I have a following two function:
$a = \mathbb{E}_x\left[\left|{\bf f}(x)^H{\bf g} \right|\right]$ and $b = \left|\mathbb{E}_x\left[{\bf f}(x) \right]^H {\bf g} \right|$, where $H$ stands for conjugate transpose operation, $x$ is a uniformly distributed random variable. Other variables are such that ${\bf f}(x) \in \mathbb{C}^{N \times 1}$ (${\bf f}$ is a function of $x$ defined as ${\bf f} = \left[e^{j\pi\cos(x)}, e^{j2\pi\cos(x)}, \dots, e^{j N\pi\cos(x)} \right]^T$) and ${\bf g} \in \mathbb{C}^{N \times 1}$.
Is $a$ is equivalent to $b$ and if yes then can someone kindly suggest how to proceed to prove it?
For a centered uniform variable, by symmetry, the expectation is zero, while the expectation of the absolute value is positive.