A coding scheme for additive noise channels with feedback--I: No bandwidth constraint

Abstract

DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:

Is feedback a performance equalizer of classic and modern codes?

“…They showed that using an ideal feedback link in a fixed-length block-coding scheme for infinite bandwidth AWGN, the probability of error decreases as a second-order exponential in the code constraint length for rates lower than capacity. Schalkwijk then extended the result in the finite bandwidth case [5]. Later, Kramer [6] and Zigangirov [7] showed for the finite and infinite bandwidth case that the above doubly exponential bounds could be replaced by k th-order exponential bounds for any k > 2 in the limit of arbitrarily large block lengths.…”

“…Elias [2] and Chang [3] showed with examples that feedback could greatly simplify error correction at rates below capacity. In 1966, Schalkwijk and Kailath [4] proposed a feedback coding scheme for additive white Gaussian noise (AWGN) channels. They showed that using an ideal feedback link in a fixed-length block-coding scheme for infinite bandwidth AWGN, the probability of error decreases as a second-order exponential in the code constraint length for rates lower than capacity.…”

Error Exponents for Variable-length block codes with feedback and cost constraints

“…For 0 < R < C(P) and P > 0, (15) can be simplified by absorbing the term ρ max e −τ ǫ(δ) into the δ of (14). This cannot be done for P = 0 since the constraint E [S τ (θ)] ≤ Pτ for all θ reduces to the unconstrained case where only zero-cost inputs are used.…”

The necessity and sufficiency of anytime capacity for control over a noisy communication link

“…However, by looking more closely at the actual tail probabilities achieved by the linear observer/controller strategy, we get a natural anytime generalization of Schalkwijk and Kailath's scheme [25] for communicating over the power constrained additive white Gaussian noise channel with noiseless feedback that achieves doubly exponential reliability. Theorem 4.6 It is possible to communicate bits reliably across a discrete-time power constrained AWGN channel with noiseless feedback at all rates R < 1 2 log 2 (1 + P σ 2 ) while achieving a g−anytime reliability of at least…”

“…Because of the rate constraint, the virtual controlŪ t takes on one of 2 R(t+1) − Rt values. For simplicity of exposition, we will ignore the integer effects and consider it to be one of 2 R values 25 …”