There is a lot of idea in this website to retrive phase of certain frequency from a noisy oscillation. For example this one: use band pass filter to filter out certain frequency and then use hilbert transform and then you can get instantaneous phase. And this one, get the angle of FFT at certain frequency and of course use circular shift before FFT. Or retrive the oscillation from continous wavelet transform.
I have this idea which sounds similiar to FFT and I want to know if it is in principle reasonable:
if you have a certain noisy data from experiment which contain several frequency components. Some of these frequency components may sit very close to each other.
I create an artifical signal which has sine waves of same frequency components. Their phase is $\omega t$. Then I shift the artifical signal corresponding to the experimental data and do FFT of the sum of these two signals at each phase shift step. Then I get a 2D plot of frequency v.s. phase shift(time delay). And I can line out certain frequency and get the phase of the oscillation.
This way sounds stupid but in practice it is intuitive and very robust. Is it reasonable to do this?
