Capacity and reliability function per fourth moment cost for WSSUS fading channels

This paper addresses the capacity of wide sense stationary uncorrelated scattering (WSSUS) fading channels. Associated with a given input signal we define a quantity called the "fourthegy" of the signal, relative to a given WUSSUS channel. The name is inspired by the fact that the measure is fourth order in the input signal amplitude. The fourthegy depends on the signal through its ambiguity function, and on the channel through a simple channel response function. The maximum possible mutual information for the channel per unit fourthegy is found. Roughly speaking, the fourthegy is a sum over time and frequency bins of the local signal energy squared. The fourthegy-to-energy ratio of direct-sequence spread spectrum signals is inversely proportional to the bandwidth. Therefore, for such signals, the capacity per unit energy (or the capacity per unit time for fixed power) tends to zero as the bandwidth increases. This does not happen to signals that are more bursty in time-frequency space, such as frequency-hopped signals or M-ary frequency shift keyed signals. A similar result was found by Gallager and Medard (see Proc. International Symposium on Information Theory'97 (ISIT), Ulm, Germany, p.471, 1997) for a less conventional channel model. Numerical evaluation of the bound shows it to be informative only for rather large bandwidths.