Gaussian Channel With Noisy Feedback and Peak Energy Constraint
TL;DR: It is shown that if the noise power in the feedback link is sufficiently small, the best error exponent for communicating an M-ary message can be strictly larger than the one without feedback.
Abstract: Optimal coding over the additive white Gaussian noise channel under the peak energy constraint is studied when there is noisy feedback over an orthogonal additive white Gaussian noise channel. As shown by Pinsker, under the peak energy constraint, the best error exponent for communicating an M-ary message, M ≥ 3, with noise-free feedback is strictly larger than the one without feedback. This paper extends Pinsker's result and shows that if the noise power in the feedback link is sufficiently small, the best error exponent for communicating an M-ary message can be strictly larger than the one without feedback. The proof involves two feedback coding schemes. One is motivated by a two-stage noisy feedback coding scheme of Burnashev and Yamamoto for binary symmetric channels, while the other is a linear noisy feedback coding scheme that extends Pinsker's noise-free feedback coding scheme. When the feedback noise power α is sufficiently small, the linear coding scheme outperforms the two-stage (nonlinear) coding scheme, and is asymptotically optimal as α tends to zero. By contrast, when α is relatively larger, the two-stage coding scheme performs better.
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TL;DR: In this article, the problem of communication over an additive white Gaussian noise (AWGN) channel with an AWGN feedback channel was studied, and a low-complexity low-delay interactive scheme that operates close to capacity for a fixed bit error probability was proposed.
Abstract: We study the problem of communication over an additive white Gaussian noise (AWGN) channel with an AWGN feedback channel. When the feedback channel is noiseless, the classic Schalkwijk–Kailath (S-K) scheme is known to achieve capacity in a simple sequential fashion, while attaining reliability superior to non-feedback schemes. In this paper, we show how simplicity and reliability can be attained even when the feedback is noisy, provided that the feedback channel is sufficiently better than the feedforward channel. Specifically, we introduce a low-complexity low-delay interactive scheme that operates close to capacity for a fixed bit error probability (e.g., $10^{-6}$ ). We then build on this scheme to provide two asymptotic constructions, one based on high dimensional lattices, and the other based on concatenated coding, that admit an error exponent significantly exceeding the best possible non-feedback exponent. Our approach is based on the interpretation of feedback transmission as a side-information problem, and employs an interactive modulo-lattice solution.
40 citations
TL;DR: In this paper, the authors investigated the achievable error probability in communication over an AWGN discrete time memoryless channel with noiseless delayless rate-limited feedback, and showed that the decay of the probability of error is at most exponential in blocklength, and obtained an upper bound for increase in the error exponent due to feedback.
Abstract: We investigate the achievable error probability in communication over an AWGN discrete time memoryless channel with noiseless delayless rate-limited feedback. For the case where the feedback rate RFB is lower than the data rate R transmitted over the forward channel, we show that the decay of the probability of error is at most exponential in blocklength, and obtain an upper bound for increase in the error exponent due to feedback. Furthermore, we show that the use of feedback in this case results in an error exponent that is at least RFB higher than the error exponent in the absence of feedback. For the case where the feedback rate exceeds the forward rate (RFB ≥ R), we propose a simple iterative scheme that achieves a probability of error that decays doubly exponentially with the codeword blocklength n. More generally, for some positive integer L, we show that a L-th order exponential error decay is achievable if RFB ≥ (L-1)R. While the above results are proved under an average feedback rate constraint, we show that all the achievability results for RFB ≥ R hold in a more restrictive case where the feedback constraint is expressed in terms of the per-channel-use feedback rate. Our results show that the error exponent as a function of RFB has a strong discontinuity at R, where it jumps from a finite value to infinity.
15 citations
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TL;DR: This paper introduces a low-complexity low-delay interactive scheme that operates close to capacity for a fixed bit error probability, and builds on this scheme to provide two asymptotic constructions that admit an error exponent significantly exceeding the best possible non-feedback exponent.
Abstract: We study the problem of communication over an additive white Gaussian noise (AWGN) channel with an AWGN feedback channel. When the feedback channel is noiseless, the classic Schalkwijk-Kailath (S-K) scheme is known to achieve capacity in a simple sequential fashion, while attaining reliability superior to non-feedback schemes. In this work, we show how simplicity and reliability can be attained even when the feedback is noisy, provided that the feedback channel is sufficiently better than the feedforward channel. Specifically, we introduce a low-complexity low-delay interactive scheme that operates close to capacity for a fixed bit error probability (e.g. $10^{-6}$). We then build on this scheme to provide two asymptotic constructions, one based on high dimensional lattices, and the other based on concatenated coding, that admit an error exponent significantly exceeding the best possible non-feedback exponent. Our approach is based on the interpretation of feedback transmission as a side-information problem, and employs an interactive modulo-lattice solution.
11 citations
Cites background from "Gaussian Channel With Noisy Feedbac..."
...I NTRODUCTION While feedback cannot increase the capacity of point-topoint memoryless channels [1], there exist noiseless feedback communication schemes that can provide a significant improvement in terms of simplicity and reliability, see e.g. [2]– [5]....
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17 Jun 2018
TL;DR: Achievable error exponent regions of a two-way additive white Gaussian noise (AWGN) channel, where two terminals exchange a fixed number of messages M, are derived and it is shown that the error exponent of the weaker channel may be improved through active feedback, at the expense of a decreased error exponent in the stronger direction.
Abstract: Achievable error exponent regions of a two-way additive white Gaussian noise (AWGN) channel, where two terminals exchange a fixed number of messages M, are derived. In particular, error exponent regions for $M$ = 2 messages under expected power and $M$ = 3 messages under almost sure power constraints are considered. For $M$ = 2 messages the use of active feedback is shown to lead to an error exponent gain over that when feedback / interaction is ignored. For $M$ = 3 messages and asymmetric channels, it is shown that the error exponent of the weaker channel may be improved through active feedback, at the expense of a decreased error exponent of the stronger direction. This may, for sufficiently asymmetric channel gains, outperform the error exponent region achieved by having both terminals operate independently of one another (ignoring the possibility of sending feedback for the other).
5 citations
Cites background or methods from "Gaussian Channel With Noisy Feedbac..."
...[6] studied the reliability for M ≥ 3 messages subject to the Peak Energy constraint (PE) on the forward channel,...
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...Then, the error exponent (30) can be derived in a similar way as P(E2) in Section II-A of [6] yielding:...
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...Noisy feedback appears to need to be much less noisy than the direct link to provide gains [6]....
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...Decoding: Following [6], terminal 2 decodes W1 immediately at the end of the transmission stage if yλ1n2 is received within a protection region....
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...Decoding: Following [6], terminal 2 decodes M1 immediately at the end of the transmission stage if y1 2 is received within a protection region....
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07 Jul 2019
TL;DR: The results show that this new VLC scheme under an almost-sure power constraint achieves an error exponent similar to an achievable exponent attained using a fixed block length scheme under a much more relaxed expected block power constraint, and is larger than that achieved by schemes without feedback.
Abstract: A one-way additive white Gaussian noise (AWGN) channel with active feedback sent over another AWGN feedback channel is considered. Achievable error exponents are presented in the finite message / zero-rate regime for a variable length coding (VLC) scheme. This coding scheme uses a form of round-robin scheduling of messages, and a simplex-based feedback code to obtain reliable feedback and remain synchronized, despite the noise in the feedback link. Our results show that this new VLC scheme under an almost-sure power constraint achieves an error exponent similar to an achievable exponent attained using a fixed block length scheme under a much more relaxed expected block power constraint, and is larger than that achieved by schemes without feedback.1
4 citations
References
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Book•
01 Jan 1965
TL;DR: Textbook on communication engineering emphasizing random processes, information and detection theory, statistical communication theory, applications, etc.
Abstract: Textbook on communication engineering emphasizing random processes, information and detection theory, statistical communication theory, applications, etc
1,519 citations
TL;DR: Upper and lower bounds are found for the error probability in decoding with optimal codes and decoding systems for a continuous channel with an additive gaussian noise and subject to an average power limitation at the transmitter.
Abstract: A study is made of coding and decoding systems for a continuous channel with an additive gaussian noise and subject to an average power limitation at the transmitter. Upper and lower bounds are found for the error probability in decoding with optimal codes and decoding systems. These bounds are close together for signaling rates near channel capacity and also for signaling rates near zero, but diverge between. Curves exhibiting these bounds are given.
836 citations
"Gaussian Channel With Noisy Feedbac..." refers background in this paper
...Wyner [3] showed that the error probability of the Schalkwijk–Kailath coding scheme [1] degrades to an exponential form under the peak energy constraint....
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TL;DR: This paper presents a coding scheme that exploits the feedback to achieve considerable reductions in coding and decoding complexity and delay over what would be needed for comparable performance with the best known (simplex) codes for the one-way channel.
Abstract: In some communication problems, it is a good assumption that the channel consists of an additive white Gaussian noise forward link and an essentially noiseless feedback link. In this paper, we study channels where no bandwidth constraint is placed on the transmitted signals. Such channels arise in space communications. It is known that the availability of the feedback link cannot increase the channel capacity of the noisy forward link, but it can considerably reduce the coding effort required to achieve a given level of performance. We present a coding scheme that exploits the feedback to achieve considerable reductions in coding and decoding complexity and delay over what would be needed for comparable performance with the best known (simplex) codes for the one-way channel. Our scheme, which was motivated by the Robbins-Monro stochastic approximation technique, can also be used over channels where the additive noise is not Gaussian but is still independent from instant to instant. An extension of the scheme for channels with limited signal bandwidth is presented in a companion paper (Part II).
579 citations
"Gaussian Channel With Noisy Feedbac..." refers background in this paper
...…access to a noisy versionỸi of Yi over the feedback (backward) additive white Gaussian noisechannel Ỹi = Yi + Z̃i, The authors are with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093 USA e-mail: (yxiang@ucsd.edu; yhk@ucsd.edu)....
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TL;DR: A sequential continuous-transmission system employing a binary symmetric forward channel and a noiseless feedback channel and its error exponent is shown to be substantially greater than the optimum block-code error exponent at all transmission rates less than channel capacity.
Abstract: The presence of a feedback channel makes possible a variety of sequential transmission procedures, each of which can be classified as either a block-transmission or a continuous-transmission scheme according to the way in which information is encoded for transmission over a noisy forward channel. A sequential continuous-transmission system employing a binary symmetric forward channel (but which is suitable for use with any discrete memoryless forward channel) and a noiseless feedback channel is described. Its error exponent is shown to be substantially greater than the optimum block-code error exponent at all transmission rates less than channel capacity. The average value and the first-order probability distribution of the effective constraint length, found by simulating the system on an IBM 709 computer, are also given.
350 citations
TL;DR: A modified Schalkwijk-Barron transmission scheme is presented for channels with noiseless feedback and the new result obtained shows that the modified scheme also attains high reliability functions for DMC's.
Abstract: A modified Schalkwijk-Barron transmission scheme is presented for channels with noiseless feedback. The modified scheme employs blockwise decision and a fixed length transmission in place of Viterbi's sequential decision feedback. The reliability functions of the modified scheme are derived for the additive white Gaussian noise (AWGN) channel and discrete memoryless channels (DMC's). For the AWGN channel the result obtained is asymptotically the same as in the case of the Schalkwijk-Barron scheme. On the other hand, the new result obtained for DMC's shows that the modified scheme also attains high reliability functions for DMC's.
159 citations