# Expectation of the trace of inverse of a Gaussian random matrix

Given a $$N×M$$ random complex gaussian matrix $$X$$ and $$N×K$$ random complex gaussian matrix $$Y$$ I'm interested in approximating the expectation expressed as: \begin{align} E[trace({(aX{X^H} + I)^{ - 1}}Y{Y^H})] \end{align} a Is a positive given variable. I know that $$XX^H$$ and $$YY^H$$ have Wishard distribution, However. I can not approximate the problem value. Thanks!

For context, this problem relates to the quality of MIMO communication link

I assume the matrices $$X$$ and $$Y$$ are independent. Since the trace commutes with the expectation value, and since the expectation value of the product of independent random variables is the product of expectation values, we have $$F(a)=\mathbb{E}\bigl[{\rm tr}\,\bigl({(aX{X^H} + I)^{ - 1}}Y{Y^H}\bigr)\bigr]={\rm tr}\bigl(\,\mathbb{E}[(aX{X^H} + I)^{ - 1}]\mathbb{E}[Y{Y^H}]\bigr).$$ The second factor is simply $$K$$ times the unit matrix, so $$F(a)=K\,{\rm tr}\,\mathbb{E}[(aX{X^H} + I)^{ - 1}].$$ We can now again exchange trace and expectation value, to rewrite this as an integral over the eigenvalues $$\mu_k$$ of $$XX^H$$, with density $$\rho(\mu)$$, $$F(a)=K\int \rho(\mu)(a\mu+1)^{-1}\,d\mu.$$ The density $$\rho(\mu)$$ is known, for large matrix size it is the Marcenko-Pastur distribution.
For the Marcenko-Pastur distribution, so for $$M\geq N\gg 1$$, I find $$F(a)=\frac{K}{2a} \left(\sqrt{a^2 (M-N)^2+2 a (M+N)+1}+a (N-M)-1\right).$$
• I am using the Wishart distribution: firstly when I use that $\mathbb{E}[YY^H]=K$ times the identity, and secondly the Marcenko-Pastur distribution is derived from the Wishart distribution. – Carlo Beenakker May 9 '20 at 6:49