Signals and Systems - Electrical Engineering

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

TheAandBcoefficients can be found (by analogy with the Laplace transform partial fraction

expansion) as

A=X(z)( 1 +0.5z−^1 )|z− (^1) =− 2 =−0.5
B=X(z)( 1 −0.5z−^1 )|z− (^1) = 2 =1.5
so that
x[n]=[−0.5(−0.5)n+1.5(0.5)n]u[n]
Consider then the partial fraction expansion in positive powers ofz. From Equation (9.32) the
proper rational functionX(z)/zcan be expanded as
X(z)
z

=

z+ 1

(z+0.5)(z−0.5)

=

C

z+0.5

+

D

z−0.5

The values ofCandDare obtained as follows:

C=

X(z)

z

(z+0.5)|z=−0.5=−0.5

D=

X(z)

z

(z−0.5)|z=0.5=1.5

We then have that

X(z)=

−0.5z

z+0.5

+

1.5z

z−0.5

which according to the table (if entries are in negative powers ofz, convert them into positive

powers ofz) we get

x[n]=[−0.5(−0.5)n+1.5(0.5)n]u[n]

which coincides with the above result.

Two simple checks on our result are given by the initial and the final value results. For the initial

value,

x[0]= 1 =lim

z→∞

X(z)

and

lim

n→∞

x[n]=lim

z→ 1

(z− 1 )X(z)= 0

Both of these check. It is important to recognize that these two checks do not guarantee that we

did not make mistakes in computing the inverse, but if the initial or the final values were not to

coincide with our results, our inverse would be wrong. n