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On the optimality of the binary reflected gray code

IEEE TRANSACTIONS ON INFORMATION THEORY 2004年第12期50卷 3170-3182页

作者： Agrell, E Lassing, J Ström, EG Ottosson, T Chalmers Univ Technol Dept Signals & Syst Commun Syst Grp Gothenburg Sweden

This paper concerns the problem of selecting a binary labeling for the signal constellation in M-PSK, M-PAM, and M-QAM communication systems. gray labelings are discussed and the original work by Frank gray is analyzed. As is noted, the number of distinct gray labelings that result in different bit-error probability grows rapidly with increasing constellation size. By introducing a recursive gray labeling construction method called expansion, the paper answers the natural question of what labeling, among all possible constellation labelings, will give the lowest possible average probability of bit errors for the considered constellations. Under certain assumptions on the channel, the answer is that the labeling proposed by gray, the binary reflected gray code, is the optimal labeling for all three constellations, which has,surprisingly, never been proved before.

关键词： average distance spectrum binary reflected gray code constellation labeling phase-shift keying (PSK) pulse-amplitude modulation (PAM) quadrature amplitude modulation (QAM)

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Flip-swap languages in binary reflected gray code order

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THEORETICAL COMPUTER SCIENCE 2022年 933卷 138-148页

作者： Sawada, Joe Williams, Aaron Wong, Dennis Univ Guelph Comp & Informat Sci Guelph ON Canada Williams Coll Comp Sci Williamstown MA 01267 USA Macao Polytech Univ Fac Appl Sci Macau Peoples R China

A flip-swap language is a set S of binary strings of length n such that S boolean OR {0(n)} is closed under two operations (when applicable): (1) Flip the leftmost 1;and (2) Swap the leftmost 1 with the bit to its right. Flip-swap languages model many combinatorial objects including necklaces, Lyndon words, prefix normal words, left factors of k-ary Dyck words, lattice paths with at most k flaws, and feasible solutions to 0-1 knapsack problems. We prove that any flip-swap language forms a cyclic 2-gray code when listed in binary reflected gray code (BRGC) order. Furthermore, a generic successor rule computes the next string when provided with a membership tester. The rule generates each string in the aforementioned flip-swap languages in O(n)-amortized per string, except for prefix normal words of length nwhich require O(n(1.864))-amortized per string. Our work generalizes results on necklaces and Lyndon words by Vajnovszki (2008) [35]. (c) 2022 Elsevier B.V. All rights reserved.

关键词： Flip-swap language binary reflected gray code BRGC Necklaces Lyndon words Dyck words gray codes

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Recursive and iterative approaches to generate rotation gray codes for stamp foldings and semi-meanders

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THEORETICAL COMPUTER SCIENCE 2025年 1031卷

作者： Liu, Bowie Wong, Dennis Lam, Chan-Tong Im, Marcus Macao Polytech Univ Fac Appl Sci Macao Special Adm Reg China Macau 999078 Macao Peoples R China

We first present a simple recursive algorithm that generates cyclic rotation gray codes for stamp foldings and semi-meanders, where consecutive strings differ by a stamp rotation. These are the first known gray codes for stamp foldings and semi-meanders, and we thus solve an open problem posted by Sawada and Li (2012) [17]. We then introduce an iterative algorithm that generates the same rotation gray codes for stamp foldings and semi-meanders. Both the recursive and iterative algorithms generate stamp foldings and semi-meanders in constant amortized time and O(n)- amortized time per string respectively, using a linear amount of memory.

关键词： Stamp foldings Meanders Semi-meanders Reflectable language binary reflected gray code gray code CAT algorithm

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gray coding for multilevel constellations in Gaussian noise

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IEEE TRANSACTIONS ON INFORMATION THEORY 2007年第1期53卷 224-235页

作者： Agrell, Erik Lassing, Johan Strom, Erik G. Ottosson, Tony Chalmers Univ Technol Commun Syst Grp Dept Signals & Syst S-41296 Gothenburg Sweden Qamcom Technol S-42130 Vastra Frolunda Sweden

The problem of finding the optimal labeling (bit-to-symbol mapping) of multilevel coherent phase shift keying (PSK), pulse amplitude modulation (PAM), and quadrature amplitude modulation (QAM) constellations with respect to minimizing the bit-error probability (BEP) over a Gaussian channel is addressed. We show that using the binary reflected gray code (BRGC) to label the signal constellation results in the lowest possible BEP for high enough signal energy-to-noise ratios and analyze what is "high enough" in this sense. It turns out that the BRGC is optimal for PSK and PAM systems whenever the target BEP is at most a few percent, which covers most systems of practical interest. New and simple closed-form expressions are presented for the BEP of PSK, PAM, and QAM using the BRGC.

关键词： binary reflected gray code bit-error probability (BEP) bit-error rate constellation labeling digital modulation gray mapping optimal labeling phase shift keying (PSK) pulse amplitude modulation (PAM) quadrature amplitude modulation (QAM)

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Whole mirror duplication-random loss model and pattern avoiding permutations

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INFORMATION PROCESSING LETTERS 2010年第11期110卷 474-480页

作者： Baril, Jean-Luc Vernay, Remi Univ Bourgogne UMR CNRS LE21 5158 F-21078 Dijon France

In this paper we study the problem of the whole mirror duplication-random loss model in terms of pattern avoiding permutations. We prove that the class of permutations obtained with this model after a given number p of duplications of the identity is the class of permutations avoiding the alternating permutations of length 2(p) + 1. We also compute the number of duplications necessary and sufficient to obtain any permutation of length n. We provide two efficient algorithms to reconstitute a possible scenario of whole mirror duplications from identity to any permutation of length n. One of them uses the well-known binary reflected gray code (gray, 1953) [10]. Other relative models are also considered. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Algorithms Combinatorial problems Pattern avoiding permutation Whole duplication-random loss model Genome Generating algorithm binary reflected gray code

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On Optimal TCM Encoders

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IEEE TRANSACTIONS ON COMMUNICATIONS 2013年第6期61卷 2178-2189页

作者： Alvarado, Alex Graell i Amat, Alexandre Brannstrom, Fredrik Agrell, Erik Univ Cambridge Dept Engn Cambridge CB2 1TN England Chalmers Dept Signals & Syst S-41296 Gothenburg Sweden

An asymptotically optimal trellis-coded modulation (TCM) encoder requires the joint design of the encoder and the binary labeling of the constellation. Since analytical approaches are unknown, the only available solution is to perform an exhaustive search over the encoder and the labeling. For large constellation sizes and/or many encoder states, however, an exhaustive search is unfeasible. Traditional TCM designs overcome this problem by using a labeling that follows the set-partitioning principle and by performing an exhaustive search over the encoders. In this paper we study binary labelings for TCM and show how they can be grouped into classes, which considerably reduces the search space in a joint design. For 8-ary constellations, the number of different binary labelings that must be tested is reduced from 8! = 40320 to 240. For the particular case of an 8-ary pulse amplitude modulation constellation, this number is further reduced to 120 and for 8-ary phase shift keying to only 30. An algorithm to generate one labeling in each class is also introduced. Asymptotically optimal TCM encoders are tabulated which are up to 0.3 dB better than the previously best known encoders.

关键词： binary reflected gray code bit-interleaved coded modulation coded modulation convolutional encoder performance bounds set-partitioning trellis-coded modulation Viterbi decoding

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Bit error probability of coherent M-ary PSK over flat Rayleigh fading channels

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ELECTRONICS LETTERS 2005年第21期41卷 1186-1187页

作者： Lassing, J Ström, EG Agrell, E Ottosson, T Qamcom Technol SE-42130 Vastra Frolunda Sweden Chalmers Univ Technol Commun Syst Grp S-41296 Gothenburg Sweden

Systems using coherent multilevel PSK to communicate over flat Rayleigh fading channels are considered. Presented is a closed-form expression for the exact bit error probability when the binary reflected gray code is used to map bits to symbols. The result is a general, simple, and correct expression, which has not been previously available in the literature. A comparison between this new expression and previous approximate expressions shows that the difference is more significant for fading channels than for Gaussian channels.

关键词： gray codes Rayleigh channels binary codes error statistics phase shiftkeying Gaussian channels binary reflected gray code bit error probability closed-formexpression coherent M-ary PSK flat Rayleigh fading channels multilevel phase shift keying

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On the Exact BER of Bit-Wise Demodulators for One-Dimensional Constellations

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IEEE TRANSACTIONS ON COMMUNICATIONS 2013年第4期61卷 1450-1459页

作者： Ivanov, Mikhail Brannstrom, Fredrik Alvarado, Alex Agrell, Erik Chalmers Univ Technol Dept Signals & Syst SE-41596 Gothenburg Sweden Univ Cambridge Dept Engn Cambridge CB2 1PZ England

The optimal bit-wise demodulator for M-ary pulse amplitude modulation (PAM) over the additive white Gaussian noise channel is analyzed in terms of uncoded bit-error rate (BER). The BER analysis is based on studying the bit patterns that form a labeling. New closed-form BER expressions for 4-PAM with any labeling are developed. Moreover, closed-form BER expressions for 11 out of 23 possible bit patterns for 8-PAM are presented, which enable us to obtain the BER for 8-PAM with some of the most popular labelings, including the binary reflected gray code and the natural binary code. Numerical results show that, regardless of the labeling, there is no difference between the optimal demodulator and the symbol-wise demodulator for any BER of practical interest (below 0.1).

关键词： Additive white Gaussian noise channel binary reflected gray code bit error probability bit-interleaved coded modulation demapper demodulator LLRs logarithmic likelihood ratio pulse-amplitude modulation uncoded transmission

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On BICM Receivers for TCM Transmission

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IEEE TRANSACTIONS ON COMMUNICATIONS 2011年第10期59卷 2692-2702页

作者： Alvarado, Alex Szczecinski, Leszek Agrell, Erik Chalmers Univ Technol Dept Signals & Syst SE-41296 Gothenburg Sweden INRS EMT Inst Natl Rech Sci Montreal PQ H5A 1K6 Canada

Recent results have shown that the performance of bit-interleaved coded modulation (BICM) using convolutional codes in nonfading channels can be significantly improved when the interleaver takes a trivial form (BICM-T), i.e., when it does not interleave the bits at all. In this paper, we give a formal explanation for these results and show that BICM-T is, in fact, the combination of a TCM transmitter and a BICM receiver. To predict the performance of BICM-T, a new type of distance spectrum for convolutional codes is introduced, analytical bounds based on this spectrum are developed, and asymptotic approximations are presented. It is shown that the free Hamming distance of the code is not the relevant optimization criterion for BICM-T. Asymptotically optimal convolutional codes for different constraint lengths are tabulated and BICM-T is shown to offer asymptotic gains of about 2 dB over traditional BICM designs based on random interleavers. The asymptotic gains over uncoded transmission are found to be the same as those obtained by Ungerboeck's one-dimensional trellis-coded modulation (1D-TCM), and therefore, in nonfading channels, BICM-T is shown to be as good as 1D-TCM.

关键词： Bit-interleaved coded Modulation binary reflected gray code coded Modulation Convolutional codes Interleaver Quadrature Amplitude Modulation Pulse Amplitude Modulation Set Partitioning Trellis coded Modulation

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Linearly Separable Subcode: A Novel Output Label with High Separability for Biometric Discretization

Linearly Separable SubCode: A Novel Output Label with High S...

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5th IEEE Conference on Industrial Electronics and Applications

作者： Lim, Meng-Hui Teoh, Andrew Beng Jin Yonsei Univ Sch Elect & Elect Engn Coll Engn Seoul 120749 South Korea

ISBN: (纸本)9781424450466

Separability in a code is crucial for guaranteeing a decent separation between multiple classes in terms of Hamming distance. In biometric discretization, this property is of particular useful for preserving dissimilarity in the continuous feature space when a feature is mapped to a code space. binary reflected gray code (BRGC), a widely employed output label in discretization schemes, however, does not appear to exhibit such a property, yielding an indefinite mapping process that impedes the overall recognition performance. This paper explores and analyzes the separability of BRGC and put forward a class of coding scheme which exhibits high separability capacity.

关键词： Biometric Discretization Separability Output Label binary reflected gray code

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