MLE parameter estimation -- confusion regarding some terms in the pdf of complex normal r.v (Part 2) This question is based on the application of the pdf which was an earlier question of mine asked here Confusion regarding pdf of circularly symmetric complex gaussian rv

If $v \sim CN(0,2\sigma^2_v)$ is a circularly complex Gaussian random variable which acts as the measurement noise in this model 
$$y_n = A + v_n \tag{1} $$ where $y$ is the observation and $A$ is a scalar unknown value which needs to be estimated. I am having a slight confusion whether there will be a 2 in the denominator of Eq(3) and Eq(4) with the $\exp(.)$ term. Based on the answer in the link, there should be no sqrt term with $\pi$ in the denominator, if $v \sim CN(0,2\sigma^2_v)$. If $v \sim N(0,\sigma^2_v)$ then there is a sqrt term.  

Can somebody please check if I have correctly written  out the log-likelihood? I think I am missing a 2 in the denominator of $\exp[.]$ term in Eq(3) but I am not quite sure.

Thank you for your time and help.

$$P_y(y_1,y_2,...,y_N) = \prod_{n=1}^N\frac{1}{2\pi \sigma^2_v} \exp \bigg(\frac{-{({y_n-A})}^H ({y_n-A})}{2\sigma^2_v} \bigg) \tag{2}$$

taking log
$$\ell   = -N\ln(2\pi\sigma^2_v) -  \frac{1}{\sigma^2_v} {\bigg[{[\sum_{n=1}^{N} {(y_n - A)}{(y_n - A)}^{\mathsf{H}}  ]}\bigg]}. \tag{3}$$
$$ = -N\ln(2\pi\sigma^2_v)- \frac{1}{2 \sigma^2_v}{\bigg[ \sum_{n=1}^{N}y_n y_n^\mathsf{H}
- 2 \sum_{n=1}^N y_n A\bigg]}  
  - \frac{1}{2 \sigma^2_v}{\bigg[  \sum_{n=1}^N   {AA}^\mathsf{H} ] \bigg]} \tag{4}$$
Hi: It's worded more clearly now in that you're estimating A and there's only one RV which makes more sense. Consider the first link I sent in the other message and go to where it says the "likelihood is". Your likelihood is quite similar except, since your variance is multiplied by 2, this causes 2 things to happen in terms of how would you change the likelihood in that document accordingly for your problem.

A) The $2 \pi \sigma^2$ that is getting raised to the $-\frac{n}{2}$ instead becomes  $4 \pi \sigma^2_{v}$

B) The denominator of the exp instead of  being $2 \sigma^2$  becomes $4 \sigma^2_{v}$. 

Make those 2 changes to that likelihood and you will have your expression. I get confused looking at your 2) because my memory of 1/2's and the squares ( in your case H's ) is blurry and it can be confusing. So better to rely on document where likelihood is correct so you can't go wrong. Keep in mind that this is under the assumption that complex RV's don't change much compared to real ones.