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0.8 The cooley-tukey fast fourier transform algorithm  (Page 2/8)

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T F : W 8 n 2 k 1 = W 0 W 0 W 0 W 1 W 0 W 2 W 0 W 3 = 1 1 1 W 1 - j 1 - j W

The twiddle factor array will always have unity in the first row and first column.

To complete [link] at this point, after the row DFT's are multiplied by the TF array, the N 1 length- N 2 DFT's of the columns are calculated. However, since the columns DFT's are oflength R M - 1 , they can be posed as a R M - 2 by R array and the process repeated, again using length- R DFT's. After M stages of length- R DFT's with TF multiplications interleaved, the DFT is complete. The flow graph of a length-2 DFT is given in Figure 1 and is called a butterfly because of its shape. The flow graph of thecomplete length-8 radix-2 FFT is shown in Figure 2 .

This figure contains four items with arrows arranged evenly and rectangularly, and arrows pointing from the leftmost items to the rightmost items. The first two items on the left read x(0) on the top and x(1) on the bottom, and are accompanied by large dots next to them, indicating a starting point for four arrows. From each dot are two arrows, one pointing directly across to the right at an item, and one pointing diagonally to the right at the other rightmost item. The top-right item that is pointed to by two arrows reads X(0) = x(0) + x(1), and the bottom-right item that is pointed to by two arrows reads X(0) = x(0) - x(1). The horizontal line pointing from the bototm-left item to the bottom-right item is labeled with a dash. The entire figure is labeled, Radix-2 butterfly. This figure contains four items with arrows arranged evenly and rectangularly, and arrows pointing from the leftmost items to the rightmost items. The first two items on the left read x(0) on the top and x(1) on the bottom, and are accompanied by large dots next to them, indicating a starting point for four arrows. From each dot are two arrows, one pointing directly across to the right at an item, and one pointing diagonally to the right at the other rightmost item. The top-right item that is pointed to by two arrows reads X(0) = x(0) + x(1), and the bottom-right item that is pointed to by two arrows reads X(0) = x(0) - x(1). The horizontal line pointing from the bototm-left item to the bottom-right item is labeled with a dash. The entire figure is labeled, Radix-2 butterfly.
A Radix-2 Butterfly
This figure is a flow graph with eight lines crossing in different directions at three points along the graph from left to right, with each beginning and end point marked with a black dot. The eight horizontal lines that flow consistently across the graph are labeled on the left in order as x_0 to x_7. At its initial point, x_0 through x_3 connect diagonally downward four rows, so that they meet x_4 through x_7 at the same horizontal level. Conversely, x_4 through x_7 connect diagonally upward to the original locations of x_0 to x_3, respectively. Directly before the first four lines connect to the last four lines, on the fifth through eighth horizontal parallels where they will connect, are small arrows indicating movement to the right, and these arrows are labeled with a dash. Directly after these lines connect on the fifth through eighth parallels are small arrows pointing to the right again, this time labeled with w^0, w^1, w^2, w^3 respectively. At this point, the figure is described about halfway across the page from left to right. The next section of crossed lines is exactly the same for the top four parallels and the bottom four parallels. The top two lines in each section connect diagonally downward two spaces, and conversely the bottom two lines connect diagonally upward the same amount. The second half of each section (the third, fourth, seventh and eight) again have arrows labeled with a dash pointing to the right, before the connected segments, and arrows pointing to the right labeled w^0 and w^2 respectively after the connections. Finally, one more set of crossed connected lines occurs, this time in an identical pattern for four groups of two lines. The lines cross this other and connect to the adjacent parallel, with odd-numbered parallels connecting downward and their even-numbered partners connecting upward. Before the connection point are again arrows pointing to the right, labeled with a dash, that occur in the second, fourth, sixth, and eighth parallels. This concludes the lines, and the labels to these lines on the right side of the figure read x_0, x_4, x_2, x_6, x_1, x_5, x_3, x_7 down the page. This figure is a flow graph with eight lines crossing in different directions at three points along the graph from left to right, with each beginning and end point marked with a black dot. The eight horizontal lines that flow consistently across the graph are labeled on the left in order as x_0 to x_7. At its initial point, x_0 through x_3 connect diagonally downward four rows, so that they meet x_4 through x_7 at the same horizontal level. Conversely, x_4 through x_7 connect diagonally upward to the original locations of x_0 to x_3, respectively. Directly before the first four lines connect to the last four lines, on the fifth through eighth horizontal parallels where they will connect, are small arrows indicating movement to the right, and these arrows are labeled with a dash. Directly after these lines connect on the fifth through eighth parallels are small arrows pointing to the right again, this time labeled with w^0, w^1, w^2, w^3 respectively. At this point, the figure is described about halfway across the page from left to right. The next section of crossed lines is exactly the same for the top four parallels and the bottom four parallels. The top two lines in each section connect diagonally downward two spaces, and conversely the bottom two lines connect diagonally upward the same amount. The second half of each section (the third, fourth, seventh and eight) again have arrows labeled with a dash pointing to the right, before the connected segments, and arrows pointing to the right labeled w^0 and w^2 respectively after the connections. Finally, one more set of crossed connected lines occurs, this time in an identical pattern for four groups of two lines. The lines cross this other and connect to the adjacent parallel, with odd-numbered parallels connecting downward and their even-numbered partners connecting upward. Before the connection point are again arrows pointing to the right, labeled with a dash, that occur in the second, fourth, sixth, and eighth parallels. This concludes the lines, and the labels to these lines on the right side of the figure read x_0, x_4, x_2, x_6, x_1, x_5, x_3, x_7 down the page.
Length-8 Radix-2 FFT Flow Graph

This flow-graph, the twiddle factor map of [link] , and the basic equation [link] should be completely understood before going further.

A very efficient indexing scheme has evolved over the years that results in a compact and efficient computer program. A FORTRANprogram is given below that implements the radix-2 FFT. It should be studied [link] to see how it implements [link] and the flow-graph representation.

N2 = N DO 10 K = 1, MN1 = N2 N2 = N2/2E = 6.28318/N1 A = 0DO 20 J = 1, N2 C = COS (A)S =-SIN (A) A = J*EDO 30 I = J, N, N1 L = I + N2XT = X(I) - X(L) X(I) = X(I) + X(L)YT = Y(I) - Y(L) Y(I) = Y(I) + Y(L)X(L) = XT*C - YT*S Y(L) = XT*S + YT*C30 CONTINUE 20 CONTINUE10 CONTINUE
A Radix-2 Cooley-Tukey FFT Program

This discussion, the flow graph of Winograd’s Short DFT Algorithms: Figure 2 and the program of [link] are all based on the input index map of Multidimensional Index Mapping: Equation 6 and [link] and the calculations are performed in-place. According to Multidimensional Index Mapping: In-Place Calculation of the DFT and Scrambling , this means the output is scrambled in bit-reversed order and should be followed by anunscrambler to give the DFT in proper order. This formulation is called a decimation-in-frequency FFT [link] , [link] , [link] . A very similar algorithm based on the output index map can be derived whichis called a decimation-in-time FFT. Examples of FFT programs are found in [link] and in the Appendix of this book.

Modifications to the basic cooley-tukey fft

Soon after the paper by Cooley and Tukey, there were improvements and extensions made. One very important discovery wasthe improvement in efficiency by using a larger radix of 4, 8 or even 16. For example, just as for the radix-2 butterfly, there areno multiplications required for a length-4 DFT, and therefore, a radix-4 FFT would have only twiddle factor multiplications. Becausethere are half as many stages in a radix-4 FFT, there would be half as many multiplications as in a radix-2 FFT. In practice, becausesome of the multiplications are by unity, the improvement is not by a factor of two, but it is significant. A radix-4 FFT is easilydeveloped from the basic radix-2 structure by replacing the length-2 butterfly by a length-4 butterfly and making a few othermodifications. Programs can be found in [link] and operation counts will be given in "Evaluation of the Cooley-Tukey FFT Algorithms" .
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Source:  OpenStax, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
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