In this problem, $\beta_u$, $w_{u,c}$ (a vector of complex elements), $x_u$ are optimization variables.
Now,
$||2\sqrt{\frac{\pi_u}{2}}\beta_u; h_{u,c}^{\rm H}w_{u,c}-\frac{1}{2\pi_u}x_u-1||_2\le h_{u,c}^{\rm H}w_{u,c}-\frac{1}{2\pi_u}x_u-1$
is a valid convex constraint, hyperbolic constraint. Here $h_{u,c}^{\rm H}w_{u,c}$ came for the original expression, $\sqrt{|h_{u,c}^{\rm H}w_{u,c}|^2}$
Now, in my problem, instead of $\sqrt{|h_{u,c}^{\rm H}w_{u,c}|^2}$, I have $\sqrt{\sum_{c=1}^{C}|h_{u,c}^{\rm H}w_{u,c}|^2}$.
$\textbf{How can I put in the form of hyperbolic constraint as above?}$
$||2\sqrt{\frac{\pi_u}{2}}\beta_u; \sqrt{\sum_{c=1}^{C}|h_{u,c}^{\rm H}w_{u,c}|^2}-\frac{1}{2\pi_u}x_u-1||_2\le \sqrt{\sum_{c=1}^{C}|h_{u,c}^{\rm H}w_{u,c}|^2}-\frac{1}{2\pi_u}x_u-1$