Equivalence between noise covariance of real and complex models of received signal
Even though I doubt your noise model, here's how you can do it:

Given the complex noise vector $\overline w$, with the covariance matrix $E[\overline w\overline w^H]=\sigma^2\overline{G}$. 

Let's separate $\overline w$ into its real and imaginary part (I use a different notation, because I think this is faster to write and read): $\overline w=w_R+jw_I$, where $w_R, w_I$ denote the real and imaginary parts of $\overline w$. Insert this into the covariance definition:

$$\begin{align}E[\overline w\overline w^H]&amp;=E[(w_R+jw_I)(w_R^T-jw_I^T)]\\&amp;=E[w_Rw_R^T+w_Iw_I^T]+2jE[w_Rw_I^T]=\overline{G}\end{align}$$

Let's compare this to the expression we get, when explicitely separating the real and imaginary parts:
$$\begin{align}E[ww^T]&amp;=\begin{pmatrix}E[w_Rw_R^T] &amp; E[w_Rw_I^T]\\E[w_Iw_R^T] &amp; E[w_Iw_I^T]\end{pmatrix}=X\end{align}.$$

Hence, in order to define $X$ (which is the matrix you are interested in), we need to know $E[w_Rw_R^T], E[w_Rw_I^T]=E[w_Iw_R^T], E[w_Iw_I^T]$.

Let's do the comparison to $E[\overline w \overline w^H]$, which we have calculated before. We know that $G_I=0$ (i.e. the imaginary part is zero). Hence, $E[w_Rw_I^T]=0$, i.e. real and imaginary part are uncorrelated. What we dont know yet are $E[w_Rw_R^T]$ and $E[w_Iw_I^T]$. We only know that $E[w_Rw_R^T]+E[w_Iw_I^T]=\overline G$. 

How to solve it? There are two possibilities:

1) (The more common assumption) We assume that real and imaginary parts of the noise have equal variance. This is often reasonble, e.g. when the noise is baseband noise and was obtained from downconversion a passband signal. Then, we have $E[w_Rw_R^T]=E[w_Iw_I^T]=\frac{1}{2}\overline{G}$.

2) Real and imaginary parts are not i.i.d. Then, the noise is not proper and the pseudocorrelation $T$ of the noise does not vanish. It is defined by $T=E[\overline w \overline w^T]$ and we calculate it to be (assuming $E[w_Iw_R^T]=0$)
$$T=E[\overline w \overline w^T]=E[w_Rw_R^T]-E[w_Iw_I^T]$$
In order to accurately describe a non-proper noise, the value of $T$ must be known. Then, we have two unknowns ($E[w_Rw_R^T]$ and $E[w_Iw_I^T]$) and two equations $$\begin{align}\overline{G}&amp;=E[w_Rw_R^T]+E[w_Iw_I^T]\\T&amp;=E[w_Rw_R^T]-E[w_Iw_I^T]\end{align}$$
from which you can get the unknown via $E[w_Rw_R^T]=\frac{1}{2}(\overline{G}+T)$ and $E[w_Iw_I^T]=\frac{1}{2}(\overline G-T)$.

Given this derivation, I still doubt your noise model. To me it does not make any sense, that in the equation $y=\overline G \overline a+\overline{w}$, the noise should have variance $\sigma^2\overline{G}$ due to ISI. Can you ensure that $\overline G$ is Hermitian?