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The split-radix fft algorithm

Recently several papers [link] , [link] , [link] , [link] , [link] have been published on algorithms to calculate a length- 2 M DFT more efficiently than a Cooley-Tukey FFT of any radix. They all havethe same computational complexity and are optimal for lengths up through 16 and until recently was thought to give the best total add-multiply countpossible for any power-of-two length. Yavne published an algorithm with the same computational complexity in 1968 [link] , but it went largely unnoticed. Johnson and Frigo have recently reported the firstimprovement in almost 40 years [link] . The reduction in total operations is only a few percent, but it is a reduction.

The basic idea behind the split-radix FFT (SRFFT) as derived by Duhamel and Hollmann [link] , [link] is the application of a radix-2 index map to the even-indexed terms and a radix-4 map to theodd- indexed terms. The basic definition of the DFT

C k = n = 0 N - 1 x n W n k

with W = e - j 2 π / N gives

C 2 k = n = 0 N / 2 - 1 [ x n + x n + N / 2 ] W 2 n k

for the even index terms, and

C 4 k + 1 = n = 0 N / 4 - 1 [ ( x n - x n + N / 2 ) - j ( x n + N / 4 - x n + 3 N / 4 ) ] W n W 4 n k

and

C 4 k + 3 = n = 0 N / 4 - 1 [ ( x n - x n + N / 2 ) + j ( x n + N / 4 - x n + 3 N / 4 ) ] W 3 n W 4 n k

for the odd index terms. This results in an L-shaped “butterfly" shown in [link] which relates a length-N DFT to one length-N/2 DFT and two length-N/4 DFT's with twiddlefactors. Repeating this process for the half and quarter length DFT's until scalars result gives the SRFFT algorithm in much thesame way the decimation-in-frequency radix-2 Cooley-Tukey FFT is derived [link] , [link] , [link] . The resulting flow graph for the algorithm calculated in place looks like a radix-2 FFT except forthe location of the twiddle factors. Indeed, it is the location of the twiddle factors that makes this algorithm use less arithmetic.The L- shaped SRFFT butterfly [link] advances the calculation of the top half by one of the M stages while the lower half, like a radix-4 butterfly, calculates two stages at once. This is illustrated for N = 8 in [link] .

This figure is an array of dots connected with diagonal and horizontal lines. There are two vertical columns of four dots followed by a third column with two dots. The first column of dots each has one horizontal line pointing across, and one diagonal line pointing downward for the first two rows and upward for the second two rows. The diagonal lines connect to dots on the second column either two rows above or two rows below their original position. To the left of the third and fourth dots on the second column, next to the horizontal lines that connect to them, are two dashes. In between the third and fourth dots of the second column is a large label, j. Further to the right, the third and fourth dots of the second column connect to the two dots in the third column. The third dot in the second column connects to the second dot in the third column, which is horizontally level with the fourth dot in the second column. The fourth dot in the second column connects to the first dot in the third column, which is horizontally level with the third dot in the second column. Along this line segment is a dash near the first dot of the third column. There are also two horizontal lines connecting the third dot in the second column with the first dot in the third column, and the fourth dot in the second column with the second dot in the third column. This figure is an array of dots connected with diagonal and horizontal lines. There are two vertical columns of four dots followed by a third column with two dots. The first column of dots each has one horizontal line pointing across, and one diagonal line pointing downward for the first two rows and upward for the second two rows. The diagonal lines connect to dots on the second column either two rows above or two rows below their original position. To the left of the third and fourth dots on the second column, next to the horizontal lines that connect to them, are two dashes. In between the third and fourth dots of the second column is a large label, j. Further to the right, the third and fourth dots of the second column connect to the two dots in the third column. The third dot in the second column connects to the second dot in the third column, which is horizontally level with the fourth dot in the second column. The fourth dot in the second column connects to the first dot in the third column, which is horizontally level with the third dot in the second column. Along this line segment is a dash near the first dot of the third column. There are also two horizontal lines connecting the third dot in the second column with the first dot in the third column, and the fourth dot in the second column with the second dot in the third column.
SRFFT Butterfly
This figure is a flow graph with eight lines crossing in different directions at three points along the graph from left to right. The eight horizontal lines that flow consistently across the graph begin with black dots. At these initial points, the first four connect diagonally downward four rows, and the last four connect diagonally upward to the first four rows. For each of the eight initial dots, there is also a horizontal line connecting these initial dots to eight vertical dots at the same point that the aforementioned diagonal lines terminate. At this point in the figure, the graph separates between a lower half and upper half. At this point in the lower half in between the dots are a label, j. The lower half continues directly from the lower four dots that were connected in the large first section, and to the right they behave in a similar fashion, with the upper two moving two spaces down diagonally, and the lower two moving two spaces up, accompanied by horizontal lines connecting the dots directly across. There is a break in horizontal movement for the lower half of the figure at this point, where the four dots are disconnected and followed by a new set of four adjacent dots. These dots are visually grouped in two sections, with more diagonal lines simply connecting each dot to one adjacent dot diagonally to the right, up or down, and two horizontal lines connecting the dots directly across. In between these two sections and the larger section to the left are two labels, one above labeled w, and the lower labeled w^3. The upper portion of the right half of the figure is not continued or connected to the large initial section. Its first portion mimics the shape of its lower neighbor, with four rows of dots, four horizontal lines to four adjacent dots, and diagonal lines moving upward and downward two spaces up or down. Connected to the lower portion of this upper section is another set of two dots, both vertically and diagonally connected to the larger section to its left. In the middle of the leftmost dots of this section is a label, j. Above and in the upper-right corner of the figure is a final disconnected set of four dots, diagonally and horizontally connected. For the entire figure, there is a dash labeling each dot that to its left is connected by a down-sloping diagonal line. This figure is a flow graph with eight lines crossing in different directions at three points along the graph from left to right. The eight horizontal lines that flow consistently across the graph begin with black dots. At these initial points, the first four connect diagonally downward four rows, and the last four connect diagonally upward to the first four rows. For each of the eight initial dots, there is also a horizontal line connecting these initial dots to eight vertical dots at the same point that the aforementioned diagonal lines terminate. At this point in the figure, the graph separates between a lower half and upper half. At this point in the lower half in between the dots are a label, j. The lower half continues directly from the lower four dots that were connected in the large first section, and to the right they behave in a similar fashion, with the upper two moving two spaces down diagonally, and the lower two moving two spaces up, accompanied by horizontal lines connecting the dots directly across. There is a break in horizontal movement for the lower half of the figure at this point, where the four dots are disconnected and followed by a new set of four adjacent dots. These dots are visually grouped in two sections, with more diagonal lines simply connecting each dot to one adjacent dot diagonally to the right, up or down, and two horizontal lines connecting the dots directly across. In between these two sections and the larger section to the left are two labels, one above labeled w, and the lower labeled w^3. The upper portion of the right half of the figure is not continued or connected to the large initial section. Its first portion mimics the shape of its lower neighbor, with four rows of dots, four horizontal lines to four adjacent dots, and diagonal lines moving upward and downward two spaces up or down. Connected to the lower portion of this upper section is another set of two dots, both vertically and diagonally connected to the larger section to its left. In the middle of the leftmost dots of this section is a label, j. Above and in the upper-right corner of the figure is a final disconnected set of four dots, diagonally and horizontally connected. For the entire figure, there is a dash labeling each dot that to its left is connected by a down-sloping diagonal line.
Length-8 SRFFT

Unlike the fixed radix, mixed radix or variable radix Cooley-Tukey FFT or even the prime factor algorithm or WinogradFourier transform algorithm , the Split-Radix FFT does not progress completely stage by stage, or, in terms of indices, does notcomplete each nested sum in order. This is perhaps better seen from the polynomial formulation of Martens [link] . Because of this, the indexing is somewhat more complicated than theconventional Cooley-Tukey program.

A FORTRAN program is given below which implements the basic decimation-in-frequency split-radix FFT algorithm. The indexingscheme [link] of this program gives a structure very similar to the Cooley-Tukey programs in [link] and allows the same modifications and improvements such as decimation-in-time, multiplebutterflies, table look-up of sine and cosine values, three real per complex multiply methods, and real data versions [link] , [link] .

SUBROUTINE FFT(X,Y,N,M) N2 = 2*NDO 10 K = 1, M-1 N2 = N2/2N4 = N2/4 E = 6.283185307179586/N2A = 0 DO 20 J = 1, N4A3 = 3*A CC1 = COS(A)SS1 = SIN(A) CC3 = COS(A3)SS3 = SIN(A3) A = J*EIS = J ID = 2*N240 DO 30 I0 = IS, N-1, ID I1 = I0 + N4I2 = I1 + N4 I3 = I2 + N4R1 = X(I0) - X(I2) X(I0) = X(I0) + X(I2)R2 = X(I1) - X(I3) X(I1) = X(I1) + X(I3)S1 = Y(I0) - Y(I2) Y(I0) = Y(I0) + Y(I2)S2 = Y(I1) - Y(I3) Y(I1) = Y(I1) + Y(I3)S3 = R1 - S2 R1 = R1 + S2S2 = R2 - S1 R2 = R2 + S1X(I2) = R1*CC1 - S2*SS1 Y(I2) =-S2*CC1 - R1*SS1X(I3) = S3*CC3 + R2*SS3 Y(I3) = R2*CC3 - S3*SS330 CONTINUE IS = 2*ID - N2 + JID = 4*ID IF (IS.LT.N) GOTO 4020 CONTINUE 10 CONTINUEIS = 1 ID = 450 DO 60 I0 = IS, N, ID I1 = I0 + 1R1 = X(I0) X(I0) = R1 + X(I1)X(I1) = R1 - X(I1) R1 = Y(I0)Y(I0) = R1 + Y(I1) 60 Y(I1) = R1 - Y(I1)IS = 2*ID - 1 ID = 4*IDIF (IS.LT.N) GOTO 50
Split-Radix FFT FORTRAN Subroutine

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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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