Signals and Systems - Electrical Engineering

458 C H A P T E R 8: Discrete-Time Signals and Systems

nExample 8.6

Determine if the signal

x[n]=

∑∞

m= 0

Xmcos(mω 0 n) ω 0 =

2 π

N 0

is periodic, and if so, determine its period.

Solution

The signalx[n] consists of the sum of a constantX 0 and cosines of frequency

mω 0 =

2 πm

N 0

m=1, 2,...

The periodicity ofx[n] depends on the periodicity of the cosines. According to the frequency of the

cosines, they are periodic of periodN 0. Thus,x[n] is periodic of periodN 0. Indeed

x[n+N 0 ]=

∑∞

m= 0

Xmcos(mω 0 (n+N 0 ))

=

∑∞

m= 0

Xmcos(mω 0 n+ 2 πm)=x[n]

n

8.2.2 Finite-Energy and Finite-Power Discrete-Time Signals

For discrete-time signals, we obtain definitions for energy and power similar to those for continuous-

time signals by replacing integrals by summations.

For a discrete-time signalx[n], we have the following definitions:

Energy: εx=

∑∞

n=−∞

|x[n]|^2 (8.9)

Power: Px= lim

N→∞

1

2 N+ 1

∑N

n=−N

|x[n]|^2 (8.10)

n x[n]is said to havefinite energyor to besquare summableifεx<∞.

n x[n]is calledabsolutely summableif

∑∞

n=−∞

|x[n]|<∞ (8.11)

n x[n]is said to havefinite powerifPx<∞.