Let $ {\bf a} \in \mathbb{C}^{N \times 1}$, where N is a positive integer. Denote the set of phasors ${\bf P}$ as ${\bf P} = \{1,\exp(j 2 \pi/M), \cdots, \exp(j 2 \pi (M-1)/M)\}$, where M is a positive integer. With the collumn vector ${\bf r} = [r_1,\cdots,r_N]^T$, find $$ {\bf r}^* = \arg \max_{{\bf r}: \; r_\ell \in {\bf P}} |{\bf r}^H {\bf a}| $$
The subscripts $( \; )^T$ and $(\;)^H$ denote the transpose and Hermitian transpose operations.
This problem can always be done numerically by a brute force search over all $M^N$ hypothesis, but is there any smart scheme that computes the solution in a smarter way than doing brute force search?
Thanks.