MLE parameter estimation -- confusion regarding some terms in the pdf of complex normal r.v (Part 2)
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.