├── SIMD
├── SIMD.cpp
├── SIMD.vcxproj.filters
├── SIMD.vcxproj
└── fft_1024.txt
├── complex_mul.png
├── SIMD.sln
├── task1_solution.md
├── .gitattributes
├── task2_solution.md
├── README.md
└── .gitignore
/SIMD/SIMD.cpp:
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https://raw.githubusercontent.com/jagger2048/fft_simd/HEAD/SIMD/SIMD.cpp
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/complex_mul.png:
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https://raw.githubusercontent.com/jagger2048/fft_simd/HEAD/complex_mul.png
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/SIMD/SIMD.vcxproj.filters:
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1 |
2 |
3 |
4 |
5 | {4FC737F1-C7A5-4376-A066-2A32D752A2FF}
6 | cpp;c;cc;cxx;def;odl;idl;hpj;bat;asm;asmx
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19 | 源文件
20 |
21 |
22 |
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/SIMD.sln:
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1 |
2 | Microsoft Visual Studio Solution File, Format Version 12.00
3 | # Visual Studio 15
4 | VisualStudioVersion = 15.0.27703.2000
5 | MinimumVisualStudioVersion = 10.0.40219.1
6 | Project("{8BC9CEB8-8B4A-11D0-8D11-00A0C91BC942}") = "SIMD", "SIMD\SIMD.vcxproj", "{139C6AB7-15A4-49CD-8B08-5FCBF60BE7E4}"
7 | EndProject
8 | Global
9 | GlobalSection(SolutionConfigurationPlatforms) = preSolution
10 | Debug|x64 = Debug|x64
11 | Debug|x86 = Debug|x86
12 | Release|x64 = Release|x64
13 | Release|x86 = Release|x86
14 | EndGlobalSection
15 | GlobalSection(ProjectConfigurationPlatforms) = postSolution
16 | {139C6AB7-15A4-49CD-8B08-5FCBF60BE7E4}.Debug|x64.ActiveCfg = Debug|x64
17 | {139C6AB7-15A4-49CD-8B08-5FCBF60BE7E4}.Debug|x64.Build.0 = Debug|x64
18 | {139C6AB7-15A4-49CD-8B08-5FCBF60BE7E4}.Debug|x86.ActiveCfg = Debug|Win32
19 | {139C6AB7-15A4-49CD-8B08-5FCBF60BE7E4}.Debug|x86.Build.0 = Debug|Win32
20 | {139C6AB7-15A4-49CD-8B08-5FCBF60BE7E4}.Release|x64.ActiveCfg = Release|x64
21 | {139C6AB7-15A4-49CD-8B08-5FCBF60BE7E4}.Release|x64.Build.0 = Release|x64
22 | {139C6AB7-15A4-49CD-8B08-5FCBF60BE7E4}.Release|x86.ActiveCfg = Release|Win32
23 | {139C6AB7-15A4-49CD-8B08-5FCBF60BE7E4}.Release|x86.Build.0 = Release|Win32
24 | EndGlobalSection
25 | GlobalSection(SolutionProperties) = preSolution
26 | HideSolutionNode = FALSE
27 | EndGlobalSection
28 | GlobalSection(ExtensibilityGlobals) = postSolution
29 | SolutionGuid = {E3E1FF75-B69C-4DC7-95F4-7833423EA8D2}
30 | EndGlobalSection
31 | EndGlobal
32 |
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/task1_solution.md:
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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 |
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/task2_solution.md:
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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 |
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/README.md:
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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 | 
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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