By Sergej Rjasanow; Olaf Steinbach

ISBN-10: 0387340416

ISBN-13: 9780387340418

The short Fourier remodel (FFT) is a mathematical procedure favourite in sign processing. This booklet makes a speciality of the applying of the FFT in numerous parts: Biomedical engineering, mechanical research, research of inventory industry facts, geophysical research, and the traditional radar communications box.

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**Example text**

33) The terms in brackets satisfy the definitions of even and odd functions, Sec. 7 43 Waveform Decomposition respectively. From Eqs. 32), the Fourier transform of Eq. 34) H(f) = R(f) + jI(f) = HeCf) + Ho(f) where He(f) = R(f) and Ho(f) = jICf). We show in Chapter 9 that decomposition can increase the speed of computation of the FFT. 11 Exponential Waveform Decomposition To demonstrate the concept of waveform decomposition, consider the exponential function [Fig. 35) e-at Following the developments leading to Eq.

22 The Fourier Transform Chap. 2 The importance of the Fourier transform pair of Eq. 44) becomes obvious in future discussions of discrete Fourier transforms. Inversion Formula Proof By means of distribution theory concepts, it is possible to derive a simple formal proof of the inversion formula of Eq. 5). Substitution of H(f) [Eq. 1)] into the inverse Fourier transform of Eq. 45) Because [Eq. t dt = 8(t) then an interchange of integration in Eq. 46) h(x)8(t - x) dx But by the definition of the impulse function of Eq.

For this reason, the latter definition of the Fourier transform pair was chosen for this book. 5 FOURIER TRANSFORM PAIRS A pictorial table of Fourier transform pairs is given in Fig. 12. This graphical and analytical catalog is by no means complete, but does contain the most frequently encountered transform pairs. ~ K I K 2Alol hit) hit) hltl hltl Time Domain t;, 1 2f~ 3 _ 0 To To To III < ItI = III > h(t) = K&(t) h(t) = K 0 0 o <0 f sin(21Tf01) 2 f 'IT 01 h(1) - 2A 0 = - 2 A h(t) = A 2 A A = IfI > fo IfI = fo IfI < fo K&(f) =0 - = H(f) = K H(f) H(f) 21TTof H(f) = 2AT" sin(2'ITTof) K K 2ATo Hltl Hltl = K W) Hill H(f) Frequency Domain en N -5T ..

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