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The Generalized Stochastic likelihood decoder: Random Coding and Expurgated Bounds

IEEE TRANSACTIONS ON INFORMATION THEORY 2017年第8期63卷 5039-5051页

作者： Merhav, Neri Technion Israel Inst Technol Dept Elect Engn IL-32000 Haifa Israel

The likelihood decoder is a stochastic decoder that selects the decoded message at random, using the posterior distribution of the true underlying message given the channel output. In this paper, we study a generalized version of this decoder, where the posterior is proportional to a general function that depends only on the joint empirical distribution of the output vector and the code word. This framework allows both mismatched versions and universal versions of the likelihood decoder, as well as the corresponding ordinary deterministic decoders, among many others. We provide a direct analysis method that yields the exact random coding exponent (as opposed to separate upper bounds and lower bounds that turn out to be compatible, which were derived earlier by Scarlett et al.). We also extend the result from pure channel coding to combined source and channel coding (random binning followed by random channel coding) with side information available to the decoder. Finally, returning to pure channel coding, we derive also an expurgated exponent for the stochastic likelihood decoder, which turns out to be at least as tight (and in some cases, strictly so) as the classical expurgated exponent of the maximum likelihood decoder, even though the stochastic likelihood decoder is suboptimal.

关键词： Stochastic decoder likelihood decoder random coding exponent expurgated exponent random binning source-channel coding

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A Lagrange-Dual Lower Bound to the Error Exponent of the Typical Random Code

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IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第6期66卷 3456-3464页

作者： Merhav, Neri Technion Israel Inst Technol IIT Andrew & Erna Viterbi Fac Elect Engn IL-3200003 Haifa Israel

A Lagrange-dual (Gallager-style) lower bound is derived for the error exponent function of the typical random code (TRC) pertaining to the i.i.d. random coding ensemble and mismatched stochastic likelihood decoding. While the original expression, derived from the method of types (the Csiszar-style expression) involves minimization over probability distributions defined on the channel input-output alphabets, the new Lagrange-dual formula involves optimization of five parameters, independently of the alphabet sizes. For both stochastic and deterministic mismatched decoding (including maximum likelihood decoding as a special case), we provide a rather comprehensive discussion on the insight behind the various ingredients of this formula and describe how its behavior varies as the coding rate exhausts the relevant range. Among other things, it is demonstrated that this expression simultaneously generalizes both the expurgated error exponent function (at zero rate) and the classical random coding exponent function at high rates, where it also meets the sphere-packing bound.

关键词： Encoding Maximum likelihood decoding Probability distribution Minimization Optimization Error probability Error exponent typical random code Lagrange duality mismatched decoder likelihood decoder

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Error Exponents of Typical Random Codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2018年第9期64卷 6223-6235页

作者： Merhav, Neri Technion Israel Inst Technol Andrew & Erna Viterbi Fac Elect Engn Techn IL-3200003 Haifa Israel

We define the error exponent of the typical random code (TRC) as the long-block limit of the negative normalized expectation of the logarithm of the error probability of the random code, as opposed to the traditional random coding error exponent, which is the limit of the negative normalized logarithm of the expectation of the error probability. For the ensemble of uniformly randomly drawn fixed composition codes, we provide exact error exponents of TRCs for a general discrete memoryless channel and a wide class of (stochastic) decoders, collectively referred to as the generalized likelihood decoder (GLD). This ensemble of fixed composition codes is shown to be no worse than any other ensemble of independent codewords that are drawn under a permutation-invariant distribution (e.g., independent identically distributed codewords). We also present relationships between the error exponent of the TRC and the ordinary random coding error exponent, as well as the expurgated exponent for the GLD. Finally, we demonstrate that our analysis technique is applicable also to more general communication scenarios, such as list decoding (for fixed-size lists) as well as the decoding with an erasure/list option in Forney's sense.

关键词： Error exponent typical random code expurgated exponent quenched average likelihood decoder

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Expurgated Bounds for the Asymmetric Broadcast Channel

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IEEE TRANSACTIONS ON INFORMATION THEORY 2019年第6期65卷 3412-3435页

作者： Averbuch, Ran Weinberger, Nir Merhav, Neri Technion Israel Inst Technol Andrew & Erna Viterbi Fac Elect Engn IL-32000 Haifa Israel MIT IDSS & LIDS 77 Massachusetts Ave Cambridge MA 02139 USA

This paper contains two main contributions concerning the expurgation of hierarchical ensembles for the asymmetric broadcast channel. The first is an analysis of the optimal maximum likelihood (ML) decoders for the weak and strong user. Two different methods of code expurgation will be used, that will provide two competing error exponents. The second is the derivation of expurgated exponents under the generalized stochastic likelihood decoder (GLD). We prove that the expurgated exponents achieved for the hierarchical ensemble under GLD decoding are at least as good as the maximum between the random coding error exponents derived in an earlier work by Averbuch and Merhav (2018) and one of our ML-based expurgated exponents.

关键词： Asymmetric broadcast channel error exponent expurgated exponent likelihood decoder

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