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

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}

• @Carlo Beenakker yes its error I will correct it, thanks Sir – hichem hb May 9 '20 at 18:30

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.