By Sergej Rjasanow; Olaf Steinbach
The quick Fourier remodel (FFT) is a mathematical process customary in sign processing. This ebook specializes in the appliance of the FFT in various parts: Biomedical engineering, mechanical research, research of inventory marketplace info, geophysical research, and the traditional radar communications box.
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Extra resources for Fast Fourier Transform and Its Applications
9) illustrated previously in Fig. 3. 10) which is identical to the Fourier transform pair of Eq. 31) illustrated in Fig. 6. Utilization of the symmetry theorem can eliminate many complicated mathematical developments; a case in point is the deVelopment of the Fourier transform pair of Eq. 31). 3 TIME AND FREQUENCY SCALING If the Fourier transform of h(t) is H(f), then the Fourier transform of h(kt), where k is a real constant greater than zero, is determined by substituting t' = kt in the Fourier integral equation: f" -00 h(kt)e-j27fft dt = foo -00 h(t')e-j27ft'(flk) dt' k = !
T dt = A f2~ Jo COS(21Tft) dt - jA = (A/21Tf) sin(21Tft) I~TO _ 2ATo sin[21T(2To)f] 21T( 2To)f f2~ Jo sin(21Tft) dt + j(A/21Tf) COS(21Tft) I~TO . 23) The Fourier Transform 16 Chap. 2 hIt) A 1 - - - -... 5 (a) General pulse waveform, (b) Fourier transform amplitude func· tion, and (c) Fourier transform phase function. (e) The amplitude spectrum is given by 2ATo . 25) Sec. 3 17 Existence of the Fourier Integral The amplitude spectrum I H(f) I and phase angle 6(f) of the Fourier transform of h(t) are shown in Figs.
1 lists various complex time functions and their respective Fourier transforms. 12 Simultaneous Fourier Transforms We can employ the relationships of Eqs. 46) to simultaneously determine the Fourier transform of two real functions. To illustrate this point, recall the linearity property of Eq. 47) Let x(t) = h(t) and y(t) = jg(t), where both h(t) and g(t) are real functions. It follows that X(f) = H(f) and Y(f) = jG(f). Because x(t) is real, then from Eqs. 48) Similarly, because y(t) is imaginary, then from Eqs.