Hi
The theoretical background probably most appropriate for the FFT (check
out some Berkeley courses for instance , as I recall some woman there
having a good web-cast about the subject) is the Electrical Engineering
interpretation and (as I'm formally educated in) the Network Theory
related to it: it's a shortened (from infinite) convolution filter.
No matter what you do with it, unless you know exactly what you're
theoretically basing yourself on, the analysis of frequencies with the
FFT will have serious limitations (and of course interesting
possibilities), and Inverse FFT-ing some data will probably have side
effects you don't count on (but of course it's fun).
The averaging of either the frequency (and don't forget the phase
information, otherwise you might get deluded that actual music can be
summed up with one phase as if it was a sine-wave) data or the IFFT-ed
sample set (which is also subject to DA converter resampling, no matter
how hard you hope the FFT will alleviate this need) generally makes of
course a somewhat smoother audio signal (as intended I suppose). But, it
doesn't solve all the problems of "hacking" transients and inaccurate
frequency components.
Lets say you FFT any signal you want (of course with no aliasing
frequencies), it is very likely that the signal won't exactly "fit" in
the length of the FFT, that is: there is a discontinuity between the
current FFT frame and the previous, ot otherwise put, if you'd want to
repeat the current FFT frame, the last sample is most likely not close
to the first sample, and higher order continuity is even more unlikely.
The frequency data associated with this will indicate all kinds of
frequency components to reflect this, and which are not coming from a
more reasonable interpretation of the input samples in frequency domain.
This will not properly resolve by averaging data in almost any case of
of musical data, at all, ever.
Also, averaging the output of the IFFT is indeed some sort of a FIR low
frequency filter which cannot exactly do what you'd normally want (a
great smooth signal with all the right properties, low distortion and
some frequency domain transform), and will always be audible.
So unless you like the effect (in some cases that is certainly
reasonable), or unless you use the FFT and some sensible frequency
domain filter response and pre- and post filter you data (like happens
in the "studio data path" example I presented) the averaging effects
coming from it and the "jaggy" signals coming out of it (in my case
Loudness curve related impulses), it isn't really going to verify some
sort of holy grail of DSP gurus (not EEs), but rather falsify such ideas!
About frequency detection accuracy: depending on what data you sent to
the FFT compuations, there are a number of bins, related to the length
of the FFT, which will not average out very well into more accurate
frequency detection, and really aren't super accurate, except for a nice
spectogram and some purposes where certain frequency ranges are
important. The FFT cannot easily be averaged into a filter longer than
it's length, and according to statistical or Information Theory rules, a
thousand or so samples simply isn't more accurate than it is.
Theo.
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Highway to Hell - Highway to Hell, AC/DC
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