[resending, I think I accidentally replied off-list]
On 1/11/2018 5:00 AM, gm wrote:
> 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.
You can analyze this by looking at the definition of the short-time
discrete Fourier transform. (Or the corresponding C code for a DFT).
Each spectral bin is the sum of samples in the windowed signal
multiplied by a complex exponential.
Off the top of my head, assuming a rectangular window, I think you'll
find that dropping every second bin in the length N spectrum gives you
the equivalent of the bin-wise sum of two length N/2 DFTs computed with
hop size N/2.
Summing adjacent bins would do something different. You could work it
out by taking the definition of the DFT and doing some algebra. I think
you'd get a spectrum with double the amplitude, frequency shifted by
half the bin-spacing. (i.e. the average of the two bin's center
frequencies).
Ross.
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