  This study introduces a new multi-antenna array synthesizer based on Slepian
functions. The synthesizer concentrates beamforming (BF) gain within a spatial
region (i.e., an angular sector), optimizing Shannon capacity of the targeted
region, which is suitable for codebook-based analog BF. Starting with the mean
capacity formula incorporating the effect of BF, Jensen inequality was used to
set upper and lower bounds of the mean capacity. Then, a novel method was
introduced by combining the two bounds into a new approximation of the mean
capacity that outperform both bounds. Finally, the approximation was formulated
to a solvable Slepian optimization problem that yielded the weights of the
synthesizer. The properties of the synthesizer were listed, including a
discussion on how it behaves by changing the width of the targeted region. The
steering method was derived, and simulation results were presented.
