  The study of the fundamental limits of covert communications, where a
transmitter Alice wants to send information to a desired recipient Bob without
detection of that transmission by an attentive and capable warden Willie, has
emerged recently as a topic of great research interest. Critical to these
analyses is a characterization of the detection problem that is presented to
Willie. Previous work has assumed that the channel distribution information
(CDI) is known to Alice, hence facilitating her characterization of Willie's
capabilities to detect the signal. However, in practice, Willie tends to be
passive and the environment heterogeneous, implying a lack of signaling
interchange between the transmitter and Willie makes it difficult if not
impossible for Alice to estimate the CDI exactly and provide covertness
guarantees. In this paper, we address this issue by developing covert
communication schemes for various assumptions on Alice's imperfect knowledge of
the CDI: 1) when the transmitter knows the channel distribution is within some
distance of a nominal channel distribution; 2) when only the mean and variance
of the channel distribution are available at Alice; 3) when Alice knows the
channel distribution is complex Gaussian but the variance is unknown. In each
case, we formulate new optimization problems to find the power allocations that
maximize covert rate subject to a covertness requirement under uncertain CDI.
Moreover, since Willie faces similar challenges as Alice in estimating the CDI,
we investigate two possible assumptions on the knowledge of the CDI at Willie:
1) CDI is known at Willie, 2) CDI is unknown at Willie. Numerical results are
presented to compare the proposed schemes from various aspects, in particular
the accuracy and efficiency of the proposed solutions for attaining desirable
covert system performance.
