Здесь приведен алгоритм БПФ для произвольного N, вычисляемый через свертку. Данный алгоритм может быть еще значительно ускорен двумя путями.
В начале функции проверяется, что
Далее, возможно усовершенствование: не надеяться на компилятор, который должен (при удачном раскладе) оптимизировать вычисления в главном цикле fft_step, а сделать оптимизацию явно. В конце этой страницы приведен оптимизированный вариант функции fft_step. Я его привел отдельно, так как он гораздо менее наглядный. Он рассчитывает на то, что серия структур ShortComplex в памяти представляет собой последовательность double, double, double,... где чередуются реальные и мнимые части.
/*
Fast Fourier Transformation
====================================================
Coded by Miroslav Voinarovsky, 2002
This source is freeware.
*/
#ifndef FFT_H_
#define FFT_H_
struct Complex;
struct ShortComplex;
/*
Fast Fourier Transformation
x: x - array of items
N: N - number of items in array
complement: false - normal (direct) transformation, true - reverse transformation
*/
extern void universal_fft(ShortComplex *x, int N, bool complement);
struct ShortComplex
{
double re, im;
inline void operator=(const Complex &y);
};
struct Complex
{
long double re, im;
inline void operator= (const Complex &y);
inline void operator= (const ShortComplex &y);
};
inline void ShortComplex::operator=(const Complex &y) { re = (double)y.re; im = (double)y.im; }
inline void Complex::operator= (const Complex &y) { re = y.re; im = y.im; }
inline void Complex::operator= (const ShortComplex &y) { re = y.re; im = y.im; }
#endif
/*
Fast Fourier Transformation
====================================================
Coded by Miroslav Voinarovsky, 2006
This source is freeware.
*/
#include "fft.h"
#include <math.h>
// This array contains values from 0 to 255 with reverse bit order
static unsigned char reverse256[]= {
0x00, 0x80, 0x40, 0xC0, 0x20, 0xA0, 0x60, 0xE0,
0x10, 0x90, 0x50, 0xD0, 0x30, 0xB0, 0x70, 0xF0,
0x08, 0x88, 0x48, 0xC8, 0x28, 0xA8, 0x68, 0xE8,
0x18, 0x98, 0x58, 0xD8, 0x38, 0xB8, 0x78, 0xF8,
0x04, 0x84, 0x44, 0xC4, 0x24, 0xA4, 0x64, 0xE4,
0x14, 0x94, 0x54, 0xD4, 0x34, 0xB4, 0x74, 0xF4,
0x0C, 0x8C, 0x4C, 0xCC, 0x2C, 0xAC, 0x6C, 0xEC,
0x1C, 0x9C, 0x5C, 0xDC, 0x3C, 0xBC, 0x7C, 0xFC,
0x02, 0x82, 0x42, 0xC2, 0x22, 0xA2, 0x62, 0xE2,
0x12, 0x92, 0x52, 0xD2, 0x32, 0xB2, 0x72, 0xF2,
0x0A, 0x8A, 0x4A, 0xCA, 0x2A, 0xAA, 0x6A, 0xEA,
0x1A, 0x9A, 0x5A, 0xDA, 0x3A, 0xBA, 0x7A, 0xFA,
0x06, 0x86, 0x46, 0xC6, 0x26, 0xA6, 0x66, 0xE6,
0x16, 0x96, 0x56, 0xD6, 0x36, 0xB6, 0x76, 0xF6,
0x0E, 0x8E, 0x4E, 0xCE, 0x2E, 0xAE, 0x6E, 0xEE,
0x1E, 0x9E, 0x5E, 0xDE, 0x3E, 0xBE, 0x7E, 0xFE,
0x01, 0x81, 0x41, 0xC1, 0x21, 0xA1, 0x61, 0xE1,
0x11, 0x91, 0x51, 0xD1, 0x31, 0xB1, 0x71, 0xF1,
0x09, 0x89, 0x49, 0xC9, 0x29, 0xA9, 0x69, 0xE9,
0x19, 0x99, 0x59, 0xD9, 0x39, 0xB9, 0x79, 0xF9,
0x05, 0x85, 0x45, 0xC5, 0x25, 0xA5, 0x65, 0xE5,
0x15, 0x95, 0x55, 0xD5, 0x35, 0xB5, 0x75, 0xF5,
0x0D, 0x8D, 0x4D, 0xCD, 0x2D, 0xAD, 0x6D, 0xED,
0x1D, 0x9D, 0x5D, 0xDD, 0x3D, 0xBD, 0x7D, 0xFD,
0x03, 0x83, 0x43, 0xC3, 0x23, 0xA3, 0x63, 0xE3,
0x13, 0x93, 0x53, 0xD3, 0x33, 0xB3, 0x73, 0xF3,
0x0B, 0x8B, 0x4B, 0xCB, 0x2B, 0xAB, 0x6B, 0xEB,
0x1B, 0x9B, 0x5B, 0xDB, 0x3B, 0xBB, 0x7B, 0xFB,
0x07, 0x87, 0x47, 0xC7, 0x27, 0xA7, 0x67, 0xE7,
0x17, 0x97, 0x57, 0xD7, 0x37, 0xB7, 0x77, 0xF7,
0x0F, 0x8F, 0x4F, 0xCF, 0x2F, 0xAF, 0x6F, 0xEF,
0x1F, 0x9F, 0x5F, 0xDF, 0x3F, 0xBF, 0x7F, 0xFF,
};
//This is minimized version of type 'complex'. All operations is inline
static long double temp;
inline void operator+=(ShortComplex &x, const Complex &y) { x.re += (double)y.re; x.im += (double)y.im; }
inline void operator-=(ShortComplex &x, const Complex &y) { x.re -= (double)y.re; x.im -= (double)y.im; }
inline void operator*=(Complex &x, const Complex &y) { temp = x.re; x.re = temp * y.re - x.im * y.im; x.im = temp * y.im + x.im * y.re; }
inline void operator*=(Complex &x, const ShortComplex &y) { temp = x.re; x.re = temp * y.re - x.im * y.im; x.im = temp * y.im + x.im * y.re; }
inline void operator/=(ShortComplex &x, double div) { x.re /= div; x.im /= div; }
inline void operator/=(Complex &x, double div) { x.re /= div; x.im /= div; }
inline void operator*=(ShortComplex&x, const ShortComplex &y) { double temp = x.re; x.re = temp * y.re - x.im * y.im; x.im = temp * y.im + x.im * y.re; }
//This is array exp(-2*pi*j/2^n) for n= 1,...,32
//exp(-2*pi*j/2^n) = Complex( cos(2*pi/2^n), -sin(2*pi/2^n) )
static Complex W2n[32]={
{-1.00000000000000000000000000000000, 0.00000000000000000000000000000000}, // W2 calculator (copy/paste) : po, ps
{ 0.00000000000000000000000000000000, -1.00000000000000000000000000000000}, // W4: p/2=o, p/2=s
{ 0.70710678118654752440084436210485, -0.70710678118654752440084436210485}, // W8: p/4=o, p/4=s
{ 0.92387953251128675612818318939679, -0.38268343236508977172845998403040}, // p/8=o, p/8=s
{ 0.98078528040323044912618223613424, -0.19509032201612826784828486847702}, // p/16=
{ 0.99518472667219688624483695310948, -9.80171403295606019941955638886e-2}, // p/32=
{ 0.99879545620517239271477160475910, -4.90676743274180142549549769426e-2}, // p/64=
{ 0.99969881869620422011576564966617, -2.45412285229122880317345294592e-2}, // p/128=
{ 0.99992470183914454092164649119638, -1.22715382857199260794082619510e-2}, // p/256=
{ 0.99998117528260114265699043772857, -6.13588464915447535964023459037e-3}, // p/(2y9)=
{ 0.99999529380957617151158012570012, -3.06795676296597627014536549091e-3}, // p/(2y10)=
{ 0.99999882345170190992902571017153, -1.53398018628476561230369715026e-3}, // p/(2y11)=
{ 0.99999970586288221916022821773877, -7.66990318742704526938568357948e-4}, // p/(2y12)=
{ 0.99999992646571785114473148070739, -3.83495187571395589072461681181e-4}, // p/(2y13)=
{ 0.99999998161642929380834691540291, -1.91747597310703307439909561989e-4}, // p/(2y14)=
{ 0.99999999540410731289097193313961, -9.58737990959773458705172109764e-5}, // p/(2y15)=
{ 0.99999999885102682756267330779455, -4.79368996030668845490039904946e-5}, // p/(2y16)=
{ 0.99999999971275670684941397221864, -2.39684498084182187291865771650e-5}, // p/(2y17)=
{ 0.99999999992818917670977509588385, -1.19842249050697064215215615969e-5}, // p/(2y18)=
{ 0.99999999998204729417728262414778, -5.99211245264242784287971180889e-6}, // p/(2y19)=
{ 0.99999999999551182354431058417300, -2.99605622633466075045481280835e-6}, // p/(2y20)=
{ 0.99999999999887795588607701655175, -1.49802811316901122885427884615e-6}, // p/(2y21)=
{ 0.99999999999971948897151921479472, -7.49014056584715721130498566730e-7}, // p/(2y22)=
{ 0.99999999999992987224287980123973, -3.74507028292384123903169179084e-7}, // p/(2y23)=
{ 0.99999999999998246806071995015625, -1.87253514146195344868824576593e-7}, // p/(2y24)=
{ 0.99999999999999561701517998752946, -9.36267570730980827990672866808e-8}, // p/(2y25)=
{ 0.99999999999999890425379499688176, -4.68133785365490926951155181385e-8}, // p/(2y26)=
{ 0.99999999999999972606344874922040, -2.34066892682745527595054934190e-8}, // p/(2y27)=
{ 0.99999999999999993151586218730510, -1.17033446341372771812462135032e-8}, // p/(2y28)=
{ 0.99999999999999998287896554682627, -5.85167231706863869080979010083e-9}, // p/(2y29)=
{ 0.99999999999999999571974138670657, -2.92583615853431935792823046906e-9}, // p/(2y30)=
{ 0.99999999999999999892993534667664, -1.46291807926715968052953216186e-9}, // p/(2y31)=
};
#define M_PI (3.1415926535897932384626433832795)
inline void complex_mul(ShortComplex *z, const ShortComplex *z1, const Complex *z2)
{
z->re = (double)(z1->re * z2->re - z1->im * z2->im);
z->im = (double)(z1->re * z2->im + z1->im * z2->re);
}
static ShortComplex *createWstore(unsigned int Nmax)
{
unsigned int N, Skew, Skew2;
ShortComplex *Wstore, *Warray, *WstoreEnd;
Complex WN, *pWN;
Skew2 = Nmax >> 1;
Wstore = new ShortComplex[Skew2];
WstoreEnd = Wstore + Skew2;
Wstore[0].re = 1.0;
Wstore[0].im = 0.0;
for(N = 4, pWN = W2n + 1, Skew = Skew2 >> 1; N <= Nmax; N += N, pWN++, Skew2 = Skew, Skew >>= 1)
{
//WN = W(1, N) = exp(-2*pi*j/N)
WN= *pWN;
for(Warray = Wstore; Warray < WstoreEnd; Warray += Skew2)
complex_mul(Warray + Skew, Warray, &WN);
}
return Wstore;
}
static void fft_step(ShortComplex *x, unsigned int T, bool complement, const ShortComplex *Wstore)
{
unsigned int Nmax, I, J, N, Nd2, k, m, Skew, mpNd2, Step;
unsigned char *Ic = (unsigned char*) &I;
unsigned char *Jc = (unsigned char*) &J;
ShortComplex S;
const ShortComplex *Warray;
Complex Temp;
Nmax = 1 << T;
//first interchanging
for(I = 1; I < Nmax - 1; I++)
{
Jc[0] = reverse256[Ic[3]];
Jc[1] = reverse256[Ic[2]];
Jc[2] = reverse256[Ic[1]];
Jc[3] = reverse256[Ic[0]];
J >>= (32 - T);
if (I < J)
{
S = x[I];
x[I] = x[J];
x[J] = S;
}
}
//main loop
for(N = 2, Nd2 = 1, Skew = Nmax >> 1, Step= 1; N <= Nmax; Nd2 = N, N += N, Skew >>= 1, Step++)
{
for(Warray = Wstore, k = 0; k < Nd2; k++, Warray += Skew)
{
for(m = k; m < Nmax; m += N)
{
Temp = *Warray;
if (complement)
Temp.im= -Temp.im;
mpNd2= m + Nd2;
Temp *= x[mpNd2];
x[mpNd2] = x[m];
x[mpNd2] -= Temp;
x[m] += Temp;
}
}
}
}
/*
x: x - array of items
N: N - number of items in array
complement: false - normal (direct) transformation, true - reverse transformation
*/
void universal_fft(ShortComplex *x, int N, bool complement)
{
ShortComplex *x_;
ShortComplex *w;
ShortComplex *Wstore;
int T;
T= (int)floor(log((double)N) / log(2.0) + 0.5);
if (1 << T == N)
{
Wstore= createWstore(N);
fft_step(x, T, complement, Wstore);
if (complement)
{
for(int i= 0; i < N; i++)
x[i]/= N;
}
delete [] Wstore;
return;
}
//find N', T
int N2= N+N;
int N_;
long double arg;
for(N_= 1, T= 0; N_ < N2; N_+= N_, T++)
{
}
//find --2pi/N/2 = pi/N
long double piN= M_PI / N;
if (complement)
piN= -piN;
//find x_[n] = x[n]*e^--2*j*pi*n*n/N/2 = x[n]*e^j*piN*n*n
x_= new ShortComplex[N_];
Complex v;
int n;
for(n= 0; n < N; ++n)
{
arg= piN*n*n;
v.re= cosl(arg);
v.im= sinl(arg);
complex_mul(x_ + n, x + n, &v);
}
for(; n < N_; ++n)
x_[n].re= x_[n].im= 0;
//find w[n] = e^-j*2*pi*(2*N-2-n)^2/N/2= e^-j*piN*(2*N-2-n)^2
w= new ShortComplex[N_];
int N22= 2*N - 2;
for(n= 0; n < N_; ++n, --N22)
{
arg= -piN*N22*N22;
w[n].re= (double)cos(arg);
w[n].im= (double)sin(arg);
}
//FFT1
Wstore= createWstore(N_);
fft_step(x_, T, false, Wstore);
//FFT2
fft_step(w, T, false, Wstore);
//svertka
for(n= 0; n < N_; ++n)
x_[n]*= w[n];
//FFT3 (complement)
fft_step(x_, T, true, Wstore);
//find X[n] = X_[n]*e^--j*2*pi*n*n/N/2 = X_[n]*e^j*piN*n*n
for(n= 0, N22= 2*N - 2; n < N; ++n, --N22)
{
arg= piN*n*n;
v.re= cosl(arg);
v.im= sinl(arg);
v/= N_;
if (complement)
v/= N;
complex_mul(x + n, x_ + N22, &v);
}
delete [] x_;
delete [] w;
delete [] Wstore;
}
Оптимизированный вариант fft_step:
static void fft_step(ShortComplex *x, unsigned int T, bool complement, const ShortComplex *Wstore)
{
unsigned int Nmax, I, J, N, Nd2, N2, k, Skew, Step;
unsigned char *Ic= (unsigned char*) &I;
unsigned char *Jc= (unsigned char*) &J;
ShortComplex S;
Nmax= 1 << T;
double cmul= complement ? -1.0 : +1.0;
//first interchanging
for(I = 1; I < Nmax - 1; I++)
{
Jc[0]= reverse256[Ic[3]];
Jc[1]= reverse256[Ic[2]];
Jc[2]= reverse256[Ic[1]];
Jc[3]= reverse256[Ic[0]];
J >>= (32 - T);
if (I < J)
{
S= x[I];
x[I]= x[J];
x[J]= S;
}
}
double *Warray;
double Wre, Wim;
double Tre, Tim;
double *arr= (double*)x;
double *arrEnd= arr + (Nmax + Nmax);
//main loop
for(N= 2, Skew= Nmax, Step= 1; N <= Nmax; N += N, Skew >>= 1, Step++)
{
N2= N + N;
Nd2= (N >> 1);
for(Warray= (double*)Wstore, k= 0; k < Nd2; k++, Warray += Skew)
{
Wre= *Warray;
Wim= cmul*Warray[1];
for(double *x1re= arr + (k + k); x1re < arrEnd; x1re+= N2)
{
double *x1im= x1re + 1;
double *x2re= x1re + N;
double *x2im= x2re + 1;
Tre = Wre * *x2re - Wim * *x2im;
Tim = Wre * *x2im + Wim * *x2re;
*x2re= *x1re - Tre;
*x2im= *x1im - Tim;
*x1re+= Tre;
*x1im+= Tim;
}
}
}
}