CHAP. 71

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THE FAST FOURIER TRANSFORM

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(b) If N = 3". then there are v stages in the radix-3 FFT. The general form of each 3-point butterfly, shown in the second stage of the flowgraph in Fig. 7-1 1, requires six multiplies (some require fewer if we do not consider multiplications by fI). Since there are N / 3 butterflies in each stage, then the total number of multiplications is

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6N log, N

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( c ) Yes, the computations may be performed in place.

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Derive a radix-3 decimation-in-frequency FFT for N = 3" a.nd draw the corresponding flowgraph for N =9.

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As with the radix-2 decimation-in-frequency FFT, with N =3", we separately evaluate the indices for which ((k))3 = 0, ((k))3 = I , and ((k))3 = 2. For ((k))3 = 0 (i.e., k is a multiple of 3),

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Separating this sum into the first N / 3 points, the second N / 3 points, and the last N / 3 points, and using the fact that W Z k = wnk, this becomes

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With a change in the indexing in the second and third sums, we have

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"+ $ Finally, because WNi, = W:/,,

and WN13 = Wi13,

which is the N/3-point DFT of the sequence in brackets. Proceeding in the same way for the samples X(3k I), we have

Finally, for the samples X(3k

+ 2) we have

THE FAST FOURIER TRANSFORM The flowgraph for a nine-point decimation-in-frequency FFT is shown below.

[CHAP. 7

Suppose that we have a number of eight-poin~ decimation-in-time FFT chips. How could these chips be used to compute a 24-point DFT

A 24-point DFT is defined by

Decimating x(n) by a factor of 3, we may decompose this DFT into three %point DFTs as follows:

Therefore, if we form the three sequences

and use the 8-point FFT chips to find the DFTs F ( k ) , G(k), and H ( k ) , the 24-point DFT of x ( n ) may be found by combining the outputs of the 8-point FFTs as follows:

Prime Factor FFT

71 .5

Find the index maps for a 21 -point prime factor FFT with N I = 7 and N 2 = 3. HOWmany multiplications are required compared to a 32-point radix-2 decimation-in-time FFT

CHAP. 71

THE FAST FOURIER TRANSFORM

For a 21-point prime factor FFT with N I = 7 and N 2 = 3, we sel A = N2 = 3 and B = N I = 7. Then, with C = N~((N;' ))N, = 15 and D = N ~ (N ( = 7, we have the following index mappings:

Thus, the two-dimensional array representation for the input is

x(7) x(14)

~ ( 1 0 ) ~ ( 1 3 ) ~ ( 1 6 ) ~ ( 1 9 ) .x(l) ~ ( 1 7 ) ~ ( 2 0 ) s(2) s(5) ~ ( 8 ) .u(l I )

and the two-dimensional array for the output is

With the prime factor FFT, there are no twiddle factors. Therefore, the only multiplications necessary are those required to compute the three 7-point DFTs, and the seven 3-point DFTs. Because each 3-point DFT requires 6 complex multiplies, and each 7-point DFT requires 42, the number of multiplies for a 2 1-point prime factor FFT is (7)(6) (3)(42) = 168. For a 32-point radix-2 FFT. on the other hand. we require

complex multiplies. Therefore. it would be more efficient to pad a 21-point sequence with zeros and compute a 32point DFT. The increasedefficiency is a result of the fact that 32 = 2' is a much more composite number than 2 1 = 7.3.

Suppose that we would like to compute a 15-point DFT of a sequence x ( n ) .

( a ) Using a mixed-radix FFT with N I = 5 and N2 = 3, the DFT is decomposed into two stages, with

the first consisting of three 5-point DFTs, and the second stage consisting of five 3-point DFTs. Make a sketch of the connections between the five- and three-point DFTs, indicating any possible twiddle factors, and the order of the inputs and outputs. (b) Repeat part ( a ) for the prime factor algorithm with N I ==5 and N 2 = 3, and determine how many complex multiplies are saved with the prime factor algorithm.

(a) Using a mixed-radix FFT with N I = 5 and N2 = 3, the index mappings for n and k are as follows: