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
10namespace Eigen {
11
12namespace internal {
13
14  // This FFT implementation was derived from kissfft http:sourceforge.net/projects/kissfft
15  // Copyright 2003-2009 Mark Borgerding
16
17template <typename _Scalar>
18struct 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
29    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
38  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
63    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
98    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
108    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
131    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
160    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
227    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
263template <typename _Scalar>
264struct kissfft_impl
265{
266  typedef _Scalar Scalar;
267  typedef std::complex<Scalar> Complex;
268
269  void clear()
270  {
271    m_plans.clear();
272    m_realTwiddles.clear();
273  }
274
275  inline
276    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
282    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
291    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
304    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
337    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
344    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
385    int PlanKey(int nfft, bool isinverse) const { return (nfft<<1) | int(isinverse); }
386
387  inline
388    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
400    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