mws_gen_fft_spe_pptdiscretefourier

# mws_gen_fft_spe_pptdiscretefourier - Discrete Fourier...

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Discrete Fourier Transform (DFT) Major: All Engineering Majors Authors: Duc Nguyen http://numericalmethods.eng.usf.edu Numerical Methods for STEM undergraduates 8/13/2009 http://numericalmethods.eng.usf.edu 1

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Discrete Fourier Transform Recalled the exponential form of Fourier series (see Eqs. 26, 28 in Ch. 11.01), one gets:  k t ikw k e C t f 0 ~ ) ( T t ikw k dt e t f T C 0 0 ) ( 1 ~ (26, repeated) (28, repeated) , ,....... , 3 , 2 , 3 2 1 t n t t t t t t t n then Eq. (26) becomes: 1 0 0 ~ ) ( N k t ikw k n n e C t f (1) If time “ ” is discretized at t http://numericalmethods.eng.usf.edu 2
Discrete Fourier Transform cont. To simplify the notation, define: n t n (2) Then, Eq. (2) 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) n ilw N n N k n ikw k N n n ilw e e C e n f 0 0 0 1 0 1 0 1 0 ~ ) (   1 0 1 0 2 ) ( ~ N n N k n N l k i k e C (4) (5) http://numericalmethods.eng.usf.edu 3

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Discrete Fourier Transform cont. Switching the order of summations on the right-hand-side of Eq.(5), one obtains:  1 0 1 0 2 ) ( 1 0 2 ~ ) ( N k N n n N l k i k N n n N il e C e n f (6) Define: 1 0 2 ) ( N n n N l k i e A (7) There are 2 possibilities for to be considered in Eq. (7) ) ( l k http://numericalmethods.eng.usf.edu 4
Discrete Fourier Transform—Case 1 Case(1) : is a multiple integer of N, such as: ; or where ) ( l k mN l k ) ( mN k ,...... 2 , 1 , 0 m Thus, Eq. (7) becomes: 1 0 1 0 2 ) 2 sin( ) 2 cos( N n N n n im mn i mn e A (8) Hence: (9) N A http://numericalmethods.eng.usf.edu 5

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## mws_gen_fft_spe_pptdiscretefourier - Discrete Fourier...

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