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I have a signal which contains sinusoidal components that oscillate at different frequencies. I think the phase of the sinusoids is changing with time. I could do a Fourier transform on small chunks of the signal and the phase of the sinusoids from that, but is there a better way to do this with a wavelet transform? I'm new to the concept of wavelets. From what I understand, it's good at doing what an FFT does but with time resolution. Could you get the phase of a sinusoid as a function of time from a wavelet transform ?

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It is indeed possible (up to a period), with complex "continuous" wavelet transforms, provide your mother wavelet ooscillate sufficiently. You can look at the scalogram modulus and its phase at the appropriate scales, as illustrated from Matlab: Wavelet coherence:

Wavelet coherence

Alternatively, you can convert your signal into an analytic one (using the Hilbert transform) and perform a real wavelet analysis.

Finally, you might dig into other linear (short-term Fourier transform) or bilinear time-frequency (Wgner-Ville, Choi-Williams) or time-scale tools.

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I would look into the short-time fourier transform. The wavelet transform is closely related to the STFT, more so than the FFT. I am not sure about getting phase information, but if you want to understand the wavelet transform more, start with the STFT.

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