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113_1_113_1_113_1_chapter16

# 113_1_113_1_113_1_chapter16 - Chapter 16 THE FAST FOURIER...

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Chapter 16 THE FAST FOURIER TRANSFORM (FFT) Copyright c 1996 by Ali H. Sayed. All rights reserved. These notes are distributed only to the students attending the undergraduate DSP course EE113 in the Electrical Engineering Department at UCLA. The notes cannot be reproduced without written consent from the instructor: Prof. A. H. Sayed, Electrical Engineering Department, UCLA, CA 90095, [email protected] As mentioned before, one of the main advantages of using the DFT in discrete-time signal processing applications is that efficient methods exist for its evaluation. The purpose of this lecture is to introduce some of these methods. The DFT and IDFT . Thus recall that the relations that define the N -point DFT and its inverse of a sequence x ( n ) are X ( k ) = N - 1 X n =0 x ( n ) e - j 2 πkn N , k = 0 , 1 , . . . , N - 1 x ( n ) = 1 N N - 1 X k =0 X ( k ) e j 2 πkn N , n = 0 , 1 , . . . , N - 1 Phase factor . For convenience of notation we shall define the complex number: W N Δ = e - j 2 π N = cos ( 2 π N ) - j sin ( 2 π N ) It corresponds to the N -th root of unity (since W N N = 1) and it satisfies the easily verifiable identities: W q N = W N/q , W q + N 2 N = - W q N , W qN/ 2 N = ( - 1) q Using W N , we can rewrite the expressions for the DFT and IDFT more compactly as follows: X ( k ) = N - 1 X n =0 x ( n ) W kn N , k = 0 , 1 , . . . , N - 1 x ( n ) = 1 N N - 1 X k =0 X ( k ) W - kn N , n = 0 , 1 , . . . , N - 1 . 171

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172 The Fast Fourier Transform (FFT) Chapter 16 Computational cost . It is clear from the definition of the DFT that computing the DFT of a sequence of length N requires N complex multiplications and ( N - 1) complex additions per coefficient X ( k ). This means that overall we need N 2 complex multi- plications and N ( N - 1) complex additions. We thus say that we need approximately 2 N 2 complex operations: N - point DFT = requires O (2 N 2 ) complex operations This cost can be reduced by employing a certain divide-and-conquer strategy that will lead us to what is known as the Fast Fourier Transform ( FFT ). There are several vari- ants of the FFT algorithm. Here we only describe the radix-2 decimation-in-timeand decimation-in-frequency versions, which are the most widely used. All other versions essentially share the same ideas and derivation. Decimation-in-time FFT . Assume x ( n ) is a sequence of length N and that N is a power of 2, say N = 2 p for some p . This requirement is not restrictive since we can always pad the sequence with zeros in order to achieve a length that is a power of 2. [By padding a sequence with zeros, we actually increase the resolution that we obtain for its DTFT .] Since N is even, we can split the sequence into two smaller sequences of size N 2 each. In one sequence we group the even-indexed samples of x ( n ) and in the other sequence we group the odd-indexed samples of x ( n ), say { x (0) , x (2) , x (4) , . . . , x ( N - 4) , x ( N - 2) } and { x (1) , x (3) , x (5) , . . . , x ( N - 3) , x ( N - 1) } . Now we can express for the DFT coefficients of x ( n ): X ( k ) = X n = even x ( n ) W kn N + X n = odd x ( n ) W kn N = N 2 - 1 X m =0 x (2 m ) W k 2 m N + N 2 - 1 X m =0 x (2 m + 1) W k (2 m +1) N = N 2 - 1 X m =0 x (2 m ) W km N/ 2 | {z } X e ( k ) + W k N N 2 - 1 X m =0 x (2 m + 1) W km N/ 2 | {z } X o ( k ) where X e ( k ) and X o ( k ) are N 2 - point DFT
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113_1_113_1_113_1_chapter16 - Chapter 16 THE FAST FOURIER...

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