audacity/src/FFT.cpp

/**********************************************************************

  FFT.cpp

  Dominic Mazzoni

  September 2000

*******************************************************************//*!

\file FFT.cpp
\brief Fast Fourier Transform routines.

  This file contains a few FFT routines, including a real-FFT
  routine that is almost twice as fast as a normal complex FFT,
  and a power spectrum routine when you know you don't care
  about phase information.

  Some of this code was based on a free implementation of an FFT
  by Don Cross, available on the web at:

    http://www.intersrv.com/~dcross/fft.html

  The basic algorithm for his code was based on Numerican Recipes
  in Fortran.  I optimized his code further by reducing array
  accesses, caching the bit reversal table, and eliminating
  float-to-double conversions, and I added the routines to
  calculate a real FFT and a real power spectrum.

*//*******************************************************************/
/*
  Salvo Ventura - November 2006
  Added more window functions:
    * 4: Blackman
    * 5: Blackman-Harris
    * 6: Welch
    * 7: Gaussian(a=2.5)
    * 8: Gaussian(a=3.5)
    * 9: Gaussian(a=4.5)
*/

#include "FFT.h"

#include <wx/intl.h>
#include <stdlib.h>
#include <stdio.h>
#include <math.h>

#include "Experimental.h"

static int **gFFTBitTable = NULL;
static const int MaxFastBits = 16;

/* Declare Static functions */
static int IsPowerOfTwo(int x);
static int NumberOfBitsNeeded(int PowerOfTwo);
static int ReverseBits(int index, int NumBits);
static void InitFFT();

int IsPowerOfTwo(int x)
{
   if (x < 2)
      return false;

   if (x & (x - 1))             /* Thanks to 'byang' for this cute trick! */
      return false;

   return true;
}

int NumberOfBitsNeeded(int PowerOfTwo)
{
   int i;

   if (PowerOfTwo < 2) {
      fprintf(stderr, "Error: FFT called with size %d\n", PowerOfTwo);
      exit(1);
   }

   for (i = 0;; i++)
      if (PowerOfTwo & (1 << i))
         return i;
}

int ReverseBits(int index, int NumBits)
{
   int i, rev;

   for (i = rev = 0; i < NumBits; i++) {
      rev = (rev << 1) | (index & 1);
      index >>= 1;
   }

   return rev;
}

void InitFFT()
{
   gFFTBitTable = new int *[MaxFastBits];

   int len = 2;
   for (int b = 1; b <= MaxFastBits; b++) {

      gFFTBitTable[b - 1] = new int[len];

      for (int i = 0; i < len; i++)
         gFFTBitTable[b - 1][i] = ReverseBits(i, b);

      len <<= 1;
   }
}

#ifdef EXPERIMENTAL_USE_REALFFTF
#include "RealFFTf.h"
#endif

void DeinitFFT()
{
   if (gFFTBitTable) {
      for (int b = 1; b <= MaxFastBits; b++) {
         delete[] gFFTBitTable[b-1];
      }
      delete[] gFFTBitTable;
   }
#ifdef EXPERIMENTAL_USE_REALFFTF
   // Deallocate any unused RealFFTf tables
   CleanupFFT();
#endif
}

inline int FastReverseBits(int i, int NumBits)
{
   if (NumBits <= MaxFastBits)
      return gFFTBitTable[NumBits - 1][i];
   else
      return ReverseBits(i, NumBits);
}

/*
 * Complex Fast Fourier Transform
 */

void FFT(int NumSamples,
         bool InverseTransform,
         float *RealIn, float *ImagIn, float *RealOut, float *ImagOut)
{
   int NumBits;                 /* Number of bits needed to store indices */
   int i, j, k, n;
   int BlockSize, BlockEnd;

   double angle_numerator = 2.0 * M_PI;
   double tr, ti;                /* temp real, temp imaginary */

   if (!IsPowerOfTwo(NumSamples)) {
      fprintf(stderr, "%d is not a power of two\n", NumSamples);
      exit(1);
   }

   if (!gFFTBitTable)
      InitFFT();

   if (!InverseTransform)
      angle_numerator = -angle_numerator;

   NumBits = NumberOfBitsNeeded(NumSamples);

   /*
    **   Do simultaneous data copy and bit-reversal ordering into outputs...
    */

   for (i = 0; i < NumSamples; i++) {
      j = FastReverseBits(i, NumBits);
      RealOut[j] = RealIn[i];
      ImagOut[j] = (ImagIn == NULL) ? 0.0 : ImagIn[i];
   }

   /*
    **   Do the FFT itself...
    */

   BlockEnd = 1;
   for (BlockSize = 2; BlockSize <= NumSamples; BlockSize <<= 1) {

      double delta_angle = angle_numerator / (double) BlockSize;

      double sm2 = sin(-2 * delta_angle);
      double sm1 = sin(-delta_angle);
      double cm2 = cos(-2 * delta_angle);
      double cm1 = cos(-delta_angle);
      double w = 2 * cm1;
      double ar0, ar1, ar2, ai0, ai1, ai2;

      for (i = 0; i < NumSamples; i += BlockSize) {
         ar2 = cm2;
         ar1 = cm1;

         ai2 = sm2;
         ai1 = sm1;

         for (j = i, n = 0; n < BlockEnd; j++, n++) {
            ar0 = w * ar1 - ar2;
            ar2 = ar1;
            ar1 = ar0;

            ai0 = w * ai1 - ai2;
            ai2 = ai1;
            ai1 = ai0;

            k = j + BlockEnd;
            tr = ar0 * RealOut[k] - ai0 * ImagOut[k];
            ti = ar0 * ImagOut[k] + ai0 * RealOut[k];

            RealOut[k] = RealOut[j] - tr;
            ImagOut[k] = ImagOut[j] - ti;

            RealOut[j] += tr;
            ImagOut[j] += ti;
         }
      }

      BlockEnd = BlockSize;
   }

   /*
      **   Need to normalize if inverse transform...
    */

   if (InverseTransform) {
      float denom = (float) NumSamples;

      for (i = 0; i < NumSamples; i++) {
         RealOut[i] /= denom;
         ImagOut[i] /= denom;
      }
   }
}

/*
 * Real Fast Fourier Transform
 *
 * This function was based on the code in Numerical Recipes in C.
 * In Num. Rec., the inner loop is based on a single 1-based array
 * of interleaved real and imaginary numbers.  Because we have two
 * separate zero-based arrays, our indices are quite different.
 * Here is the correspondence between Num. Rec. indices and our indices:
 *
 * i1  <->  real[i]
 * i2  <->  imag[i]
 * i3  <->  real[n/2-i]
 * i4  <->  imag[n/2-i]
 */

void RealFFT(int NumSamples, float *RealIn, float *RealOut, float *ImagOut)
{
#ifdef EXPERIMENTAL_USE_REALFFTF
   // Remap to RealFFTf() function
   int i;
   HFFT hFFT = GetFFT(NumSamples);
   float *pFFT = new float[NumSamples];
   // Copy the data into the processing buffer
   for(i=0; i<NumSamples; i++)
      pFFT[i] = RealIn[i];

   // Perform the FFT
   RealFFTf(pFFT, hFFT);

   // Copy the data into the real and imaginary outputs
   for(i=1;i<(NumSamples/2);i++) {
      RealOut[i]=pFFT[hFFT->BitReversed[i]  ];
      ImagOut[i]=pFFT[hFFT->BitReversed[i]+1];
   }
   // Handle the (real-only) DC and Fs/2 bins
   RealOut[0] = pFFT[0];
   RealOut[i] = pFFT[1];
   ImagOut[0] = ImagOut[i] = 0;
   // Fill in the upper half using symmetry properties
   for(i++ ; i<NumSamples; i++) {
      RealOut[i] =  RealOut[NumSamples-i];
      ImagOut[i] = -ImagOut[NumSamples-i];
   }
   delete [] pFFT;
   ReleaseFFT(hFFT);

#else

   int Half = NumSamples / 2;
   int i;

   float theta = M_PI / Half;

   float *tmpReal = new float[Half];
   float *tmpImag = new float[Half];

   for (i = 0; i < Half; i++) {
      tmpReal[i] = RealIn[2 * i];
      tmpImag[i] = RealIn[2 * i + 1];
   }

   FFT(Half, 0, tmpReal, tmpImag, RealOut, ImagOut);

   float wtemp = float (sin(0.5 * theta));

   float wpr = -2.0 * wtemp * wtemp;
   float wpi = -1.0 * float (sin(theta));
   float wr = 1.0 + wpr;
   float wi = wpi;

   int i3;

   float h1r, h1i, h2r, h2i;

   for (i = 1; i < Half / 2; i++) {

      i3 = Half - i;

      h1r = 0.5 * (RealOut[i] + RealOut[i3]);
      h1i = 0.5 * (ImagOut[i] - ImagOut[i3]);
      h2r = 0.5 * (ImagOut[i] + ImagOut[i3]);
      h2i = -0.5 * (RealOut[i] - RealOut[i3]);

      RealOut[i] = h1r + wr * h2r - wi * h2i;
      ImagOut[i] = h1i + wr * h2i + wi * h2r;
      RealOut[i3] = h1r - wr * h2r + wi * h2i;
      ImagOut[i3] = -h1i + wr * h2i + wi * h2r;

      wr = (wtemp = wr) * wpr - wi * wpi + wr;
      wi = wi * wpr + wtemp * wpi + wi;
   }

   RealOut[0] = (h1r = RealOut[0]) + ImagOut[0];
   ImagOut[0] = h1r - ImagOut[0];

   delete[]tmpReal;
   delete[]tmpImag;
#endif //EXPERIMENTAL_USE_REALFFTF
}

#ifdef EXPERIMENTAL_USE_REALFFTF
/*
 * InverseRealFFT
 *
 * This function computes the inverse of RealFFT, above.
 * The RealIn and ImagIn is assumed to be conjugate-symmetric
 * and as a result the output is purely real.
 * Only the first half of RealIn and ImagIn are used due to this
 * symmetry assumption.
 */
void InverseRealFFT(int NumSamples, float *RealIn, float *ImagIn, float *RealOut)
{
   // Remap to RealFFTf() function
   int i;
   HFFT hFFT = GetFFT(NumSamples);
   float *pFFT = new float[NumSamples];
   // Copy the data into the processing buffer
   for(i=0; i<(NumSamples/2); i++)
      pFFT[2*i  ] = RealIn[i];
   if(ImagIn == NULL) {
      for(i=0; i<(NumSamples/2); i++)
         pFFT[2*i+1] = 0;
   } else {
      for(i=0; i<(NumSamples/2); i++)
         pFFT[2*i+1] = ImagIn[i];
   }
   // Put the fs/2 component in the imaginary part of the DC bin
   pFFT[1] = RealIn[i];

   // Perform the FFT
   InverseRealFFTf(pFFT, hFFT);

   // Copy the data to the (purely real) output buffer
   ReorderToTime(hFFT, pFFT, RealOut);

   delete [] pFFT;
   ReleaseFFT(hFFT);
}
#endif // EXPERIMENTAL_USE_REALFFTF

/*
 * PowerSpectrum
 *
 * This function computes the same as RealFFT, above, but
 * adds the squares of the real and imaginary part of each
 * coefficient, extracting the power and throwing away the
 * phase.
 *
 * For speed, it does not call RealFFT, but duplicates some
 * of its code.
 */

void PowerSpectrum(int NumSamples, float *In, float *Out)
{
#ifdef EXPERIMENTAL_USE_REALFFTF
   // Remap to RealFFTf() function
   int i;
   HFFT hFFT = GetFFT(NumSamples);
   float *pFFT = new float[NumSamples];
   // Copy the data into the processing buffer
   for(i=0; i<NumSamples; i++)
      pFFT[i] = In[i];

   // Perform the FFT
   RealFFTf(pFFT, hFFT);

   // Copy the data into the real and imaginary outputs
   for(i=1;i<NumSamples/2;i++) {
      Out[i]= (pFFT[hFFT->BitReversed[i]  ]*pFFT[hFFT->BitReversed[i]  ])
         + (pFFT[hFFT->BitReversed[i]+1]*pFFT[hFFT->BitReversed[i]+1]);
   }
   // Handle the (real-only) DC and Fs/2 bins
   Out[0] = pFFT[0]*pFFT[0];
   Out[i] = pFFT[1]*pFFT[1];
   delete [] pFFT;
   ReleaseFFT(hFFT);

#else // EXPERIMENTAL_USE_REALFFTF

   int Half = NumSamples / 2;
   int i;

   float theta = M_PI / Half;

   float *tmpReal = new float[Half];
   float *tmpImag = new float[Half];
   float *RealOut = new float[Half];
   float *ImagOut = new float[Half];

   for (i = 0; i < Half; i++) {
      tmpReal[i] = In[2 * i];
      tmpImag[i] = In[2 * i + 1];
   }

   FFT(Half, 0, tmpReal, tmpImag, RealOut, ImagOut);

   float wtemp = float (sin(0.5 * theta));

   float wpr = -2.0 * wtemp * wtemp;
   float wpi = -1.0 * float (sin(theta));
   float wr = 1.0 + wpr;
   float wi = wpi;

   int i3;

   float h1r, h1i, h2r, h2i, rt, it;

   for (i = 1; i < Half / 2; i++) {

      i3 = Half - i;

      h1r = 0.5 * (RealOut[i] + RealOut[i3]);
      h1i = 0.5 * (ImagOut[i] - ImagOut[i3]);
      h2r = 0.5 * (ImagOut[i] + ImagOut[i3]);
      h2i = -0.5 * (RealOut[i] - RealOut[i3]);

      rt = h1r + wr * h2r - wi * h2i;
      it = h1i + wr * h2i + wi * h2r;

      Out[i] = rt * rt + it * it;

      rt = h1r - wr * h2r + wi * h2i;
      it = -h1i + wr * h2i + wi * h2r;

      Out[i3] = rt * rt + it * it;

      wr = (wtemp = wr) * wpr - wi * wpi + wr;
      wi = wi * wpr + wtemp * wpi + wi;
   }

   rt = (h1r = RealOut[0]) + ImagOut[0];
   it = h1r - ImagOut[0];
   Out[0] = rt * rt + it * it;

   rt = RealOut[Half / 2];
   it = ImagOut[Half / 2];
   Out[Half / 2] = rt * rt + it * it;

   delete[]tmpReal;
   delete[]tmpImag;
   delete[]RealOut;
   delete[]ImagOut;
#endif // EXPERIMENTAL_USE_REALFFTF
}

/*
 * Windowing Functions
 */

int NumWindowFuncs()
{
   return eWinFuncCount;
}

const wxChar *WindowFuncName(int whichFunction)
{
   switch (whichFunction) {
   default:
   case eWinFuncRectangular:
      return _("Rectangular");
   case eWinFuncBartlett:
      return wxT("Bartlett");
   case eWinFuncHamming:
      return wxT("Hamming");
   case eWinFuncHanning:
      return wxT("Hanning");
   case eWinFuncBlackman:
      return wxT("Blackman");
   case eWinFuncBlackmanHarris:
      return wxT("Blackman-Harris");
   case eWinFuncWelch:
      return wxT("Welch");
   case eWinFuncGaussian25:
      return wxT("Gaussian(a=2.5)");
   case eWinFuncGaussian35:
      return wxT("Gaussian(a=3.5)");
   case eWinFuncGaussian45:
      return wxT("Gaussian(a=4.5)");
   }
}

void NewWindowFunc(int whichFunction, int NumSamples, bool extraSample, float *in)
{
   if (extraSample)
      --NumSamples;

   switch (whichFunction) {
   default:
      fprintf(stderr, "FFT::WindowFunc - Invalid window function: %d\n", whichFunction);
      break;
   case eWinFuncRectangular:
      // Multiply all by 1.0f -- do nothing
      break;

   case eWinFuncBartlett:
   {
      // Bartlett (triangular) window
      const int nPairs = (NumSamples - 1) / 2; // whether even or odd NumSamples, this is correct
      const float denom = NumSamples / 2.0f;
      in[0] = 0.0f;
      for (int ii = 1;
           ii <= nPairs; // Yes, <=
           ++ii) {
         const float value = ii / denom;
         in[ii] *= value;
         in[NumSamples - ii] *= value;
      }
      // When NumSamples is even, in[half] should be multiplied by 1.0, so unchanged
      // When odd, the value of 1.0 is not reached
   }
      break;
   case eWinFuncHamming:
   {
      // Hamming
      const double multiplier = 2 * M_PI / NumSamples;
      static const double coeff0 = 0.54, coeff1 = -0.46;
      for (int ii = 0; ii < NumSamples; ++ii)
         in[ii] *= coeff0 + coeff1 * cos(ii * multiplier);
   }
      break;
   case eWinFuncHanning:
   {
      // Hanning
      const double multiplier = 2 * M_PI / NumSamples;
      static const double coeff0 = 0.5, coeff1 = -0.5;
      for (int ii = 0; ii < NumSamples; ++ii)
         in[ii] *= coeff0 + coeff1 * cos(ii * multiplier);
   }
      break;
   case eWinFuncBlackman:
   {
      // Blackman
      const double multiplier = 2 * M_PI / NumSamples;
      const double multiplier2 = 2 * multiplier;
      static const double coeff0 = 0.42, coeff1 = -0.5, coeff2 = 0.08;
      for (int ii = 0; ii < NumSamples; ++ii)
         in[ii] *= coeff0 + coeff1 * cos(ii * multiplier) + coeff2 * cos(ii * multiplier2);
   }
      break;
   case eWinFuncBlackmanHarris:
   {
      // Blackman-Harris
      const double multiplier = 2 * M_PI / NumSamples;
      const double multiplier2 = 2 * multiplier;
      const double multiplier3 = 3 * multiplier;
      static const double coeff0 = 0.35875, coeff1 = -0.48829, coeff2 = 0.14128, coeff3 = -0.01168;
      for (int ii = 0; ii < NumSamples; ++ii)
         in[ii] *= coeff0 + coeff1 * cos(ii * multiplier) + coeff2 * cos(ii * multiplier2) + coeff3 * cos(ii * multiplier3);
   }
      break;
   case eWinFuncWelch:
   {
      // Welch
      const float N = NumSamples;
      for (int ii = 0; ii < NumSamples; ++ii) {
         const float iOverN = ii / N;
         in[ii] *= 4 * iOverN * (1 - iOverN);
      }
   }
      break;
   case eWinFuncGaussian25:
   {
      // Gaussian (a=2.5)
      // Precalculate some values, and simplify the fmla to try and reduce overhead
      static const double A = -2 * 2.5*2.5;
      const float N = NumSamples;
      for (int ii = 0; ii < NumSamples; ++ii) {
         const float iOverN = ii / N;
         // full
         // in[ii] *= exp(-0.5*(A*((ii-NumSamples/2)/NumSamples/2))*(A*((ii-NumSamples/2)/NumSamples/2)));
         // reduced
         in[ii] *= exp(A * (0.25 + (iOverN * iOverN) - iOverN));
      }
   }
      break;
   case eWinFuncGaussian35:
   {
      // Gaussian (a=3.5)
      static const double A = -2 * 3.5*3.5;
      const float N = NumSamples;
      for (int ii = 0; ii < NumSamples; ++ii) {
         const float iOverN = ii / N;
         in[ii] *= exp(A * (0.25 + (iOverN * iOverN) - iOverN));
      }
   }
      break;
   case eWinFuncGaussian45:
   {
      // Gaussian (a=4.5)
      static const double A = -2 * 4.5*4.5;
      const float N = NumSamples;
      for (int ii = 0; ii < NumSamples; ++ii) {
         const float iOverN = ii / N;
         in[ii] *= exp(A * (0.25 + (iOverN * iOverN) - iOverN));
      }
   }
      break;
   }

   if (extraSample && whichFunction != eWinFuncRectangular) {
      double value = 0.0;
      switch (whichFunction) {
      case eWinFuncHamming:
         value = 0.08;
         break;
      case eWinFuncGaussian25:
         value = exp(-2 * 2.5 * 2.5 * 0.25);
         break;
      case eWinFuncGaussian35:
         value = exp(-2 * 3.5 * 3.5 * 0.25);
         break;
      case eWinFuncGaussian45:
         value = exp(-2 * 4.5 * 4.5 * 0.25);
         break;
      default:
         break;
      }
      in[NumSamples] *= value;
   }
}

// See cautions in FFT.h !
void WindowFunc(int whichFunction, int NumSamples, float *in)
{
   bool extraSample = false;
   switch (whichFunction)
   {
   case eWinFuncHamming:
   case eWinFuncHanning:
   case eWinFuncBlackman:
   case eWinFuncBlackmanHarris:
      extraSample = true;
      break;
   default:
      break;
   case eWinFuncBartlett:
      // PRL:  Do nothing here either
      // But I want to comment that the old function did this case
      // wrong in the second half of the array, in case NumSamples was odd
      // but I think that never happened, so I am not bothering to preserve that
      break;
   }
   NewWindowFunc(whichFunction, NumSamples, extraSample, in);
}

void DerivativeOfWindowFunc(int whichFunction, int NumSamples, bool extraSample, float *in)
{
   if (eWinFuncRectangular == whichFunction)
   {
      // Rectangular
      // There are deltas at the ends
      --NumSamples;
      // in[0] *= 1.0f;
      for (int ii = 1; ii < NumSamples; ++ii)
         in[ii] = 0.0f;
      in[NumSamples] *= -1.0f;
      return;
   }

   if (extraSample)
      --NumSamples;

   double A;
   switch (whichFunction) {
   case eWinFuncBartlett:
   {
      // Bartlett (triangular) window
      // There are discontinuities in the derivative at the ends, and maybe at the midpoint
      const int nPairs = (NumSamples - 1) / 2; // whether even or odd NumSamples, this is correct
      const float value = 2.0f / NumSamples;
      in[0] *=
         // Average the two limiting values of discontinuous derivative
         value / 2.0f;
      for (int ii = 1;
         ii <= nPairs; // Yes, <=
         ++ii) {
         in[ii] *= value;
         in[NumSamples - ii] *= -value;
      }
      if (NumSamples % 2 == 0)
         // Average the two limiting values of discontinuous derivative
         in[NumSamples / 2] = 0.0f;
      if (extraSample)
         in[NumSamples] *=
         // Average the two limiting values of discontinuous derivative
            -value / 2.0f;
      else
         // Halve the multiplier previously applied
         // Average the two limiting values of discontinuous derivative
         in[NumSamples - 1] *= 0.5f;
   }
      break;
   case eWinFuncHamming:
   {
      // Hamming
      // There are deltas at the ends
      const double multiplier = 2 * M_PI / NumSamples;
      static const double coeff0 = 0.54, coeff1 = -0.46 * multiplier;
      in[0] *= coeff0;
      if (!extraSample)
         --NumSamples;
      for (int ii = 0; ii < NumSamples; ++ii)
         in[ii] *= - coeff1 * sin(ii * multiplier);
      if (extraSample)
         in[NumSamples] *= - coeff0;
      else
         // slightly different
         in[NumSamples] *= - coeff0 - coeff1 * sin(NumSamples * multiplier);
   }
      break;
   case eWinFuncHanning:
   {
      // Hanning
      const double multiplier = 2 * M_PI / NumSamples;
      const double coeff1 = -0.5 * multiplier;
      for (int ii = 0; ii < NumSamples; ++ii)
         in[ii] *= - coeff1 * sin(ii * multiplier);
      if (extraSample)
         in[NumSamples] = 0.0f;
   }
      break;
   case eWinFuncBlackman:
   {
      // Blackman
      const double multiplier = 2 * M_PI / NumSamples;
      const double multiplier2 = 2 * multiplier;
      const double coeff1 = -0.5 * multiplier, coeff2 = 0.08 * multiplier2;
      for (int ii = 0; ii < NumSamples; ++ii)
         in[ii] *= - coeff1 * sin(ii * multiplier) - coeff2 * sin(ii * multiplier2);
      if (extraSample)
         in[NumSamples] = 0.0f;
   }
      break;
   case eWinFuncBlackmanHarris:
   {
      // Blackman-Harris
      const double multiplier = 2 * M_PI / NumSamples;
      const double multiplier2 = 2 * multiplier;
      const double multiplier3 = 3 * multiplier;
      const double coeff1 = -0.48829 * multiplier,
         coeff2 = 0.14128 * multiplier2, coeff3 = -0.01168 * multiplier3;
      for (int ii = 0; ii < NumSamples; ++ii)
         in[ii] *= - coeff1 * sin(ii * multiplier) - coeff2 * sin(ii * multiplier2) - coeff3 * sin(ii * multiplier3);
      if (extraSample)
         in[NumSamples] = 0.0f;
   }
      break;
   case eWinFuncWelch:
   {
      // Welch
      const float N = NumSamples;
      const float NN = NumSamples * NumSamples;
      for (int ii = 0; ii < NumSamples; ++ii) {
         in[ii] *= 4 * (N - ii - ii) / NN;
      }
      if (extraSample)
         in[NumSamples] = 0.0f;
      // Average the two limiting values of discontinuous derivative
      in[0] /= 2.0f;
      in[NumSamples - 1] /= 2.0f;
   }
      break;
   case eWinFuncGaussian25:
      // Gaussian (a=2.5)
      A = -2 * 2.5*2.5;
      goto Gaussian;
   case eWinFuncGaussian35:
      // Gaussian (a=3.5)
      A = -2 * 3.5*3.5;
      goto Gaussian;
   case eWinFuncGaussian45:
      // Gaussian (a=4.5)
      A = -2 * 4.5*4.5;
      goto Gaussian;
   Gaussian:
   {
      // Gaussian (a=2.5)
      // There are deltas at the ends
      const float invN = 1.0f / NumSamples;
      const float invNN = invN * invN;
      // Simplify formula from the loop for ii == 0, add term for the delta
      in[0] *= exp(A * 0.25) * (1 - invN);
      if (!extraSample)
         --NumSamples;
      for (int ii = 1; ii < NumSamples; ++ii) {
         const float iOverN = ii * invN;
         in[ii] *= exp(A * (0.25 + (iOverN * iOverN) - iOverN)) * (2 * ii * invNN - invN);
      }
      if (extraSample)
         in[NumSamples] *= exp(A * 0.25) * (invN - 1);
      else {
         // Slightly different
         const float iOverN = NumSamples * invN;
         in[NumSamples] *= exp(A * (0.25 + (iOverN * iOverN) - iOverN)) * (2 * NumSamples * invNN - invN - 1);
      }
   }
      break;
   default:
      fprintf(stderr, "FFT::DerivativeOfWindowFunc - Invalid window function: %d\n", whichFunction);
   }
}