Approximating expectation of the trace of inverse of a Gaussian random matrix combination

Question

In order to characterize the performances of MIMO systems that depend directly on the distribution of the eigenvalues of random Hermitian matrix so I would like to feature the quality of some particular case that is defined as. For a given random complex gaussian matrix A $(N, M)$ and $B(N, K)$ Gaussian matrix and constant parameter $\alpha$, where ${\tilde A}$ indicate the Moore–Penrose inverse of $A$. The problem consists to find the expectations defined as:

\begin{align} E[\operatorname{trace}((aI + \frac{{\tilde A\,B{B^H}{{\tilde A}^H}}}{{\operatorname{trace}(\tilde A\,B{B^H}{{\tilde A}^H})}})({B^H}B))]\end{align}

Answer 1

A few remarks: $AA^H$ is invertible, the MP inverse $\tilde{A}=A^H(AA^H)^{-1}$; so the trace in the denominator is $${\rm tr}\,\tilde{A}BB^H\tilde{A}^H={\rm tr}\,A^H(AA^H)^{-1}BB^H(AA^H)^{-1}A={\rm tr}\,BB^H(AA^H)^{-1}.$$ The expectation value of this trace follows from your previous question and answer.

Now the full expectation value in this new question is unlikely to have a closed-form answer for any $N,M,K$, but for large values you can decouple the averages in numerator and denominator, because the trace is "self-averaging", meaning that you can replace it by its expectation value.

If this large $N,M,K$ limit is of interest, I may try to develop this a bit further.