Signals and Systems - Electrical Engineering

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)

(10.8)