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1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2009 Mark Borgerding mark a borgerding net
5 //
6 // This Source Code Form is subject to the terms of the Mozilla
7 // Public License v. 2.0. If a copy of the MPL was not distributed
8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9 
10 namespace Eigen {
11 
12 namespace internal {
13 
14   // This FFT implementation was derived from kissfft http:sourceforge.net/projects/kissfft
15   // Copyright 2003-2009 Mark Borgerding
16 
17 template <typename _Scalar>
18 struct kiss_cpx_fft
19 {
20   typedef _Scalar Scalar;
21   typedef std::complex<Scalar> Complex;
22   std::vector<Complex> m_twiddles;
23   std::vector<int> m_stageRadix;
24   std::vector<int> m_stageRemainder;
25   std::vector<Complex> m_scratchBuf;
26   bool m_inverse;
27 
28   inline
make_twiddleskiss_cpx_fft29     void make_twiddles(int nfft,bool inverse)
30     {
31       m_inverse = inverse;
32       m_twiddles.resize(nfft);
33       Scalar phinc =  (inverse?2:-2)* acos( (Scalar) -1)  / nfft;
34       for (int i=0;i<nfft;++i)
35         m_twiddles[i] = exp( Complex(0,i*phinc) );
36     }
37 
factorizekiss_cpx_fft38   void factorize(int nfft)
39   {
40     //start factoring out 4's, then 2's, then 3,5,7,9,...
41     int n= nfft;
42     int p=4;
43     do {
44       while (n % p) {
45         switch (p) {
46           case 4: p = 2; break;
47           case 2: p = 3; break;
48           default: p += 2; break;
49         }
50         if (p*p>n)
51           p=n;// impossible to have a factor > sqrt(n)
52       }
53       n /= p;
54       m_stageRadix.push_back(p);
55       m_stageRemainder.push_back(n);
56       if ( p > 5 )
57         m_scratchBuf.resize(p); // scratchbuf will be needed in bfly_generic
58     }while(n>1);
59   }
60 
61   template <typename _Src>
62     inline
workkiss_cpx_fft63     void work( int stage,Complex * xout, const _Src * xin, size_t fstride,size_t in_stride)
64     {
65       int p = m_stageRadix[stage];
66       int m = m_stageRemainder[stage];
67       Complex * Fout_beg = xout;
68       Complex * Fout_end = xout + p*m;
69 
70       if (m>1) {
71         do{
72           // recursive call:
73           // DFT of size m*p performed by doing
74           // p instances of smaller DFTs of size m,
75           // each one takes a decimated version of the input
76           work(stage+1, xout , xin, fstride*p,in_stride);
77           xin += fstride*in_stride;
78         }while( (xout += m) != Fout_end );
79       }else{
80         do{
81           *xout = *xin;
82           xin += fstride*in_stride;
83         }while(++xout != Fout_end );
84       }
85       xout=Fout_beg;
86 
87       // recombine the p smaller DFTs
88       switch (p) {
89         case 2: bfly2(xout,fstride,m); break;
90         case 3: bfly3(xout,fstride,m); break;
91         case 4: bfly4(xout,fstride,m); break;
92         case 5: bfly5(xout,fstride,m); break;
93         default: bfly_generic(xout,fstride,m,p); break;
94       }
95     }
96 
97   inline
bfly2kiss_cpx_fft98     void bfly2( Complex * Fout, const size_t fstride, int m)
99     {
100       for (int k=0;k<m;++k) {
101         Complex t = Fout[m+k] * m_twiddles[k*fstride];
102         Fout[m+k] = Fout[k] - t;
103         Fout[k] += t;
104       }
105     }
106 
107   inline
bfly4kiss_cpx_fft108     void bfly4( Complex * Fout, const size_t fstride, const size_t m)
109     {
110       Complex scratch[6];
111       int negative_if_inverse = m_inverse * -2 +1;
112       for (size_t k=0;k<m;++k) {
113         scratch[0] = Fout[k+m] * m_twiddles[k*fstride];
114         scratch[1] = Fout[k+2*m] * m_twiddles[k*fstride*2];
115         scratch[2] = Fout[k+3*m] * m_twiddles[k*fstride*3];
116         scratch[5] = Fout[k] - scratch[1];
117 
118         Fout[k] += scratch[1];
119         scratch[3] = scratch[0] + scratch[2];
120         scratch[4] = scratch[0] - scratch[2];
121         scratch[4] = Complex( scratch[4].imag()*negative_if_inverse , -scratch[4].real()* negative_if_inverse );
122 
123         Fout[k+2*m]  = Fout[k] - scratch[3];
124         Fout[k] += scratch[3];
125         Fout[k+m] = scratch[5] + scratch[4];
126         Fout[k+3*m] = scratch[5] - scratch[4];
127       }
128     }
129 
130   inline
bfly3kiss_cpx_fft131     void bfly3( Complex * Fout, const size_t fstride, const size_t m)
132     {
133       size_t k=m;
134       const size_t m2 = 2*m;
135       Complex *tw1,*tw2;
136       Complex scratch[5];
137       Complex epi3;
138       epi3 = m_twiddles[fstride*m];
139 
140       tw1=tw2=&m_twiddles[0];
141 
142       do{
143         scratch[1]=Fout[m] * *tw1;
144         scratch[2]=Fout[m2] * *tw2;
145 
146         scratch[3]=scratch[1]+scratch[2];
147         scratch[0]=scratch[1]-scratch[2];
148         tw1 += fstride;
149         tw2 += fstride*2;
150         Fout[m] = Complex( Fout->real() - Scalar(.5)*scratch[3].real() , Fout->imag() - Scalar(.5)*scratch[3].imag() );
151         scratch[0] *= epi3.imag();
152         *Fout += scratch[3];
153         Fout[m2] = Complex(  Fout[m].real() + scratch[0].imag() , Fout[m].imag() - scratch[0].real() );
154         Fout[m] += Complex( -scratch[0].imag(),scratch[0].real() );
155         ++Fout;
156       }while(--k);
157     }
158 
159   inline
bfly5kiss_cpx_fft160     void bfly5( Complex * Fout, const size_t fstride, const size_t m)
161     {
162       Complex *Fout0,*Fout1,*Fout2,*Fout3,*Fout4;
163       size_t u;
164       Complex scratch[13];
165       Complex * twiddles = &m_twiddles[0];
166       Complex *tw;
167       Complex ya,yb;
168       ya = twiddles[fstride*m];
169       yb = twiddles[fstride*2*m];
170 
171       Fout0=Fout;
172       Fout1=Fout0+m;
173       Fout2=Fout0+2*m;
174       Fout3=Fout0+3*m;
175       Fout4=Fout0+4*m;
176 
177       tw=twiddles;
178       for ( u=0; u<m; ++u ) {
179         scratch[0] = *Fout0;
180 
181         scratch[1]  = *Fout1 * tw[u*fstride];
182         scratch[2]  = *Fout2 * tw[2*u*fstride];
183         scratch[3]  = *Fout3 * tw[3*u*fstride];
184         scratch[4]  = *Fout4 * tw[4*u*fstride];
185 
186         scratch[7] = scratch[1] + scratch[4];
187         scratch[10] = scratch[1] - scratch[4];
188         scratch[8] = scratch[2] + scratch[3];
189         scratch[9] = scratch[2] - scratch[3];
190 
191         *Fout0 +=  scratch[7];
192         *Fout0 +=  scratch[8];
193 
194         scratch[5] = scratch[0] + Complex(
195             (scratch[7].real()*ya.real() ) + (scratch[8].real() *yb.real() ),
196             (scratch[7].imag()*ya.real()) + (scratch[8].imag()*yb.real())
197             );
198 
199         scratch[6] = Complex(
200             (scratch[10].imag()*ya.imag()) + (scratch[9].imag()*yb.imag()),
201             -(scratch[10].real()*ya.imag()) - (scratch[9].real()*yb.imag())
202             );
203 
204         *Fout1 = scratch[5] - scratch[6];
205         *Fout4 = scratch[5] + scratch[6];
206 
207         scratch[11] = scratch[0] +
208           Complex(
209               (scratch[7].real()*yb.real()) + (scratch[8].real()*ya.real()),
210               (scratch[7].imag()*yb.real()) + (scratch[8].imag()*ya.real())
211               );
212 
213         scratch[12] = Complex(
214             -(scratch[10].imag()*yb.imag()) + (scratch[9].imag()*ya.imag()),
215             (scratch[10].real()*yb.imag()) - (scratch[9].real()*ya.imag())
216             );
217 
218         *Fout2=scratch[11]+scratch[12];
219         *Fout3=scratch[11]-scratch[12];
220 
221         ++Fout0;++Fout1;++Fout2;++Fout3;++Fout4;
222       }
223     }
224 
225   /* perform the butterfly for one stage of a mixed radix FFT */
226   inline
bfly_generickiss_cpx_fft227     void bfly_generic(
228         Complex * Fout,
229         const size_t fstride,
230         int m,
231         int p
232         )
233     {
234       int u,k,q1,q;
235       Complex * twiddles = &m_twiddles[0];
236       Complex t;
237       int Norig = static_cast<int>(m_twiddles.size());
238       Complex * scratchbuf = &m_scratchBuf[0];
239 
240       for ( u=0; u<m; ++u ) {
241         k=u;
242         for ( q1=0 ; q1<p ; ++q1 ) {
243           scratchbuf[q1] = Fout[ k  ];
244           k += m;
245         }
246 
247         k=u;
248         for ( q1=0 ; q1<p ; ++q1 ) {
249           int twidx=0;
250           Fout[ k ] = scratchbuf[0];
251           for (q=1;q<p;++q ) {
252             twidx += static_cast<int>(fstride) * k;
253             if (twidx>=Norig) twidx-=Norig;
254             t=scratchbuf[q] * twiddles[twidx];
255             Fout[ k ] += t;
256           }
257           k += m;
258         }
259       }
260     }
261 };
262 
263 template <typename _Scalar>
264 struct kissfft_impl
265 {
266   typedef _Scalar Scalar;
267   typedef std::complex<Scalar> Complex;
268 
clearkissfft_impl269   void clear()
270   {
271     m_plans.clear();
272     m_realTwiddles.clear();
273   }
274 
275   inline
fwdkissfft_impl276     void fwd( Complex * dst,const Complex *src,int nfft)
277     {
278       get_plan(nfft,false).work(0, dst, src, 1,1);
279     }
280 
281   inline
fwd2kissfft_impl282     void fwd2( Complex * dst,const Complex *src,int n0,int n1)
283     {
284         EIGEN_UNUSED_VARIABLE(dst);
285         EIGEN_UNUSED_VARIABLE(src);
286         EIGEN_UNUSED_VARIABLE(n0);
287         EIGEN_UNUSED_VARIABLE(n1);
288     }
289 
290   inline
inv2kissfft_impl291     void inv2( Complex * dst,const Complex *src,int n0,int n1)
292     {
293         EIGEN_UNUSED_VARIABLE(dst);
294         EIGEN_UNUSED_VARIABLE(src);
295         EIGEN_UNUSED_VARIABLE(n0);
296         EIGEN_UNUSED_VARIABLE(n1);
297     }
298 
299   // real-to-complex forward FFT
300   // perform two FFTs of src even and src odd
301   // then twiddle to recombine them into the half-spectrum format
302   // then fill in the conjugate symmetric half
303   inline
fwdkissfft_impl304     void fwd( Complex * dst,const Scalar * src,int nfft)
305     {
306       if ( nfft&3  ) {
307         // use generic mode for odd
308         m_tmpBuf1.resize(nfft);
309         get_plan(nfft,false).work(0, &m_tmpBuf1[0], src, 1,1);
310         std::copy(m_tmpBuf1.begin(),m_tmpBuf1.begin()+(nfft>>1)+1,dst );
311       }else{
312         int ncfft = nfft>>1;
313         int ncfft2 = nfft>>2;
314         Complex * rtw = real_twiddles(ncfft2);
315 
316         // use optimized mode for even real
317         fwd( dst, reinterpret_cast<const Complex*> (src), ncfft);
318         Complex dc = dst[0].real() +  dst[0].imag();
319         Complex nyquist = dst[0].real() -  dst[0].imag();
320         int k;
321         for ( k=1;k <= ncfft2 ; ++k ) {
322           Complex fpk = dst[k];
323           Complex fpnk = conj(dst[ncfft-k]);
324           Complex f1k = fpk + fpnk;
325           Complex f2k = fpk - fpnk;
326           Complex tw= f2k * rtw[k-1];
327           dst[k] =  (f1k + tw) * Scalar(.5);
328           dst[ncfft-k] =  conj(f1k -tw)*Scalar(.5);
329         }
330         dst[0] = dc;
331         dst[ncfft] = nyquist;
332       }
333     }
334 
335   // inverse complex-to-complex
336   inline
invkissfft_impl337     void inv(Complex * dst,const Complex  *src,int nfft)
338     {
339       get_plan(nfft,true).work(0, dst, src, 1,1);
340     }
341 
342   // half-complex to scalar
343   inline
invkissfft_impl344     void inv( Scalar * dst,const Complex * src,int nfft)
345     {
346       if (nfft&3) {
347         m_tmpBuf1.resize(nfft);
348         m_tmpBuf2.resize(nfft);
349         std::copy(src,src+(nfft>>1)+1,m_tmpBuf1.begin() );
350         for (int k=1;k<(nfft>>1)+1;++k)
351           m_tmpBuf1[nfft-k] = conj(m_tmpBuf1[k]);
352         inv(&m_tmpBuf2[0],&m_tmpBuf1[0],nfft);
353         for (int k=0;k<nfft;++k)
354           dst[k] = m_tmpBuf2[k].real();
355       }else{
356         // optimized version for multiple of 4
357         int ncfft = nfft>>1;
358         int ncfft2 = nfft>>2;
359         Complex * rtw = real_twiddles(ncfft2);
360         m_tmpBuf1.resize(ncfft);
361         m_tmpBuf1[0] = Complex( src[0].real() + src[ncfft].real(), src[0].real() - src[ncfft].real() );
362         for (int k = 1; k <= ncfft / 2; ++k) {
363           Complex fk = src[k];
364           Complex fnkc = conj(src[ncfft-k]);
365           Complex fek = fk + fnkc;
366           Complex tmp = fk - fnkc;
367           Complex fok = tmp * conj(rtw[k-1]);
368           m_tmpBuf1[k] = fek + fok;
369           m_tmpBuf1[ncfft-k] = conj(fek - fok);
370         }
371         get_plan(ncfft,true).work(0, reinterpret_cast<Complex*>(dst), &m_tmpBuf1[0], 1,1);
372       }
373     }
374 
375   protected:
376   typedef kiss_cpx_fft<Scalar> PlanData;
377   typedef std::map<int,PlanData> PlanMap;
378 
379   PlanMap m_plans;
380   std::map<int, std::vector<Complex> > m_realTwiddles;
381   std::vector<Complex> m_tmpBuf1;
382   std::vector<Complex> m_tmpBuf2;
383 
384   inline
PlanKeykissfft_impl385     int PlanKey(int nfft, bool isinverse) const { return (nfft<<1) | int(isinverse); }
386 
387   inline
get_plankissfft_impl388     PlanData & get_plan(int nfft, bool inverse)
389     {
390       // TODO look for PlanKey(nfft, ! inverse) and conjugate the twiddles
391       PlanData & pd = m_plans[ PlanKey(nfft,inverse) ];
392       if ( pd.m_twiddles.size() == 0 ) {
393         pd.make_twiddles(nfft,inverse);
394         pd.factorize(nfft);
395       }
396       return pd;
397     }
398 
399   inline
real_twiddleskissfft_impl400     Complex * real_twiddles(int ncfft2)
401     {
402       std::vector<Complex> & twidref = m_realTwiddles[ncfft2];// creates new if not there
403       if ( (int)twidref.size() != ncfft2 ) {
404         twidref.resize(ncfft2);
405         int ncfft= ncfft2<<1;
406         Scalar pi =  acos( Scalar(-1) );
407         for (int k=1;k<=ncfft2;++k)
408           twidref[k-1] = exp( Complex(0,-pi * (Scalar(k) / ncfft + Scalar(.5)) ) );
409       }
410       return &twidref[0];
411     }
412 };
413 
414 } // end namespace internal
415 
416 } // end namespace Eigen
417 
418 /* vim: set filetype=cpp et sw=2 ts=2 ai: */
419