  In pursuance of the unused spectrum in higher frequencies, millimeter wave
(mmWave) bands have a pivotal role. However, the high path-loss and poor
scattering associated with mmWave communications highlight the necessity of
employing effective beamforming techniques. In order to efficiently search for
the beam to serve a user and to jointly serve multiple users it is often
required to use a composite beam which consists of multiple disjoint lobes. A
composite beam covers multiple desired angular coverage intervals (ACIs) and
ideally has maximum and uniform gain (smoothness) within each desired ACI,
negligible gain (leakage) outside the desired ACIs, and sharp edges. We propose
an algorithm for designing such ideal composite codebook by providing an
analytical closed-form solution with low computational complexity. There is a
fundamental trade-off between the gain, leakage and smoothness of the beams.
Our design allows to achieve different values in such trade-off based on
changing the design parameters. We highlight the shortcomings of the uniform
linear arrays (ULAs) in building arbitrary composite beams. Consequently, we
use a recently introduced twin-ULA (TULA) antenna structure to effectively
resolve these inefficiencies. Numerical results are used to validate the
theoretical findings.
