Coding Schemes and Asymptotic Capacity for the Gaussian Broadcast and Interference Channels With Feedback

Michael Christoph Gastpar, Amos Lapidoth
Institute of Electrical and Electronics Engineers, 2014
Article

Résumé

A coding scheme is proposed for the memoryless Gaussian broadcast channel with correlated noises and feedback. For all noise correlations other than ±1, the gap between the sumrate that the scheme achieves and the full-cooperation bound vanishes as the signal-to-noise ratio tends to infinity. When the correlation coefficient is −1, the gains afforded by feedback are unbounded and the prelog is doubled. When the correlation coefficient is +1, we demonstrate a dichotomy that if the noise variances are equal, then feedback is useless, and otherwise, feedback affords unbounded rate gains and doubles the prelog. The unbounded feedback gains, however, require perfect (noiseless) feedback. When the feedback links are noisy, the feedback gains are bounded, unless the feedback noise decays to zero sufficiently fast with the signal-to-noise ratio. Extensions to more receivers are also discussed as is the memoryless Gaussian interference channel with feedback.

Official source

https://infoscience.epfl.ch/record/195742?ln=fr

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Suppose Q is a family of discrete memoryless channels. An unknown member of Q will be available, with perfect, causal output feedback for communication. Is there a coding scheme (possibly with variable transmission time) that can achieve the Burnashev error exponent uniformly over Q ? For two families of channels we show that the answer is yes. Furthermore, for each of these two classes, in addition to achieve the maximum error exponent, it is possible to uniformly attain any given fraction of the channel capacity. Therefore, in terms of achievable rates and delay, there are situations in which the knowledge of the channel becomes irrelevant. In the second part of the thesis, we show that for arbitrary sets of channels the Burnashev error exponent cannot in general be uniformly achieved. In particular we give a sufficient condition for a pair of channels so that no coding strategy reaches Burnashev's exponent simultaneously on both channels. As a third part we study a scenario where communication is carried by first testing the channel by means of a training sequence, then coding according to the channel estimate. We provide an upper bound on the maximum achievable error exponent of such coding schemes. This bound is typically much lower than the maximum achievable error exponent over a channel with feedback. For example in the case of binary symmetric channels this bound has a slope that vanishes at capacity. This result suggests that in terms of error exponent, a good universal feedback scheme combines channel estimation with information delivery, rather than separating them. In the final chapter, we address the question of communicating quickly and reliably. We consider a simple situation of two message communication over a known channel with feedback. We propose a simple decoding rule, and show that it minimizes a weighted combination of the probability of error and decoding delay for a certain range of crossover probabilities and combination weights.

EPFL2005

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Suppose Q is a family of discrete memoryless channels. An unknown member of Q is available with perfect (causal) feedback for communication. A recent result (A. Tchamkerten and I.E. Telatar) shows the existence, for certain families of channels (e.g. binary symmetric channels and Z channels), of coding schemes that achieve Burnashev's exponent universally over these families. In other words, in certain cases, there is no loss in the error exponent by ignoring the channel: transmitter and receiver can design optimal blind coding schemes that perform as well as the best feedback coding schemes tuned for the channel under use. Here we study the situation where communication is carried by first testing the channel by means of a training sequence, then coding the information according to the channel estimate. We provide an upper bound on the maximum achievable error exponent of any such scheme. If we consider binary symmetric channels and Z channels this bound is much lower than Burnashev's exponent. This suggests that in terms of error exponent, a good universal feedback scheme entangles channel estimation with information delivery, rather than separating them.

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Approximate Feedback Capacity of the Gaussian Multicast Channel

Michael Christoph Gastpar, Naveen Goela

We characterize the capacity region to within log {2(M − 1)} bits/s/Hz for the M -transmitter K -receiver Gaus- sian multicast channel with feedback where each receiver wishes to decode every message from the M transmitters. Extending Cover-Leung’s achievable scheme intended for (M, K) = (2, 1), we show that this generalized scheme achieves the cutset-based outer bound within log {2(M − 1)} bits per transmitter for all channel parameters. In contrast to the capacity in the non- feedback case, the feedback capacity improves upon the naive intersection of the feedback capacities of K individual multiple access channels. We find that feedback provides unbounded multiplicative gain at high signal-to-noise ratios as was shown in the Gaussian interference channel. To complement the results, we establish the exact feedback capacity of the Avestimehr-Diggavi- Tse deterministic model, from which we make the observation that feedback can also be beneficial for function computation.

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