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Register a new userNew Formulas of Feedback Capacity for AGN Channels with Memory: A Time-Domain Sufficient Statistic Approach
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Charalambos Charalambous D.
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In the recent paper [1] it is shown, via an application example, that the Cover and Pombra [2] characterization of the $n-$block or transmission feedback capacity formula, of additive Gaussian noise (AGN) channels, is the subject of much confusion in the literature, with redundant incorrect results. The main objective of this paper is to clarify the main points of confusion and remove any further ambiguity. The first part of the paper applies time-domain methods, to derive for a first time, equivalent sequential characterizations of the Cover and Pombra characterization of feedback capacity of AGN channels driven by nonstationary and nonergodic Gaussian noise. The optimal channel input processes of the new equivalent sequential characterizations are expressed as functionals of a sufficient statistic and a Gaussian orthogonal innovations process. From the new representations follows that the Cover and Pombra $n-$block capacity formula is expressed as a functional of two generalized matrix difference Riccati equations (DRE) of filtering theory of Gaussian systems, contrary to results that appeared in the literature. In the second part of the paper the existence of the asymptotic limit of the $n-$block feedback capacity formula is shown to be equivalent to the convergence properties of solutions of the two generalized DREs. Further, necessary and or sufficient conditions are identified for existence of the asymptotic limits, for stable and unstable Gaussian noise, when the optimal input distributions are time-invariant, but not necessarily stationary. The paper contains an in depth analysis, with examples, of the specific technical issues, which are overlooked in past literature [3-7], that studied the AGN channel of [2], for stationary noises.
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In this paper we derive closed-form formulas of feedback capacity and nonfeedback achievable rates, for Additive Gaussian Noise (AGN) channels driven by nonstationary autoregressive moving average (ARMA) noise (with unstable one poles and zeros), based on time-invariant feedback codes and channel input distributions. From the analysis and simulations follows the surprising observations, (i) the use of time-invariant channel input distributions gives rise to multiple regimes of capacity that depend on the parameters of the ARMA noise, which may or may not use feedback, (ii) the more unstable the pole (resp. zero) of the ARMA noise the higher (resp. lower) the feedback capacity, (iii) certain conditions, known as detectability and stabilizability are necessary and sufficient to ensure the feedback capacity formulas and nonfeedback achievable rates {it are independent of the initial state of the ARMA noise}. Another surprizing observation is that Kims cite{kim2010} characterization of feedback capacity which is developed for stable ARMA noise, if applied to the unstable ARMA noise, gives a lower value of feedback capacity compared to our feedback capacity formula.
The main objective of this paper is to derive a new sequential characterization of the Cover and Pombra cite{cover-pombra1989} characterization of the $n-$finite block or transmission feedback information ($n$-FTFI) capacity, which clarifies several issues of confusion and incorrect interpretation of results in literature. The optimal channel input processes of the new equivalent sequential characterizations are expressed as functionals of a sufficient statistic and a Gaussian orthogonal innovations process. From the new representations follows that the Cover and Pombra characterization of the $n-$FTFI capacity is expressed as a functional of two generalized matrix difference Riccati equations (DRE) of filtering theory of Gaussian systems. This contradicts results which are redundant in the literature, and illustrates the fundamental complexity of the feedback capacity formula.
The two-receiver broadcast packet erasure channel with feedback and memory is studied. Memory is modeled using a finite-state Markov chain representing a channel state. Two scenarios are considered: (i) when the transmitter has causal knowledge of the channel state (i.e., the state is visible), and (ii) when the channel state is unknown at the transmitter, but observations of it are available at the transmitter through feedback (i.e., the state is hidden). In both scenarios, matching outer and inner bounds on the rates of communication are derived and the capacity region is determined. It is shown that similar results carry over to channels with memory and delayed feedback and memoryless compound channels with feedback. When the state is visible, the capacity region has a single-letter characterization and is in terms of a linear program. Two optimal coding schemes are devised that use feedback to keep track of the sent/received packets via a network of queues: a probabilistic scheme and a deterministic backpressure-like algorithm. The former bases its decisions solely on the past channel state information and the latter follows a max-weight queue-based policy. The performance of the algorithms are analyzed using the frameworks of rate stability in networks of queues, max-flow min-cut duality in networks, and finite-horizon Lyapunov drift analysis. When the state is hidden, the capacity region does not have a single-letter characterization and is, in this sense, uncomputable. Approximations of the capacity region are provided and two optimal coding algorithms are outlined. The first algorithm is a probabilistic coding scheme that bases its decisions on the past L acknowledgments and its achievable rate region approaches the capacity region exponentially fast in L. The second algorithm is a backpressure-like algorithm that performs optimally in the long run.
The two-receiver broadcast packet erasure channel with feedback and memory is studied. Memory is modeled using a finite-state Markov chain representing a channel state. The channel state is unknown at the transmitter, but observations of this hidden Markov chain are available at the transmitter through feedback. Matching outer and inner bounds are derived and the capacity region is determined. The capacity region does not have a single-letter characterization and is, in this sense, uncomputable. Approximations of the capacity region are provided and two optimal coding algorithms are outlined. The first algorithm is a probabilistic coding scheme that bases its decisions on the past L feedback sequences. Its achievable rate-region approaches the capacity region exponentially fast in L. The second algorithm is a backpressure-like algorithm that performs optimally in the long run.
The feedback sum-rate capacity is established for the symmetric $J$-user Gaussian multiple-access channel (GMAC). The main contribution is a converse bound that combines the dependence-balance argument of Hekstra and Willems (1989) with a variant of the factorization of a convex envelope of Geng and Nair (2014). The converse bound matches the achievable sum-rate of the Fourier-Modulated Estimate Correction strategy of Kramer (2002).
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