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I have a real signal f sampled at 96Hz composed of clusters of harmonics scattered tigthly around evenly central frequencies close to 1Hz and 2Hz (multiples of 0.966Hz) that I'll call features. These are modulated in amplitude and phase by a slower nonstatinary system $C(t)$ that is approximately constant over each feature but different between features. The features also interact quadratically to form additional overtone features at 3Hz and 4Hz that are of interest. Because of the time variation from $C(t)$ and some noise, analysis is nonstationary and signal analysis limited to features. However, the bandwidth of each feature is always small (10%) compared to the distance between octaves, so the problem is like modal decomposition but with some known structure and the abstraction of each feature as a single modulated harmonic should be correct.

I would like to characterize the octaval features as individual harmonics with amplitude and phase. What are my options? I know that an L1-normalized cmorB-C wavelet (B=4 or 8 and C=0.96) centered at the desired central frequencies does OK, compromising between overlap between $\omega_3$ and $\omega_4$ and time resolution issues with $\omega_1$. B=8 seems to do particularly well for frequencies like $\omega_3$ and $\omega_4$, because the wavelet is barely resolved enough in frequency to separate the features and thus wide and flat enough to integrate within-band components.

But then there remains $\omega_1$, where the wavelet response is neither wide nor narrlow compared to the effective width of the feature. One option would be to simply use a different Morlet wavelet with B=2.5, which would make it time resolved at $\omega_1$ and barely resolved at the feature level. I think this is common practice but it feels like I am undoing the wavelet to be more like a short term Fourier analysis.

Alternatively, I could integrate the wavelet over the effective width of the feature. I have seen this done with power, presumably using a different normalization. Is there a way to do this integration and produce an effective amplitude and phase?

An alternative based on focusing the signal that would preserve amplitude and phase analogous to synchrosqueezing?

Eli S
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    Avoid pywt. It helps to include 1-3 signals of interest (e.g. Google Drive, ufile, Dropbox). – OverLordGoldDragon May 03 '23 at 16:21
  • Thanks @OverLordGoldDragon. I presume ssqueezepy is a good choice? I'm happy to compare to ssqueezepy. I can probably figure out how the parameter B relates to the width parameter there (mu?) but is the central frequency option (parameter C in pywavelets) something that has an analog in ssqueezepy? If I want to center on 0.996 Hz do I just choose my scale very surgically? – Eli S May 03 '23 at 18:57
  • Yes, ssqueezepy will do. Side comment, another thing about the site is to ask non-trivial questions as separate posts rather than in comments, and I'd qualify the "pywt vs ssqueezepy" as such. I say side comment since, I tried this before and found pywt way too cryptic, and it's impossible to reproduce accurately anyway since that'd make ssq broken too. A quick way is to compare outputs (abs) on test signals, and use scales='log' in ssq if very low freqs are of interest; this helps reproduce the resolution tradeoff. 996, can, dunno if need. – OverLordGoldDragon May 04 '23 at 14:10

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