In the origin paper on Synchrosqueezing Wavelet Transform, the phase transform, used to extract the instantaneous frequency of a signal $f(t)$, is defined as

$$ \omega (a, b) = -j[W_\psi f(a, b)]^{-1} \frac{\partial}{\partial b} W_\psi f(a, b), \tag{1} $$

where $W_\psi f$ is the CWT of $f(t)$. $\psi$ is assumed analytic, or having support on non-negative frequencies: $\hat \psi (\omega < 0) = 0$ (hat = $\mathcal F$). Then, $(1)$ can be interpreted in an example:

Differentiating w.r.t. $b$, $j\Omega$ is extracted from blue, the rest (red) kept the same. $\Omega$ is thus obtained by dividing out red and $j$, which is what $(1)$ does. But what if the CWT never yielded blue, or a term from which $\Omega$ is 'nicely' extracted? $f$ isn't nice at all in practice. Then a more general, log-derivative, or $f'/f$, interpretation is needed:

this is the infinitesimal relative change in $f$; that is, the infinitesimal absolute change in $f$, namely $f'$, scaled by the current value of $f$

But that's too general. Is there a more specific interpretation in context? i.e. extracting instantaneous frequency via CWT, or more generally log-derivative of a frequency-domain representation.


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