I am studying an optimization problem
\begin{equation} \mathbf{w}^* = \text{argmax} \sum_{d=1}^D \log \bigg( \frac{|\mathbf{f}_d^H\mathbf{w}|^2+c_1}{|\mathbf{f}_d^H\mathbf{w}|^2+c_2} \bigg)\\ \\ \text{subject to} \quad |\mathbf{h}_k^H\mathbf{w}|^2\;\ge c_3,\qquad k=1,\ldots,K\\ |\mathbf{w}_k^H\mathbf{w}|^2 \;=\;1 \end{equation}
where all the bold-faced letters are N×1 vectors and the c's are scalar constants.
I tried to use several approximations and manipulations to put the problem in the form of a SOCP or a SDP (since I think those are the most probable programs that the problem would simplify to) but I could not. I would appreciate any help or hint on how to get a solution.