Suppose $W(t)$ is a standard Brownian Motion (or a Wiener Process), thus for $0\le s<t$, we have $$W(t)-W(s)\sim N(0,\sigma^2(t-s)), W(0)=0$$
Now define another stochastic process $A(t)$ as $$A(t)=cos(\omega_0t+W(t))$$ where $\omega_0$ is a positive constant.
Here is my question: How to calculate the power spectrum density (PSD) of $A(t)$.
The following is what I have tried:
- Firstly I tried to solve this problem with Wiener-Khintchine theorem. But unfortunately, as I have derived, $A(t)$ is not a stationary process and thus Wiener-Khintchine theorem is not applicable.
- Then I tried to perform a direct computation of the power spectral density from the definition as the limit of the magnitude of the windowed Fourier transform. $$S_{A}(f) = \lim_{\tau \to \infty} \frac{1}{\tau} \int_{t_1=0}^{\tau} \int_{t_2=0}^{\tau} e^{i 2\pi f (t_1-t_2)} \langle A(t_1)A(t_2) \rangle dt_1 dt_2$$ $$S_{A}(f) = \lim_{\tau \to \infty} \frac{1}{\tau} \int_{t_1=0}^{\tau} \int_{t_2=0}^{\tau} e^{i 2\pi f (t_1-t_2)} \langle cos(\omega_0t_1+W(t_1))cos(\omega_0t_2+W(t_2)) \rangle dt_1 dt_2$$
As a result, I get stuck here and don't know to solve this integration.