Let $\textbf{a}$ be in the random vector in the form of $$\textbf{a} = \begin{bmatrix} e^{-j k_1 cos(\theta)} & e^{-j k_2 cos(\theta)} & \cdots & e^{-j k_N cos(\theta)} \end{bmatrix}$$ where $k$s are all constants and $\theta$ is a random variable with known distribution.
How do I go on evaluating $$E_\theta[\textbf{a}^H X \textbf{a}]$$ where $X$ is a complex matrix independent of $\theta$.
\begin{eqnarray} \mathbb{E}_\theta[{\bf a}^H~ {\bf X}~{\bf a}] &=& \mathbb{E}_\theta \left[\sum_{mn}a_m^*X_{mn}a_{mn} \right]\\ &=&\sum_{mn}X_{mn}\mathbb{E}_{\theta}[e^{jk_m\cos\theta} e^{-jk_n\cos\theta}] \\ &=& \sum_{mn}X_{mn} \mathbb{E}_\theta[e^{jk_{mn}\cos\theta}] ~~~\mbox{with}~~~k_{km} = k_m - k_n \end{eqnarray}
The problem is reduced to calculating this last expected value, but that depends on the the PDF of $\theta$
$$ \mathbb{E}_\theta[e^{jk_{mn}\cos\theta}] = \int {\rm d}\theta~ e^{jk_{mn}\cos\theta}f(\theta) $$