574 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems
n Eigenfunctions and the DTFT.The frequency representation of a discrete-time linear time-invariant
(LTI) system is shown to be the DTFT of the impulse response of the system. Indeed, according to
the eigenfunction property of LTI systems, if the input of such a system is a complex exponential,
x[n]=ejω^0 n, the steady-state output, calculated with the convolution sum, is given by
y[n]=
∑
k
h[k]x[n−k]=
∑
k
h[k]ejω^0 (n−k)=ejω^0 nH(ejω^0 ) (10.6)
where
H(ejω^0 )=
∑
k
h[k]e−jω^0 k (10.7)
or the DTFT of the impulse responseh[n] of the system computed atω=ω 0. As with continuous-
time systems, the system needs to be bounded-input bounded-output (BIBO) stable. Without the
stability of the system, there is no guarantee that there will be a steady-state response.
nExample 10.1
Consider the noncausal signalx[n]=α|n|with|α|<1. Determine its DTFT. Use the obtained DTFT
to find
∑∞
n=−∞
α|n|
Solution
The Z-transform ofx[n] is
X(z)=
∑∞
n= 0
αnz−n+
∑∞
m= 0
αmzm− 1
=
1
1 −αz−^1
+
1
1 −αz
− 1 =
1 −α^2
1 −α(z+z−^1 )+α^2
where the first term has an ROC of|z|>|α|, and the ROC of the second term is|z|< 1 /|α|. Thus,
the region of convergence ofX(z)is
ROC: |α|<|z|<
1
|α|
and that includes the unit circle. Thus, the DTFT is
X(ejω)=
1 −α^2
( 1 +α^2 )− 2 αcos(ω)