Page 35 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

(4
Fig. 1-24

1.6. Find the even and odd components of x(r) = eJ'.

Let x,(r) and x,(I) be the even and odd components of ei', respectively.

eJ' =x,(I) +x,(I)
From Eqs. (1.5) and (1.6) and using Euler's formula, we obtain

x,( I) = $(eJr + e-J') = cos I

x,,(I) = f(ei'-e-j') =jsint

Show that the product of two even signals or of two odd signals is an even signal and
that the product of an even and an odd signaI is an odd signal.
Let x(t) =xl(t)x2(t). If XJI) and x2(l) are both even, then

x(-l) =x,(-I)X,(-t) =xI(I)x2(t) =x(t)

and x(t) is even. If x,(t) and x2(t) are both odd, then

x(-I) =x,(-I)x,(-I) = -x,(t)[-x2(t)] =x1(t)x2(t) =x(t)

and x(t) is even. If x,(t) is even and x2(f) is odd, then

and X(I) is odd. Note that in the above proof, variable I represents either a continuous or a
discrete variable.