Markov Coding Research Articles

A symmetric Markov coding and the ergodic theorem for actions of Fuchsian Groups

A symmetric Markov coding and the ergodic theorem for actions of Fuchsian Groups

Dividing the grids of compressed sensing for channel estimation and investigating Markov codes

Dividing the grids of compressed sensing for channel estimation and investigating Markov codes

Achievable Rates for Gaussian Degraded Relay Channels With Non-Vanishing Error Probabilities

This paper revisits the Gaussian degraded relay channel, where the link that carries information from the source to the destination is a physically degraded version of the link that carries information from the source to the relay. The source and the relay are subject to expected power constraints. The $\varepsilon $ -capacity of the channel is characterized and it is strictly larger than the capacity for any $\varepsilon >0$ , which implies that the channel does not possess the strong converse property. The proof of the achievability part is based on several key ideas: block Markov coding, which is used in the classical decode-forward strategy, power control for Gaussian channels under expected power constraints, and a careful scaling between the block size and the total number of block uses. The converse part is proved by first establishing two non-asymptotic lower bounds on the error probability, which are derived from the type-II errors of some binary hypothesis tests. Subsequently, each lower bound is simplified by conditioning on an event related to the power of some linear combination of the codewords transmitted by the source and the relay. Lower and upper bounds on the second-order term of the optimal coding rate are also obtained.

Link Regimes Analysis for Partial Decode-Forward Two-Way Relay Transmission

We propose a composite decode-forward (DF) scheme for the two-way relay channel in the full-duplex mode by combining coherent, independent, and partial relaying strategies. The relay partially decodes each user’s information in each block and forwards this information coherently with the source user to the destination user in the next block as in block Markov coding. In addition, the relay independently broadcasts a binning index of both users’ decoded information parts in the next block as in independent network coding. Each technique has a different impact on the relay power usage and the rate region. We further consider the independent and partial DF scheme for its more practical channel state information requirements, and derive in closed-form link regimes when this scheme achieves a strictly larger rate region than just time sharing between its constituent techniques, direct transmission, and independent DF relaying, and when it reduces to a simpler scheme. The analytical approach is based on maximizing the weighted composite DF sum rate and comparing with the outermost time-sharing line connecting corner points of rate regions of the constituent techniques. Numerical results demonstrate significant rate gains by performing link adaptation of the composite scheme based on the identified link regimes.

Link-State Optimized Decode-Forward Transmission for Two-Way Relaying

We establish a novel link-state regime result for a composite decode-forward (DF) two-way relaying scheme with a direct link. During transmission, our scheme combines block Markov coding and an independent coding scheme that resembles network coding at the relay. A developed novel approach optimizes the composite technique by analyzing the dual variable space to identify link-state regimes in which a particular combination of transmission techniques is optimal. Our results expose an interesting trend: when the user-to-relay link is marginally stronger than the direct link, independent coding is optimal and the relay can conserve power. However, for larger user-to-relay link gains, the relay must use full power and supplement independent coding with block Markov coding to achieve the largest rate region. For Rayleigh fading links, we demonstrate that relay power savings are achievable in most node configurations. The link-state regimes are further applied to perform link adaptation in fading to illustrate significant data rate gains over direct transmission even under a more practical, long-term link state information. These link-state regime results are useful for the application of two-way DF relaying in practice.

Link Regime and Power Savings of Decode-Forward Relaying in Fading Channels

In this paper, we re-examine the relay channel under the decode-forward (DF) strategy. Contrary to the established belief that block Markov coding is always the rate-optimal DF strategy, under certain channel conditions (a link regime), independent signaling between the source and relay achieves the same transmission rate without requiring coherent channel phase information. Further, this independent signaling regime allows the relay to conserve power. As such, we design a composite DF relaying strategy that achieves the same rate as block Markov DF but with less required relay power. The finding is attractive from the link adaptation perspective to adapt relay coding and relay power according to the link state. We examine this link adaptation in fading under both perfect channel state information (CSI) and practical CSI, in which nodes have perfect receive and long-term transmit CSI, and derive the corresponding relay power savings in both cases. We also derive the outage probability of the composite relaying scheme which adapts the signaling to the link regime. Through simulation, we expose a novel trade-off for relay placement showing that the relay conserves the most power when closer to the destination but achieves the most rate gain when closer to the source.

Time-Asynchronous Gaussian Multiple Access Relay Channel With Correlated Sources

We study the transmission of a set of correlated sources ${(U_{1},\ldots ,U_{K})}$ over a Gaussian multiple access relay channel with time asynchronism between the encoders. We assume that the maximum possible offset $ { \mathsf {d_{max}}(n)}$ between the transmitters grows without bound as the block length ${n \rightarrow \infty }$ , while the relative ratio $ {{ \mathsf {d_{max}}(n) / n}}$ of the maximum possible offset to the block length asymptotically vanishes. For such a joint source-channel coding problem and under specific gain conditions, we derive necessary and sufficient conditions for reliable communications and show that separate source and channel coding achieves optimal performance. In particular, we first derive a general outer bound on the source entropy content for all channel gains as our main result. Then, using Slepian–Wolf source coding combined with the channel coding scheme on top of block Markov coding, we show that the thus achieved inner bound matches the outer bound. As a corollary, we also address the problem of sending a pair of correlated sources over a two-user interference channel in the same context.

A Note on the Broadcast Channel With Stale State Information at the Transmitter

This paper shows that the Maddah-Ali–Tse (MAT) scheme, which achieves the symmetric capacity of two example broadcast channels with strictly causal state information at the transmitter, is a simple special case of the Shayevitz–Wigger (SW) scheme for the broadcast channel with generalized feedback, which involves block Markov coding, Gray–Wyner compression, superposition coding, and Marton coding. Focusing on the class of symmetric broadcast channels with state, we derive an expression for the maximum achievable symmetric rate using the SW scheme. We show that the MAT results for the two-receiver case can be recovered by evaluating this expression for the special case in which superposition coding and Marton coding are not used. We then introduce a new broadcast channel example that shares many features of the MAT examples. We show that another special case of our maximum symmetric rate expression in which superposition coding is also used attains a higher symmetric rate than the MAT scheme. The symmetric capacity of this new example is not known, however.

Simple C*-algebras arising from certain Markov codes

Simple C*-algebras arising from certain Markov codes

On the Multiple-Access Channel With Common Rate-Limited Feedback

This paper studies the multiple-access channel (MAC) with rate-limited feedback. The channel output is encoded into one stream of bits, which is provided causally to the two users at the channel input. An achievable rate region for this setup is derived, based on superposition of information, block Markov coding, and coding with various degrees of side information for the feedback link. The suggested region coincides with the Cover-Leung inner bound for large feedback rates. The result is then extended for cases where there is only a feedback link to one of the transmitters, and for a more general case where there are two separate feedback links to both transmitters. We compute achievable regions for the Gaussian MAC and for the binary erasure MAC. The Gaussian region is computed for the case of common rate-limited feedback, whereas the region for the binary erasure MAC is computed for one-sided feedback. It is known that for the latter, the Cover-Leung region is tight, and we obtain results that coincide with the feedback capacity region for high feedback rates.

Wiretap Channel With Causal State Information

A lower bound on the secrecy capacity of the wiretap channel with state information available causally at both the encoder and the decoder is established. The lower bound is shown to be strictly larger than that for the noncausal case by Liu and Chen. Achievability is proved using block Markov coding, Shannon strategy, and key generation from common state information. The state sequence available at the end of each block is used to generate a key to enhance the transmission rate of the confidential message in the following block. An upper bound on the secrecy capacity when the state is available noncausally at the encoder and the decoder is established and is shown to coincide with the aforementioned lower bound for several classes of wiretap channels with state.

An Achievable Rate Region for the Cognitive Interference Channel With Causal Bidirectional Cooperation

In this paper, we consider a two-transmitter-two-receiver cognitive interference channel with causal bidirectional cooperation (CIFC-CBC), where the transmitters operate in a full-duplex mode and cooperate with each other in a causal way. Cooperation between the transmitters is realized using a strategy that combines both decode-and-forward (DF) and compress-and-forward (CF) schemes. We derive an achievable rate region for the considered system based on a proposed coding scheme that jointly implements rate splitting, block Markov coding, superposition coding, Gel'fand-Pinsker coding, and backward decoding. We then apply the derived rate region result to Gaussian channels and show via numerical results that the derived rate region extends the rate regions achieved by pure DF- and pure CF-based cooperation.

On energy efficiency and optimum resource allocation of relay transmissions in the low-power regime

Relay links are expected to play a critical role in the design of wireless networks. This paper investigates the energy efficiency of relay communications in the low-power regime under two different scenarios: when the relay has unlimited power supply and when it has limited power supply. A system with a source node, a destination node, and a single relay operating in the time division duplex (TDD) mode was considered. Analysis and simulations are used to compare the energy required for transmitting one information bit in three different relay schemes: amplify and forward (AnF), decode and forward (DnF), and block Markov coding (BMC). Relative merits of these relay schemes in comparison with direct transmissions (direct Tx) are discussed. The optimal allocation of power and transmission time between source and relay is also studied.

Expanding maps of an interval with holes

We study a class of open chaotic dynamical systems. Consider an expanding map of an interval from which a few small open subintervals are removed (thus creating ‘holes’). Almost every point of the original interval then eventually escapes through the holes, so there can be no absolutely continuous invariant measures. We construct a so-called conditionally invariant measure that is equivalent to the Lebesgue measure. Our measure is unique and naturally generates an invariant measure, which is singular. These results generalize early work by Pianigiani, Yorke, Collet, Martinez, and Schmidt, who studied similar maps under an additional Markov assumption. We do not assume any Markov property here and use ‘bounded variation’ techniques rather than Markov coding. Our results supplement those of Keller, who studied analytic interval maps with holes by using different techniques.

A dimension formula for Bernoulli convolutions

We present a “dynamical” approach to the study of the singularity of infinitely convolved Bernoulli measuresvβ, for β the golden section. We introducevβ as the transverse measure of the maximum entropy measure μ on the repelling set invariant for contracting maps of the square, the “fat baker's” transformation. Our approach strongly relies on the Markov structure of the underlying dynamical system. Indeed, if β=golden mean, the fat baker's transformation has a very simple Markov coding. The “ambiguity” (of order two) of this coding, which appears when projecting on the line, due to passages for the central, overlapping zone, can be expressed by means of products of matrices (of order two). This product has a Markov distribution inherited by the Markov structure of the map. The dimension of the projected measure is therefore associated to the growth of this product; our dimension formula appears in a natural way as a version of the Furstenberg-Guivarch formula. Our technique provides an explicit dimension formula and, most important, provides a formalism well suited for the multifractal analysis of this measure, as we will show in a forthcoming paper.

Circular codes, loop counting, and zeta-functions

We prove a simple formula for the zeta-function of coded systems generated by circular codes (and more generally by circular Markov codes). We apply this to the loop counting method for determining the topological entropy of a subshift of finite type, to the zeta-function of the Dyck-shift over 2 N symbols, and to the zeta-function of a subshift of finite ype which is obtained from a full shift by deleting one block of arbitrary length.

Geometrical Markov coding of geodesics on surfaces of constant negative curvature

AbstractA natural geometrical representation of the geodesic flow on a surface M of constant negative curvature is given in which the base transformation is the shift on a (finite type) space of shortest words relative to a fixed generating set for π1(M) and the height function is the hyperbolic distance across a fundamental region for π1(M). This representation is obtained by comparing cutting sequences on M with generalised continued fraction expansions of endpoints on ℝ