# 0.4 Streaming fft  (Page 8/19)

 Page 8 / 19

## Hard-coded four-step

This section presents an implementation of the four-step algorithm  [link] that leverages hard-coded sub-transforms to compute larger transforms. The implementation uses an implicit memory transpose (along with vector register transposes) and scales particularly well with VL. In contrast to the fully hard-coded implementation in theprevious section, the four-step implementation requires no new leaf primitives as VL increases, i.e., the code is much the same when $VL>1$ as it is when $VL=1$ .

## The four-step algorithm

A transform of size $N$ is decomposed into a two-dimensional array of size ${n}_{1}×{n}_{2}$ where $N={n}_{1}{n}_{2}$ . Selecting ${n}_{1}={n}_{2}=\sqrt{N}$ (or close) often obtains the best performance results  [link] . When either of the factors is larger than the other, it is the larger of the two factors that will determine performance, because the larger factor effectively brings the memory wall closer. The four steps of the algorithm are:

1. Compute ${n}_{1}$ FFTs of length ${n}_{2}$ along the columns of the array;
2. Multiply each element of the array with ${\omega }_{N}^{ij}$ , where $i$ and $j$ are the array coordinates;
3. Transpose the array;
4. Compute ${n}_{2}$ FFTs of length ${n}_{1}$ along the columns of the array.

For this out-of-place implementation, steps 2 and 3 are performed as part of step 1. Step 1 reads data from the input array and computes the FFTs, but before storing the data in the final pass, it is multiplied by the twiddle factors from step 2. After this, the data is stored to rows in the output array, and thus the transpose of step 3 is performed implicitly. Step 4 is then computed as usual: FFTs are computed along the columns of the output array.

This method of computing the four-step algorithm in two steps requires only minor modifications in order to support multiple vector lengths: with $VL>1$ , multiple columns are read and computed in parallel without modification of the code, but before storing multiple columns of data to rows, a register transpose is required.

## Vector length 1

When $VL=1$ , three hard-coded FFTs are elaborated.

1. FFT of length ${n}_{2}$ with stride ${n}_{1}×2$ for the first column of step 1;
2. FFT of length ${n}_{2}$ with stride ${n}_{1}×2$ and twiddle multiplications on outputs – for all other columns of step 1;
3. FFT of length ${n}_{1}$ with stride ${n}_{2}×2$ for columns in step 4.

In order to generate the code for the four-step sub-transforms, some minor modifications are made to the fully hard-coded code generator that was presented in the previous section.

The first FFT is used to handle the first column of step 1, where there are no twiddle factor multiplications because one of the array coordinates for step 2 is zero, and thus ${\omega }_{N}^{0}$ is unity. This FFT may be elaborated as in "Vector length 1" with the addition of a stride factor for the input address calculation. The second FFT is elaborated as per the first FFT, but with the addition of twiddle factor multiplications oneach register prior to the store operations. The third FFT is elaborated as per the first FFT, but with strided input and output addresses.

const SFFT_D __attribute__ ((aligned(32))) *LUT; const SFFT_D *pLUT;void sfft_dcf64_fs_x1_0(sfft_plan_t *p, const void *vin, void *vout){   const SFFT_D *in = vin;  SFFT_D *out = vout;   SFFT_R r0,r1,r2,r3,r4,r5,r6,r7;  L_4(in+0,in+64,in+32,in+96,&r0,&r1,&r2,&r3);   L_2(in+16,in+80,in+112,in+48,&r4,&r5,&r6,&r7);   K_0(&r0,&r2,&r4,&r6);   S_4(r0,r2,r4,r6,out+0,out+4,out+8,out+12);  K_N(VLIT2(0.7071,0.7071),VLIT2(0.7071,-0.7071),&r1,&r3,&r5,&r7);   S_4(r1,r3,r5,r7,out+2,out+6,out+10,out+14);} void sfft_dcf64_fs_x1_n(sfft_plan_t *p, const void *vin, void *vout){  const SFFT_D *in = vin;   SFFT_D *out = vout;  SFFT_R r0,r1,r2,r3,r4,r5,r6,r7;   L_4(in+0,in+64,in+32,in+96,&r0,&r1,&r2,&r3);   L_2(in+16,in+80,in+112,in+48,&r4,&r5,&r6,&r7);   K_0(&r0,&r2,&r4,&r6);   r2 = MUL(r2,LOAD(pLUT+4),LOAD(pLUT+6));  r4 = MUL(r4,LOAD(pLUT+12),LOAD(pLUT+14));   r6 = MUL(r6,LOAD(pLUT+20),LOAD(pLUT+22));  S_4(r0,r2,r4,r6,out+0,out+4,out+8,out+12);   K_N(VLIT2(0.7071,0.7071),VLIT2(0.7071,-0.7071),&r1,&r3,&r5,&r7);   r1 = MUL(r1,LOAD(pLUT+0),LOAD(pLUT+2));  r3 = MUL(r3,LOAD(pLUT+8),LOAD(pLUT+10));  r5 = MUL(r5,LOAD(pLUT+16),LOAD(pLUT+18));   r7 = MUL(r7,LOAD(pLUT+24),LOAD(pLUT+26));  S_4(r1,r3,r5,r7,out+2,out+6,out+10,out+14);   pLUT += 28;} void sfft_dcf64_fs_x2(sfft_plan_t *p, const void *vin, void *vout){  const SFFT_D *in = vin;   SFFT_D *out = vout;  SFFT_R r0,r1,r2,r3,r4,r5,r6,r7;   L_4(in+0,in+64,in+32,in+96,&r0,&r1,&r2,&r3);   L_2(in+16,in+80,in+112,in+48,&r4,&r5,&r6,&r7);   K_0(&r0,&r2,&r4,&r6);   S_4(r0,r2,r4,r6,out+0,out+32,out+64,out+96);  K_N(VLIT2(0.7071,0.7071),VLIT2(0.7071,-0.7071),&r1,&r3,&r5,&r7);   S_4(r1,r3,r5,r7,out+16,out+48,out+80,out+112);} void sfft_dcf64_fs(sfft_plan_t *p, const void *vin, void *vout) {  const SFFT_D *in = vin;   SFFT_D *out = vout;  pLUT =  LUT;   int i;  sfft_dcf64_fs_x1_0(p, in, out);   for(i=1;i<8;i++) sfft_dcf64_fs_x1_n(p, in+(i*2), out+(i*16));   for(i=0;i<8;i++) sfft_dcf64_fs_x2(p, out+(i*2), out+(i*2)); } Hard-coded four-step VL-1 size-64 FFT 

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By OpenStax By Brooke Delaney By Madison Christian By Ali Sid By Madison Christian By JavaChamp Team By Yasser Ibrahim By CB Biern By Rohini Ajay By