I have a signal $s$, of which the PSD I call as $S$.
So,
$$ S(\omega) = \frac{1}{N} \left|\sum_{n = 0}^{N-1} s(n) \exp(-j\omega n)\right|^2 $$
I have a closed form expectation of the above mentioned $S$:
$$ \mathbb{E}[S(\omega)] = F(\Theta) $$ where $F$ is the expectation with some parameters $\Theta$.
However, the signal is corrupted with a white Gaussian noise of variance $\sigma_n^2$
$z = s + n$, where $n \sim \mathcal{CN}(0, \sigma_n^2)$.
I want to write the probability distribution of $p(\Theta | Z )$ , if the PSD of $z$ is $Z$.
I know that the PSD is exponentially distributed at each frequency point. So, in principle, $S$ is distributed exponentially with expectation $F$ and similarly, the noise is also exponentially distributed. I have used a probability that I write below, however, it gives me a big variance in my estimates of $\Theta$, although the bias is good enough.
$$ \log(p(\Theta | Z )) = -\sum_{\omega} \left( L \log(F + \sigma_n^2) + \frac{\sum_{l= 1}^{L} Z_l(\omega)}{F + \sigma_n^2} \right) $$
where $Z_l$ are just different realizations of $Z$. Am I doing something wrong?
