I have a complex sequence $z(t)$ in time which I know to be a Gaussian process. I read that the complex Gaussian process is not only characterized by the covariance, but also the pseudo-covariance matrix. However, I do not have expression for either of this. What I know is that the power spectral density (PSD) of $z(t)$, let's call it $Z(f)$ is:
$$ Z(f) = \frac{A}{\sqrt{2 \pi \sigma_v^2}} \exp\left(-\frac{\left(\mu_v - f\right)^2}{2\sigma_v^2}\right) + P_N $$
where $\mu_v$ and $\sigma_v$ are parameters of the PSD and $P_N$ is a noise spectral density. I assume that the quantities $A$ and $P_N$ are known. So, the hyper-parameters of the model are $\Theta = [\mu_v, \sigma_v]$.
To obtain a Gaussian process like log-likelihood to optimize for the hyper-parameters, I always used the following expression:
$$ \log(p(z | \Theta)) = -\frac{1}{2} \left( z^H K^{-1}(\Theta) z + |K(\Theta)| + n \log{\left(2\pi\right)} \right) $$ The $K(\Theta)$ is the auto-covariance in 2D that only depends on the time difference between the samples of $z$ and can be obtained functionally as the inverse Fourier Transform of the PSD.
$$ K(t_p, t_q) = A \exp\left(-2\pi^2(t_p-t_q)^2\sigma_v^2\right) \exp(-j2\pi\mu_v(t_p-t_q)) $$
It is a complex function where $j = \sqrt{-1}$.
When I read about complex Gaussian processes, I see they use covariance and pseudo-covariance to formulate an even complicated likelihood function. Is this a special case. Is the likelihood I am using a correct one? Like is it a circular or a proper case of the Gaussian process? I fail to see it or derive it.
In addition, can I use these simple posteriors for this?
$$ \mathbf{m^{\text{post}}(\mathbf{t^{'}})} = K(\mathbf{t}, \mathbf{t}^{'})^H K^{-1}(\mathbf{t}, \mathbf{t})\mathbf{z} $$ $$ K^{\text{post}}(\mathbf{t}^{'}, \mathbf{t}^{'}) = K(\mathbf{t}^{'}, \mathbf{t^{'}}) - K(\mathbf{t}, \mathbf{t}^{'})^H K^{-1}(\mathbf{t}, \mathbf{t}) K(\mathbf{t}, \mathbf{t}^{'}) $$
Also, is there a way to find the covariance and pseudo-covariance from the PSD directly?