[music-dsp] Is FFT filtering? (was Newbie question about fre quency analysis)

[Frederick Umminger]

On Wed, 28 November 2001, Yaakov Stein wrote:
> Let's clarify the situation.
> 
> The N-point FFT is precisely equivalent to a bank of N/2 band-pass filters
> with a particular (sinc) skirt followed by N/2 mixers to bring the results
> down
> to zero frequency. Alternatively, the FFT is a bank of N/2 mixers that bring
> 
> different parts of the signal down to zero frequency followed by the same
> (complex) low-pass filter.

snip

> So it is still NOT one sample at a time. 

You left out the decimation of each channel. If you do not decimate, then the FFT _is_ being done one sample at a time. Alternatively, by doing the FFT in blocks spaced n samples apart you are implicitly decimating each channel by a factor of n after performing the filtering.

The "mixing" of each channel down to the vicinity of zero frequency is usually done through the magic of the aliasing caused by this decimation.

This may seem like a small point, but for many kinds of processing (e.g. anything nonlinear) it is really important to be aware of this decimation, the potential for aliasing, and the bandwidth restrictions that result. You don't want the decimation to put too much aliasing noise in each channel (implying requirements for the low-pass filter = window), and you want each channel oversampled enough that any nonlinear processing of that channel will not generate too much new aliasing noise (implying requirements for the amount of overlap).
 
-Frederick Umminger

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