mws_gen_fft_spe_fastfourierdevl

# mws_gen_fft_spe_fastfourierdevl - 11.05.1 Chapter 11.05...

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Unformatted text preview: 11.05.1 Chapter 11.05 Informal Development of Fast Fourier Transform (FFT) Introduction Recalled the DFT pairs of Equations (22) and (23) (of Chapter 11.04) and swapping the indices k n , one obtains: ∑ − = = − = 1 2 ) ( ~ N k k N w in n e k f C π (1) ∑ − = = = 1 2 ~ 1 ) ( N n k N w in n e C N k f π (2) where 1 ,... 3 , 2 , 1 , , − = N k n (3) While the above DFT pairs of equations are convenient for computer implementation, they still require substantial computation effort. The objective of this chapter, therefore, is to develop the improved version of DFT (namely Fast Fourier Transform, or FFT) so that much larger sampling data can be handled more efficiently. Let N i e W π 2 − = (hence 1 ) 2 sin( ) 2 cos( 2 = − = = − π π π i e W i N ) (4) Then Equation (1) and Equation (2) become ∑ − = = = 1 ) ( ) ( ~ ~ N k nk n W k f n C C (5) ∑ − = − = 1 ~ 1 ) ( N n nk n W C N k f It should be emphasized here that in performing interpolation, one usually has to solve a system of equations to determine the unknown coefficients of the linear combination of basis functions that fit the given data. For example, if 4 = N , then one need to solve the following system (see the second part of Equation (5)), for obtaining { } C ~ , with a given vector { } f . = − − − − − − − − − ) 3 ( ) 2 ( ) 1 ( ) ( ) 3 ( ~ ) 2 ( ~ ) 1 ( ~ ) ( ~ 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) 11.05.2 Chapter 11.05 However, the inverse of the above coefficient matrix can be easily obtained as = − − − − − − − − − − 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 Thus, the unknown vector { } C ~ can be computed as matrix times vector operations, as following: Assuming , 2 4 ) 2 ( = = = r N then (see the first part of Equation (5)) = ) 3 ( ) 2 ( ) 1 ( ) ( ) 3 ( ~ ) 2 ( ~ ) 1 ( ~ ) ( ~ ) 3 )( 3 ( ) 2 )( 3 ( ) 1 )( 3 ( ) )( 3 ( ) 3 )( 2 ( ) 2 )( 2 ( ) 1 )( 2 ( ) )( 2 ( ) 3 )( 1 ( ) 2 )( 1 ( ) 1 )( 1 ( ) )( 1 ( ) 3 )( ( ) 2 )( ( ) 1 )( ( ) )( ( f f f f W W W W W W W W W W W W W W W W C C C C (6) = ) 3 ( ) 2 ( ) 1 ( ) ( ) 3 ( ~ ) 2 ( ~ ) 1 ( ~ ) ( ~ 9 6 3 6 4...
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mws_gen_fft_spe_fastfourierdevl - 11.05.1 Chapter 11.05...

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