MLE formulation -- confusion regarding the terms in the equation (Part1)
Hi: I don't have time to carefully look at everything right now and I'm unfamiliar with circular complex random variables. But, if it's similar to regular normal rv's and the two variables are independent , then their sum has mean zero and their variance is $ 2 \sigma^2 + 2 \sigma^2_{v}$. So, the likelihood  can be written having mean zero and that variance.  But it's not clear to me what you're estimating. There aren't any coefficients in the model ( implied one's I guess somewhere or maybe you are doing bayesian ? ) so what is actually observed ? $x_{n}$ or
 $v_{n}$. That has to be known if you want to write the likelihood correctly.
If $x$ is viewed as fixed and known and not a random variable, then what you wrote is missing a factor of 2 in the factor before the exp. The $2 \pi$ has to be multiplied by the variance which is $2 \sigma^2_{v}$. Also, it's not clear to me how X can be viewed as fixed since you said in the question that it was an RV also. If it is an RV and also not observed, then that has to be accounted for in the likelihood.  That's all I can tell you and even what I'm saying may not be correct if complex means something totally different. Good luck.