## Is there any relationship between the autocorrelation of the instantaneous phase extracted from signal and the autocorrelation of the signal in time domain

Started by 4 weeks ago3 replieslatest reply 4 weeks ago122 views

Hello,

Here is my question and I would be really grateful if you could help me understand this.  In DSP, for extracting the instantaneous phase of a signal one could use Hilbert Transform.  Let the signal be $x(t)$.  I would like to know if there is any relationship between the "autocorrelation" in the signal x(t) and the autocorrelation in the "instantaneous phase" time series $\phi(t)$.  That is, is there any analytical way to find a relation between the ACF of $\phi(t)$ and ACF of $x(t)$; where $\phi(t)$ being the instantaneous phase of the x(t).

Next question is that if one can infer that non-stationarity of a signal x(t) results in non-stationarity of $\phi(t)$?

I appreciate your thoughts and help.  Happy Holidays and Happy New Year.

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Not sure if this is what you want, but this is an impulse response magnitude (green) and phase (red). I've annotated a few values of phase, just for reference.

Below them is the autocorrelation of each, plotted on a different timebase.

There seems to be a distinct similarity between the two.

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Dear @MarkSitkowski,

Thank you so much for taking the time to help.  This demonstrates the idea behind, but I am seeking if there is anyway to analytically establish the mathematical proof behind it to figure out as to whether this is generalizable.

For example, I want to know the feasibility of finding a relation between the followings:

$R_{xx}(\tau) = \sum_{\tau=-\infty}^{+\infty}x[m]x[m-\tau]$

$R_{\phi\phi}(\tau) = \sum_{\tau=-\infty}^{+\infty}\phi[m]\phi[m-\tau]$

given that $\phi[t] = atan(\frac{Im(z[t])}{Re(z[t])})$ where z[t] is the analytical signal of x[t].

Again, thank you Mark for your time and kindness to provide a reply.

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