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

Question

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 $$\ell = -N\ln(2\pi\sigma^2_v) - \frac{1}{\sigma^2_v} {\bigg[{[\sum_{n=1}^{N} {(y_n - A)}{(y_n - A)}^{\mathsf{H}} ]}\bigg]}. \tag{3}$$ $$ = -N\ln(2\pi\sigma^2_v)- \frac{1}{2 \sigma^2_v}{\bigg[ \sum_{n=1}^{N}y_n y_n^\mathsf{H} - 2 \sum_{n=1}^N y_n A\bigg]} - \frac{1}{2 \sigma^2_v}{\bigg[ \sum_{n=1}^N {AA}^\mathsf{H} ] \bigg]} \tag{4}$$

Answer 1

I looked this up on Wikipedia: A complex Gaussian random variable $V = \mathfrak{Re}(V)+j\mathfrak{Im}(V)$ is said to be zero mean circularly symmetric $\mathcal{CN}(0,\Gamma)$ if the random vector $[\mathfrak{Re}(V),\mathfrak{Im}(V)]$ is a Gaussian random vector with mean $[0,0]$ and covariance matrix $ \frac{1}{2}\begin{bmatrix} \mathfrak{Re}(\Gamma) & -\mathfrak{Im}(\Gamma) \\ \mathfrak{Im}(\Gamma) & \mathfrak{Re}(\Gamma) \end{bmatrix} = \frac{1}{2}\begin{bmatrix} 2\sigma_v^2 & 0 \\ 0 & 2\sigma_v^2 \end{bmatrix} = \begin{bmatrix} \sigma_v^2 & 0 \\ 0 & \sigma_v^2 \end{bmatrix}. $

For your question, you need to know how to write the density function of each of the $Y_i$'s. Since $Y_i = A + V_i$, the random vector $[\mathfrak{Re}(Y_i),\mathfrak{Im}(Y_i)]$ is a 2D Gaussian vector distributed according to $\mathcal{N}\left(\begin{bmatrix} \mathfrak{Re}(A) \\ \mathfrak{Im}(A) \end{bmatrix},\begin{bmatrix} \sigma_v^2 & 0 \\ 0 & \sigma_v^2 \end{bmatrix}\right)$ i.e. the density function can be written as \begin{eqnarray} \frac{1}{\sigma_v\sqrt{2\pi}}\exp\left(-{\frac{(w-\mathfrak{Re}(A))^2}{2\sigma_v^2}}\right) &\cdot& \frac{1}{\sigma_v\sqrt{2\pi}}\exp\left(-{\frac{(x-\mathfrak{Im}(A))^2}{2\sigma_v^2}}\right)\\& = & \frac{1}{2\pi\sigma_v^2}\exp\left(-\frac{(w-\mathfrak{Re}(A))^2+(x-\mathfrak{Im}(A))^2}{2\sigma_v^2}\right) \\ &=& \frac{1}{2\pi\sigma_v^2} \exp\left(-\frac{(y_i-A)\overline{(y_i-A)}}{2\sigma_v^2}\right) \end{eqnarray} where $y_i=w+jx$ is a "placeholder" for the complex random variable $Y_i$ and the "overbar" denotes complex conjugate i.e. $\bar y=w-jx.$

Finally, we are ready to write the density function of the complex random vector $[Y_1,\ldots, Y_N]$. We note that due to the circular symmetry of each of the components, it is same as the density of the real valued Gaussian random vector $[\mathfrak{Re}(Y_1),\mathfrak{Im}(Y_1),\ldots, \mathfrak{Re}(Y_N),\mathfrak{Im}(Y_N)].$

The likelihood function can now be written as $$ p(Y_1,\ldots,Y_N)=\prod_{i=1}^N \frac{1}{2\pi\sigma_v^2} \exp\left(-\frac{(y_i-A)\overline{(y_i-A)}}{2\sigma_v^2}\right) $$ and the negative-log-likelihood becomes \begin{eqnarray} -\log p(Y_1,\ldots,Y_N)&=& N\log(2\pi\sigma_v^2) + \frac{1}{2\sigma_v^2} \sum_{i=1}^N (y_i-A)\overline{(y_i-A)} \\ &=& N\log(2\pi\sigma_v^2) + \frac{1}{2\sigma_v^2} \sum_{i=1}^N (y_i-A)\overline{(y_i-A)} \end{eqnarray}

Answer 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.