I am reading a paper and got stuck on a formula stated without elaboration.
The formula stated that for a model:
$$X=\sqrt \rho SH+W$$
with $S$ is random complex $T\times M$.
$H$ is random complex $M\times N$ having elements i.i.d and circular symmetric normal $CN(0,1)$.
and $W$ is random complex $T\times N$ having elements i.i.d and $CN(0,1)$.
The paper states that given $S$, $X$ is conditional complex normal with PDF: $$f(X|S)=(1/\pi)^{NT} \times \frac{\exp{[tr(X^+(\rho SS^+ + I)^{-1}X)]}}{(\det(\rho X X^+ + I))^N}$$
with $tr$ means trace of matrix, and $^+$ is conjungate transpose.
I am trying to prove this formula without success.
I use the general matrix normal distribution that can be found here https://en.wikipedia.org/wiki/Matrix_normal_distribution for a zero mean $X$
Given $S$, with notation $\Bbb{E}$ for expectance, $\Re$ is real(.), $\Im$ is imag(.). If $X=A+jB$,
set $Y=\begin{bmatrix} A \\ B \\ \end{bmatrix} $
$\Gamma = \Bbb{E}[XX^+|S]=\rho SS^+ + I$
$C = \Bbb{E}[XX^T] = 0$ (circularly symmetric)
$\Gamma_T = \Bbb{E}[X^+X]=\rho\Bbb{E}[H^+ SS^+H] + I$
$C_T = \Bbb{E}[X^TX] = 0$ (circularly symmetric)
Because $C=0$ and $C_T=0$, we have
$\Bbb{E}[AA^T] = 0.5 \times \Re(\Gamma)$
$\Bbb{E}[AB^T] = 0.5 \times \Im(-\Gamma)$
$\Bbb{E}[BA^T] = 0.5 \times \Im(\Gamma)$
$\Bbb{E}[BB^T] = 0.5 \times \Re(\Gamma)$
$\Bbb{E}[A^TA] = \Bbb{E}[B^TB] = 0.5 \times \Re(\Gamma_T)$
Then
$U \times tr(V)=\Bbb{E}[YY^T|S]=0.5 \times \begin{bmatrix} \Re(\Gamma) & -\Im(\Gamma) \\ \Im(\Gamma) & \Re(\Gamma) \\ \end{bmatrix} $
$V \times tr(U)=\Bbb{E}[Y^TY|S]=\Re(\Gamma_T)$
Then $U^{-1} = 2\times tr(V) \begin{bmatrix} \Re(\Gamma^{-1}) & -\Im(\Gamma^{-1}) \\ \Im(\Gamma^{-1}) & \Re(\Gamma^{-1}) \\ \end{bmatrix}$
With simple manipulation
$Y^TU^{-1}Y = \begin{bmatrix} A^T & B^T \end{bmatrix} \begin{bmatrix} \Re(\Gamma^{-1}) & -\Im(\Gamma^{-1}) \\ \Im(\Gamma^{-1}) & \Re(\Gamma^{-1}) \\ \end{bmatrix} \begin{bmatrix} A^T \\ B^T\\ \end{bmatrix}$
$X^+\Gamma^{-1}X = (A^T-jB^T)(\Re(\Gamma^{-1})+j\Im(\Gamma^{-1}))(A+jB)=(2tr(V))^{-1} \times Y^TU^{-1}Y + j(..)$
Consider only the case that $X^+\Gamma^{-1}X$ is real (a classic assumption), $2tr(V)\times X^+\Gamma^{-1}X = Y^TU^{-1}Y$
The inner part of exp in denominator of the PDF becomes
$-0.5tr(V^{-1}Y^TU^{-1}Y)=-tr(V)tr(V^{-1}X^+\Gamma^{-1}X)$
In comparing with the stated formula
$-tr(X^+\Gamma^{-1}X)$
I got stuck here. It seems that the V matrix part vanishes in the stated formula but I don't know how. I tried using the zero pseudo-covariance characteristic of circular symmetric variable but no success. Can anyone point me in the right direction ?
Thanks a lot.