Expurgated Error Exponent Research Articles

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

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.

Error Exponents of Typical Random Trellis Codes

In continuation to an earlier work, where error exponents of typical random codes were studied in the context of general block coding, with no underlying structure, here we carry out a parallel study on typical random, time–varying trellis codes, focusing on a certain range of low rates. By analyzing an upper bound to the error probability of the typical random trellis code, using the method of types, we first derive a Csiszar–style error exponent formula (with respect to the constraint length), which allows to characterize properties of good codes and dominant error events. We also derive a Gallager–style form, which turns out to be related to the expurgated error exponent. The main result is further extended to channels with memory and mismatch.

Information-Theoretic Characterization of MIMO Systems With Multiple Rayleigh Scattering

We present an information-theoretic analysis of a point-to-point multiple-input multiple-output (MIMO) link affected by Rayleigh fading and multiple scattering, under perfect channel state information at the receiver. Unlike previous work addressing this setting, we investigate the random coding error exponent, its associated cutoff rate and the expurgated error exponent, and derive closed-form expressions for them. Moreover, leveraging the average mutual information expression presented by Akemann et al. , we derive another important metric, namely, the sum rate, under linear receive processing and independent stream decoding. In particular, we characterize the performance of the minimum mean squared error receiver in closed form, and that of the zero forcing receiver by resorting to bounding techniques. The bulk of the work relies on results about finite-dimensional random matrix products, a number of which are novel and detailed in the Appendices. The analysis, validated through numerical results, highlights the severe degradation in the performance of linear receivers due to multi-fold scattering. It also unveils the performance trend of multiple scattering MIMO channels as a function of the number of antennas and the number of scattering stages.

Error exponents analysis of dual‐hop η ‐ μ and κ ‐ μ fading channel with amplify‐and‐forward relaying

In this study, the authors investigate the Gallager's error exponents of dual-hop amplify-and-forward systems over generalised η-μ and κ-μ fading channels, two versatile channel models which encompass a number of popular fading channels such as Rayleigh, Rician, Nakagami-m, Hoyt and one-sided Gaussian fading channels. The authors present new analytical expressions for the probability density function of the end-to-end signal-to-noise-ratio (SNR) of the system. These analytical expressions are then applied to analyse the system performance through the study of Gallager's exponents, which are classical tight bounds of error exponents and present the tradeoff between practical information rate and the reliability of communication. Two types of Gallager's exponents, namely, random coding error exponent and expurgated error exponent, are studied. Based on the newly derived analytical expressions, the authors provide an efficient method to compute the required codeword length to achieve a predefined upper bound of error probability. In addition, the analytical expressions are derived for the cutoff rate and ergodic capacity of the system. Moreover, simplified expressions are presented at the high SNR regime. All the analytical results are verified via Monte–Carlo simulations.

Error exponents for multi-keyhole MIMO channels

Along with the channel capacity, the error exponent is one of the most important information-theoretic measures of reliability, because it sets ultimate bounds on the performance of communication systems employing codes of finite complexity. In this paper, we derive closed-form expressions for the Gallager random coding and expurgated error exponents for multi-keyhole multiple-input multiple-output (MIMO) channels, which provide insights into a fundamental tradeoff between the communication reliability and information rate. We investigate the effect of keyholes on the error exponents and cutoff rate. Moreover, without an extensive Monte-Carlo simulation we can easily compute the codeword length necessary to achieve a predefined error probability at a given rate, which quantifies the effects of the number of antennas, channel coherence time, and the number of keyholes. In addition, we derive exact closed-form expressions for the ergodic capacity and cutoff rate based on the easily computable Meijer G-function. Finally, we extend our study to Rayleigh-product MIMO channels and keyhole MIMO channels.

Group Codes Outperform Binary-Coset Codes on Nonbinary Symmetric Memoryless Channels

Typical minimum distances and error exponents are analyzed on the 8-PSK Gaussian channel for two capacity-achieving code ensembles with different algebraic structure. It is proved that the typical group code over the cyclic group of order eight achieves both the Gilbert-Varshamov bound and the expurgated error exponent. On the other hand, the typical binary-coset codes (under any labeling) is shown to be bounded away both from the Gilbert-Varshamov bound (at any rate) and the expurgated exponent (at low rates). The reason for this phenomenon is shown to rely on the symmetry structure of the 8-PSK constellation, which is known to match the cyclic group of order eight, but not the direct product of three copies of the binary group. The presented results indicate that designing group codes matching the symmetry of the channel guarantees better typical-code performance than designing codes whose algebraic structure does not match the channel. This contrasts the well-known fact that the average binary-coset code achieves both the capacity and the random-coding error exponent of any discrete memoryless channel.

Random codes: minimum distances and error exponents

Minimum distances, distance distributions, and error exponents on a binary-symmetric channel (BSC) are given for typical codes from Shannon's random code ensemble and for typical codes from a random linear code ensemble. A typical random code of length N and rate R is shown to have minimum distance N/spl delta//sub GV/(2R), where /spl delta//sub GV/(R) is the Gilbert-Varshamov (GV) relative distance at rate R, whereas a typical linear code (TLC) has minimum distance N/spl delta//sub GV/(R). Consequently, a TLC has a better error exponent on a BSC at low rates, namely, the expurgated error exponent.