  Narrow beams are key to wireless communications in millimeter wave frequency
bands. Beam alignment (BA) allows the base station (BS) to adjust the direction
and width of the beam used for communication. During BA, the BS transmits a
number of scanning beams covering different angular regions. The goal is to
minimize the expected width of the uncertainty region (UR) that includes the
angle of departure of the user. Conventionally, in interactive BA, it is
assumed that the feedback corresponding to each scanning packet is received
prior to transmission of the next one. However, in practice, the feedback delay
could be larger because of propagation or system constraints. This paper
investigates BA strategies that operate under arbitrary fixed feedback delays.
This problem is analyzed through a source coding prospective where the feedback
sequences are viewed as source codewords. It is shown that these codewords form
a codebook with a particular characteristic which is used to define a new class
of codes called d-unimodal codes. By analyzing the properties of these codes, a
lower bound on the minimum achievable expected beamwidth is provided. The
results reveal potential performance improvements in terms of the BA duration
it takes to achieve a fixed expected width of the UR over the state-of-the-art
BA methods which do not consider the effect of delay.
