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CHAP. 71
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THE FAST FOURIER TRANSFORM
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Fig. 7-10.
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FFT algorithm for N = 12.
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For some values of N , with the appropriate index mapping, it is possible to completely eliminate the twiddle factors. These mapping have the form
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where A, B , C, and D are integers, and ((.))N denotes the evaluation of Ihe index modulo N. If N = N I . N . and if NI and N2 are relativelyprime (i.e., they have no common factors), the twiddle factors may be eliminated with the appropriate values for A, B , C, and D. The requirements on these numbers are as follows: I. 2. All numbers between 0 and N and k2 are varied.
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I for n and k must appear uniquely as n I and
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The numbers A, B , C, and D are such that
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( A n l + R n r ) t C k ~ fDk2)
- W;:kl -
WnzP
The second condition requires that
Finding a set of numbers that satisfies these two conditions falls in the domain of number- theory, which will not be considered here. However, one set of numbers that satisfies these conditions is
where ( ( N ; ' ) ) ~ ~ denotes the multiplicative inverse of NI modulo N2. For example, if N = 12 with N I = 3 and Nz = 4, ( ( 4 ~ ' ) =~I because ((4 . ) = 1 and ((3-1))4 = 3 because ((3 . 3))3 = I .
THE FAST FOURIER TRANSFORM
[CHAP. 7
EXAMPLE 7.4.1 A 12-point prime factor algorithm with N I = 3 and N 2 = 4 is as follows. With A = Nz = 4 and B = N I = 3 , and with C = N ~ ( ( N ~ ' ) = 4, and U = N I ( ( ~ ; l ) ) N z9. Thus, the index mappings for n and k are )N =
and the two-dimensional array representation for the input is
The representation for X ( k ) is therefore
Thus, the DFT is evaluated by lint computing the three-point DFT of each row of the input array, followed by the four-point DFT of each column. The following figure shows how the four-point DFTs are interconnected to the three-point DFTs.
Because a 4-point DFT does not require any multiplications (see Prob. 7.1 1 ), and because each 3-point DFT requires only 4 complex multiplications, the 12-point prime factor algorithm requires 16 complex multiplies. For a mixed-radix FFT, there are, in addition, six twiddle factors. The cost for eliminating these six multiplications is an increase in complexity in indexing and in programming.
CHAP. 71
THE FAST FOURIER TRANSFORM
Solved Problems
Radix-2 FFT Algorithms 7.1
Assume that a complex multiply takes 1 p s and that the amount of time to compute a DFT is determined by the amount of time it takes to perform all of the multiplications. (a) How much time does it take to compute a 1024-point DFT directly (b) How much time is required if an FFT is used
(c) Repeat parts (a) and (b) for a 4096-point DFT.
(a) Including possible multiplications by fI, computing an N-point DFT directly requires N complex multipli' cations. If it takes I p s per complex multiply, the direct evaluation of a 1024-point DFT requires
(b) With a radix-2 FFT, the number of complex multiplications is approximately (N/2) log, N which, for N = 1024, is equal to 5 120. Therefore, the amount of time to compute a 1024-point DFT using an FFT is
If the length of the DFT is increased by a factor of 4 to N = 4096, the number of multiplications necessary to compute the DFT directly increases by a factor of 16. Therefore, the time required to evaluate the DFT directly is 1 ~ ~ =.16.78 S 7 If, on the other hand, an FFT is used, the number of multiplications is 2,048 . log, 4,096 = 24,576 and the amount of time to evaluate the DFT is
A complex-valued sequence x ( n ) of length N = 8 192 is to be convolved with a complex-valued sequence h ( n ) of length L = 5 12.
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