mws_gen_fft_spe_discretefourier

mws_gen_fft_spe_discretefourier - Chapter 11.04 Discrete...

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11.04.1 Chapter 11.04 Discrete Fourier Transform Introduction Recalled the exponential form of Fourier series (see Equations 18 and 20 from Chapter 11.02), −∞ = = k t ikw k e C t f 0 ~ ) ( (18, Ch. 11.02) × = T t ikw k dt e t f T C 0 0 ) ( 1 ~ (20, Ch. 11.02) While the above integral can be used to compute k C ~ , it is more preferable to have a discretized formula version to compute k C ~ . Furthermore, the Discrete Fourier Transform (or DFT) [1–5] will also facilitate the development of much more efficient algorithms for Fast Fourier Transform (or FFT), to be discussed in Chapters 11.05 and 11.06. Derivations of DFT Formulas If time “ t ” is discretized at , ,....... , 3 , 2 , 3 2 1 t n t t t t t t t n = = = = Then Equation (18, of Chapter 11.02) becomes = = 1 0 0 ~ ) ( N k t ikw k n n e C t f (1) To simplify the notation, define n t n = (2) Then, Equations (1) can be written as = = 1 0 0 ~ ) ( N k n ikw k e C n f (3) In the above formula, “ n ” is an integer counter. However, ) ( n f and n t do NOT have to be integer numbers. Multiplying both sides of Equation (3) by n ilw e 0 , and performing the summation on “ n ”, one obtains ( note: l = integer number)
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