Autocorrelation function of a Gaussian random signal

Question

Im a beginner at signal processing and I've gotten the following question in an exercise:

"Write down the equation for a Gaussian probability density distribution and relate the different variables to the autocorrelation function of a Gaussian random signal"

The first part of the question is seems pretty straightforward and I write the Gaussian PDF as: $$ f_x(x) = \frac{1}{\sigma_x \sqrt{2 \pi}} \ \ \text{exp}\bigg[ - \frac{(x-m_x)^2}{2 \sigma_x ^2} \bigg] $$

But the second part i dont know how to aproach it. How do I find the autocorrelation function $R(\tau)$ of a Gaussian random signal?
Do I find it as: $$ R(\tau) = E[x(t)x(t-\tau)] $$ and then calulate the integral? Or is the autocorrelation function known? I could not find the formual for the autocorrelation function anywhere except using chatGPT which i dont really trust.

Answer 1

You're missing some necessary information to get the autocorrelation:

$$ R_{xx}(\tau) = \operatorname{E}\Big\{ x(t)x(t+\tau) \Big\} $$

You have

$$ f_{x(t)}(\alpha) = \frac{1}{\sqrt{2 \pi} \sigma_x} \ \ e^{- \frac{(\alpha-\mu_x)^2}{2 \sigma_x ^2} } $$

That's not enough. You need, additionally, this conditional probability:

$$ f_{x(t+\tau)x(t)}(\alpha,\beta) = f_{x(t+\tau)}\big(\alpha|\beta \big) \cdot f_{x(t)}(\beta)$$

One possibility might be:

$$\begin{align} f_{x(t+\tau)x(t)}(\alpha,\beta) &= f_{x(t+\tau)}\big(\alpha|\beta \big) \cdot f_{x(t)}(\beta) \\ \\ &= \frac{1}{\sqrt{2 \pi \left(\sigma_x^2 - R_{xx}(\tau)\right)}} e^{-\frac12 \frac{\left(\alpha-\beta\sigma_x^{-2}R_{xx}(\tau)\right)^2}{\sigma_x^2 - R_{xx}(\tau)}} \ \cdot \ \frac{1}{\sqrt{2 \pi} \sigma_x} e^{-\frac12 \left(\frac{\beta}{\sigma_x}\right)^2} \\ \end{align}$$

In this answer I show how you go from those dependent probabilities to the autocorrelation function.