Stochastic Ordering of Fading Channels Through the Shannon Transform

A new stochastic order between two fading distributions is introduced. A fading channel dominates another in the ergodic capacity ordering sense, if the Shannon transform of the first is greater than that of the second at all values of average signal to noise ratio. It is shown that some parametric fading models such as the Nakagami-m, Rician, and Hoyt are distributions that are monotonic in their line of sight parameters with respect to the ergodic capacity order. Some operations under which the ergodic capacity order is preserved are also discussed. Through these properties of the ergodic capacity order, it is possible to compare under two different fading scenarios, the ergodic capacity of a composite system involving multiple fading links with coding/decoding capabilities only at the transmitter/receiver. Such comparisons can be made even in cases when a closed form expression for the ergodic capacity of the composite system is not analytically tractable. Applications to multiple access channels, and extensions to multiple-input multiple-output (MIMO) systems are also discussed.