[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).

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