  We study the information-theoretic limits of joint communication and sensing
when the sensing task is modeled as the estimation of a discrete channel state
fixed during the transmission of an entire codeword. This setting captures
scenarios in which the time scale over which sensing happens is significantly
slower than the time scale over which symbol transmission occurs. The tradeoff
between communication and sensing then takes the form of a tradeoff region
between the rate of reliable communication and the state detection-error
exponent. We investigate such tradeoffs for both mono-static and bi-static
scenarios, in which the sensing task is performed at the transmitter or
receiver, respectively. In the mono-static case, we develop an exact
characterization of the tradeoff in open-loop, when the sensing is not used to
assist the communication. We also show the strict improvement brought by a
closed-loop operation, in which the sensing informs the communication. In the
bi-static case, we develop an achievable tradeoff region that highlights the
fundamentally different nature of the bi-static scenario. Specifically, the
rate of communication plays a key role in the characterization of the tradeoff
and we show how joint strategies, which simultaneously estimate message and
state, outperform successive strategies, which only estimate the state after
decoding the transmitted message.
