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Separating this sum into the first N/2 points and the last N / 2 points, and using the fact that becomes
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With a change in the indexing on the second sum we have
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which is the N /2-point DFT of the sequence that is formed by adding the first N/2 points of x(n) to the last N 12. Proceeding in the same way for the odd samples of X(k) leads to
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A flowgraph illustrating this first stage of decimation is shown in Fig. 7-7. As with the decimation-in-time FFT, the decimation may be continued until only two-point DFTs remain. A complete eight-point decimation-infrequency FFT is shown in Fig. 7-8. The complexity of the decimation-in-frequency FFT is the same as the decimation-in-time, and the computations may be performed in place. Finally, note that although the input sequence x(n) is in normal order, the frequency samples X(k) are in bit-reversed order.
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CHAP. 71
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THE FAST FOURIER TRANSFORM
Fig. 7-7. An eight-point decimation-in-frequency FFT algorithm after the first stage of' decimation.
Fig. 7-8. Eight-point radix-2 decimation-in-frequency FFT.
7.3 FFT ALGORITHMS FOR COMPOSITE N
It is not always possible to work with sequences whose length is a power of 2. However, efficient computation of the DFT is still possible if the sequence length may be written as a product of factors. For example, suppose that N may be factored as follows:
N=N,.N2
We then decompose x ( n ) into N 2 sequences of length N I and arrange these sequences in an array as follows:
THE FAST FOURIER TRANSFORM
[CHAP. 7
EXAMPLE 7.3.1 For a sequence of length N = 15. with N I = 3 and N2 = 5, the sequence x ( n ) may be decimated into five sequences of length three, and these sequences may then be arranged in a two-dimensional array as follows:
Alternatively, if we let N I = 5 and N 2 = 3, ~ ( n may be decimated into three sequences of length five and arranged in a ) two-dimensional array of three rows and five columns,
By defining index maps for n and k as follows,
the N -point DFT may be expressed as
Note that the inner summation,
is the N I-point DFT of the sequence x(N2nI n2), which is row n2 of the two-dimensional array in Eq. (7.5). Computing the N I -point DFT of each row of the array produces another array,
consisting of the complex numbers G(n2, k~). Note that because the data in row n2 is not needed after the NI-point DFT of x(N2nI n2) is computed, G(n2, k l ) may be stored in the same row (i.e., the computations may be performed in place).
CHAP. 71
THE FAST FOURIER TRANSFORM
The next step in the evaluation of X ( k ) in Eq. (7.7) is to multiply by the twiddle factors w,$"*:
The final step is to compute the N2-point DFT of the columns of the array ;(n2,k l ) :
The DFT coefficients are then read out row-wise from the two-dimensional array:
A pictorial representation of this decomposition is shown in Fig. 7-9 for N = 15.
Fig. 7-9. Computation of a 15-point DFT with N I = 3 and N2 = 5 using 3-point and 5-point DFTs.
EXAMPLE 7.3.2 Suppose that we want to compute the 12-point DFT of x ( n ) . With N I = 3 and N 2 = 4, the first step is to form a two-dimensional array consisting of N I = 3 columns and N2 = 4 rows,
and compute the DFT of each row,
For example, the DFT of the first row is
THE FAST FOURIER TRANSFORM
[CHAP. 7
The next step is to multiply each term by the appropriate twiddle factor. The array of factors is
[; ;I $1
w;: w , :
This produces the array C ( n 2 ,k , ) :
The final step is to compute the DFT of each column:
This results in the Howgraph shown in Fig. 7-10. Note that because N 2 can be factored, N Z = 2 x 2, the four-point DFTs of the columns of G(nz,, ) may be evaluated using two-point DFTs. For example, if the first column is arranged in a k two-dimensional array,
after taking the two-point DFTs of the rows, the terms are multiplied by the twiddle factors
and then the two-point DFTs of the columns are computed.
Up to this point, we have only assumed that N could be factored as N = N I . N Z I t is possible, however, that either or both of these factors could be factored further. What is important for the FFT algorithm to be efficient is that N be a highly composite number:
In this case, it is possible to define multidimensional index maps for ti and k as follows,
and the development of the FFT algorithm proceeds as described above. If N = R " , the corresponding FFT algorithm is called a Radix-R algorithm. If the factors are not equal, the FFT is called a mixed-radix algorithm.
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