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


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.

  • $\begingroup$ 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... $\endgroup$
    – User327201
    Commented Oct 20, 2022 at 9:41
  • $\begingroup$ I have a similar gut feeling. Showing this, if possible, won't be easy. An entry point could be to start with a monochromatic wave packet, arguing that any signal can be expressed as a sum of sines, and then add degrees of freedom successively. $\endgroup$
    – Max
    Commented Oct 20, 2022 at 10:07

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