We study a basic joint communication and sensing setup from an information theoretic perspective. A transmitter sends an encoded message through a pair of noisy channels connected to a receiver and a sensor. The receiver is interested in decoding the message from its noisy observation. The sensor has access to the message as side information, and instead is interested in estimating an unknown yet fixed binary state of its channel. This basic setup models scenarios where a wireless transceiver broadcasts an information bearing waveform, to be decoded by possibly more than one receiver, and then observes a modulated echo of the waveform from which it wishes to detect an element or object in the environment. We consider binary symmetric and Gaussian settings with multiplicative binary states, and establish the fundamental trade-off between reliable message communication and efficient state detection in these settings. This trade-off is captured by the message communication rate against the state detection error exponent. The results give insights into the benefits of carrying out communication and sensing jointly.
