<< Chapter < Page Chapter >> Page >

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

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fast fourier transforms' conversation and receive update notifications?

Ask