By E. Oran Brigham
Here's a new booklet that identifies and translates the fundamental fundamentals of the quick Fourier rework (FFT). It hyperlinks in a unified presentation the Fourier rework, discrete Fourier remodel, FFT, and primary purposes of the FFT. The FFT is turning into a major analytical instrument in such varied fields as linear structures, optics, chance concept, quantum physics, antennas, and sign research, yet there has constantly been an issue of speaking its basics. hence the purpose of this ebook is to supply a readable and practical therapy of the FFT and its major functions. In his Preface the writer explains the association of his subject matters, "... each significant idea is built by way of a three-stage sequential procedure. First, the concept that is brought via an intuitive improvement that's frequently pictorial and nature. moment, a non-sophisticated (but completely sound) mathematical remedy is constructed to aid the intuitive arguments. The 3rd level includes sensible examples designed to check and extend the idea that being mentioned. it truly is felt that this three-step process supplies which means in addition to mathematical substance to the elemental houses of the FFT. --- from book's dustjacket
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Additional info for The Fast Fourier Transform: An Introduction to Its Theory and Application
EXAMPLE To =4 h(1) (2-17) 2-4 Consider the function (2-20) (2-22) 18 Chap. 2 THE FOURIER TRANSFORM Sec. 2-3 THE FOUJUER TRANSFORM 19 By means of Condition 2, the Fourier transform pair HIli hIlI· 2Afo sin 1:°11 2Af, sin (2n/ot) o 2n/ot 01 B ~f H(/) = A 1/1 0 (2-27) has been established and is illustrated in Fig. 2-5. , 210 Condition 3. Although not specifically stated, all functions for which Conditions I and 2 hold are assumed to be of bounded variation; that is, they can be represented by a curve of finite length in any finite time interval.
From Condition 2 the Fourier transform of h(t) exists and is given by f~ 2Af, sin (2ft/ot) e- 12• /1 dt H(f) = _~ ~ f~ sin (2n/ot)[cos (2n/t) n _~ t = 1 2n/ot 0 j sin (2n/t)] dt = ~ f~ sin (2n/ot) cos (2n/t) dt n _~ t (2-23) The imaginary term integrates to zero since the integrand term is an odd function. 1. 1. S(t)e- 12 • ,t dt = Ke O = K r~ [~]eI2,'t d/ = r~ K cos ~2n/t) d/ + j f~ K sin (2nft) d/ (2-30) 2nt (2-31) Because the integrand of the second integral is an odd function, the integral is zero; the first integral is meaningless unless it is interpreted in the sense of distribution theory.
Fl·H(f) (3-20) The· time-shifted Fourier transform pair is 2A h(1 - 10 ) EXAMPLE ·To H(f)e-/hfl' (3-21) 3-6 A pictorial description of this pair is illustrated in Fig. 3-4. As shown, timeshifting results in a change in the phase angle (J(f) = tan-I[/(f)/R(f)]. Note that time-shifting does not alter the magnitude of the Fourier transform. ;Hl(f) (3-22) 4h(4t) H(~ 4 4A where H(f) has been assumed (0 be real for simplicity. These results are easily extended to the case of H(f), a complex function.
The Fast Fourier Transform: An Introduction to Its Theory and Application by E. Oran Brigham