Let $\boldsymbol{H} \in \mathbb{C}^{m \times n}$ be a matrix and x be a zero-mean signal vector $x \in \mathbb{C}^{n \times 1}$ that has a covariance matrix $\boldsymbol{\Sigma}_{x x}$.
For a known H, I want to: obtain $\boldsymbol{\Sigma}_{x x}$ to maximize $\mathbb{E}_{x}\left\{\|\boldsymbol{H} \boldsymbol{x}\|_{2}^{2}\right\}$ subject to $\operatorname{trace}\left\{\Sigma_{xx}\right\}=\alpha>0$ Here is my attempt: $$ \begin{aligned} &\mathbb{E}\|H x\|_{2}^{2}\\ &\mathbb{E}\left(\operatorname{tr}\left({x}^{H} H^{H} \cdot {H} \cdot{x}\right)\right\}\\ &\mathbb{E}\left\{\operatorname{tr}\left\{H \cdot{x}\left(x^{H} H^{H}\right)\right]\right.\\ &\operatorname{tr}\left\{H \cdot\left\{\mathbb{E}(x x^{H})\right\} H^{H}\right\} \end{aligned} $$ $$tr \left\{H \cdot \boldsymbol{\Sigma}_{x x}\cdot H^{H}\right\} $$ Now I am stuck at this point and a I don't know how to proceed or make use of $\operatorname{trace}\left\{\Sigma_{xx}\right\}=\alpha>0$.
Any suggestions ??