For Given a $N×M$ random complex gaussian matrix $X$ where $M=XX^H$, let $\lambda_1>\lambda_2>....>\lambda_N$ be the ordered eigenvalues of $M$ my objective is to get an estimation of \begin{align} f = {(\sum\limits_{i = 1}^{Min(N,M)} {\frac{1}{{\sqrt {{\lambda _i}} }}} )^2} \end{align}

Let me take $N< M$. For $N\gg 1$ the fluctuations in the sum $\sum_i \lambda_i^{-1/2}$ are smaller than the expectation value by a factor $1/N$, so we may estimate $$\mathbb{E}[f]\approx\left(\mathbb{E}\left[\sum_i\lambda_i^{-1/2}\right]\right)^2=N^2\left(\int \rho_{\rm MP}(\lambda)\lambda^{-1/2}d\lambda\right)^2,$$ with $\rho_{\rm MP}(\lambda)$ the Marchenko-Pastur distribution.

For $N=M$ the expectation value of $f$ is divergent.