Link between maximum delay of cross-correlation and phase velocity

Question

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. $|X(\omega)|=1$?

Answer 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.