16 FFT is very small. What you will find is FFTs smaller than say 64 will be hard coded assembler with no loops to get the highest possible performance. This means they can be highly susceptible to variations in instruction sets, compiler optimisations, even 64 or 32bit words.
What happens when you run a test of FFT sizes from 16 -> 1048576 in powers of 2? I say this as a particular hard-coded asm routine on Linux might not be the best optimized for your machine, whereas you might have been lucky on the Windows implementation for that particular size. A comparison of all sizes in this range will give you a better indication of the Linux vs. Windows performance.
Have you calibrated FFTW? When first run FFTW guesses the fastest implementation per machine, however if you have special instruction sets, or a particular sized cache or other processor features then these can have a dramatic effect on execution speed. As a result performing a calibration will test the speed of various FFT routines and choose the fastest per size for your specific hardware. Calibration involves repeatedly computing the plans and saving the FFTW "Wisdom" file generated. The saved calibration data (this is a lengthy process) can then be re-used. I suggest doing it once when your software starts up and re-using the file each time. I have noticed 4-10x performance improvements for certain sizes after calibrating!
Below is a snippet of code I have used to calibrate FFTW for certain sizes. Please note this code is pasted verbatim from a DSP library I worked on so some function calls are specific to my library. I hope the FFTW specific calls are helpful.
// Calibration FFTW
void DSP::forceCalibration(void)
{
// Try to import FFTw Wisdom for fast plan creation
FILE *fftw_wisdom = fopen("DSPDLL.ftw", "r");
// If wisdom does not exist, ask user to calibrate
if (fftw_wisdom == 0)
{
int iStatus2 = AfxMessageBox("FFTw not calibrated on this machine."\
"Would you like to perform a one-time calibration?\n\n"\
"Note:\tMay take 40 minutes (on P4 3GHz), but speeds all subsequent FFT-based filtering & convolution by up to 100%.\n"\
"\tResults are saved to disk (DSPDLL.ftw) and need only be performed once per machine.\n\n"\
"\tMAKE SURE YOU REALLY WANT TO DO THIS, THERE IS NO WAY TO CANCEL CALIBRATION PART-WAY!",
MB_YESNO | MB_ICONSTOP, 0);
if (iStatus2 == IDYES)
{
// Perform calibration for all powers of 2 from 8 to 4194304
// (most heavily used FFTs - for signal processing)
AfxMessageBox("About to perform calibration.\n"\
"Close all programs, turn off your screensaver and do not move the mouse in this time!\n"\
"Note:\tThis program will appear to be unresponsive until the calibration ends.\n\n"
"\tA MESSAGEBOX WILL BE SHOWN ONCE THE CALIBRATION IS COMPLETE.\n");
startTimer();
// Create a whole load of FFTw Plans (wisdom accumulates automatically)
for (int i = 8; i <= 4194304; i *= 2)
{
// Create new buffers and fill
DSP::cFFTin = new fftw_complex[i];
DSP::cFFTout = new fftw_complex[i];
DSP::fconv_FULL_Real_FFT_rdat = new double[i];
DSP::fconv_FULL_Real_FFT_cdat = new fftw_complex[(i/2)+1];
for(int j = 0; j < i; j++)
{
DSP::fconv_FULL_Real_FFT_rdat[j] = j;
DSP::cFFTin[j][0] = j;
DSP::cFFTin[j][1] = j;
DSP::cFFTout[j][0] = 0.0;
DSP::cFFTout[j][1] = 0.0;
}
// Create a plan for complex FFT.
// Use the measure flag to get the best possible FFT for this size
// FFTw "remembers" which FFTs were the fastest during this test.
// at the end of the test, the results are saved to disk and re-used
// upon every initialisation of the DSP Library
DSP::pCF = fftw_plan_dft_1d
(i, DSP::cFFTin, DSP::cFFTout, FFTW_FORWARD, FFTW_MEASURE);
// Destroy the plan
fftw_destroy_plan(DSP::pCF);
// Create a plan for real forward FFT
DSP::pCF = fftw_plan_dft_r2c_1d
(i, fconv_FULL_Real_FFT_rdat, fconv_FULL_Real_FFT_cdat, FFTW_MEASURE);
// Destroy the plan
fftw_destroy_plan(DSP::pCF);
// Create a plan for real inverse FFT
DSP::pCF = fftw_plan_dft_c2r_1d
(i, fconv_FULL_Real_FFT_cdat, fconv_FULL_Real_FFT_rdat, FFTW_MEASURE);
// Destroy the plan
fftw_destroy_plan(DSP::pCF);
// Destroy the buffers. Repeat for each size
delete [] DSP::cFFTin;
delete [] DSP::cFFTout;
delete [] DSP::fconv_FULL_Real_FFT_rdat;
delete [] DSP::fconv_FULL_Real_FFT_cdat;
}
double time = stopTimer();
char * strOutput;
strOutput = (char*) malloc (100);
sprintf(strOutput, "DSP.DLL Calibration complete in %d minutes, %d seconds\n"\
"Please keep a copy of the DSPDLL.ftw file in the root directory of your application\n"\
"to avoid re-calibration in the future\n", (int)time/(int)60, (int)time%(int)60);
AfxMessageBox(strOutput);
isCalibrated = 1;
// Save accumulated wisdom
char * strWisdom = fftw_export_wisdom_to_string();
FILE *fftw_wisdomsave = fopen("DSPDLL.ftw", "w");
fprintf(fftw_wisdomsave, "%s", strWisdom);
fclose(fftw_wisdomsave);
DSP::pCF = NULL;
DSP::cFFTin = NULL;
DSP::cFFTout = NULL;
fconv_FULL_Real_FFT_cdat = NULL;
fconv_FULL_Real_FFT_rdat = NULL;
free(strOutput);
}
}
else
{
// obtain file size.
fseek (fftw_wisdom , 0 , SEEK_END);
long lSize = ftell (fftw_wisdom);
rewind (fftw_wisdom);
// allocate memory to contain the whole file.
char * strWisdom = (char*) malloc (lSize);
// copy the file into the buffer.
fread (strWisdom,1,lSize,fftw_wisdom);
// import the buffer to fftw wisdom
fftw_import_wisdom_from_string(strWisdom);
fclose(fftw_wisdom);
free(strWisdom);
isCalibrated = 1;
return;
}
}
The secret sauce is to create the plan using the FFTW_MEASURE flag, which specifically measures hundreds of routines to find the fastest for your particular type of FFT (real, complex, 1D, 2D) and size:
DSP::pCF = fftw_plan_dft_1d (i, DSP::cFFTin, DSP::cFFTout,
FFTW_FORWARD, FFTW_MEASURE);
Finally, all benchmark tests should also be performed with a single FFT Plan stage outside of execute, called from code that is compiled in release mode with optimizations on and detached from the debugger. Benchmarks should be performed in a loop with many thousands (or even millions) of iterations and then take the average run time to compute the result. As you probably know the planning stage takes a significant amount of time and the execute is designed to be performed multiple times with a single plan.
