By Martin S. Roden (Auth.)
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Sample text
F{k)G{-k)dk. 57) is one form of Parseval's formula. It can be made to relate to energy by further taking the special case of, g(í)=/*(í), where the asterisk (*) represents complex conjugate. We can show that, if g{t) = / * ( 0 , then, G{w) =F *(-ω). 58) Plugging these relationships into Parseval's formula ( E q . 59) We have used the fact that the product of a function with its complex conjugate is equal to the magnitude of the function, squared. , dissipated in 1-ohm resistor if the signal represented the voltage across the resistor (or the current through it).
F o r this region, the transform of cos Wot is non-zero only at the point ω = ω©. T h e only case for which we have previously experienced the definition of frequency is that in which the time function is a pure sinusoid. Since for the pure sinusoid, the positive "ω''-axis appears to have meaning when interpreted as frequency, we shall consider ourselves justified in calling "ω" a frequency variable. W e shall continue to do this until such time that a contradiction arises (as it turns out, none will, since for other than a pure sinusoid, we will define frequency in terms of the Fourier Transform).
2 with 2π ωο = -^Γ. This transform is sketched in Fig. 31. 2π , ' Τ ' Ill - ω ο Fig. 14. The Fourier Transform of a periodic train of impulses is itself a periodic train of impulses! This is the only case we will encounter in which the Fourier Transform and the original time function r e semble each other. This result will be used as a basis for an intuitive proof of the sampling theorem, a remarkable result which we shall now present. " Stated in a diflFerent w a y , / ( f ) can be uniquely determined from its values at a sequence of equidistant points in time.