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January 26, 2006

Lessons in FFT

   I\'ve been working on a project that requires a low-pass filter on sampled waveforms.  There is a generic low-pass algorithm:
       W[n] = C*S[n]+(1-C)*S[n-1]
     S = Input buffer
     W = Output buffer
     C = Filter coefficient
   This a is fast and functional low-pass filter, but introduces phase-shift into the signal.  The signal distortion is too great to be unusable with the filter coefficient we required.
   Another way of doing a low-pass filter is to do so in the frequency domain.  This requires a discrete Fourier transform (DFT) on the waveform, nulling out the high frequencies and an inverse DFT.  We had done this successfully with some C code we found on the net.  Since this was implemented in a hurry, it was hardly efficient.  I knew that implementations must exist that took advantage of specialized CPU instructions such as MMX, SSE and 3dnow!, all of which the CPU I\'m working with supports.  SSE and 3dnow!, in particular, could be of advantage to an FFT because they added scalar floating-point hardware acceleration.  
   In my quest for a better algorithm, I kept coming back to FFTW (Fastest Fourier Transform in the West), an DFT library developed at MIT (Massachusetts Institute of Technology).  At first, I shied away from this library, as it seemed to be far more then I was looking for; capable of discrete Fourier transforms on multi-dimensional arrays of complex data.  The mer task of transforming a one dimensional array of real data seemed insulting to such a powerful tool.  Why would the writers bother concentrating optimization effort to such a simple task?  As I found out, however, a good number of applications used FFTW; applications that needed a fast, simple DFT.  Specifically, the LAME mp3 encoder library.  Clearly, that project needs something quick.
   As my implementation started, I had a good number of problems.  First of all, I had no idea how my existing FFT algorithm was going about it\'s busness.  I knew the FFT algorithm returned data in the frequency domain, and I understood the filtering function.  However, reading about FFTs, it was clear to me the transform had to be returning complex data.  Complex numbers are acually two numbers: a real part, and an imaginary part.  It is typically expressed as ( a + b*i ), where \"a\" is the real part, \"b\" is the imaginary part, and \"i\" is the square root of -1.
   There was a second problem as well.  The FFT algorithm was also offset the data by 1 element and prepended \"0\" to the front before the transform.  FFTW stated specifically it did not do this.  This turned out not to be much of an issue at all.  The biggest problem was figuring out what output was created by the phantom FFT.
   The third problem wasn\'t much of an issue either, once I understood what was happening.  The output of the FFTW was not normalized.  Thus, one could not transform the input data to frequency and back again without an intermediate step.  This step simply involves dividing all the elements in the sample by the number of samples.
   It turns out the problem was in the ordering.  The phantom algorithm returned complex data, as it must.  The data was arranged r[0],i[0],r[1]i[1]...r[n],i[n].  FFTW\'s transform function for real data returns it\'s output r[0],r[1]...r[n],i[n]...i[1],i[0].  So, rather then ever other element being the imaginary part, the 2nd half of the output was the imaginary part in reverse order.  I can only assume this was arranged this way because it was the most efficient for the algorithm.  After I understood the layout of the two FFT algorithm\'s output, I succeeded in modifying the filtering portion of the low-pass filter code to deal with this layout.
   The results, a 15x speed improvement over the previous algorithm.  Considering the fact this is a real-time application, that\'s a world of difference.
   Pictured is the isthmus at Riverside Park in Beloit.

2 comments have been made.

From greg838

April 06, 2006 at 3:50 AM

From greg838

April 06, 2006 at 3:55 AM

Sorry for my previous post, it was an error. It seems that I'm trying to do the same thing, but I don't understand something : In order to filter your image, you should take the module of the fftw and threshold it to remove the high frequency. That means you get real data. In order to compute your reverse fftw, you use fftw_r2r? I will be interested in seeing the filtering algorithm.
sorry for my bad english...
Greetings
greg
   I visited Tazz and Tami today and we chatted about up photography most of the evening.  Tazz let me try out his new 17-85MM f4-5.6 IS lens to which I was pleased.  A little slower at the bottom end then my 17-55mm f3.5-5.6 and noticeably heavier.  But the auto-focus was extremely fast and quiet.  The 85mm top end is nice as well; a discernible improvement over the 55mm top end of my default lens.  In all I think it\'s a nice lens.
   A lamp at Tazz and Tami\'s house.  
   Finally got around to writing the code for adding comments to pictures.  That puts the image gallery script at about 98% functional.  There\'s still no code for child safety (i.e. galleries alright and not recommended for young viewers) and I want to add exif information from the pictures to the database.
   Lisa and Vinny.
   Although a solid 30 degrees outside, I decided I wanted to take a bike ride.  After layering up and putting air in the rear tire, I departed for a half hour tour of Riverside Park.  This is likely the coldest I\'ve ever bothered trying to ride, but went alright.  Aside from needing something to cover my face a little better, I didn\'t get too cold.  197 pictures taken, 22 proofed and 2.35 miles ridden.
   Watertower Park in Beloit.
   It was found the Indigo Dragon wasn\'t backing up the SQL databases.  This is done by gziping all the files in /var/lib/mysql and placing them in the web site\'s directory.  By simply existing in the web sites directory, the data is automatically backed up to several locations; the Red-Dragon, the Iron-Dragon and the external 250 GB backup drive.  We found this out when we wanted to do some work on the site with the Iron Dragon, only to discover the database wasn\'t present.
   Kaytee P. at last nights post Mountain Lions\' show gathering.