15FFT - MAE591 Random Data FFT It is convenient to start...

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MAE591 Random Data FFT It is convenient to start with n = 0 to define the finite Fourier transform of sampled data, i.e. () = = 1 0 ) 2 exp ) ( ) , ( N n fnh j n x h T f X π At the discrete frequencies of Nh k T k f k = = , 1 ,...... , 3 , 2 , 1 , 0 = N k , the discrete Fourier transform (DFT) is often defined as 1 = = = 1 0 ) ( ) ( ) , ( N n k k kn W n x h T f X X , 1 ,...... , 3 , 2 , 1 , 0 = N k . and the inverse discrete Fourier transform (IDFT) as = = = N k k n kn W X N t n x x 1 ) ( 1 ) ( , 1 ,...... , 3 , 2 , 1 , 0 = N n where h has been included with to have a scale factor of unity before the summation. Here ) ( k f X π = N kn j kn 2 exp ) ( W , having the properties such as i integer for iN W u W N u W v W u W v u W 1 ) ( ), ( ) ( ), ( ) ( ) ( = = + = + . Note that N 2 multiply-adds operations are needed for computation of N -point discrete Fourier transforms. Fast Fourier Transform (FFT) can reduce the computational effort to 2 pN multiply-adds operations, where N = 2 p . Thus the speed ratio between the FFT and the converntional DFT becomes p N 4 .
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15FFT - MAE591 Random Data FFT It is convenient to start...

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