Link between maximum delay of cross-correlation and phase velocity

I try to express the relation between:

• the delay obtained by maximizing the cross-correlation between two signals $$x$$ and $$y$$ noted $$\tau_{x,y}$$ and
• the phase velocity noted $$v_p(\omega)$$

In the case where:

• The signals $$x$$ and $$y$$ are spaced by a constant distance $$\Delta$$
• The signal $$y$$ is a shifted version - at speed $$v_p(\omega)$$ - of the signal $$x$$, i.e.$$Y(\omega)=X(\omega)e^{-j\omega\Delta/v_p(\omega)}$$ ; with $$X(\omega)$$ and $$Y(\omega)$$ the Fourier transforms of $$x$$ and $$y$$

With these assumptions, the delay that corresponds to the maximum correlation between $$x$$ and $$y$$ is expressed as:

$$\tau_{x,y} = \text{max}_t \int_{-\infty}^{\infty} |X(\omega)|^2e^{j\omega(t+\Delta/v_p(\omega))}d\omega$$

I can't manage to go any further. Is it possible to go further and express $$\tau_{x,y}$$ as a function of $$v_p(\omega)$$? Perhaps by adding the assumption of white signal, i.e. $$|X(\omega)|=1$$?

For periodic power signals, the relationship $$\tau_{x,y} = \max_t(R_{xy})$$ is not unequivocal. Thus, restricting the considered signals to finite energy signals is required. But even then, it will probably be impossible to find an elegant functional relationship between $$\tau_{x,y}$$ and $$v_p$$, as it strongly depends on the signals involved. The trivial case without dispersion is simple and you might find classes of signals with a mathematically elegant relationship even with dispersion but in general it will not get simpler than the equation you derived. I would love to be proved wrong, as this is quite interesting.
• Thank you for your help. I have the feeling that for finite energy signals, the delay obtained by correlation can be considered as a weighted mean of the delays associated to the phase velocity, i.e. $$\tau_{x,y} \approx \frac{\int_{-\infty}^{\infty} |X(\omega)|^2 \frac{\Delta}{v_p(\omega)}}{\int_{-\infty}^{\infty} |X(\omega)|^2}$$ But I don't know how to show it... Oct 20, 2022 at 9:41