mws_gen_fft_spe_ppttheoreticalfourier

mws_gen_fft_spe_ppttheoreticalfourier - NumericalMethods...

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Numerical Methods Fast Fourier Transform    Part: Theoretical Development of  Fast Fourier Transform http://numericalmethods.eng.usf.edu ( 29 4 2 2 = = N
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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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Major: All Engineering Majors Authors: Duc Nguyen http://numericalmethods.eng.usf.edu Numerical Methods for STEM undergraduates http://numericalmethods.eng.usf.edu 11/01/11 5 Chapter 11.06: Theoretical  Development of Fast Fourier Transform Lecture # 16  ( 29 4 2 2 = = N
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Theoretical Development of FFT For the case of                    r N 2 = Recall Equation (5) from Chapter 11.05 Informal  Development of FFT, - = = = 1 0 ) ( ) ( ~ ~ N k nk n W k f n C C where N i e W π 2 - = 11/01/11 6 (11.65) Consider the case                         .   In this case, we can express    and    as 2-bit binary numbers:  k n 4 2 2 2 = = = r N ) 1 , 1 ( ), 0 , 1 ( ), 1 , 0 ( ), 0 , 0 ( ) , ( ) 3 , 2 , 1 , 0 ( 0 1 = = = k k k (1) (2)
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Theoretical Development cont. Eqs. (1) and (2) can also be expressed in compact forms, as following 0 0 1 1 0 1 2 2 2 k k k k k + = + = 0 0 1 1 0 1 2 2 2 n n n n n + = + = 1 or , 0 , , , 0 1 0 1 = n n k k where  (3) (4) In the new notations, Eq.(11.65) becomes ∑ ∑ = = + + = 1 0 0 1 0 1 ) 0 1 2 )( 0 1 2 ( 0 1 0 1 ) , ( ) , ( ~ k k k k n n W k k f n n C (5) 11/01/11 7
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                                            http://numericalmethods.eng.usf.edu 8 Consider ( 29 ( 29 ( 29 ( 29 0 0 1 2 1 2 0 1 2 0 1 2 0 1 2 * k n n k n n k k n n W W W + + + + = ( 29 0 0 1 2 1 0 2 1 1 4 k n n k n k n W W W + = ( 29 0 0 1 2 1 0 2 1 1 4 ] [ k n n k n k n W W W + = Theoretical Development cont.
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Theoretical Development cont. Notice that [ ] 4 2 4 = - N i e W π 4 4 2 = - i e 1 ) 2 sin( ) 2 cos( 2 = - = = - i e i Hence Eq. (5) can be simplified to = + = = 1 0 0 0 ) 0 1 2 ( 1 0 1 ) 1 0 2 ( 0 1 0 1 ~ ) , ( ) , ( k k n n k k n W W k k f n n C (7) 11/01/11 9
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http://numericalmethods.eng.usf.edu 10 Define = = 1 0 1 ) 1 0 2 ( 0 1 0 0 1 ) , ( ) , ( k
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mws_gen_fft_spe_ppttheoreticalfourier - NumericalMethods...

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