Thanks for your time
My question rephrased:
Lets assume a spectrum of size N, can you create a meaningfull spectrum
of size N/2
by simply adding every other bin together?
Neglecting the artefacts of the forward transform, lets say an
artificial spectrum
(or a spectrum after peak picking that discards the region around the peaks)
Lets say two sinusoids in two adjacent bins, will summing them into a
single bin of a half sized spectrum
make sense and represent them adequately?
In my limited understanding, yes, but I am not sure, and would like to
know why not
if that is not the case.
I'm not sure what "compress" means in this context, nor am I sure what
"fall together" means. But here's some points to note:
A steady state sine wave in the time domain will be transformed by a
short-time fourier transform into a spectral peak, convolved (in the
frequency domain) by the spectrum of the analysis envelope. If you
know that all of your inputs are sine waves, then you can perform
"spectral peak picking" (AKA MQ analysis) and reduce your signal to a
list of sine waves and their frequencies and phases -- this is the
sinusoidal component of Serra's SMS (explained in the pdf linked above).
Note that since a sinusoid ends up placing non-zero values in every
FFT bin, you'd need to account for that in your spectral estimation,
which basic MQ does not -- hence it does not perfectly estimate the
sinusoids.
In any case, most signals are not sums of stationary sinusoids. And
since signals are typically buried in noise, or superimposed on top of
each other, so the problem is not well posed. For two very simple
examples: consider two stable sine waves at 440Hz and 441Hz -- you
will need a very long FFT to distinguish this from a single
amplitude-modulated sine wave? or consider a sine wave plus white
noise -- the accuracy of frequency and phase recovery will depend on
how much input you have to work with.
I think by "compression" you mean "represent sparsely" (i.e. with some
reduced representation.) The spectral modeling approach is to "model"
the signal by assuming it has some particular structure (e.g.
sinusoids+noise, or sinusoids+transients+noise) and then work out how
to extract this structure from the signal (or to reassemble it for
synthesis).
An alternative (more mathematical) approach is to simply assume that
the signal is sparse in some (unknown) domain. It turns out that if
your signal is sparse, you can apply a constrained random
dimensionality reduction to the signal and not lose any information.
This is the field of compressed sensing. Note that in this case, you
haven't recovered any structure.
Ross
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