Page 15 - Schaum's Outline of Theory and Problems of Signals and Systems

4 SlGNALS AND SYSTEMS [CHAP. 1

Any signal x(t) or x[n] can be expressed as a sum of two signals, one of which is even
and one of which is odd. That is,

where xe(t) = ${x(t) +x(-t)] even part of x(t)
(1.5)
xe[n] = i{x[n] +x[-n]) even part of x[n]
x,(t) = ${x(t) -x(-t)) odd part of x(t )
( 1.6 )
x,[n] = ${x[n] -x[-n]) odd part of x[n]
Note that the product of two even signals or of two odd signals is an even signal and
that the product of an even signal and an odd signal is an odd signal (Prob. 1.7).

F. Periodic and Nonperiodic Signals:
A continuous-time signal x(t) is said to be periodic with period T if there is a positive
nonzero value of T for which
x(t + T) =x(t) all t (1.7)
An example of such a signal is given in Fig. 1-3(a). From Eq. (1.7) or Fig. 1-3(a) it follows
that

for all t and any integer m. The fundamental period T, of x(t) is the smallest positive
value of T for which Eq. (1.7) holds. Note that this definition does not work for a constant

(b)
Fig. 1-3 Examples of periodic signals.