1 | // This file is part of Eigen, a lightweight C++ template library
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2 | // for linear algebra.
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3 | //
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4 | // Copyright (C) 2009 Mark Borgerding mark a borgerding net
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5 | //
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6 | // This Source Code Form is subject to the terms of the Mozilla
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7 | // Public License v. 2.0. If a copy of the MPL was not distributed
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8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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9 |
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10 | namespace Eigen {
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11 |
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12 | namespace internal {
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13 |
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14 | // This FFT implementation was derived from kissfft http:sourceforge.net/projects/kissfft
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15 | // Copyright 2003-2009 Mark Borgerding
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16 |
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17 | template <typename _Scalar>
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18 | struct kiss_cpx_fft
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19 | {
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20 | typedef _Scalar Scalar;
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21 | typedef std::complex<Scalar> Complex;
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22 | std::vector<Complex> m_twiddles;
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23 | std::vector<int> m_stageRadix;
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24 | std::vector<int> m_stageRemainder;
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25 | std::vector<Complex> m_scratchBuf;
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26 | bool m_inverse;
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27 |
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28 | inline
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29 | void make_twiddles(int nfft,bool inverse)
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30 | {
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31 | using std::acos;
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32 | m_inverse = inverse;
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33 | m_twiddles.resize(nfft);
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34 | Scalar phinc = (inverse?2:-2)* acos( (Scalar) -1) / nfft;
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35 | for (int i=0;i<nfft;++i)
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36 | m_twiddles[i] = exp( Complex(0,i*phinc) );
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37 | }
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38 |
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39 | void factorize(int nfft)
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40 | {
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41 | //start factoring out 4's, then 2's, then 3,5,7,9,...
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42 | int n= nfft;
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43 | int p=4;
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44 | do {
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45 | while (n % p) {
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46 | switch (p) {
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47 | case 4: p = 2; break;
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48 | case 2: p = 3; break;
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49 | default: p += 2; break;
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50 | }
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51 | if (p*p>n)
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52 | p=n;// impossible to have a factor > sqrt(n)
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53 | }
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54 | n /= p;
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55 | m_stageRadix.push_back(p);
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56 | m_stageRemainder.push_back(n);
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57 | if ( p > 5 )
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58 | m_scratchBuf.resize(p); // scratchbuf will be needed in bfly_generic
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59 | }while(n>1);
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60 | }
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61 |
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62 | template <typename _Src>
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63 | inline
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64 | void work( int stage,Complex * xout, const _Src * xin, size_t fstride,size_t in_stride)
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65 | {
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66 | int p = m_stageRadix[stage];
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67 | int m = m_stageRemainder[stage];
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68 | Complex * Fout_beg = xout;
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69 | Complex * Fout_end = xout + p*m;
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70 |
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71 | if (m>1) {
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72 | do{
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73 | // recursive call:
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74 | // DFT of size m*p performed by doing
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75 | // p instances of smaller DFTs of size m,
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76 | // each one takes a decimated version of the input
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77 | work(stage+1, xout , xin, fstride*p,in_stride);
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78 | xin += fstride*in_stride;
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79 | }while( (xout += m) != Fout_end );
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80 | }else{
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81 | do{
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82 | *xout = *xin;
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83 | xin += fstride*in_stride;
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84 | }while(++xout != Fout_end );
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85 | }
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86 | xout=Fout_beg;
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87 |
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88 | // recombine the p smaller DFTs
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89 | switch (p) {
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90 | case 2: bfly2(xout,fstride,m); break;
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91 | case 3: bfly3(xout,fstride,m); break;
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92 | case 4: bfly4(xout,fstride,m); break;
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93 | case 5: bfly5(xout,fstride,m); break;
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94 | default: bfly_generic(xout,fstride,m,p); break;
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95 | }
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96 | }
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97 |
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98 | inline
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99 | void bfly2( Complex * Fout, const size_t fstride, int m)
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100 | {
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101 | for (int k=0;k<m;++k) {
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102 | Complex t = Fout[m+k] * m_twiddles[k*fstride];
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103 | Fout[m+k] = Fout[k] - t;
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104 | Fout[k] += t;
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105 | }
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106 | }
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107 |
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108 | inline
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109 | void bfly4( Complex * Fout, const size_t fstride, const size_t m)
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110 | {
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111 | Complex scratch[6];
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112 | int negative_if_inverse = m_inverse * -2 +1;
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113 | for (size_t k=0;k<m;++k) {
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114 | scratch[0] = Fout[k+m] * m_twiddles[k*fstride];
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115 | scratch[1] = Fout[k+2*m] * m_twiddles[k*fstride*2];
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116 | scratch[2] = Fout[k+3*m] * m_twiddles[k*fstride*3];
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117 | scratch[5] = Fout[k] - scratch[1];
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118 |
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119 | Fout[k] += scratch[1];
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120 | scratch[3] = scratch[0] + scratch[2];
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121 | scratch[4] = scratch[0] - scratch[2];
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122 | scratch[4] = Complex( scratch[4].imag()*negative_if_inverse , -scratch[4].real()* negative_if_inverse );
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123 |
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124 | Fout[k+2*m] = Fout[k] - scratch[3];
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125 | Fout[k] += scratch[3];
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126 | Fout[k+m] = scratch[5] + scratch[4];
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127 | Fout[k+3*m] = scratch[5] - scratch[4];
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128 | }
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129 | }
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130 |
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131 | inline
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132 | void bfly3( Complex * Fout, const size_t fstride, const size_t m)
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133 | {
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134 | size_t k=m;
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135 | const size_t m2 = 2*m;
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136 | Complex *tw1,*tw2;
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137 | Complex scratch[5];
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138 | Complex epi3;
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139 | epi3 = m_twiddles[fstride*m];
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140 |
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141 | tw1=tw2=&m_twiddles[0];
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142 |
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143 | do{
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144 | scratch[1]=Fout[m] * *tw1;
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145 | scratch[2]=Fout[m2] * *tw2;
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146 |
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147 | scratch[3]=scratch[1]+scratch[2];
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148 | scratch[0]=scratch[1]-scratch[2];
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149 | tw1 += fstride;
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150 | tw2 += fstride*2;
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151 | Fout[m] = Complex( Fout->real() - Scalar(.5)*scratch[3].real() , Fout->imag() - Scalar(.5)*scratch[3].imag() );
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152 | scratch[0] *= epi3.imag();
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153 | *Fout += scratch[3];
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154 | Fout[m2] = Complex( Fout[m].real() + scratch[0].imag() , Fout[m].imag() - scratch[0].real() );
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155 | Fout[m] += Complex( -scratch[0].imag(),scratch[0].real() );
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156 | ++Fout;
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157 | }while(--k);
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158 | }
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159 |
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160 | inline
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161 | void bfly5( Complex * Fout, const size_t fstride, const size_t m)
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162 | {
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163 | Complex *Fout0,*Fout1,*Fout2,*Fout3,*Fout4;
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164 | size_t u;
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165 | Complex scratch[13];
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166 | Complex * twiddles = &m_twiddles[0];
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167 | Complex *tw;
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168 | Complex ya,yb;
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169 | ya = twiddles[fstride*m];
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170 | yb = twiddles[fstride*2*m];
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171 |
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172 | Fout0=Fout;
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173 | Fout1=Fout0+m;
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174 | Fout2=Fout0+2*m;
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175 | Fout3=Fout0+3*m;
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176 | Fout4=Fout0+4*m;
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177 |
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178 | tw=twiddles;
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179 | for ( u=0; u<m; ++u ) {
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180 | scratch[0] = *Fout0;
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181 |
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182 | scratch[1] = *Fout1 * tw[u*fstride];
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183 | scratch[2] = *Fout2 * tw[2*u*fstride];
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184 | scratch[3] = *Fout3 * tw[3*u*fstride];
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185 | scratch[4] = *Fout4 * tw[4*u*fstride];
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186 |
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187 | scratch[7] = scratch[1] + scratch[4];
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188 | scratch[10] = scratch[1] - scratch[4];
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189 | scratch[8] = scratch[2] + scratch[3];
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190 | scratch[9] = scratch[2] - scratch[3];
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191 |
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192 | *Fout0 += scratch[7];
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193 | *Fout0 += scratch[8];
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194 |
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195 | scratch[5] = scratch[0] + Complex(
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196 | (scratch[7].real()*ya.real() ) + (scratch[8].real() *yb.real() ),
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197 | (scratch[7].imag()*ya.real()) + (scratch[8].imag()*yb.real())
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198 | );
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199 |
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200 | scratch[6] = Complex(
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201 | (scratch[10].imag()*ya.imag()) + (scratch[9].imag()*yb.imag()),
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202 | -(scratch[10].real()*ya.imag()) - (scratch[9].real()*yb.imag())
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203 | );
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204 |
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205 | *Fout1 = scratch[5] - scratch[6];
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206 | *Fout4 = scratch[5] + scratch[6];
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207 |
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208 | scratch[11] = scratch[0] +
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209 | Complex(
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210 | (scratch[7].real()*yb.real()) + (scratch[8].real()*ya.real()),
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211 | (scratch[7].imag()*yb.real()) + (scratch[8].imag()*ya.real())
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212 | );
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213 |
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214 | scratch[12] = Complex(
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215 | -(scratch[10].imag()*yb.imag()) + (scratch[9].imag()*ya.imag()),
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216 | (scratch[10].real()*yb.imag()) - (scratch[9].real()*ya.imag())
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217 | );
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218 |
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219 | *Fout2=scratch[11]+scratch[12];
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220 | *Fout3=scratch[11]-scratch[12];
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221 |
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222 | ++Fout0;++Fout1;++Fout2;++Fout3;++Fout4;
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223 | }
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224 | }
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225 |
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226 | /* perform the butterfly for one stage of a mixed radix FFT */
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227 | inline
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228 | void bfly_generic(
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229 | Complex * Fout,
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230 | const size_t fstride,
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231 | int m,
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232 | int p
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233 | )
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234 | {
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235 | int u,k,q1,q;
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236 | Complex * twiddles = &m_twiddles[0];
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237 | Complex t;
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238 | int Norig = static_cast<int>(m_twiddles.size());
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239 | Complex * scratchbuf = &m_scratchBuf[0];
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240 |
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241 | for ( u=0; u<m; ++u ) {
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242 | k=u;
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243 | for ( q1=0 ; q1<p ; ++q1 ) {
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244 | scratchbuf[q1] = Fout[ k ];
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245 | k += m;
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246 | }
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247 |
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248 | k=u;
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249 | for ( q1=0 ; q1<p ; ++q1 ) {
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250 | int twidx=0;
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251 | Fout[ k ] = scratchbuf[0];
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252 | for (q=1;q<p;++q ) {
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253 | twidx += static_cast<int>(fstride) * k;
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254 | if (twidx>=Norig) twidx-=Norig;
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255 | t=scratchbuf[q] * twiddles[twidx];
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256 | Fout[ k ] += t;
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257 | }
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258 | k += m;
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259 | }
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260 | }
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261 | }
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262 | };
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263 |
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264 | template <typename _Scalar>
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265 | struct kissfft_impl
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266 | {
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267 | typedef _Scalar Scalar;
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268 | typedef std::complex<Scalar> Complex;
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269 |
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270 | void clear()
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271 | {
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272 | m_plans.clear();
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273 | m_realTwiddles.clear();
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274 | }
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275 |
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276 | inline
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277 | void fwd( Complex * dst,const Complex *src,int nfft)
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278 | {
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279 | get_plan(nfft,false).work(0, dst, src, 1,1);
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280 | }
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281 |
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282 | inline
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283 | void fwd2( Complex * dst,const Complex *src,int n0,int n1)
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284 | {
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285 | EIGEN_UNUSED_VARIABLE(dst);
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286 | EIGEN_UNUSED_VARIABLE(src);
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287 | EIGEN_UNUSED_VARIABLE(n0);
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288 | EIGEN_UNUSED_VARIABLE(n1);
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289 | }
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290 |
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291 | inline
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292 | void inv2( Complex * dst,const Complex *src,int n0,int n1)
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293 | {
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294 | EIGEN_UNUSED_VARIABLE(dst);
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295 | EIGEN_UNUSED_VARIABLE(src);
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296 | EIGEN_UNUSED_VARIABLE(n0);
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297 | EIGEN_UNUSED_VARIABLE(n1);
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298 | }
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299 |
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300 | // real-to-complex forward FFT
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301 | // perform two FFTs of src even and src odd
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302 | // then twiddle to recombine them into the half-spectrum format
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303 | // then fill in the conjugate symmetric half
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304 | inline
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305 | void fwd( Complex * dst,const Scalar * src,int nfft)
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306 | {
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307 | if ( nfft&3 ) {
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308 | // use generic mode for odd
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309 | m_tmpBuf1.resize(nfft);
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310 | get_plan(nfft,false).work(0, &m_tmpBuf1[0], src, 1,1);
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311 | std::copy(m_tmpBuf1.begin(),m_tmpBuf1.begin()+(nfft>>1)+1,dst );
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312 | }else{
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313 | int ncfft = nfft>>1;
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314 | int ncfft2 = nfft>>2;
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315 | Complex * rtw = real_twiddles(ncfft2);
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316 |
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317 | // use optimized mode for even real
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318 | fwd( dst, reinterpret_cast<const Complex*> (src), ncfft);
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319 | Complex dc = dst[0].real() + dst[0].imag();
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320 | Complex nyquist = dst[0].real() - dst[0].imag();
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321 | int k;
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322 | for ( k=1;k <= ncfft2 ; ++k ) {
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323 | Complex fpk = dst[k];
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324 | Complex fpnk = conj(dst[ncfft-k]);
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325 | Complex f1k = fpk + fpnk;
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326 | Complex f2k = fpk - fpnk;
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327 | Complex tw= f2k * rtw[k-1];
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328 | dst[k] = (f1k + tw) * Scalar(.5);
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329 | dst[ncfft-k] = conj(f1k -tw)*Scalar(.5);
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330 | }
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331 | dst[0] = dc;
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332 | dst[ncfft] = nyquist;
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333 | }
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334 | }
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335 |
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336 | // inverse complex-to-complex
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337 | inline
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338 | void inv(Complex * dst,const Complex *src,int nfft)
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339 | {
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340 | get_plan(nfft,true).work(0, dst, src, 1,1);
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341 | }
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342 |
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343 | // half-complex to scalar
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344 | inline
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345 | void inv( Scalar * dst,const Complex * src,int nfft)
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346 | {
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347 | if (nfft&3) {
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348 | m_tmpBuf1.resize(nfft);
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349 | m_tmpBuf2.resize(nfft);
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350 | std::copy(src,src+(nfft>>1)+1,m_tmpBuf1.begin() );
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351 | for (int k=1;k<(nfft>>1)+1;++k)
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352 | m_tmpBuf1[nfft-k] = conj(m_tmpBuf1[k]);
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353 | inv(&m_tmpBuf2[0],&m_tmpBuf1[0],nfft);
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354 | for (int k=0;k<nfft;++k)
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355 | dst[k] = m_tmpBuf2[k].real();
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356 | }else{
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357 | // optimized version for multiple of 4
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358 | int ncfft = nfft>>1;
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359 | int ncfft2 = nfft>>2;
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360 | Complex * rtw = real_twiddles(ncfft2);
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361 | m_tmpBuf1.resize(ncfft);
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362 | m_tmpBuf1[0] = Complex( src[0].real() + src[ncfft].real(), src[0].real() - src[ncfft].real() );
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363 | for (int k = 1; k <= ncfft / 2; ++k) {
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364 | Complex fk = src[k];
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365 | Complex fnkc = conj(src[ncfft-k]);
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366 | Complex fek = fk + fnkc;
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367 | Complex tmp = fk - fnkc;
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368 | Complex fok = tmp * conj(rtw[k-1]);
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369 | m_tmpBuf1[k] = fek + fok;
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370 | m_tmpBuf1[ncfft-k] = conj(fek - fok);
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371 | }
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372 | get_plan(ncfft,true).work(0, reinterpret_cast<Complex*>(dst), &m_tmpBuf1[0], 1,1);
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373 | }
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374 | }
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375 |
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376 | protected:
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377 | typedef kiss_cpx_fft<Scalar> PlanData;
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378 | typedef std::map<int,PlanData> PlanMap;
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379 |
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380 | PlanMap m_plans;
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381 | std::map<int, std::vector<Complex> > m_realTwiddles;
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382 | std::vector<Complex> m_tmpBuf1;
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383 | std::vector<Complex> m_tmpBuf2;
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384 |
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385 | inline
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386 | int PlanKey(int nfft, bool isinverse) const { return (nfft<<1) | int(isinverse); }
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387 |
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388 | inline
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389 | PlanData & get_plan(int nfft, bool inverse)
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390 | {
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391 | // TODO look for PlanKey(nfft, ! inverse) and conjugate the twiddles
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392 | PlanData & pd = m_plans[ PlanKey(nfft,inverse) ];
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393 | if ( pd.m_twiddles.size() == 0 ) {
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394 | pd.make_twiddles(nfft,inverse);
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395 | pd.factorize(nfft);
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396 | }
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397 | return pd;
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398 | }
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399 |
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400 | inline
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401 | Complex * real_twiddles(int ncfft2)
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402 | {
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403 | using std::acos;
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404 | std::vector<Complex> & twidref = m_realTwiddles[ncfft2];// creates new if not there
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405 | if ( (int)twidref.size() != ncfft2 ) {
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406 | twidref.resize(ncfft2);
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407 | int ncfft= ncfft2<<1;
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408 | Scalar pi = acos( Scalar(-1) );
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409 | for (int k=1;k<=ncfft2;++k)
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410 | twidref[k-1] = exp( Complex(0,-pi * (Scalar(k) / ncfft + Scalar(.5)) ) );
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411 | }
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412 | return &twidref[0];
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413 | }
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414 | };
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415 |
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416 | } // end namespace internal
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417 |
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418 | } // end namespace Eigen
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419 |
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420 | /* vim: set filetype=cpp et sw=2 ts=2 ai: */
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