A proof of the strong converse theorem for Gaussian multiple access channels

Fong, Silas L.; Tan, Vincent Y. F.

doi:10.1109/itwf.2015.7360756

Cited by 11 publications

(12 citation statements)

References 16 publications

(19 reference statements)

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“…Proof: Appendix D. 5) In [24], Fong and Tan derive a converse for the Gaussian MAC with second-order term O( √ n log n)1. This converse does not match the second-order term in the achievability bounds proven in this paper.…”

Section: An Rcu Bound and Its Analysis For The Gaussian Multiplementioning

confidence: 99%

Gaussian Multiple and Random Access in the Finite Blocklength Regime

Yavas

Kostina

Effros

2020

2020 IEEE International Symposium on Information Theory (ISIT)

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This paper presents finite-blocklength achievability bounds for the Gaussian multiple access channel (MAC) and random access channel (RAC) under average-error and maximal-power constraints. Using random codewords uniformly distributed on a sphere and a maximum likelihood decoder, the derived MAC bound on each transmitter's rate matches the MolavianJazi-Laneman bound (2015) in its first-and secondorder terms, improving the remaining terms to 1 2 log n n + O 1 n bits per channel use. The result then extends to a RAC model in which neither the encoders nor the decoder knows which of K possible transmitters are active. In the proposed rateless coding strategy, decoding occurs at a time nt that depends on the decoder's estimate t of the number of active transmitters k. Single-bit feedback from the decoder to all encoders at each potential decoding time ni, i ≤ t, informs the encoders when to stop transmitting. For this RAC model, the proposed code achieves the same first-, second-, and third-order performance as the best known result for the Gaussian MAC in operation. Index Terms-Gaussian multiple access channel, Gaussian random access channel, third-order asymptotics, finite blocklength, maximum likelihood decoder, spherical distribution.

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Section: An Rcu Bound and Its Analysis For The Gaussian Multiplementioning

confidence: 99%

Gaussian Multiple and Random Access in the Finite Blocklength Regime

Yavas

Kostina

Effros

2020

2020 IEEE International Symposium on Information Theory (ISIT)

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“…Without feedback, inner bounds for the second-order coding rates for the Gaussian MAC were independently established by Scarlett, Martinez, and Guillén i Fàbregas [27] and MolavianJazi and Laneman [28]. The strong converse, together with (non-matching) outer bounds for the second-order coding rates, was established by Fong and Tan [29]. For the Gaussian MAC with degraded message sets, the complete second-order asymptotics was derived by Scarlett and Tan [30].…”

Section: B Related Workmentioning

confidence: 99%

“…We ignore integer constraints on the number of codewords Mn in(29). We simply set Mn to the nearest integer to the number on the right-hand-side of (29).…”

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confidence: 99%

On Gaussian Channels With Feedback Under Expected Power Constraints and With Non-Vanishing Error Probabilities

Truong

Fong

Tan

2017

IEEE Trans. Inform. Theory

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In this paper, we consider single-and multi-user Gaussian channels with feedback under expected power constraints and with non-vanishing error probabilities. In the first of two contributions, we study asymptotic expansions for the additive white Gaussian noise (AWGN) channel with feedback under the average error probability formalism. By drawing ideas from Gallager and Nakiboglu's work for the direct part and the meta-converse for the converse part, we establish the ε-capacity and show that it depends on ε in general and so the strong converse fails to hold. Furthermore, we provide bounds on the second-order term in the asymptotic expansion. We show that for any positive integer L, the second-order term is bounded between a term proportional to − ln (L) n (where ln (L) (·) is the L-fold nested logarithm function) and a term proportional to + √ n ln n where n is the blocklength. The lower bound on the second-order term shows that feedback does provide an improvement in the maximal achievable rate over the case where no feedback is available. In our second contribution, we establish the ε-capacity region for the AWGN multiple access channel (MAC) with feedback under the expected power constraint by combining ideas from hypothesis testing, information spectrum analysis, Ozarow's coding scheme, and power control.

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“…An alternative strong converse proof was presented by Ahlswede in [8]; this proof used Augustin's converse argument [9] in place of the blowing-up lemma, followed by a more refined wringing step. A strong converse for the Gaussian MAC was proved in [10], using an argument based on that of [8].…”

Section: Introductionmentioning

confidence: 99%

A Second-Order Converse Bound for the Multiple-Access Channel via Wringing Dependence

Kosut¹

2020

Preprint

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A new converse bound is presented for the two-user multiple-access channel under the average probability of error constraint. This bound shows that for most channels of interest, the second-order coding rate-that is, the difference between the best achievable rates and the asymptotic capacity region as a function of blocklength n with fixed probability of error-is O(1/ √ n) bits per channel use. The principal tool behind this converse proof is a new measure of dependence between two random variables called wringing dependence, as it is inspired by Ahlswede's wringing technique. The O(1/ √ n) gap is shown to hold for any channel satisfying certain regularity conditions, which includes all discrete-memoryless channels and the Gaussian multiple-access channel. Exact upper bounds as a function of the probability of error are proved for the coefficient in the O(1/ √ n) term, although for most channels they do not match existing achievable bounds.This result asserts that achievable second-order bounds of [13]-[18] are order-optimal; that is, the gap between the capacity region and the blocklength-n achievable region, in either direction, is at most O(1/ √ n). We provide a specific upper bound on the coefficient in the O(1/ √ n) term, although for most channels it does not match the achievability bounds. The main difficulty in proving a second-order converse for the MAC is to properly deal with the independence between the transmitters. The problem variant with degraded message sets, as studied in [19], [20], seems to be easier precisely because the transmitted signals are not independent. The independence that is inherent to the standard MAC prohibits many of the methods to prove second-order converses for the point-to-point channel; for example, one cannot restrict the inputs to a fixed type (empirical distribution), which is one of the steps in the point-to-point converse in [12], since imposing a fixed joint type on the two input signals creates dependence. An alternative approach adopted in [23] to prove second-order converses uses the notion of reverse hypercontractivity. This technique provides a strengthening of Fano's inequality, wherein the coding rate is upper bounded by the mutual information plus an O(1/ √ n) error term. However, this technique relies on the geometric average error criterion, which is stronger than the usual average error criterion (but weaker than the maximal error criterion). The method of [23] can be applied to the average error criterion by first expurgating the code-i.e., removing some of the codewords with the largest probability of error. However, with the MAC, we cannot just expurgate codewords, we must expurgate codeword pairs, which again introduces some dependence between inputs. For this reason, reverse hypercontractivity can be viewed as a replacement for the blowing-up lemma or Augustin's converse, but does not remove the need for wringing. Interestingly, the technique that we use here seems to be related to hypercontractivity; see Sec. III-D for more details.To handle the ind...

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A proof of the strong converse theorem for Gaussian multiple access channels

Cited by 11 publications

References 16 publications

Gaussian Multiple and Random Access in the Finite Blocklength Regime

Gaussian Multiple and Random Access in the Finite Blocklength Regime

On Gaussian Channels With Feedback Under Expected Power Constraints and With Non-Vanishing Error Probabilities

A Second-Order Converse Bound for the Multiple-Access Channel via Wringing Dependence

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