Asynchronous Capacity per Unit Cost

The capacity per unit cost, or equivalently minimum cost to transmit one bit, is a well-studied quantity. It has been studied under the assumption of full synchrony between the transmitter and the receiver. In many applications, such as sensor networks, transmissions are very bursty, with small amounts of bits arriving infrequently at random times. In such scenarios, the cost of acquiring synchronization is significant and one is interested in the fundamental limits on communication without assuming a priori synchronization. In this paper, we show that the minimum cost to transmit B bits of information asynchronously is (B + \bar{H})k_sync, where k_sync is the synchronous minimum cost per bit and \bar{H} is a measure of timing uncertainty equal to the entropy for most reasonable arrival time distributions. This result holds when the transmitter can stay idle at no cost and is a particular case of a general result which holds for arbitrary cost functions.