kissfft_impl.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Mark Borgerding mark a borgerding net
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
 
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
 
namespace Eigen {
 
namespace internal {
 
// This FFT implementation was derived from kissfft http:sourceforge.net/projects/kissfft
// Copyright 2003-2009 Mark Borgerding
 
template <typename Scalar_>
struct kiss_cpx_fft {
  typedef Scalar_ Scalar;
  typedef std::complex<Scalar> Complex;
  std::vector<Complex> m_twiddles;
  std::vector<int> m_stageRadix;
  std::vector<int> m_stageRemainder;
  std::vector<Complex> m_scratchBuf;
  bool m_inverse;
 
  static const Scalar m_pi4;  // constant pi / 4
 
  inline void make_twiddles(int nfft, bool inverse) {
    using numext::cos;
    using numext::sin;
    m_inverse = inverse;
    m_twiddles.resize(nfft);
    Scalar phinc = m_pi4 / nfft;
    Scalar flip = inverse ? Scalar(1) : Scalar(-1);
    m_twiddles[0] = Complex(Scalar(1), Scalar(0));
    if ((nfft & 1) == 0) m_twiddles[nfft / 2] = Complex(Scalar(-1), Scalar(0));
    int i = 1;
    for (; i * 8 < nfft; ++i) {
      Scalar c = Scalar(cos(i * 8 * phinc));
      Scalar s = Scalar(sin(i * 8 * phinc));
      m_twiddles[i] = Complex(c, s * flip);
      m_twiddles[nfft - i] = Complex(c, -s * flip);
    }
    for (; i * 4 < nfft; ++i) {
      Scalar c = Scalar(cos((2 * nfft - 8 * i) * phinc));
      Scalar s = Scalar(sin((2 * nfft - 8 * i) * phinc));
      m_twiddles[i] = Complex(s, c * flip);
      m_twiddles[nfft - i] = Complex(s, -c * flip);
    }
    for (; i * 8 < 3 * nfft; ++i) {
      Scalar c = Scalar(cos((8 * i - 2 * nfft) * phinc));
      Scalar s = Scalar(sin((8 * i - 2 * nfft) * phinc));
      m_twiddles[i] = Complex(-s, c * flip);
      m_twiddles[nfft - i] = Complex(-s, -c * flip);
    }
    for (; i * 2 < nfft; ++i) {
      Scalar c = Scalar(cos((4 * nfft - 8 * i) * phinc));
      Scalar s = Scalar(sin((4 * nfft - 8 * i) * phinc));
      m_twiddles[i] = Complex(-c, s * flip);
      m_twiddles[nfft - i] = Complex(-c, -s * flip);
    }
  }
 
  void factorize(int nfft) {
    // start factoring out 4's, then 2's, then 3,5,7,9,...
    int n = nfft;
    int p = 4;
    do {
      while (n % p) {
        switch (p) {
          case 4:
            p = 2;
            break;
          case 2:
            p = 3;
            break;
          default:
            p += 2;
            break;
        }
        if (p * p > n) p = n;  // impossible to have a factor > sqrt(n)
      }
      n /= p;
      m_stageRadix.push_back(p);
      m_stageRemainder.push_back(n);
      if (p > 5) m_scratchBuf.resize(p);  // scratchbuf will be needed in bfly_generic
    } while (n > 1);
  }
 
  template <typename Src_>
  inline void work(int stage, Complex *xout, const Src_ *xin, size_t fstride, size_t in_stride) {
    int p = m_stageRadix[stage];
    int m = m_stageRemainder[stage];
    Complex *Fout_beg = xout;
    Complex *Fout_end = xout + p * m;
 
    if (m > 1) {
      do {
        // recursive call:
        // DFT of size m*p performed by doing
        // p instances of smaller DFTs of size m,
        // each one takes a decimated version of the input
        work(stage + 1, xout, xin, fstride * p, in_stride);
        xin += fstride * in_stride;
      } while ((xout += m) != Fout_end);
    } else {
      do {
        *xout = *xin;
        xin += fstride * in_stride;
      } while (++xout != Fout_end);
    }
    xout = Fout_beg;
 
    // recombine the p smaller DFTs
    switch (p) {
      case 2:
        bfly2(xout, fstride, m);
        break;
      case 3:
        bfly3(xout, fstride, m);
        break;
      case 4:
        bfly4(xout, fstride, m);
        break;
      case 5:
        bfly5(xout, fstride, m);
        break;
      default:
        bfly_generic(xout, fstride, m, p);
        break;
    }
  }
 
  inline void bfly2(Complex *Fout, const size_t fstride, int m) {
    for (int k = 0; k < m; ++k) {
      Complex t = Fout[m + k] * m_twiddles[k * fstride];
      Fout[m + k] = Fout[k] - t;
      Fout[k] += t;
    }
  }
 
  inline void bfly4(Complex *Fout, const size_t fstride, const size_t m) {
    Complex scratch[6];
    int negative_if_inverse = m_inverse * -2 + 1;
    for (size_t k = 0; k < m; ++k) {
      scratch[0] = Fout[k + m] * m_twiddles[k * fstride];
      scratch[1] = Fout[k + 2 * m] * m_twiddles[k * fstride * 2];
      scratch[2] = Fout[k + 3 * m] * m_twiddles[k * fstride * 3];
      scratch[5] = Fout[k] - scratch[1];
 
      Fout[k] += scratch[1];
      scratch[3] = scratch[0] + scratch[2];
      scratch[4] = scratch[0] - scratch[2];
      scratch[4] = Complex(scratch[4].imag() * negative_if_inverse, -scratch[4].real() * negative_if_inverse);
 
      Fout[k + 2 * m] = Fout[k] - scratch[3];
      Fout[k] += scratch[3];
      Fout[k + m] = scratch[5] + scratch[4];
      Fout[k + 3 * m] = scratch[5] - scratch[4];
    }
  }
 
  inline void bfly3(Complex *Fout, const size_t fstride, const size_t m) {
    size_t k = m;
    const size_t m2 = 2 * m;
    Complex *tw1, *tw2;
    Complex scratch[5];
    Complex epi3;
    epi3 = m_twiddles[fstride * m];
 
    tw1 = tw2 = &m_twiddles[0];
 
    do {
      scratch[1] = Fout[m] * *tw1;
      scratch[2] = Fout[m2] * *tw2;
 
      scratch[3] = scratch[1] + scratch[2];
      scratch[0] = scratch[1] - scratch[2];
      tw1 += fstride;
      tw2 += fstride * 2;
      Fout[m] = Complex(Fout->real() - Scalar(.5) * scratch[3].real(), Fout->imag() - Scalar(.5) * scratch[3].imag());
      scratch[0] *= epi3.imag();
      *Fout += scratch[3];
      Fout[m2] = Complex(Fout[m].real() + scratch[0].imag(), Fout[m].imag() - scratch[0].real());
      Fout[m] += Complex(-scratch[0].imag(), scratch[0].real());
      ++Fout;
    } while (--k);
  }
 
  inline void bfly5(Complex *Fout, const size_t fstride, const size_t m) {
    Complex *Fout0, *Fout1, *Fout2, *Fout3, *Fout4;
    size_t u;
    Complex scratch[13];
    Complex *twiddles = &m_twiddles[0];
    Complex *tw;
    Complex ya, yb;
    ya = twiddles[fstride * m];
    yb = twiddles[fstride * 2 * m];
 
    Fout0 = Fout;
    Fout1 = Fout0 + m;
    Fout2 = Fout0 + 2 * m;
    Fout3 = Fout0 + 3 * m;
    Fout4 = Fout0 + 4 * m;
 
    tw = twiddles;
    for (u = 0; u < m; ++u) {
      scratch[0] = *Fout0;
 
      scratch[1] = *Fout1 * tw[u * fstride];
      scratch[2] = *Fout2 * tw[2 * u * fstride];
      scratch[3] = *Fout3 * tw[3 * u * fstride];
      scratch[4] = *Fout4 * tw[4 * u * fstride];
 
      scratch[7] = scratch[1] + scratch[4];
      scratch[10] = scratch[1] - scratch[4];
      scratch[8] = scratch[2] + scratch[3];
      scratch[9] = scratch[2] - scratch[3];
 
      *Fout0 += scratch[7];
      *Fout0 += scratch[8];
 
      scratch[5] = scratch[0] + Complex((scratch[7].real() * ya.real()) + (scratch[8].real() * yb.real()),
                                        (scratch[7].imag() * ya.real()) + (scratch[8].imag() * yb.real()));
 
      scratch[6] = Complex((scratch[10].imag() * ya.imag()) + (scratch[9].imag() * yb.imag()),
                           -(scratch[10].real() * ya.imag()) - (scratch[9].real() * yb.imag()));
 
      *Fout1 = scratch[5] - scratch[6];
      *Fout4 = scratch[5] + scratch[6];
 
      scratch[11] = scratch[0] + Complex((scratch[7].real() * yb.real()) + (scratch[8].real() * ya.real()),
                                         (scratch[7].imag() * yb.real()) + (scratch[8].imag() * ya.real()));
 
      scratch[12] = Complex(-(scratch[10].imag() * yb.imag()) + (scratch[9].imag() * ya.imag()),
                            (scratch[10].real() * yb.imag()) - (scratch[9].real() * ya.imag()));
 
      *Fout2 = scratch[11] + scratch[12];
      *Fout3 = scratch[11] - scratch[12];
 
      ++Fout0;
      ++Fout1;
      ++Fout2;
      ++Fout3;
      ++Fout4;
    }
  }
 
  /* perform the butterfly for one stage of a mixed radix FFT */
  inline void bfly_generic(Complex *Fout, const size_t fstride, int m, int p) {
    int u, k, q1, q;
    Complex *twiddles = &m_twiddles[0];
    Complex t;
    int Norig = static_cast<int>(m_twiddles.size());
    Complex *scratchbuf = &m_scratchBuf[0];
 
    for (u = 0; u < m; ++u) {
      k = u;
      for (q1 = 0; q1 < p; ++q1) {
        scratchbuf[q1] = Fout[k];
        k += m;
      }
 
      k = u;
      for (q1 = 0; q1 < p; ++q1) {
        int twidx = 0;
        Fout[k] = scratchbuf[0];
        for (q = 1; q < p; ++q) {
          twidx += static_cast<int>(fstride) * k;
          if (twidx >= Norig) twidx -= Norig;
          t = scratchbuf[q] * twiddles[twidx];
          Fout[k] += t;
        }
        k += m;
      }
    }
  }
};
 
template <typename _Scalar>
const typename kiss_cpx_fft<_Scalar>::Scalar kiss_cpx_fft<_Scalar>::m_pi4 =
    numext::atan(kiss_cpx_fft<_Scalar>::Scalar(1));
 
template <typename Scalar_>
struct kissfft_impl {
  typedef Scalar_ Scalar;
  typedef std::complex<Scalar> Complex;
 
  void clear() {
    m_plans.clear();
    m_realTwiddles.clear();
  }
 
  inline void fwd(Complex *dst, const Complex *src, int nfft) { get_plan(nfft, false).work(0, dst, src, 1, 1); }
 
  inline void fwd2(Complex *dst, const Complex *src, int n0, int n1) {
    EIGEN_UNUSED_VARIABLE(dst);
    EIGEN_UNUSED_VARIABLE(src);
    EIGEN_UNUSED_VARIABLE(n0);
    EIGEN_UNUSED_VARIABLE(n1);
  }
 
  inline void inv2(Complex *dst, const Complex *src, int n0, int n1) {
    EIGEN_UNUSED_VARIABLE(dst);
    EIGEN_UNUSED_VARIABLE(src);
    EIGEN_UNUSED_VARIABLE(n0);
    EIGEN_UNUSED_VARIABLE(n1);
  }
 
  // real-to-complex forward FFT
  // perform two FFTs of src even and src odd
  // then twiddle to recombine them into the half-spectrum format
  // then fill in the conjugate symmetric half
  inline void fwd(Complex *dst, const Scalar *src, int nfft) {
    if (nfft & 3) {
      // use generic mode for odd
      m_tmpBuf1.resize(nfft);
      get_plan(nfft, false).work(0, &m_tmpBuf1[0], src, 1, 1);
      std::copy(m_tmpBuf1.begin(), m_tmpBuf1.begin() + (nfft >> 1) + 1, dst);
    } else {
      int ncfft = nfft >> 1;
      int ncfft2 = nfft >> 2;
      Complex *rtw = real_twiddles(ncfft2);
 
      // use optimized mode for even real
      fwd(dst, reinterpret_cast<const Complex *>(src), ncfft);
      Complex dc(dst[0].real() + dst[0].imag());
      Complex nyquist(dst[0].real() - dst[0].imag());
      int k;
      for (k = 1; k <= ncfft2; ++k) {
        Complex fpk = dst[k];
        Complex fpnk = conj(dst[ncfft - k]);
        Complex f1k = fpk + fpnk;
        Complex f2k = fpk - fpnk;
        Complex tw = f2k * rtw[k - 1];
        dst[k] = (f1k + tw) * Scalar(.5);
        dst[ncfft - k] = conj(f1k - tw) * Scalar(.5);
      }
      dst[0] = dc;
      dst[ncfft] = nyquist;
    }
  }
 
  // inverse complex-to-complex
  inline void inv(Complex *dst, const Complex *src, int nfft) { get_plan(nfft, true).work(0, dst, src, 1, 1); }
 
  // half-complex to scalar
  inline void inv(Scalar *dst, const Complex *src, int nfft) {
    if (nfft & 3) {
      m_tmpBuf1.resize(nfft);
      m_tmpBuf2.resize(nfft);
      std::copy(src, src + (nfft >> 1) + 1, m_tmpBuf1.begin());
      for (int k = 1; k < (nfft >> 1) + 1; ++k) m_tmpBuf1[nfft - k] = conj(m_tmpBuf1[k]);
      inv(&m_tmpBuf2[0], &m_tmpBuf1[0], nfft);
      for (int k = 0; k < nfft; ++k) dst[k] = m_tmpBuf2[k].real();
    } else {
      // optimized version for multiple of 4
      int ncfft = nfft >> 1;
      int ncfft2 = nfft >> 2;
      Complex *rtw = real_twiddles(ncfft2);
      m_tmpBuf1.resize(ncfft);
      m_tmpBuf1[0] = Complex(src[0].real() + src[ncfft].real(), src[0].real() - src[ncfft].real());
      for (int k = 1; k <= ncfft / 2; ++k) {
        Complex fk = src[k];
        Complex fnkc = conj(src[ncfft - k]);
        Complex fek = fk + fnkc;
        Complex tmp = fk - fnkc;
        Complex fok = tmp * conj(rtw[k - 1]);
        m_tmpBuf1[k] = fek + fok;
        m_tmpBuf1[ncfft - k] = conj(fek - fok);
      }
      get_plan(ncfft, true).work(0, reinterpret_cast<Complex *>(dst), &m_tmpBuf1[0], 1, 1);
    }
  }
 
 protected:
  typedef kiss_cpx_fft<Scalar> PlanData;
  typedef std::map<int, PlanData> PlanMap;
 
  PlanMap m_plans;
  std::map<int, std::vector<Complex> > m_realTwiddles;
  std::vector<Complex> m_tmpBuf1;
  std::vector<Complex> m_tmpBuf2;
 
  inline int PlanKey(int nfft, bool isinverse) const { return (nfft << 1) | int(isinverse); }
 
  inline PlanData &get_plan(int nfft, bool inverse) {
    // TODO look for PlanKey(nfft, ! inverse) and conjugate the twiddles
    PlanData &pd = m_plans[PlanKey(nfft, inverse)];
    if (pd.m_twiddles.size() == 0) {
      pd.make_twiddles(nfft, inverse);
      pd.factorize(nfft);
    }
    return pd;
  }
 
  inline Complex *real_twiddles(int ncfft2) {
    using std::acos;
    std::vector<Complex> &twidref = m_realTwiddles[ncfft2];  // creates new if not there
    if ((int)twidref.size() != ncfft2) {
      twidref.resize(ncfft2);
      int ncfft = ncfft2 << 1;
      Scalar pi = acos(Scalar(-1));
      for (int k = 1; k <= ncfft2; ++k) twidref[k - 1] = exp(Complex(0, -pi * (Scalar(k) / ncfft + Scalar(.5))));
    }
    return &twidref[0];
  }
};
 
}  // end namespace internal
 
}  // end namespace Eigen