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All that I heard so far about phase spectrum - that it is very hard to process. Are there any methods to do that, and is it possible to extract information about the noise from phase spectrum?

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you can look at

https://math.stackexchange.com/questions/1290620/what-is-the-importance-of-phase-spectrum-in-fourier-transform

the short answer is that for noise, phase is random. There shouldn't be any structure to exploit, while for signal, there often is.

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