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Fig. 77. An eightpoint decimationinfrequency FFT algorithm after the first stage of' decimation.
Fig. 78. Eightpoint radix2 decimationinfrequency 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 twodimensional 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 ) twodimensional 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 Ipoint DFT of the sequence x(N2nI n2), which is row n2 of the twodimensional 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 NIpoint 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 N2point DFT of the columns of the array ;(n2,k l ) : The DFT coefficients are then read out rowwise from the twodimensional array: A pictorial representation of this decomposition is shown in Fig. 79 for N = 15.
Fig. 79. Computation of a 15point DFT with N I = 3 and N2 = 5 using 3point and 5point DFTs.
EXAMPLE 7.3.2 Suppose that we want to compute the 12point DFT of x ( n ) . With N I = 3 and N 2 = 4, the first step is to form a twodimensional 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. 710. Note that because N 2 can be factored, N Z = 2 x 2, the fourpoint DFTs of the columns of G(nz,, ) may be evaluated using twopoint DFTs. For example, if the first column is arranged in a k twodimensional array, after taking the twopoint DFTs of the rows, the terms are multiplied by the twiddle factors
and then the twopoint 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 RadixR algorithm. If the factors are not equal, the FFT is called a mixedradix algorithm.

