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mws_gen_fft_spe_pptdiscretefourier

# mws_gen_fft_spe_pptdiscretefourier - NumericalMethods Part:...

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Numerical Methods Discrete Fourier Transform    Part: Discrete Fourier Transform  http://numericalmethods.eng.usf.edu

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For more details on this topic  Go to  http://numericalmethods.eng.usf.edu Click on Keyword Click on Discrete Fourier Transform
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Chapter 11.04 : Discrete Fourier  Transform (DFT)   Major: All Engineering Majors Authors: Duc Nguyen http://numericalmethods.eng.usf.edu Numerical Methods for STEM undergraduates 12/09/11 http://numericalmethods.eng.usf.edu 5 Lecture # 8

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Discrete Fourier Transform Recalled the exponential form of Fourier series  (see Eqs. 39, 41 in Ch. 11.02), one gets: -∞ = = k t ikw k e C t f 0 ~ ) ( { } - × = T t ikw k dt e t f T C 0 0 ) ( 1 ~    (39, repeated) (41, repeated)                                             http://numericalmethods.eng.usf.edu 6
http://numericalmethods.eng.usf.edu 7 , ,....... , 3 , 2 , 3 2 1 t n t t t t t t t n = = = = then Eq. (39) becomes: - = = 1 0 0 ~ ) ( N k n t ikw k n e C t f (1) If time “   ” is discretized at  t Discrete Fourier Transform

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Discrete Fourier Transform cont. To simplify the notation, define: n t n = (2) Then, Eq. (1) can be written as: - = = 1 0 0 ~ ) ( N k n ikw k e C n f (3) Multiplying both sides of Eq. (3) by  n ilw e 0 - , and performing the summation on “  ”, one n obtains (note:  l = integer number)                                             http://numericalmethods.eng.usf.edu 8
http://numericalmethods.eng.usf.edu 9 n ilw N n N k n ikw k N n n ilw e e C e n f 0 1 0 1 0 0 1 0 0 ~ ) ( - - = - = - = - ∑ ∑ × = × ∑ ∑ - = - = - = 1 0 1 0 2 ) ( ~ N n N k n N l k i k e C π (4) (5) Discrete Fourier Transform cont.

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Discrete Fourier Transform cont. Switching the order of summations on the
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mws_gen_fft_spe_pptdiscretefourier - NumericalMethods Part:...

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