mws_gen_fft_spe_pptfastfourierdevl

mws_gen_fft_spe_pptfastfourierdevl - Numerical Methods Fast...

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Numerical Methods Fast Fourier Transform Part: Informal Development of Fast 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 Fast Fourier Transform
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Chapter 11.05: Informal Development of Fast Fourier Transform Major: All Engineering Majors Authors: Duc Nguyen http://numericalmethods.eng.usf.edu Numerical Methods for STEM undergraduates http://numericalmethods.eng.usf.edu 9/4/2010 5 Lecture # 11
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Informal Development of Fast Fourier Transform Recall the DFT pairs of Equations (20) and (21) of Chapter 11.04 and swapping the indexes , one obtains n k = = = 1 0 2 0 ) ( ~ N k k N w in n e k f C π = = = 1 0 2 0 ~ 1 ) ( N n k N w in n e C N k f (1) (2) where 1 ,..., 3 , 2 , 1 , 0 , = N k n (3) N i e W 2 = ( ) 1 2 = = i N e W hence Let (4) 6 http://numericalmethods.eng.usf.edu
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Informal Development cont. Then Eq. (1) and Eq. (2) become = = = 1 0 ) ( ) ( ~ ~ N k nk n W k f n C C = = 1 0 ~ 1 ) ( N n nk n W C N k f (5) Assuming , then ) 2 ( 2 4 = = = r N = ) 3 ( ) 2 ( ) 1 ( ) 0 ( ) 3 ( ~ ) 2 ( ~ ) 1 ( ~ ) 0 ( ~ 1 1 1 1 1 1 1 1 9 6 3 6 4 2 3 2 1 f f f f C C C C W W W W W W W W W N (5A) 7 http://numericalmethods.eng.usf.edu
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Informal Development cont. To obtain the above unknown vector for a given vector , the coefficient matrix can be easily converted as { } f { } C ~ = 9 6 3 6 4 2 3 2 1 1 9 6 3 6 4 2 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 W W W W W W W W W W W W W W W W W W N 8 http://numericalmethods.eng.usf.edu
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http://numericalmethods.eng.usf.edu 9 Hence, the unknown vector can be computed as with matrix vector operations, as following { } C ~ = ) 3 ( ) 2 ( ) 1 ( ) 0 ( ) 3 ( ~ ) 2
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida - Tampa.

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mws_gen_fft_spe_pptfastfourierdevl - Numerical Methods Fast...

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