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Hi all, and apologies in advance for the long-winded question. 

I have recently implemented a fast convolution algorithm using the FFT/iFFT to convert input
signal and filter into the frequency domain, multiply them out, and iFFT them back to the time
domain to get my result. It all seems to work fine, however, I recently got a little confused
on one point. 

I need to determine the gain that my filter will be applying to the signal. I've done this by
importing my coefficients into Matlab and calling freqz (sig proc toolbox, I think). It works
fine and I get gains goings from about 50 dB at DC to around 0 dB at Nyquist (fs/2). I get the
same result if I call fft and plot the magnitude of the coefficients.

That all seemed to make perfect sense. Then I thought I'd have a look at the spectrum of my
filter in an Audio Editor like Sound Forge or GoldWave. When I look at it there, the gains are
down by about 95 dB. Now my filter is very long (impulse response reverb); in this case around
62,000 samples. I can account for the difference by assuming that these audio editors, when
they calculate the spectrum, do the divide by N (FFT/iFFT length) in the FFT, not the iFFT,
since 20 * log10(62000) is around the 95 dB (unless this is a weird coincidence). I know that
Matlab's FFT/iFFT combo does the divide by N in the iFFT (which, I believe, is where it's
normally done (e.g., Oppenheim/Schafer), and where I do it in my implementation. Is that a
reasonable conclusion?

In any case, what is the real gain of the filter? If I would do my divide by N in the FFT
instead, or perhaps divide by sqrt(N) in both FFT and iFFT, I would see a different magnitude
response for this filter. My understanding is that for the FFT to conserve energy, you need to
normalize by sqrt(N), yet I don't see that happening in the Matlab case. I might have thought
that doing so would give a magnitude response for which you could conclude that those values
are the gain of the filter.

More importantly, wouldn't the result you get in doing fast convolution differ depending on
where the FFT/IFFT 1/N normalization was done? Is it perhaps a given in the definition of it
that the iFFT will do the 1/N normalization? I can say that when I do it this way, the result
is correct (i.e., if I do a time domain convolution I get the same result as with the fast
convolution in the freq domain).

I know I'm a bit confused on this, just not sure which part(s) I've got wrong.

Thanks very much,

Ron
	

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