Generalizing Capacity: New Definitions and Capacity Theorems for Composite Channels

Abstract

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.

Toward Massive, Ultrareliable, and Low-Latency Wireless Communication With Short Packets

“…(3). For example, for a single-antenna system, the outage probability as a function of the rate R is given by [20], [19], [17] P out (R) = P log 1 + |H| 2 ρ < R…”

Outage Probability Under Channel Distribution Uncertainty

“…While these theorems treat all channels in the class equally and build a code that performs well on any such channel, the corresponding capacity is typically limited by the worst channel in the class and may be low, even though most channels in class are good and the worst channel is realized with low probability, i.e., it is a conservative performance indicator. To avoid this problem, a concept of composite channel has been introduced [8], [20], where each channel in a class has associated probability measure so that bad low-probability channels do not penalize significantly the performance metric. The corresponding channel capacity theorems can be proved via the concept of information density [18], [20] or using the compound channel approach [3], [12].…”

“…To avoid this problem, a concept of composite channel has been introduced [8], [20], where each channel in a class has associated probability measure so that bad low-probability channels do not penalize significantly the performance metric. The corresponding channel capacity theorems can be proved via the concept of information density [18], [20] or using the compound channel approach [3], [12]. Another possibility to model the uncertainty of CSI is to assume that the transmitter knows only the channel distribution but not the channel itself.…”

“…4 With a slight modification in notations, this block-fading model can also be extended to the case where each codeword sees a finite number of channel realizations, e.g., as in [12] and [19], and our results will hold in that case as well. 5 It can be further shown that the outage probability is the best achievable average codeword error probability [18]- [20]. probability for a given and the minimization is over all possible distributions of the input satisfying the power constraint.…”

“…Operational meaning of the outage capacity/probability follows from the compound channel capacity theorems [6]- [8], [12], [19] (which guarantee an existence of a code that works on every channel in the no-outage set); see also [18] and [20] for a modern treatment using the concept of information density.…”