-2
$\begingroup$

Is there a solution to the expected value/variance for a Gaussian with random phase:

$$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$

?

For $t=0$, the solution is for example given here: Resultant probability distribution when taking the cosine of gaussian distributed variable

$\endgroup$
1
$\begingroup$

This probably isn't a good question for mathoverflow but is more suited for math.stackexchange (which is for less advanced topics)

That said, here is the answer. You just reexpress the cosine function as a sum of two complex exponentials, and find that your answer can be expressed using the Fourier transform of the Gaussian density function. That fourier transform is known: it's another Gaussian, and so you have the solution to your question

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.