Signals and Systems - Electrical Engineering

9.5 One-Sided Z-Transform Inverse 551

− 1 −0.5 0

(a) (b)

0.5 1

− 2

−1.5

− 1

−0.5

0

0.5

1

1.5

2

22

Real part

Imaginary part

0 2 4 6 8 10

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

n

x[

n]

FIGURE 9.10
(a) Poles and zeros ofX(z)and (b) inverse Z-transformx[n].

Ifx[n]has a one-sided Z-transformX(z), thenx[n−N]has the following one-sided Z-transform:

Z[x[n−N]]=z−NX(z)+x[−1]z−N+^1 +x[−2]z−N+^2 +···+x[−N] (9.35)

Indeed, we have that

Z(x[n−N])=

∑∞

n= 0

x[n−N]z−n=

∑∞

m=−N

x[m]z−(m+N)

=z−N

∑∞

m= 0

x[m]z−m+

∑−^1

m=−N

x[m]z−(m+N)

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

where we first letm=n−Nand then separated the sum into two, one corresponding to the
Z-transform ofx[n] multiplied byz−N(the delay on the signal) and a second sum that corresponds
to initial values{x[i],−N≤i≤− 1 }.