  The ability of Multiple-Input Multiple-Output (MIMO) radar systems to adapt
waveforms across antennas allows flexibility in the transmit beampattern
design. In cognitive radar, a popular cost function is to minimize the
deviation against an idealized beampattern (which is arrived at with knowledge
of the environment). The optimization of the transmit beampattern becomes
particularly challenging in the presence of practical constraints on the
transmit waveform. One of the hardest of such constraints is the non-convex
constant modulus constraint, which has been the subject of much recent work. In
a departure from most existing approaches, we develop a solution that involves
direct optimization over the non-convex complex circle manifold. That is, we
derive a new projection, descent, and retraction (PDR) update strategy that
allows for monotonic cost function improvement while maintaining feasibility
over the complex circle manifold (constant modulus set). For quadratic cost
functions (as is the case with beampattern deviation), we provide analytical
guarantees of monotonic cost function improvement along with proof of
convergence to a local minima. We evaluate the proposed PDR algorithm against
other candidate MIMO beampattern design methods and show that PDR can
outperform competing wideband beampattern design methods while being
computationally less expensive. Finally, orthogonality across antennas is
incorporated in the PDR framework by adding a penalty term to the beampattern
cost function. Enabled by orthogonal waveforms, robustness to target direction
mismatch is also demonstrated.
