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258 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5
5.36. Using the frequency convolution theorem (5.59), derive the modulation theorem
(5.148).
From Eq. (5.144) we have
cos w,t - a6(w - 0,) + a6(0 + w0)
By the frequency convolution theorem (5.59) we have
1
~(t) cos 0,t - -X(0)* [aS(w - 00) + d(0 + w~)]
2lT
= +X(O - 0,) + $X(w + 0,)
The last equality follows from Eq. (2.59).
5.37. Verify Parseval's relation (5.63), that is,
From the frequency convolution theorem (5.59) we have
Setting w = 0, we get
By changing the dummy variable of integration, we obtain
5.38. Prove Parseval's identity [Eq. (5.6411 or Parseval's theorem for the Fourier transform,
that is,
By definition (5.31) we have
where * denotes the complex conjugate. Thus,
x*(t) -X*(-w)
