Capacity of a Relay Acoustic Channel

A relay acoustic link consisting of N hops spanning a fixed distance d is considered. The capacity of the relay link is found to increase with the number of hops as N^gamma, where gamma is a positive constant less than 1. At the same time, the power required to span the link decreases, but because each additional relay introduces a fixed cost, the number of relays in a practical system is limited. Taking this fact into account, an overall cost is defined, based either on a power, or an energy per bit criterion. Minimization of the cost function provides an analytical solution for the optimal number of relays. The cost of relaying is also evident in the delay, as each relay introduces a delay proportional to the data packet length. However, the distance between relays becomes shorter as more relays are added, thus supporting transmission at a higher rate. Assuming transmission at a rate equal to the channel capacity, the overall link delay is found to behave as N^{1 -gamma}. Compared to a situation in which transmission rate is chosen irrespective of the number of relays (a linear increase of the delay with N), rate adjustment offers an improved delay performance.