EE3TP4_FFT_v2_Lecture 31

EE3TP4_FFT_v2_Lecture 31 - Discrete Fourier Transform DFT...

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Discrete Fourier Transform X [ k ]= n = 0 N 1 x [ n ] e j2π kn / N k = 0, 1, 2, . .. , N 1 x [ n ]= 1 N k = 0 N 1 X [ k ] e j2π kn / N n = 0, 1, 2, . .. , N 1 DFT Inverse DFT

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Fast Fourier Transform (FFT) • The FFT is simply an algorithm for efficiently computing the DFT! X [ k ]= n = 0 N 1 x [ n ] e j2π kn / N k = 0, 1, 2, . .. , N 1 Computing the DFT directly from the above definition requires N 2 multiplications, and, N 2 additions. Note that these are complex additions and multiplications! How can this be done more efficiently?
Note: The most common FFT algorithm requires that the number of samples put into it be a power of two … later we’ll talk about how to “zero-pad” up to a power of two! N 2 log 2 N multiplications, and N 2 log 2 N additions. Similar ideas hold for computing the inverse DFT. FFT Complexity Using the FFT requires about

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Note log scales The following plot shows the drastic improvement the FFT gives over the DFT:
X [ k ]= n = 0 N 1 x [ n ] e j2π kn / N k = 0, 1, 2, . .. , N 1 DFT Computation Speedup e.g., consider the case for k = 1, X [ 1 ]= n = 0 N 1 x [ n ] e j2π n / N N = 8 Im Re “unit circle” j j 1 1 x [ 0 ] x [ 1

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This note was uploaded on 02/01/2011 for the course ECE 3TP4 taught by Professor Drterrencetodd during the Fall '10 term at McMaster University.

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EE3TP4_FFT_v2_Lecture 31 - Discrete Fourier Transform DFT...

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