├── .gitattributes ├── .gitignore ├── README.md ├── SIMD.sln ├── SIMD ├── SIMD.cpp ├── SIMD.vcxproj ├── SIMD.vcxproj.filters └── fft_1024.txt ├── complex_mul.png ├── task1_solution.md └── task2_solution.md /.gitattributes: -------------------------------------------------------------------------------- 1 | ############################################################################### 2 | # Set default behavior to automatically normalize line endings. 3 | ############################################################################### 4 | * text=auto 5 | 6 | ############################################################################### 7 | # Set default behavior for command prompt diff. 8 | # 9 | # This is need for earlier builds of msysgit that does not have it on by 10 | # default for csharp files. 11 | # Note: This is only used by command line 12 | ############################################################################### 13 | #*.cs diff=csharp 14 | 15 | ############################################################################### 16 | # Set the merge driver for project and solution files 17 | # 18 | # Merging from the command prompt will add diff markers to the files if there 19 | # are conflicts (Merging from VS is not affected by the settings below, in VS 20 | # the diff markers are never inserted). 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Turn it on by uncommenting the 52 | # entries below. 53 | ############################################################################### 54 | #*.doc diff=astextplain 55 | #*.DOC diff=astextplain 56 | #*.docx diff=astextplain 57 | #*.DOCX diff=astextplain 58 | #*.dot diff=astextplain 59 | #*.DOT diff=astextplain 60 | #*.pdf diff=astextplain 61 | #*.PDF diff=astextplain 62 | #*.rtf diff=astextplain 63 | #*.RTF diff=astextplain 64 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | ## Ignore Visual Studio temporary files, build results, and 2 | ## files generated by popular Visual Studio add-ons. 3 | 4 | # User-specific files 5 | *.suo 6 | *.user 7 | *.userosscache 8 | *.sln.docstates 9 | 10 | # User-specific files (MonoDevelop/Xamarin Studio) 11 | *.userprefs 12 | 13 | # Build results 14 | [Dd]ebug/ 15 | [Dd]ebugPublic/ 16 | [Rr]elease/ 17 | [Rr]eleases/ 18 | x64/ 19 | x86/ 20 | bld/ 21 | [Bb]in/ 22 | [Oo]bj/ 23 | [Ll]og/ 24 | 25 | # Visual Studio 2015 cache/options directory 26 | .vs/ 27 | # Uncomment if you have tasks that create the project's static files in wwwroot 28 | #wwwroot/ 29 | 30 | # MSTest test Results 31 | [Tt]est[Rr]esult*/ 32 | [Bb]uild[Ll]og.* 33 | 34 | # NUNIT 35 | *.VisualState.xml 36 | TestResult.xml 37 | 38 | # Build Results of an ATL Project 39 | [Dd]ebugPS/ 40 | [Rr]eleasePS/ 41 | dlldata.c 42 | 43 | # DNX 44 | project.lock.json 45 | project.fragment.lock.json 46 | artifacts/ 47 | 48 | *_i.c 49 | *_p.c 50 | *_i.h 51 | *.ilk 52 | *.meta 53 | *.obj 54 | *.pch 55 | *.pdb 56 | *.pgc 57 | *.pgd 58 | *.rsp 59 | *.sbr 60 | *.tlb 61 | *.tli 62 | *.tlh 63 | *.tmp 64 | *.tmp_proj 65 | *.log 66 | *.vspscc 67 | *.vssscc 68 | .builds 69 | *.pidb 70 | *.svclog 71 | *.scc 72 | 73 | # Chutzpah Test files 74 | _Chutzpah* 75 | 76 | # Visual C++ cache files 77 | ipch/ 78 | *.aps 79 | *.ncb 80 | *.opendb 81 | *.opensdf 82 | *.sdf 83 | *.cachefile 84 | *.VC.db 85 | *.VC.VC.opendb 86 | 87 | # Visual Studio profiler 88 | *.psess 89 | *.vsp 90 | *.vspx 91 | *.sap 92 | 93 | # TFS 2012 Local Workspace 94 | $tf/ 95 | 96 | # Guidance Automation Toolkit 97 | *.gpState 98 | 99 | # ReSharper is a .NET coding add-in 100 | _ReSharper*/ 101 | *.[Rr]e[Ss]harper 102 | *.DotSettings.user 103 | 104 | # JustCode is a .NET coding add-in 105 | .JustCode 106 | 107 | # TeamCity is a build add-in 108 | _TeamCity* 109 | 110 | # DotCover is a Code Coverage Tool 111 | *.dotCover 112 | 113 | # NCrunch 114 | _NCrunch_* 115 | .*crunch*.local.xml 116 | nCrunchTemp_* 117 | 118 | # MightyMoose 119 | *.mm.* 120 | AutoTest.Net/ 121 | 122 | # Web workbench (sass) 123 | .sass-cache/ 124 | 125 | # Installshield output folder 126 | [Ee]xpress/ 127 | 128 | # DocProject is a documentation generator add-in 129 | DocProject/buildhelp/ 130 | DocProject/Help/*.HxT 131 | DocProject/Help/*.HxC 132 | DocProject/Help/*.hhc 133 | DocProject/Help/*.hhk 134 | DocProject/Help/*.hhp 135 | DocProject/Help/Html2 136 | DocProject/Help/html 137 | 138 | # Click-Once directory 139 | publish/ 140 | 141 | # Publish Web Output 142 | *.[Pp]ublish.xml 143 | *.azurePubxml 144 | # TODO: Comment the next line if you want to checkin your web deploy settings 145 | # but database connection strings (with potential passwords) will be unencrypted 146 | #*.pubxml 147 | *.publishproj 148 | 149 | # Microsoft Azure Web App publish settings. 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Backup files are not needed, 207 | # because we have git ;-) 208 | _UpgradeReport_Files/ 209 | Backup*/ 210 | UpgradeLog*.XML 211 | UpgradeLog*.htm 212 | 213 | # SQL Server files 214 | *.mdf 215 | *.ldf 216 | 217 | # Business Intelligence projects 218 | *.rdl.data 219 | *.bim.layout 220 | *.bim_*.settings 221 | 222 | # Microsoft Fakes 223 | FakesAssemblies/ 224 | 225 | # GhostDoc plugin setting file 226 | *.GhostDoc.xml 227 | 228 | # Node.js Tools for Visual Studio 229 | .ntvs_analysis.dat 230 | 231 | # Visual Studio 6 build log 232 | *.plg 233 | 234 | # Visual Studio 6 workspace options file 235 | *.opt 236 | 237 | # Visual Studio LightSwitch build output 238 | **/*.HTMLClient/GeneratedArtifacts 239 | **/*.DesktopClient/GeneratedArtifacts 240 | **/*.DesktopClient/ModelManifest.xml 241 | **/*.Server/GeneratedArtifacts 242 | **/*.Server/ModelManifest.xml 243 | _Pvt_Extensions 244 | 245 | # Paket dependency manager 246 | .paket/paket.exe 247 | paket-files/ 248 | 249 | # FAKE - F# Make 250 | .fake/ 251 | 252 | # JetBrains Rider 253 | .idea/ 254 | *.sln.iml 255 | 256 | # CodeRush 257 | .cr/ 258 | 259 | # Python Tools for Visual Studio (PTVS) 260 | __pycache__/ 261 | *.pyc -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # fft_simd 2 | A simple demo shows how to use the SIMD,*Single Instruction Multiple Data*, to optimize and accelerate the FFT algorithm. 3 | 4 | - [x] **Task 1.** Implement standard DFT algorithm with C++. See [task1_solution](https://github.com/jagger2048/fft_simd/blob/master/task1_solution.md) 5 | 6 | - [x] **Task 2.** Rewrite the DFT with Cooley–Tukey FFT algorithm. See [task2_solution](https://github.com/jagger2048/fft_simd/blob/master/task2_solution.md) 7 | 8 | - [x] **Task 3.** Ues SIMD to accelerate the FFT alogrithm. See blews in this page. 9 | 10 | - [x] **Task 4.** Make a time consuming contrast among these algorithms.See the table as blews: 11 | 12 | | Algorithm | Time consum | 13 | | ------------- | ----------------------------------------------- | 14 | | Standard dft | 606 ms | 15 | | FFT | 74 ms (out-off-place algorithm) ,7ms (in-place) | 16 | | FFT with SIMD | 1.011 ms | 17 | 18 | ---------- 19 | 20 | ### **Introduction to SIMD** 21 | 22 | Referecnr [the wiki of SIMD](https://en.wikipedia.org/wiki/SIMD). 23 | 24 | ### **SIMD instructions chosen** 25 | 26 | We use SSE2 in this project. SSE2 (Streaming SIMD Extensions 2) is one of the Intel SIMD, it supports two double-precision-floate operations as its MMX rigister are extended from 64 to 128 bits. 27 | 28 | 29 | 30 | ### **SIMD instructions exercise** 31 | 32 | To be familiar with SSE2 instructions, we use SSE2 to compute the complex multiplication as an exercise. 33 | 34 | Complex multiplication can be described as follows: 35 | 36 | 37 | $$ 38 | if A = a+b*j,B=c+d*j,then,A*B=(ac-bd)+(bc+ad)*j 39 | $$ 40 | 41 | 42 | We use a figure to summary and described the complex multiplication using SSE2: 43 | 44 | ![complex_mul](https://github.com/jagger2048/fft_simd/blob/master/complex_mul.png) 45 | 46 | You can find the code in the **complex_mul(double * A, double * B)** function in the project. 47 | 48 | 49 | 50 | ### **Using SIMD to accelerate the FFT** 51 | 52 | The FFT algorithm was introduced in [task2_solution](), reviewing the codes, we can find that the combing part consum most time. So we can use SIMD to rewrite this part. Blew is the orginal code: 53 | 54 | ```c++ 55 | ... 56 | // combine the odd and even parts 57 | complex e = X[k]; // even 58 | complex o = X[k + N / 2]; // odd 59 | // w is the "twiddle-factor" 60 | complex w = exp(complex(0, -2.*PI*k / N)); 61 | X[k] = e + w * o; 62 | X[k + N / 2] = e - w * o; 63 | ... 64 | ``` 65 | 66 | Well, parameters e,o and w are complex type, and our goal becomes using SIMD to process a complex addition, a complex subtraction and a complex multiplication. 67 | 68 | - First, using a struct type ,***comp***,to replace the C++ built-in complex type. 69 | 70 | ```c++ 71 | struct comp { 72 | double real =0; 73 | double imag = 0; 74 | }; 75 | ``` 76 | 77 | The comp struct pack two double-type value, real part and imag part. So we can use a *comp-pointer* to access a complex number. In fact, a *comp* takes up 8 bytes,128 bit, and it can be loaded into a SSE2's MMX rigister. 78 | 79 | - Rewrite the original codes with SSE2 instructions. 80 | 81 | ```c++ 82 | __m128d o = _mm_load_pd( (double *)&X[k + N/2 ] ); // odd 83 | double cc = cos(-2.*PI*k / N); 84 | double ss = sin(-2.*PI*k / N); 85 | __m128d wr = _mm_set_pd( cc,cc ); // cc 86 | __m128d wi = _mm_set_pd( ss, ss ); // dd 87 | // compute the w*o 88 | wr = _mm_mul_pd(o,wr); // ac|bc 89 | __m128d n1 = _mm_shuffle_pd(o,o,_MM_SHUFFLE2(0,1) ); // invert 90 | wi = _mm_mul_pd(n1, wi); // bd|ad 91 | n1 = _mm_sub_pd(wr, wi); // ac-bd|x 92 | wr = _mm_add_pd(wr, wi); // x|bc+ad 93 | n1 = _mm_shuffle_pd(n1,wr,_MM_SHUFFLE2(1,0));// select ac-bd|bc+ad 94 | 95 | o = _mm_load_pd((double *)&X[k]); // load even part 96 | wr = _mm_add_pd(o, n1); // compute even part, X_e + w * X_o; 97 | wi = _mm_sub_pd(o, n1); // compute odd part, X_e - w * X_o; 98 | _mm_store_pd((double *)&X[k],wr); 99 | _mm_store_pd((double *)&X[k + N/2],wi); 100 | ``` 101 | 102 | We use 4 MMX rigisters, and cost 4 load/store , 2 set ,6 ops( +-*/) , 2 reorder operations. It's obvious that the time consuming of 1024-point FFT was decreased from 7ms to 1.01ms ! 103 | 104 | 105 | 106 | ---------------------- 107 | 108 | ### Some useful link:. 109 | 110 | **FFT:** 111 | 112 | https://en.wikipedia.org/wiki/Fast_Fourier_transform Introduction to FFT 113 | 114 | https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm Cooley–Tukey FFT algorithm 115 | 116 | **SIMD:** 117 | 118 | SIMD and sse2 introduction: 119 | 120 | ​ https://www.intel.com/content/www/us/en/support/articles/000005779/processors.html 121 | 122 | **Intel(R) C++ Compiler Intrinsics Reference - Intel® Software:** 123 | 124 | ​ https://software.intel.com/sites/default/files/ee/11/6373 125 | 126 | **Streaming SIMD Extensions 2 Instructions:** 127 | 128 | ​ https://msdn.microsoft.com/zh-cn/library/kcwz153a%28v=vs.110%29.aspx?f=255&MSPPError=-2147217396 129 | 130 | Intrinsics Guide: 131 | 132 | ​ https://software.intel.com/sites/landingpage/IntrinsicsGuide/#cats=Load&expand=3296 133 | 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-------------------------------------------------------------------------------- https://raw.githubusercontent.com/jagger2048/fft_simd/7929762d5c4caffc5b7246f5dcab6919c4e6a711/complex_mul.png -------------------------------------------------------------------------------- /task1_solution.md: -------------------------------------------------------------------------------- 1 | # Task1 solution 2 | 3 | - Task 1. Implement standard DFT algorithm with C++. 4 | 5 | As we all know, The DFT, *discrete Fourier transform* transforms, can be defined by follow formula: 6 | 7 | $$ 8 | {\begin{aligned}X_{k}&=\sum _{n=0}^{N-1}x_{n}\cdot e^{-{\frac {2\pi i}{N}}kn}=\sum _{n=0}^{N-1}x_{n}\cdot [\cos(2\pi kn/N)-i\cdot \sin(2\pi kn/N)]\end{aligned}} 9 | $$ 10 | The last expression follows from the previous one by Euler's formula, which can caculate the exponent using cosine operates. We can implement the DFT formula with C++ as belows: 11 | 12 | ```c++ 13 | int standard_fft(vector fft_in, vector> &fft_out) 14 | { 15 | // Standard fft algorithm without any optimization. 16 | // Ref:https://en.wikipedia.org/wiki/DFT 17 | 18 | int N = fft_in.size(); 19 | for (size_t k = 0; k != N; ++k) { 20 | //complex x_n(0, 0); 21 | double re = 0; // the real part 22 | double im = 0; // the imag part 23 | for (size_t n = 0; n != N; ++n) { 24 | re += fft_in[n] * cos(-double(k * 2 * PI * n / N)); // Euler's formula： exp( i*x ) = cos_x + i*sin_x 25 | im += fft_in[n] * sin(-double(k * 2 * PI * n / N)); 26 | //x_n += fft_in[n] * exp( (double)k *-1i * 2.0*PI* (double)( n )/double( N) ); 27 | // be careful for the division in C++ 28 | } 29 | 30 | fft_out.push_back(complex(re, im)); 31 | } 32 | return 0; 33 | } 34 | ``` 35 | 36 | Ps. The number of dft is the fft_i's size. 37 | 38 | The standard DFT algorithm takes a long time to compute the retult and it's inefficient in real application.So we should consider the other faster algorithm , the famous solution is Cooley–Tukey FFT algorithm. 39 | 40 | See [Task2: Cooley–Tukey FFT algorithm](https://github.com/jagger2048/fft_simd/blob/master/task2_solution.md). 41 | -------------------------------------------------------------------------------- /task2_solution.md: -------------------------------------------------------------------------------- 1 | # Task2 solution 2 | 3 | - Task 2. Rewrite the DFT with Cooley–Tukey FFT algorithm. 4 | 5 | Cooley–Tukey FFT algorithm is the most common [fast Fourier transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform) (FFT) algorithm. Our goals is to replace the standard DFT algorithm with FFT. In general, the DFT algoritm can be described as follows formula: 6 | $$ 7 | {\displaystyle {\begin{matrix}X_{k}=\underbrace {\sum \limits _{m=0}^{N/2-1}x_{2m}e^{-{\frac {2\pi i}{N/2}}mk}} _{\mathrm {DFT\;of\;even-indexed\;part\;of\;} x_{n}}{}+e^{-{\frac {2\pi i}{N}}k}\underbrace {\sum \limits _{m=0}^{N/2-1}x_{2m+1}e^{-{\frac {2\pi i}{N/2}}mk}} _{\mathrm {DFT\;of\;odd-indexed\;part\;of\;} x_{n}}=E_{k}+e^{-{\frac {2\pi i}{N}}k}O_{k}.\end{matrix}}} 8 | $$ 9 | The input sequeue's DFT, $ X_{k} $ , can be separed into two parts , the even part and oven part, and compute the two parts' DFT respectively .Then finally,combine them you will get the orginal sequeue's DFT. The process continues until the length of sequeue as small as possible ,at that moment subsequeue's DFT can be computed easily. We can reference the code blows and learn how the FFT is going: 10 | 11 | ```c++ 12 | void fft2(complex* X, int N) { 13 | if (N < 2) { 14 | // bottom of recursion. 15 | // Do nothing here, because already X[0] = x[0] 16 | } 17 | else { 18 | separate(X, N); // all evens to lower half, all odds to upper half 19 | fft2(X, N / 2); // recurse even items 20 | fft2(X + N / 2, N / 2); // recurse odd items 21 | // combine results of two half recursions 22 | for (int k = 0; k e = X[k]; // even 25 | complex o = X[k + N / 2]; // odd 26 | // w is the "twiddle-factor" 27 | complex w = exp(complex(0, -2.*PI*k / N)); 28 | X[k] = e + w * o; 29 | X[k + N / 2] = e - w * o; 30 | /* 31 | comp e = X[k]; // even 32 | comp o = X[k + N / 2]; // odd 33 | complex ww = exp(complex(0, -2.*PI*k / N)); 34 | comp w; 35 | w.real = ww.real(); 36 | w.imag = ww.imag(); 37 | comp wo = complex_mul_c(w, o); 38 | X[k].real = e.real + wo.real; 39 | X[k].imag = e.imag + wo.imag; 40 | X[k + N / 2].real = e.real - wo.real; 41 | X[k + N / 2].imag = e.imag - wo.imag; 42 | */ 43 | } 44 | } 45 | } 46 | ``` 47 | 48 | With this FFT algorithm,it cost 7ms to compute a 1024 points DFT .Our processor is Intel(R) Core I5-3470. And it is faster than the standard DFT algorithm at the same condition, almost 600ms. 49 | 50 | In the next section we will use SIMD to accelerate the FFT alogrithm above. 51 | 52 | See [Task3: Accelerate the FFT alogrithm](https://github.com/jagger2048/fft_simd/blob/master/README.md). 53 | 54 | 55 | 56 | _______ 57 | 58 | Ps: At the begining I writed a FFT algorithm myself, which is a out-off-place algorithm and it cost 70ms to compute a 1024-point DFT. For learning SIMD operate , I chosed a faster and easy to understand FFT algorithm from the wiki . 59 | 60 | Ref: https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm 61 | --------------------------------------------------------------------------------