Page 271 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

From Eq. (5.53) we have
X( -t) -X(-w)
Thus, if x(t) is real, then by Eq. (5.61~) we have
x( -1) -X(-0) =X*(w)

(a) From Eq. (5.154)

Thus, by Eq. (5.158) we obtain

(b) From Eq. (5.155)
1
e-"'u(t) - -
a+jw
Thus, by Eq. (5.158) we get

5.41. Consider a real signal x(t ) and let

X(o) = F[x(t)] = A(w) + jB(o)

where x,(t) and x,(t) are the even and odd components of x(t), respectively. Show
that

x&) +No) (5.161~)
x,( t ) -jB(o ) (5.161b)
From Eqs. (1.5) and (1.6) we have
x,(t) = f [.(I) + X( - t)]

x,(t) = f[x(t) -x(-t)]
Now if x(t) is real, then by Eq. (5.158) we have
X(t) HX(O) =A(@) + jB(w)
x(-t) HX(-W) =X*(W) =A(w) -jB(w)
Thus, we conclude that

x,(t) -~x(w) + ;X*(O) =A(w)
x,(t) t--, ~x(w) - $x*(o) = jB(w)
Equations (5.161~) and (5.161b) show that the Fourier transform of a real even signal is a real
function of o, and that of a real odd signal is an imaginary function of w, respectively.