Ergodicity of Markov channels

Abstract

A Markov channel is a discrete information channel that includes as special cases the finite state channels and finite state codes of information theory. Kieffer and Rahe proved that one-sided and two-sided Markov channels have the following property: If the input source to a Markov channel is asymptotically mean stationary (AMS), then so is the resulting input-output process and hence the ergodic theorem and the Shannon-McMillan-Breiman theorem hold for the input-output process. Kieffer and Rahe also provided a sufficient condition for any AMS ergodic source to yield an AMS ergodic input-output process. New conditions for a Markov channel to have this ergodicity property are presented and discussed here. Several relations are developed among various classes of channels, including weakly ergodic, indecomposable, and strongly mixing channels. Some connections between Markov channels and the theory of nonhomogeneous Markov chains are also discussed. G IVEN AN INFORMATION SOURCE (a discretetime random process) and a noisy channel (essentially a regular conditional probability measure describing a probability measure on output sequences given an input sequence), information about the source can be communicated to a receiver by first encoding the source sequence into a channel input sequence and decoding the channel output sequence into a reproduction sequence observed by the receiver. Assume that we have some measure of the quality of the reproduction sequence, that is, how well it approximates the original source sequence. The coding theorems of information theory quantify the theoretically optimum performance that can be achieved using the given source and given channel with any encoder and decoder within some constrained class, where "optimum" means that the system has the minimum possible average distortion. The design algorithms of information theory are methods for actually designing codes that work well, ideally not too badly in comparison with the theoretical optimum. The proofs of coding theorems rest primarily on the ergodic theorem and on the Shannon-McMillan-Breiman theorem. They also generally require that the appropriate sample averages converge to constants and Manuscript

Capacity Analysis of Discrete Energy Harvesting Channels

“…Due to the technical nature, however, the exposition of the underlying stationarity and ergodicity theory is deferred to the Appendices. Such a theory is largely based on the theory of Markov channels developed in [25], [26] and the ergodic theory of stationary Markov chains in [27].…”

“…To apply the Shannon-McMillan-Breiman theorem in our channel models, we need to derive the required stationarity and ergodicity conditions. For that purpose in the appendices of this paper we introduce the theory of stationarity and ergodicity for Markov channels, mostly established by [25], [26] (also see Gray's books [36], [41]). It turns out, however, that some results in Gray et al [26] are inaccurate and/or not properly proved; thus before making use of them we must fix these issues first.…”

“…For that purpose in the appendices of this paper we introduce the theory of stationarity and ergodicity for Markov channels, mostly established by [25], [26] (also see Gray's books [36], [41]). It turns out, however, that some results in Gray et al [26] are inaccurate and/or not properly proved; thus before making use of them we must fix these issues first. In addition, besides the existing theory we also want to develop some extended results tailored for our own purposes, especially for the application in the finite state channels arising from energy harvesting systems.…”

“…Throughout the appendices we follow the notations in [25], [26], which uses a convention different from our main text. The reason is, we want to make the notations consistent with the related literature to facilitate a coherent understanding of the material.…”

Tighter Bounds on the Capacity of Finite-State Channels Via Markov Set-Chains

“…It is worth noting that (1) is valid for any initial state ; moreover, (1) is also valid whether or not the transmitter and the receiver know the initial state. Define It was shown in [3] that Therefore, we have (4) for all . Note that (4) provides computable finite-letter upper and lower bounds on the channel capacity, and for indecomposable channels, the bounds are asymptotically tight as .…”