  In this paper, we study optimal waveform design to maximize mutual
information (MI) for a joint communication and (radio) sensing (JCAS, a.k.a.,
radar-communication) multi-input multi-output (MIMO) downlink system. We
consider a typical packet-based signal structure which includes training and
data symbols. We first derive the conditional MI for both sensing and
communication under correlated channels by considering the training overhead
and channel estimation error (CEE). Then, we derive a lower bound for the
channel estimation error and optimize the power allocation between the training
and data symbols to minimize the CEE. Based on the optimal power allocation, we
provide optimal waveform design methods for three scenarios, including
maximizing MI for communication only and for sensing only, and maximizing a
weighted sum MI for both communication and sensing. We also present extensive
simulation results that provide insights on waveform design and validate the
effectiveness of the proposed designs.
