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I am solving a problem where the noise in the system is given by a Wiener process $$\Phi_{L}(t) = \int_0^t \Phi'_{L}(\tau)d\tau $$ where $\Phi'_L(t)$ is modelled as a zero-mean white Gaussian process with a power spectral density (PSD) of $$S_{\Phi'_{L}}(\omega) = 2\pi\Delta\nu $$ After doing some calculation, I have arrived at a variable $I(t)$ that looks like this

$$I(t) = \cos(\Phi_L(t-t_{+}) - \Phi_L(t-t_0))$$

I want to find the auto-correlation of the function I(t) now. $$R = <I(t).I(t+\tau)>$$ $t_+$ and $t_0$ are constant. I am trying to solve the auto-correlation but somehow, now able to make use of the $\Phi_L(t)$ relation with $\Delta\nu$. I am not sure how can I go about it. Any help?

I am able to reach till this step

$$R = \frac{1}{2} \big\langle \cos(\Delta\Phi_1 - \Delta\Phi_2) + \cos(\Delta\Phi_1 + \Delta\Phi_2) \big\rangle $$ $$\Delta\Phi_1 = \Phi_L(t - t_+) - \Phi_L(t - t_0), \quad \Delta\Phi_2 = \Phi_L(t + \tau - t_+) - \Phi_L(t + \tau - t_0). $$

SiPh
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  • Should the $\phi'_L(t)$ in the integral have $\tau$ as an argument instead of $t$? – LSK21 Dec 10 '24 at 08:57
  • Also, judging by your physics tag, I am assuming that $\langle \cdot \rangle$ signifies an expectation, correct? – LSK21 Dec 10 '24 at 09:03
  • @LSK21 Thanks. Yes on both counts. I have corrected the integral variable. – SiPh Dec 10 '24 at 09:09
  • This might be helpful: https://math.stackexchange.com/questions/3121914/variance-of-sine-and-cosine-of-a-random-variable – LSK21 Dec 10 '24 at 13:29

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