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00001 // This file is part of Eigen, a lightweight C++ template library 00002 // for linear algebra. 00003 // 00004 // Copyright (C) 2009 Mark Borgerding mark a borgerding net 00005 // 00006 // This Source Code Form is subject to the terms of the Mozilla 00007 // Public License v. 2.0. If a copy of the MPL was not distributed 00008 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 00009 00010 namespace Eigen { 00011 00012 namespace internal { 00013 00014 // This FFT implementation was derived from kissfft http:sourceforge.net/projects/kissfft 00015 // Copyright 2003-2009 Mark Borgerding 00016 00017 template <typename _Scalar> 00018 struct kiss_cpx_fft 00019 { 00020 typedef _Scalar Scalar; 00021 typedef std::complex<Scalar> Complex; 00022 std::vector<Complex> m_twiddles; 00023 std::vector<int> m_stageRadix; 00024 std::vector<int> m_stageRemainder; 00025 std::vector<Complex> m_scratchBuf; 00026 bool m_inverse; 00027 00028 inline 00029 void make_twiddles(int nfft,bool inverse) 00030 { 00031 using std::acos; 00032 m_inverse = inverse; 00033 m_twiddles.resize(nfft); 00034 Scalar phinc = (inverse?2:-2)* acos( (Scalar) -1) / nfft; 00035 for (int i=0;i<nfft;++i) 00036 m_twiddles[i] = exp( Complex(0,i*phinc) ); 00037 } 00038 00039 void factorize(int nfft) 00040 { 00041 //start factoring out 4's, then 2's, then 3,5,7,9,... 00042 int n= nfft; 00043 int p=4; 00044 do { 00045 while (n % p) { 00046 switch (p) { 00047 case 4: p = 2; break; 00048 case 2: p = 3; break; 00049 default: p += 2; break; 00050 } 00051 if (p*p>n) 00052 p=n;// impossible to have a factor > sqrt(n) 00053 } 00054 n /= p; 00055 m_stageRadix.push_back(p); 00056 m_stageRemainder.push_back(n); 00057 if ( p > 5 ) 00058 m_scratchBuf.resize(p); // scratchbuf will be needed in bfly_generic 00059 }while(n>1); 00060 } 00061 00062 template <typename _Src> 00063 inline 00064 void work( int stage,Complex * xout, const _Src * xin, size_t fstride,size_t in_stride) 00065 { 00066 int p = m_stageRadix[stage]; 00067 int m = m_stageRemainder[stage]; 00068 Complex * Fout_beg = xout; 00069 Complex * Fout_end = xout + p*m; 00070 00071 if (m>1) { 00072 do{ 00073 // recursive call: 00074 // DFT of size m*p performed by doing 00075 // p instances of smaller DFTs of size m, 00076 // each one takes a decimated version of the input 00077 work(stage+1, xout , xin, fstride*p,in_stride); 00078 xin += fstride*in_stride; 00079 }while( (xout += m) != Fout_end ); 00080 }else{ 00081 do{ 00082 *xout = *xin; 00083 xin += fstride*in_stride; 00084 }while(++xout != Fout_end ); 00085 } 00086 xout=Fout_beg; 00087 00088 // recombine the p smaller DFTs 00089 switch (p) { 00090 case 2: bfly2(xout,fstride,m); break; 00091 case 3: bfly3(xout,fstride,m); break; 00092 case 4: bfly4(xout,fstride,m); break; 00093 case 5: bfly5(xout,fstride,m); break; 00094 default: bfly_generic(xout,fstride,m,p); break; 00095 } 00096 } 00097 00098 inline 00099 void bfly2( Complex * Fout, const size_t fstride, int m) 00100 { 00101 for (int k=0;k<m;++k) { 00102 Complex t = Fout[m+k] * m_twiddles[k*fstride]; 00103 Fout[m+k] = Fout[k] - t; 00104 Fout[k] += t; 00105 } 00106 } 00107 00108 inline 00109 void bfly4( Complex * Fout, const size_t fstride, const size_t m) 00110 { 00111 Complex scratch[6]; 00112 int negative_if_inverse = m_inverse * -2 +1; 00113 for (size_t k=0;k<m;++k) { 00114 scratch[0] = Fout[k+m] * m_twiddles[k*fstride]; 00115 scratch[1] = Fout[k+2*m] * m_twiddles[k*fstride*2]; 00116 scratch[2] = Fout[k+3*m] * m_twiddles[k*fstride*3]; 00117 scratch[5] = Fout[k] - scratch[1]; 00118 00119 Fout[k] += scratch[1]; 00120 scratch[3] = scratch[0] + scratch[2]; 00121 scratch[4] = scratch[0] - scratch[2]; 00122 scratch[4] = Complex( scratch[4].imag()*negative_if_inverse , -scratch[4].real()* negative_if_inverse ); 00123 00124 Fout[k+2*m] = Fout[k] - scratch[3]; 00125 Fout[k] += scratch[3]; 00126 Fout[k+m] = scratch[5] + scratch[4]; 00127 Fout[k+3*m] = scratch[5] - scratch[4]; 00128 } 00129 } 00130 00131 inline 00132 void bfly3( Complex * Fout, const size_t fstride, const size_t m) 00133 { 00134 size_t k=m; 00135 const size_t m2 = 2*m; 00136 Complex *tw1,*tw2; 00137 Complex scratch[5]; 00138 Complex epi3; 00139 epi3 = m_twiddles[fstride*m]; 00140 00141 tw1=tw2=&m_twiddles[0]; 00142 00143 do{ 00144 scratch[1]=Fout[m] * *tw1; 00145 scratch[2]=Fout[m2] * *tw2; 00146 00147 scratch[3]=scratch[1]+scratch[2]; 00148 scratch[0]=scratch[1]-scratch[2]; 00149 tw1 += fstride; 00150 tw2 += fstride*2; 00151 Fout[m] = Complex( Fout->real() - Scalar(.5)*scratch[3].real() , Fout->imag() - Scalar(.5)*scratch[3].imag() ); 00152 scratch[0] *= epi3.imag(); 00153 *Fout += scratch[3]; 00154 Fout[m2] = Complex( Fout[m].real() + scratch[0].imag() , Fout[m].imag() - scratch[0].real() ); 00155 Fout[m] += Complex( -scratch[0].imag(),scratch[0].real() ); 00156 ++Fout; 00157 }while(--k); 00158 } 00159 00160 inline 00161 void bfly5( Complex * Fout, const size_t fstride, const size_t m) 00162 { 00163 Complex *Fout0,*Fout1,*Fout2,*Fout3,*Fout4; 00164 size_t u; 00165 Complex scratch[13]; 00166 Complex * twiddles = &m_twiddles[0]; 00167 Complex *tw; 00168 Complex ya,yb; 00169 ya = twiddles[fstride*m]; 00170 yb = twiddles[fstride*2*m]; 00171 00172 Fout0=Fout; 00173 Fout1=Fout0+m; 00174 Fout2=Fout0+2*m; 00175 Fout3=Fout0+3*m; 00176 Fout4=Fout0+4*m; 00177 00178 tw=twiddles; 00179 for ( u=0; u<m; ++u ) { 00180 scratch[0] = *Fout0; 00181 00182 scratch[1] = *Fout1 * tw[u*fstride]; 00183 scratch[2] = *Fout2 * tw[2*u*fstride]; 00184 scratch[3] = *Fout3 * tw[3*u*fstride]; 00185 scratch[4] = *Fout4 * tw[4*u*fstride]; 00186 00187 scratch[7] = scratch[1] + scratch[4]; 00188 scratch[10] = scratch[1] - scratch[4]; 00189 scratch[8] = scratch[2] + scratch[3]; 00190 scratch[9] = scratch[2] - scratch[3]; 00191 00192 *Fout0 += scratch[7]; 00193 *Fout0 += scratch[8]; 00194 00195 scratch[5] = scratch[0] + Complex( 00196 (scratch[7].real()*ya.real() ) + (scratch[8].real() *yb.real() ), 00197 (scratch[7].imag()*ya.real()) + (scratch[8].imag()*yb.real()) 00198 ); 00199 00200 scratch[6] = Complex( 00201 (scratch[10].imag()*ya.imag()) + (scratch[9].imag()*yb.imag()), 00202 -(scratch[10].real()*ya.imag()) - (scratch[9].real()*yb.imag()) 00203 ); 00204 00205 *Fout1 = scratch[5] - scratch[6]; 00206 *Fout4 = scratch[5] + scratch[6]; 00207 00208 scratch[11] = scratch[0] + 00209 Complex( 00210 (scratch[7].real()*yb.real()) + (scratch[8].real()*ya.real()), 00211 (scratch[7].imag()*yb.real()) + (scratch[8].imag()*ya.real()) 00212 ); 00213 00214 scratch[12] = Complex( 00215 -(scratch[10].imag()*yb.imag()) + (scratch[9].imag()*ya.imag()), 00216 (scratch[10].real()*yb.imag()) - (scratch[9].real()*ya.imag()) 00217 ); 00218 00219 *Fout2=scratch[11]+scratch[12]; 00220 *Fout3=scratch[11]-scratch[12]; 00221 00222 ++Fout0;++Fout1;++Fout2;++Fout3;++Fout4; 00223 } 00224 } 00225 00226 /* perform the butterfly for one stage of a mixed radix FFT */ 00227 inline 00228 void bfly_generic( 00229 Complex * Fout, 00230 const size_t fstride, 00231 int m, 00232 int p 00233 ) 00234 { 00235 int u,k,q1,q; 00236 Complex * twiddles = &m_twiddles[0]; 00237 Complex t; 00238 int Norig = static_cast<int>(m_twiddles.size()); 00239 Complex * scratchbuf = &m_scratchBuf[0]; 00240 00241 for ( u=0; u<m; ++u ) { 00242 k=u; 00243 for ( q1=0 ; q1<p ; ++q1 ) { 00244 scratchbuf[q1] = Fout[ k ]; 00245 k += m; 00246 } 00247 00248 k=u; 00249 for ( q1=0 ; q1<p ; ++q1 ) { 00250 int twidx=0; 00251 Fout[ k ] = scratchbuf[0]; 00252 for (q=1;q<p;++q ) { 00253 twidx += static_cast<int>(fstride) * k; 00254 if (twidx>=Norig) twidx-=Norig; 00255 t=scratchbuf[q] * twiddles[twidx]; 00256 Fout[ k ] += t; 00257 } 00258 k += m; 00259 } 00260 } 00261 } 00262 }; 00263 00264 template <typename _Scalar> 00265 struct kissfft_impl 00266 { 00267 typedef _Scalar Scalar; 00268 typedef std::complex<Scalar> Complex; 00269 00270 void clear() 00271 { 00272 m_plans.clear(); 00273 m_realTwiddles.clear(); 00274 } 00275 00276 inline 00277 void fwd( Complex * dst,const Complex *src,int nfft) 00278 { 00279 get_plan(nfft,false).work(0, dst, src, 1,1); 00280 } 00281 00282 inline 00283 void fwd2( Complex * dst,const Complex *src,int n0,int n1) 00284 { 00285 EIGEN_UNUSED_VARIABLE(dst); 00286 EIGEN_UNUSED_VARIABLE(src); 00287 EIGEN_UNUSED_VARIABLE(n0); 00288 EIGEN_UNUSED_VARIABLE(n1); 00289 } 00290 00291 inline 00292 void inv2( Complex * dst,const Complex *src,int n0,int n1) 00293 { 00294 EIGEN_UNUSED_VARIABLE(dst); 00295 EIGEN_UNUSED_VARIABLE(src); 00296 EIGEN_UNUSED_VARIABLE(n0); 00297 EIGEN_UNUSED_VARIABLE(n1); 00298 } 00299 00300 // real-to-complex forward FFT 00301 // perform two FFTs of src even and src odd 00302 // then twiddle to recombine them into the half-spectrum format 00303 // then fill in the conjugate symmetric half 00304 inline 00305 void fwd( Complex * dst,const Scalar * src,int nfft) 00306 { 00307 if ( nfft&3 ) { 00308 // use generic mode for odd 00309 m_tmpBuf1.resize(nfft); 00310 get_plan(nfft,false).work(0, &m_tmpBuf1[0], src, 1,1); 00311 std::copy(m_tmpBuf1.begin(),m_tmpBuf1.begin()+(nfft>>1)+1,dst ); 00312 }else{ 00313 int ncfft = nfft>>1; 00314 int ncfft2 = nfft>>2; 00315 Complex * rtw = real_twiddles(ncfft2); 00316 00317 // use optimized mode for even real 00318 fwd( dst, reinterpret_cast<const Complex*> (src), ncfft); 00319 Complex dc = dst[0].real() + dst[0].imag(); 00320 Complex nyquist = dst[0].real() - dst[0].imag(); 00321 int k; 00322 for ( k=1;k <= ncfft2 ; ++k ) { 00323 Complex fpk = dst[k]; 00324 Complex fpnk = conj(dst[ncfft-k]); 00325 Complex f1k = fpk + fpnk; 00326 Complex f2k = fpk - fpnk; 00327 Complex tw= f2k * rtw[k-1]; 00328 dst[k] = (f1k + tw) * Scalar(.5); 00329 dst[ncfft-k] = conj(f1k -tw)*Scalar(.5); 00330 } 00331 dst[0] = dc; 00332 dst[ncfft] = nyquist; 00333 } 00334 } 00335 00336 // inverse complex-to-complex 00337 inline 00338 void inv(Complex * dst,const Complex *src,int nfft) 00339 { 00340 get_plan(nfft,true).work(0, dst, src, 1,1); 00341 } 00342 00343 // half-complex to scalar 00344 inline 00345 void inv( Scalar * dst,const Complex * src,int nfft) 00346 { 00347 if (nfft&3) { 00348 m_tmpBuf1.resize(nfft); 00349 m_tmpBuf2.resize(nfft); 00350 std::copy(src,src+(nfft>>1)+1,m_tmpBuf1.begin() ); 00351 for (int k=1;k<(nfft>>1)+1;++k) 00352 m_tmpBuf1[nfft-k] = conj(m_tmpBuf1[k]); 00353 inv(&m_tmpBuf2[0],&m_tmpBuf1[0],nfft); 00354 for (int k=0;k<nfft;++k) 00355 dst[k] = m_tmpBuf2[k].real(); 00356 }else{ 00357 // optimized version for multiple of 4 00358 int ncfft = nfft>>1; 00359 int ncfft2 = nfft>>2; 00360 Complex * rtw = real_twiddles(ncfft2); 00361 m_tmpBuf1.resize(ncfft); 00362 m_tmpBuf1[0] = Complex( src[0].real() + src[ncfft].real(), src[0].real() - src[ncfft].real() ); 00363 for (int k = 1; k <= ncfft / 2; ++k) { 00364 Complex fk = src[k]; 00365 Complex fnkc = conj(src[ncfft-k]); 00366 Complex fek = fk + fnkc; 00367 Complex tmp = fk - fnkc; 00368 Complex fok = tmp * conj(rtw[k-1]); 00369 m_tmpBuf1[k] = fek + fok; 00370 m_tmpBuf1[ncfft-k] = conj(fek - fok); 00371 } 00372 get_plan(ncfft,true).work(0, reinterpret_cast<Complex*>(dst), &m_tmpBuf1[0], 1,1); 00373 } 00374 } 00375 00376 protected: 00377 typedef kiss_cpx_fft<Scalar> PlanData; 00378 typedef std::map<int,PlanData> PlanMap; 00379 00380 PlanMap m_plans; 00381 std::map<int, std::vector<Complex> > m_realTwiddles; 00382 std::vector<Complex> m_tmpBuf1; 00383 std::vector<Complex> m_tmpBuf2; 00384 00385 inline 00386 int PlanKey(int nfft, bool isinverse) const { return (nfft<<1) | int(isinverse); } 00387 00388 inline 00389 PlanData & get_plan(int nfft, bool inverse) 00390 { 00391 // TODO look for PlanKey(nfft, ! inverse) and conjugate the twiddles 00392 PlanData & pd = m_plans[ PlanKey(nfft,inverse) ]; 00393 if ( pd.m_twiddles.size() == 0 ) { 00394 pd.make_twiddles(nfft,inverse); 00395 pd.factorize(nfft); 00396 } 00397 return pd; 00398 } 00399 00400 inline 00401 Complex * real_twiddles(int ncfft2) 00402 { 00403 using std::acos; 00404 std::vector<Complex> & twidref = m_realTwiddles[ncfft2];// creates new if not there 00405 if ( (int)twidref.size() != ncfft2 ) { 00406 twidref.resize(ncfft2); 00407 int ncfft= ncfft2<<1; 00408 Scalar pi = acos( Scalar(-1) ); 00409 for (int k=1;k<=ncfft2;++k) 00410 twidref[k-1] = exp( Complex(0,-pi * (Scalar(k) / ncfft + Scalar(.5)) ) ); 00411 } 00412 return &twidref[0]; 00413 } 00414 }; 00415 00416 } // end namespace internal 00417 00418 } // end namespace Eigen 00419 00420 /* vim: set filetype=cpp et sw=2 ts=2 ai: */