Let's say I have a signal $y(t) = Acos(2\pi f_c t)$, where $f_c$ is the carrier frequency and $t$ is the independent variable. Since I work with discrete signals i sample this signal with a sampling rate $f_s = 100f_c$, so I obtain $y[n] = \big\{ t \rightarrow \frac{n}{100f_c}\big\} = Acos(\frac{\pi}{50} n)$. Now if we consider $x= \frac{\pi}{50}n$, we have $y = Acos(x)$ where $x \sim unif[-\pi,\pi]$ and $A$ is deterministic. Which is the probability density funciton of variable $y$, i.e. $f_y(y)$ ?
Thanks a lot.
There is no chance that $|y|>|A|$ because $y=A\cos x$ and $\cos$ only takes values between $-1$ and $1$. This means $f_y(y)=0$ when $|y|>|A|$. For those $y$ with $|y|\leq |A|$ we have $$f_y(y)= \frac{1}{\pi|A|}\frac{1}{\sqrt{1-(y/A)^2}}.$$
You can derive this expression by carefully applying the change of variables formula for integrals. You need to be careful because if $y=a$, then there are two possibilities for $x$, $\arccos (a/A)$ and $-\arccos (a/A)$, and these values occur with equal probability due to the fact that $\cos$ is an even function.