Victoria Kostina

Fixed-Length Lossy Compression in the Finite Blocklength Regime

This paper studies the minimum achievable source coding rate as a function of blocklength <i>n</i> and probability ϵ that the distortion exceeds a given level <i>d</i> . Tight general achievability and converse bounds are derived that hold at arbitrary fixed blocklength. For stationary memoryless sources with separable distortion, the minimum rate achievable is shown to be closely approximated by <i>R</i>(<i>d</i>) + √<i>V</i>(<i>d</i>)/(<i>n</i>) <i>Q</i>^-1(ϵ), where <i>R</i>(<i>d</i>) is the rate-distortion function, <i>V</i>(<i>d</i>) is the rate dispersion, a characteristic of the source which measures its stochastic variability, and <i>Q</i>^-1(·) is the inverse of the standard Gaussian complementary cumulative distribution function.

Lossy Joint Source-Channel Coding in the Finite Blocklength Regime

This paper finds new tight finite-blocklength bounds for the best achievable lossy joint source-channel code rate, and demonstrates that joint source-channel code design brings considerable performance advantage over a separate one in the nonasymptotic regime. A joint source-channel code maps a block of k source symbols onto a length-n channel codeword, and the fidelity of reproduction at the receiver end is measured by the probability e that the distortion exceeds a given threshold d. For memoryless sources and channels, it is demonstrated that the parameters of the best joint source-channel code must satisfy nC - kR(d) ≈ √(nV + k V(d)) Q-1(e), where C and V are the channel capacity and channel dispersion, respectively; R(d) and V(d) are the source rate-distortion and rate-dispersion functions; and Q is the standard Gaussian complementary cumulative distribution function. Symbol-by-symbol (uncoded) transmission is known to achieve the Shannon limit when the source and channel satisfy a certain probabilistic matching condition. In this paper, we show that even when this condition is not satisfied, symbol-by-symbol transmission is, in some cases, the best known strategy in the nonasymptotic regime.

Error Rates of the Maximum-Likelihood Detector for Arbitrary Constellations: Convex/Concave Behavior and Applications

Motivated by a recent surge of interest in convex optimization techniques, convexity/concavity properties of error rates of the maximum likelihood detector operating in the AWGN channel are studied and extended to frequency-flat slow-fading channels. Generic conditions are identified under which the symbol error rate (SER) is convex/concave for arbitrary multidimensional constellations. In particular, the SER is convex in SNR for any one- and two-dimensional constellation, and also in higher dimensions at high SNR. Pairwise error probability and bit error rate are shown to be convex at high SNR, for arbitrary constellations and bit mapping. Universal bounds for the SER first and second derivatives are obtained, which hold for arbitrary constellations and are tight for some of them. Applications of the results are discussed, which include optimum power allocation in spatial multiplexing systems, optimum power/time sharing to decrease or increase (jamming problem) error rate, an implication for fading channels (¿fading is never good in low dimensions¿) and optimization of a unitary-precoded OFDM system. For example, the error rate bounds of a unitary-precoded OFDM system with QPSK modulation, which reveal the best and worst precoding, are extended to arbitrary constellations, which may also include coding. The reported results also apply to the interference channel under Gaussian approximation, to the bit error rate when it can be expressed or approximated as a nonnegative linear combination of individual symbol error rates, and to coded systems.

On optimum power allocation for the V-BLAST

A unified analytical framework for optimum power allocation in the unordered V-BLAST algorithm and its comparative performance analysis are presented. Compact closed-form approximations for the optimum power allocation are derived, based on average total and block error rates. The choice of the criterion has little impact on the power allocation and, overall, the optimum strategy is to allocate more power to lower step transmitters and less to higher ones. High-SNR approximations for optimized average block and total error rates are given. The SNR gain of optimization is rigorously defined and studied using analytical tools, including lower and upper bounds, high and low SNR approximations. The gain is upper bounded by the number of transmit antennas, for any modulation format and type of fading channel. While the average optimization is less complex than the instantaneous one, its performance is almost as good at high SNR. A measure of robustness of the optimized algorithm is introduced and evaluated. The optimized algorithm is shown to be robust to perturbations in individual and total transmit powers. Based on the algorithm robustness, a pre-set power allocation is suggested as a low-complexity alternative to the other optimization strategies, which exhibits only a minor loss in performance over the practical SNR range.

Lossy joint source-channel coding in the finite blocklength regime

This paper finds new tight finite-blocklength bounds for the best achievable lossy joint source-channel code rate, and demonstrates that joint source-channel code design brings considerable performance advantage over a separate one in the nonasymptotic regime. A joint source-channel code maps a block of <i>k</i> source symbols onto a length-<i>n</i> channel codeword, and the fidelity of reproduction at the receiver end is measured by the probability ε that the distortion exceeds a given threshold <i>d</i>. For memoryless sources and channels, it is demonstrated that the parameters of the best joint source-channel code must satisfy <i>nC</i> - <i>kR</i>(<i>d</i>) ≈ √(<i>nV</i> + <i>k V</i>(<i>d</i>)) <i>Q</i>^-1(ε), where <i>C</i> and <i>V</i> are the channel capacity and channel dispersion, respectively; <i>R</i>(<i>d</i>) and <i>V</i>(<i>d</i>) are the source rate-distortion and rate-dispersion functions; and <i>Q</i> is the standard Gaussian complementary cumulative distribution function. Symbol-by-symbol (uncoded) transmission is known to achieve the Shannon limit when the source and channel satisfy a certain probabilistic matching condition. In this paper, we show that even when this condition is not satisfied, symbol-by-symbol transmission is, in some cases, the best known strategy in the nonasymptotic regime.

Fixed-length lossy compression in the finite blocklength regime: Discrete memoryless sources

This paper studies the minimum achievable source coding rate as a function of blocklength n and tolerable distortion level d. Tight general achievability and converse bounds are derived that hold at arbitrary fixed blocklength. For stationary memoryless sources with separable distortion, the minimum rate achievable is shown to be q closely approximated by equation, where R(d) is the rate-distortion function, V (d) is the rate dispersion, a characteristic of the source which measures its stochastic variability, Q−1 (·) is the inverse of the standard Gaussian complementary cdf, and ε is the probability that the distortion exceeds d. The new bounds and the second-order approximation of the minimum achievable rate are evaluated for the discrete memoryless source with symbol error rate distortion. In this case, the second-order approximation reduces to equation if the source is non-redundant.

Channels With Cost Constraints: Strong Converse and Dispersion

This paper shows the strong converse and the dispersion of memoryless channels with cost constraints and performs a refined analysis of the third-order term in the asymptotic expansion of the maximum achievable channel coding rate, showing that it is equal to (1/2)((log n)/n) in most cases of interest. The analysis is based on a nonasymptotic converse bound expressed in terms of the distribution of a random variable termed the b-tilted information density, which plays a role similar to that of the d-tilted information in lossy source coding. We also analyze the fundamental limits of lossy joint-source-channel coding over channels with cost constraints.

Channels with cost constraints: Strong converse and dispersion

This paper shows the strong converse and the dispersion of memoryless channels with cost constraints. The analysis is based on a new non-asymptotic converse bound expressed in terms of the distribution of a random variable termed the b-tilted information density, which plays a role similar to that of the information density in channel coding without cost constraints. We also analyze the fundamental limits of lossy joint-source-channel coding over channels with cost constraints.

Data compression with low distortion and finite blocklength

This paper considers lossy source coding of n-dimensional continuous memoryless sources with low mean-square error distortion and shows a simple, explicit approximation to the minimum source coding rate. More precisely, a nonasymptotic version of Shannons lower bound is presented. Lattice quantizers are shown to approach that lower bound, provided that the source density is smooth enough and the distortion is low, which implies that fine multidimensional lattice coverings are nearly optimal in the rate-distortion sense even at finite n. The achievability proof technique avoids both the usual random coding argument and the simplifying assumption of the presence of a dither signal.

Optimum Power and Rate Allocation for Coded V-BLAST: Average Optimization

An analytical framework for performance analysis and optimization of coded V-BLAST is developed. Average power and/or rate allocations to minimize the outage probability as well as their robustness and dual problems are investigated. Compact, closed-form expressions for the optimum allocations and corresponding system performance are given. The uniform power allocation is shown to be near optimum in the low outage regime in combination with the optimum rate allocation. The average rate allocation provides the largest performance improvement (extra diversity gain), and the average power allocation offers a modest SNR gain limited by the number of transmit antennas but does not increase the diversity gain. The dual problems are shown to have the same solutions as the primal ones. All these allocation strategies are shown to be robust. The reported results also apply to coded multiuser detection and channel equalization systems relying on successive interference cancellation.