1.3 Continuous-Time Signals 77
Ifx( 0 )=2, we have
xe(t)=0.5[x(t)+x(−t)]
=
cos( 4 t) t> 0
cos( 4 t) t< 0
2 t= 0
while the odd component is the same. The even component has a discontinuity att=0. n
1.3.3 Periodic and Aperiodic Signals
A useful characterization of signals is whether they areperiodicoraperiodic(nonperiodic).
An analog signalx(t)is periodic if
n it is defined for all possible values oft,−∞<t<∞, and
n there is a positive real valueT 0 , theperiodofx(t), such that
x(t+kT 0 )=x(t) (1.9)
for any integerk.
The period ofx(t)is the smallest possible value ofT 0 > 0 that makes the periodicity possible. Thus, although
NT 0 for an integerN> 1 is also a period ofx(t)it should not be considered the period.
Remarks
n The infinite support and the unique characteristic of the period make periodic signals nonexistent in
practical applications. Despite this, periodic signals are of great significance in the Fourier representation
of signals and in their processing, as we will see later. The representation of aperiodic signals is obtained
from that of periodic signals, and the response of systems to periodic sinusoids is fundamental in the theory
of linear systems.
n Although seemingly redundant, the first part of the definition of a periodic signal indicates that it is not
possible to have a nonzero periodic signal with a finite support (i.e., the analog signal is zero outside an
interval t∈[t 1 ,t 2 ]). This first part of the definition is needed for the second part to make sense.
n It is exasperating to find the period of a constant signal x(t)=A; visually x(t)is periodic but its period
is not clear. Any positive value could be considered the period, but none will be taken. The reason is that
x(t)=A=Acos( 0 t)or of zero frequency, and as such its period is not determined since we would have
to divide by zero—not permitted. Thus, a constant signal is a periodic signal of nondefinable period!
nExample 1.8
Consider the analog sinusoid
x(t)=Acos( 0 t+θ) −∞<t<∞
Determine the period of this signal, and indicate for what frequency 0 the period ofx(t)is not
clearly defined.
