I want to compute the expectation of a generalized Rayleigh quotient, i.e.,
$$\mathbb{E}_{\mathbf{x}} \bigg[ \frac{\mathbf{x}^{\mathrm{H}} \mathbf{A} \mathbf{x}}{\mathbf{x}^{\mathrm{H}} \mathbf{B} \mathbf{x}} \bigg]$$
where $\mathbf{x} \in \mathcal{CN} (\mathbf{0}, \mathbf{I})$ and $\mathbf{A}, \mathbf{B}$ are hermitian, positive definite matrices (they are known, thus I know the eigenvalue decomposition for each of them).
- Tentative solution:
I can write $\tilde{\mathbf{x}} = \mathbf{B}^{\frac{1}{2}} \mathbf{x}$, i.e., $\tilde{\mathbf{x}} \in \mathcal{CN} (\mathbf{0}, \mathbf{B})$, and obtain
$$\mathbb{E}_{\mathbf{x}} \bigg[ \frac{\mathbf{x}^{\mathrm{H}} \mathbf{A} \mathbf{x}}{\mathbf{x}^{\mathrm{H}} \mathbf{B} \mathbf{x}} \bigg] = \mathbb{E}_{\tilde{\mathbf{x}}} \bigg[ \frac{\tilde{\mathbf{x}}^{\mathrm{H}} (\mathbf{B}^{- \frac{1}{2}} \mathbf{A} \mathbf{B}^{- \frac{1}{2}}) \tilde{\mathbf{x}}}{\tilde{\mathbf{x}}^{\mathrm{H}} \tilde{\mathbf{x}}} \bigg] = \mathbb{E}_{\tilde{\mathbf{x}}} \bigg[ \frac{\tilde{\mathbf{x}}^{\mathrm{H}}}{\|\tilde{\mathbf{x}}\|} (\mathbf{B}^{- \frac{1}{2}} \mathbf{A} \mathbf{B}^{- \frac{1}{2}}) \frac{\tilde{\mathbf{x}}}{\|\tilde{\mathbf{x}}\|} \bigg].$$
Now I no longer have a generalized Rayleigh quotient with non-normalized random vectors with covariance $\mathbf{I}$ but a quadratic form where the random vectors are normalized and have covariance matrix $\mathbf{B}$, which I'm not sure is any simpler.
Please refer also to my other question here: Expectation of ratio of quadratic products