Having One's Cake and Eating it Too: The Problem of Joint Communication and Sensing

Traditionally, channel estimation has been undertaken only in service of better data communication. However, a number of problem frameworks (sonar, cognitive radio, digital watermarking) require the reconstruction of a transmitted message as well as estimating properties of the channel over which the message was transmitted. We abstract these scenarios to one wherein a source sends a message to the destination and the destination endeavors to both decode the message and estimate the channel to some fidelity. Three key cases of channel knowledge at the transmitter are considered: the transmitter is oblivious to the channel; the transmitter has strictly causal channel state information and the transmitter has causal channel state information. We establish that the distortion constraint at the destination is equivalent to an additional cost constraint on the input source distribution. We define a new capacity-distortion function which characterizes the fundamental tradeoff between transmission rate and state estimation distortion. We compute the capacity-distortion function for the three cases of channel knowledge at the transmitter. The results provide some interesting insights as to how practical encoding should be designed in order to achieve our goals; in particular, separated training and data signaling is suboptimal.