Generalizing Capacity: New Definitions and Capacity Theorems for Composite Channels

Abstract

We consider three capacity definitions for composite channels with channel side information at the receiver. A composite channel consists of a collection of different channels with a distribution characterizing the probability that each channel is in operation. The Shannon capacity of a channel is the highest rate asymptotically achievable with arbitrarily small error probability. Under this definition, the transmission strategy used to achieve the capacity must achieve arbitrarily small error probability for all channels in the collection comprising the composite channel. The resulting capacity is dominated by the worst channel in its collection, no matter how unlikely that channel is. We, therefore, broaden the definition of capacity to allow for some outage. The capacity versus outage is the highest rate asymptotically achievable with a given probability of decoder-recognized outage. The expected capacity is the highest average rate asymptotically achievable with a single encoder and multiple decoders, where channel side information determines the channel in use. The expected capacity is a generalization of capacity versus outage since codes designed for capacity versus outage decode at one of two rates (rate zero when the channel is in outage and the target rate otherwise) while codes designed for expected capacity can decode at many rates. Expected capacity equals Shannon capacity for channels governed by a stationary ergodic random process but is typically greater for general channels. The capacity versus outage and expected capacity definitions relax the constraint that all transmitted information must be decoded at the receiver. We derive channel coding theorems for these capacity definitions through information density and provide numerical examples to highlight their connections and differences. We also discuss the implications of these alternative capacity definitions for end-to-end distortion, source-channel coding, and separation.

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