FFTW: Inverse of forward fft not equal to original function

Question

I'm trying to use FFTW to compute fast summations, and I've run into an issue:

int numFreq3 = numFreq*numFreq*numFreq;

FFTW_complex* dummy_sq_fft = (FFTW_complex*)FFTW_malloc( sizeof(FFTW_complex)*numFreq3 );

FFTW_complex* dummy_sq = (FFTW_complex*)FFTW_malloc( sizeof(FFTW_complex)*numFreq3 );

FFTW_complex* orig = (FFTW_complex*)FFTW_malloc( sizeof(FFTW_complex)*numFreq3 );

FFTW_plan dummyPlan = FFTW_plan_dft_3d( numFreq, numFreq, numFreq,
                  orig, dummy_sq_fft,
                  FFTW_FORWARD, FFTW_MEASURE );

FFTW_plan dummyInvPlan = FFTW_plan_dft_3d( numFreq, numFreq, numFreq,
                      dummy_sq_fft, dummy_sq,
                      FFTW_BACKWARD, FFTW_MEASURE );

for(int i= 0; i < numFreq3; i++) {
  orig[ i ][ 0 ] = sparseProfile02[ 0 ][ i ][ 0 ];
  //img. part == 0
  orig[ i ][ 1 ]  = sparseProfile02[ 0 ][ i ] [ 1 ];
}

FFTW_execute(dummyPlan);
FFTW_execute(dummyInvPlan);

int count = 0; 
for(int i=0; i<numFreq3; i++) {
  double factor = dummy_sq[ i ][ 0 ]/sparseProfile02[ 0 ][ i ][ 0 ];

  if(factor < 0) {
    count++;
  }
}

std::cout<<"Count "<<count<<"\n";

FFTW_free(dummy_sq_fft);
FFTW_free(dummy_sq);
FFTW_destroy_plan(dummyPlan);
FFTW_destroy_plan(dummyInvPlan);

(Here sparseProfile02[0] is of type FFTW_complex*, and contains only positive real data.)

Since we have dummy_sq = IFFT(FFT(sparseProfile02[ 0 ])), we must have dummy_sq = n^3*sparseProfile02. But this is true only some of the time; in fact, values on the dummy_sq grid are negative whenever corresponding values on the sparseProfile02 grid are zero (but not vice-versa). Does anyone know why this is happening?

Answer 1

At the risk of dabbling in necromancy, you should note in the fftw docs (here) that it expressly states that fftw does not normalize, thus the result of the inverse transform of a previously transformed signal will be the original signal scaled by 'n' or the length of the signal.

Could be the problem.

Answer 2

The FFTs (forward and inverse) have rounding error, and I think this is what's biting you. In general, you shouldn't expect a zero to stay exactly zero through your process (although it could be zero for trivial test cases). In your test loop, is

fabs(dummy_sq[i][0] - numFreq*numFreq*numFreq*sparseProfile02[0][i][0])

large relative to the magnitude of your data?

As a really simple (pathological) example with just a 1D FFT of size 2 with real values:

ifft(fft([1e20, 1.0])) != [2e20, 2.0]

The 1.0 is lost in the 1e20 with a double precision FFT.

It's also possible you're getting some NaNs for factor when you divide by a zero sample in sparseProfile02.