Page 232 - Schaum's Outline of Theory and Problems of Signals and Systems
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CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS 221
Equation (5.58) is referred to as the time convolution theorem, and it states that convolu-
tion in the time domain becomes multiplication in the frequency domain (Prob. 5.31). As
in the case of the Laplace transform, this convolution property plays an important role in
the study of continuous-time LTI systems (Sec. 5.5) and also forms the basis for our
discussion of filtering (Sec. 5.6).
K. Multiplication:
The multiplication property (5.59) is the dual property of Eq. (5.58) and is often referred
to as the frequency convolution theorem. Thus, multiplication in the time domain becomes
convolution in the frequency domain (Prob. 5.35).
L. Additional Properties:
If x(t) is real, let
where x,( t ) and xo(t) are the even and odd components of x( t 1, respectively. Let
Then
Equation (5.61~) is the necessary and sufficient condition for x( t ) to be real (Prob. 5.39).
Equations (5.61b) and (5.61~) show that the Fourier transform of an even signal is a real
function of o and that the Fourier transform of an odd signal is a pure imaginary function
of w.
M. Parseval's Relations:
