# Log derivative interpretation

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.