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Register a new userFeedback Capacity Formulas of AGN Channels Driven by Nonstationary Autoregressive Moving Average Noise
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Stelios Louka
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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.
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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.
The zero-error feedback capacity of the Gelfand-Pinsker channel is established. It can be positive even if the channels zero-error capacity is zero in the absence of feedback. Moreover, the error-free transmission of a single bit may require more than one channel use. These phenomena do not occur when the state is revealed to the transmitter causally, a case that is solved here using Shannon strategies. Cost constraints on the channel inputs or channel states are also discussed, as is the scenario where---in addition to the message---also the state sequence must be recovered.
Flat-fading channels that are correlated in time are considered under peak and average power constraints. For discrete-time channels, a new upper bound on the capacity per unit time is derived. A low SNR analysis of a full-scattering vector channel is used to derive a complimentary lower bound. Together, these bounds allow us to identify the exact scaling of channel capacity for a fixed peak to average ratio, as the average power converges to zero. The upper bound is also asymptotically tight as the average power converges to zero for a fixed peak power. For a continuous time infinite bandwidth channel, Viterbi identified the capacity for M-FSK modulation. Recently, Zhang and Laneman showed that the capacity can be achieved with non-bursty signaling (QPSK). An additional contribution of this paper is to obtain similar results under peak and average power constraints.
It is known that for a discrete channel with correlated additive noise, the ordinary capacity with or without feedback both equal $ log q-mathcal{H} (Z) $, where $ mathcal{H}(Z) $ is the entropy rate of the noise process $ Z $ and $ q $ is the alphabet size. In this paper, a class of finite-state additive noise channels is introduced. It is shown that the zero-error feedback capacity of such channels is either zero or $C_{0f} =log q -h (Z) $, where $ h (Z) $ is the {em topological entropy} of the noise process. A topological condition is given when the zero-error capacity is zero, with or without feedback. Moreover, the zero-error capacity without feedback is lower-bounded by $ log q-2 h (Z) $. We explicitly compute the zero-error feedback capacity for several examples, including channels with isolated errors and a Gilbert-Elliot channel.
Jolfaei et al. used feedback to create transmit signals that are simultaneously useful for multiple users in a broadcast channel. Later, Georgiadis and Tassiulas studied erasure broadcast channels with feedback, and presented the capacity region under certain assumptions. These results provided the fundamental ideas used in communication protocols for networks with delayed channel state information. However, to the best of our knowledge, the capacity region of erasure broadcast channels with feedback and with a common message for both receivers has never been presented. This latter problem shows up as a sub-problem in many multi-terminal communication networks such as the X-Channel, and the two-unicast problem. In this work, we present the capacity region of the two-user erasure broadcast channels with delayed feedback, private messages, and a common message. We consider arbitrary and possibly correlated erasure distributions. We develop new outer-bounds that capture feedback and quantify the impact of delivering a common message on the capacity region. We also propose a transmission strategy that achieves the outer-bounds. Our transmission strategy differs from prior results in that to achieve the capacity, it creates side-information at the weaker user such that the decodability is ensured even if we multicast the common message with a rate higher than its link capacity.
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