## 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 10 months ago3 replieslatest reply 10 months ago135 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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