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% ( I ) = 5 - 5 j and f ( 9 ) = 5
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+ 5 j , with X(k) = 0 fork = 0 and k = 2,. . . ,8.
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f ( 0 ) = 4, f ( k ) = 2/(1 - e-j"'I4) fork = 1. 3 , 5 , 7 , and f ( k ) = 0 fork = 2,4.6.
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I ( n ) is conjugate symmetric, l ( n )= 2 * ( N - n).
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( a ) X(k) = 1
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,kk '
X(l)=X(7)= x ( n ) = S(n)
$ andX(3)= X ( 9 ) =
cos(3nn/5).
+ i+
k X(k) = ~ 6 ( -
x ( n ) = ;[S(n - 3) (a) I
+ S(n - 13)]+ 51-S(n - 5) + S(n - 1 I ) ] .
(I.)
+ X(k). ( h ) W j k X ( ( 4 k))4. Re[X(k)]. y ( n ) = -S(n) + 26(n - I) + 2S(n - 7 ) + 6(n - 8).
w ( n ) = 0. x ( ~ J " ~ /+ (- 1)' N) X ( k ) = X(N
~r(2N-k)lN 1-
- k).
(a) x ( n ) = S(n) S(n - 2) - S(n - 5 ) + 6(n - 8). ( h ) x ( n ) = 6(n - 2) - S(n - 8).
Y ( k ) = $ [ ~ ( ( k I)), + X ( ( k + l ) ) N ] f ~ ~ O - 1. k5N ~
y ( n ) = [ 6 , 5 , 4 . 3 , 2 , 2 , 2 , 3 , 4 , 51. Nx((N - n ) ) ~ .
CHAP. 61
THE DFT
[2, I , 1, 1, 11. 17 DFTs and 1 6 IDFTs for overlap-add and the same for overlap save.
35 5 n 5 127.
A f = 2 kHz.
Y ( k ) = 0.42X(k)- 0.25[X((k- 1 ) ) ~+ X((k
+ I ) ) N ] + 0.04[X((k--
2 ) ) ~ X((k
+ 2)),v].
7
The Fast Fourier Transform
7.1 INTRODUCTION
In Chap. 6 we saw that the discrete Fourier transform (DFT) could be used to perform convolutions. In this chapter we look at the computational requirements of the DFT and derive some fast algorithms for computing the DFT. These algorithms are known, generically, asfast Fourier fransforms (FFTs). We begin with the radix-2 decimation-in-time FFT, an algorithm published in 1965 by Cooley and Tukey. We then look at mixed-radix FFT algorithms and the prime factor FFT.
RADIX-2 FFT ALGORITHMS
The N -point DFT of an N -point sequence s ( n ) is
Because x(n) may be either real or complex, evaluating X(k) requires on the order of N complex multiplications and N complex additions for each value of k. Therefore, because there are N values of X(k), computing an N-point DFT requires N* complex multiplications and additions. The basic strategy that is used in the FFT algorithm is one of "divide and conquer." which involves decomposing an N-point DFT into successively smaller DFTs. To see how this works, suppose that the length of x(n) is even (i.e., N is divisible by 2). If x(n) is decin~aredinto two sequences of length N/2, computing the N/2-point DFT of each of these sequences requires approximately ( N 12)' multiplications and the same number of additions. Thus, the two DFTs require 2 ( ~ / 2 ) '= { N' multiplies and adds. Therefore, if it is possible to find the N-point DFT of s ( n ) from these two N/2-point DFTS in fewer than N 2 / 2 operations, a savings has been realized.
7.2.1 Decimation-in-Time FFT
The decimation-in-time FFT algorithm is based on splitting (decimating) x(n) into smaller sequences and finding X ( k ) from the DFTs of these decimated sequences. This section describes how this decimation leads to an efficient algorithm when the sequence length is a power of 2. Let x(n) be a sequence of length N = 2", and suppose that x(n) is split (decimated) into two subsequences, each of length N/2. As illustrated in Fig. 7-1, the first sequence, ~ ( I z is, formed from the even-index terms, )
and the second, h(n), is formed from the odd-index terms, h(n) = x(2n
+ 1)
n = 0, I .
.. . , - -I N
In terms of these sequences, the N -point DFT of x(n) is
tr=O
n even
CHAP. 71
THE FAST FOURIER TRANSFORM
Because
wiik= wik, Eq. ( 7 . 2 )may be written as ,
Note that the first term is the N/2-point DFT of g ( n ) , and the second is the N 12-point DFT of h(n):
Although the N/2-point DFTs of g ( n ) and h ( n ) are sequences of length NL2, the periodicity of the complex exponentials allows us to write
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