title: DSP - Filtering in the frequency domain via FFT
body: <p>I've been playing around a little with the Exocortex implementation of the FFT, but I'm having some problems.</p>
  
  <p>Whenever I modify the amplitudes of the frequency bins before calling the iFFT the resulting signal contains some clicks and pops, especially when low frequencies are present in the signal (like drums or basses). However, this does not happen if I attenuate all the bins by the same factor.</p>
  
  <p>Let me put an example of the output buffer of a 4-sample FFT:</p>
  
  <pre><code>// Bin 0 (DC)
  FFTOut[0] = 0.0000610351563
  FFTOut[1] = 0.0
  
  // Bin 1
  FFTOut[2] = 0.000331878662
  FFTOut[3] = 0.000629425049
  
  // Bin 2
  FFTOut[4] = -0.0000381469727
  FFTOut[5] =  0.0
  
  // Bin 3, this is the first and only negative frequency bin.
  FFTOut[6] =  0.000331878662
  FFTOut[7] = -0.000629425049
  </code></pre>
  
  <p>The output is composed of pairs of floats, each representing the real and imaginay parts of a single bin. So, bin 0 (array indexes 0, 1) would represent the real and imaginary parts of the DC frequency. As you can see, bins 1 and 3 both have the same values, (except for the sign of the Im part), so I guess bin 3 is the first negative frequency, and finally indexes (4, 5) would be the last positive frequency bin.</p>
  
  <p>Then to attenuate the frequency bin 1 this is what I do:</p>
  
  <pre><code>// Attenuate the 'positive' bin
  FFTOut[2] *= 0.5;
  FFTOut[3] *= 0.5;
  
  // Attenuate its corresponding negative bin.
  FFTOut[6] *= 0.5;
  FFTOut[7] *= 0.5;
  </code></pre>
  
  <p>For the actual tests I'm using a 1024-length FFT and I always provide all the samples so no 0-padding is needed. </p>
  
  <pre><code>// Attenuate
  var halfSize = fftWindowLength / 2;
  float leftFreq = 0f;
  float rightFreq = 22050f; 
  for( var c = 1; c &lt; halfSize; c++ )
  {
      var freq = c * (44100d / halfSize);
  
      // Calc. positive and negative frequency indexes.
      var k = c * 2;
      var nk = (fftWindowLength - c) * 2;
  
      // This kind of attenuation corresponds to a high-pass filter.
      // The attenuation at the transition band is linearly applied, could
      // this be the cause of the distortion of low frequencies?
      var attn = (freq &lt; leftFreq) ? 
                      0 : 
                      (freq &lt; rightFreq) ? 
                          ((freq - leftFreq) / (rightFreq - leftFreq)) :
                          1;
  
      // Attenuate positive and negative bins.
      mFFTOut[ k ] *= (float)attn;
      mFFTOut[ k + 1 ] *= (float)attn;
      mFFTOut[ nk ] *= (float)attn;
      mFFTOut[ nk + 1 ] *= (float)attn;
  }
  </code></pre>
  
  <p>Obviously I'm doing something wrong but can't figure out what.</p>
  
  <p>I don't want to use the FFT output as a means to generate a set of FIR coefficients since I'm trying to implement a very basic dynamic equalizer.</p>
  
  <p>What's the correct way to filter in the frequency domain? what I'm missing?</p>
  
  <p>Also, is it really needed to attenuate negative frequencies as well? I've seen an FFT implementation where neg. frequency values are zeroed before synthesis.</p>
  
  <p>Thanks in advance.</p>