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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Reply by MarkSitkowskiDecember 28, 2019

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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Reply by h2oDecember 28, 2019

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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Reply by Y(J)SDecember 28, 2019

It depends on what you are assuming about your signal. 

Remember that the phase signal is only half the picture (and if by \phi(t) with a small "p" you mean the phase after removing the linear trend, it is even less than half).

So, a signal could be nonstationary due to nonstationary amplitude, despite having completely stationary phase (even linear phase).

Regarding the autocorrelations, think of a narrowband signal FM signal. You will see a peak corresponding to the carrier frequency, which has nothing to do with small "p" instantaneous phase. But if the modulation is itself sinusoidal (i.e., the phase is sinusoidal and thus has a autocorrelation peak at the modulation frequency) you will see autocorrelation peaks in the signal related to this.

More generally, write the integral INT x(t) x(t-\tau) dt = INT A(t) e^{i \phi(t)} A(t-\tau) e^(i \phi(t-\tau)} dt . If the As are not time dependent you are left with the integral INT e^{i (\phi(t) - \phi(t-\tau))} dt which shows that the difference appears in the exponent. For small \tau you can expand the exponent and after the constant term you get a difference integral and not a square difference (which is related to the product).