  This paper studies a robust design problem for far-field line-of-sight (LOS)
channels where phase errors are present. Compared with the commonly used
additive error model, the phase error model is more suitable for capturing the
uncertainty in an LOS channel, as the dominant source of uncertainty lies in
the phase. We consider a multiple-input single-output (MISO) multicast
scenario, in which our goal is to design a beamformer that minimizes the
transmit power while satisfying probabilistic signal-to-noise ratio (SNR)
constraints. The probabilistic constraints give rise to a new computational
challenge, as they involve random trigonometric forms. In this work, we propose
to first approximate the random trigonometric form by its second-order Taylor
expansion and then tackle the resulting random quadratic form using a
Bernstein-type inequality. The advantage of such an approach is that an
approximately optimal beamformer can be obtained using the standard
semidefinite relaxation technique. In the simulations, we first show that if a
non-robust design (i.e., one that does not take phase errors into account) is
used, then the whole system may collapse. We then show that our proposed method
is less conservative than the existing robust design based on Gaussian
approximation and thus requires a lower power budget.
