can be found by inspection. Indeed, if the given Z-transform

is

X(z)=

∑N

n= 0

x[n]z−n

=x[0]+x[1]z−^1 +x[2]z−^2 +···+x[N]z−N (9.31)

by the definition of the Z-transform,x[k] is the coefficient of the monomialz−kfork=0, 1,...,N;

thus the inverse Z-transform is given by the sequence{x[0],x[1],...,x[n]}. For instance, if we have a

Z-transform

X(z)= 1 + 2 z−^10 + 3 z−^20

the inverse is a sequence

x[n]=δ[n]+ 2 δ[n−10]+ 3 δ[n−20]

so thatx[0]=1,x[10]=2,x[20]=3, andx[n]=0 forn6=0, 10, 20, respectively. In this case it

makes sense to do this becauseNis finite, but ifN→∞, this way of finding the inverse Z-transform

might not be very practical. In that case, thelong-divisionmethod and thepartial fraction expan-

sionmethod, which we consider next, are more appropriate. In this section we will consider the

inverse of one-sided Z-transforms, and in the next section we consider the inverse of two-sided

transforms.

9.5.1 Long-Division Method

When a rational functionX(z)=B(z)/A(z), having as ROC the outside of a circle of radiusR(i.e.,x[n]is causal),

is expressed as

X(z)=x[0]+x[1]z−^1 +x[2]z−^2 +···

then the inverse is the sequence{x[0],x[1],x[2],...}, or

x[n]=x[0]δ[n]+x[1]δ[n−1]+x[2]δ[n−2]+···

To find the inverse we simply divide the polynomialB(z)byA(z)to obtain a possible infinite-order

polynomial in negative powers ofz−^1. The coefficients of this polynomial are the inverse values. The

disadvantage of this method is that it does not provide a closed-form solution, unless there is a clear

connection between the terms of the sequence. But this method is useful when we are interested in

finding some of the initial values of the sequencex[n].