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Error Rates of the Maximum-Likelihood Detector for Arbitrary Constellations: Convex/Concave Behavior and Applications
- IEEE TRANSACTIONS ON INFORMATION THEORY
, 2010
"... 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 ..."
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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 Outage and Error Rate Analysis of the Ordered V-BLAST
- IEEE Transactions on Wireless Communications
, 2008
"... Abstract — Outage and error rate performance of the ordered BLAST with more than 2 transmit antennas is evaluated for i.i.d. Rayleigh fading channels. A number of lower and upper bounds on the 1 st step outage probability at any SNR are derived, which are further used to obtain accurate approximatio ..."
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Abstract — Outage and error rate performance of the ordered BLAST with more than 2 transmit antennas is evaluated for i.i.d. Rayleigh fading channels. A number of lower and upper bounds on the 1 st step outage probability at any SNR are derived, which are further used to obtain accurate approximations to average block and total error rates. For m Tx antennas, the effect of the optimal ordering at the first step is an m-fold SNR gain. As m increases to infinity, the BLER decreases to zero, which is a manifestation of the space-time autocoding effect in the V-BLAST. While the sub-optimal ordering (based on the before-projection SNR) suffers a few dB SNR penalty compared to the optimal one, it has a lower computational complexity and a 3 dB SNR gain compared to the unordered V-BLAST and can be an attractive solution for low-complexity/low-energy systems. Uncoded D-BLAST exhibits the same outage and error rate performance as that of the V-BLAST. An SNR penalty of the linear receiver interfaces compared to the BLAST is also analytically evaluated. Index Terms — Multi-antenna (MIMO) system, V-BLAST, performance analysis, autocoding effect I.
Optimum Power and Rate Allocation for Coded V-BLAST
, 902
"... Abstract—An analytical framework for minimizing the outage probability of a coded spatial multiplexing system while keeping the rate close to the capacity is developed. Based on this framework, specific strategies of optimum power and rate allocation for the coded V-BLAST architecture are obtained a ..."
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Abstract—An analytical framework for minimizing the outage probability of a coded spatial multiplexing system while keeping the rate close to the capacity is developed. Based on this framework, specific strategies of optimum power and rate allocation for the coded V-BLAST architecture are obtained and its performance is analyzed. A fractional waterfilling algorithm, which is shown to optimize both the capacity and the outage probability of the coded V-BLAST, is proposed. Compact, closedform expressions for the optimum allocation of the average power are given. The uniform allocation of average power is shown to be near optimum at moderate to high SNR for the coded V-BLAST with the average rate allocation (when per-stream rates are set to match the per-stream capacity). The results reported also apply to multiuser detection and channel equalization relying on successive interference cancelation. Index Terms—Multi-antenna (MIMO) system, spatial multiplexing, coded V-BLAST, power/rate allocation, waterfilling, performance analysis I.
Error Rates of Capacity-Achieving Codes Are Convex
"... Abstract — Motivated by a wide-spread use of convex optimization techniques, convexity properties of bit error rate of the maximum likelihood detector operating in the AWGN channel are studied for arbitrary constellations and bit mappings, which also includes coding under maximum-likelihood decoding ..."
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Cited by 1 (1 self)
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Abstract — Motivated by a wide-spread use of convex optimization techniques, convexity properties of bit error rate of the maximum likelihood detector operating in the AWGN channel are studied for arbitrary constellations and bit mappings, which also includes coding under maximum-likelihood decoding. Under this generic setting, the pairwise probability of error and bit error rate are shown to be convex functions of the SNR and noise power in the high SNR/low noise regime with explicitlydetermined boundary. Any code, including capacity-achieving ones, whose decision regions include the hardened noise spheres (from the noise sphere hardening argument in the channel coding theorem) satisfies this high SNR requirement and thus has convex error rates in both SNR and noise power. We conjecture that all capacity-achieving codes have convex error rates. I.
On Convexity of Error Rates in Digital Communications
- AVAILABLE AT HTTP://ARXIV.ORG/ABS/1304.8102). 2013 IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY45
, 2013
"... Convexity properties of error rates of a class of decoders, including the maximum-likelihood/min-distance one as a special case, are studied for arbitrary constellations, bit mapping, and coding. Earlier results obtained for the additive white Gaussian noise channel are extended to a wide class of n ..."
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Convexity properties of error rates of a class of decoders, including the maximum-likelihood/min-distance one as a special case, are studied for arbitrary constellations, bit mapping, and coding. Earlier results obtained for the additive white Gaussian noise channel are extended to a wide class of noise densities, including unimodal and spherically invariant noise. Under these broad conditions, symbol and bit error rates are shown to be convex functions of the signal-to-noise ratio (SNR) in the high-SNR regime with an explicitly determined threshold, which depends only on the constellation dimension-ality and minimum distance, thus enabling an application of the powerful tools of convex optimization to such digital communi-cation systems in a rigorous way. It is the decreasing nature of the noise power density around the decision region boundaries that ensures the convexity of symbol error rates in the general case. The known high/low-SNR bounds of the convexity/concavity regions are tightened and no further improvement is shown to be possible in general. The high-SNR bound fits closely into the channel coding theorem: all codes, including capacity-achieving ones, whose decision regions include the hardened noise spheres (from the noise sphere hardening argument in the channel coding theorem), satisfy this high-SNR requirement and thus has convex error rates in both SNR and noise power. We conjecture that all capacity-achieving codes have convex error rates. Convexity prop-erties in signal amplitude and noise power are also investigated. Some applications of the results are discussed. In particular, it is shown that fading is convexity-preserving and is never good in low dimensions under spherically invariant noise, which may also include any linear diversity combining.
Performance Analysis of V-BLAST with Optimum Power Allocation
"... Abstract—Comprehensive performance analysis of the unordered V-BLAST algorithm with various power allocation strategies is presented, which makes use of analytical tools and resorts to Monte-Carlo simulations for validation purposes only. High-SNR approximations for the optimized average block and t ..."
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Abstract—Comprehensive performance analysis of the unordered V-BLAST algorithm with various power allocation strategies is presented, which makes use of analytical tools and resorts to Monte-Carlo simulations for validation purposes only. High-SNR approximations for the 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 transmitters, for any modulation format and any type of fading. This upper bound is achieved at high SNR by the considered optimization strategies. 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, including compact closed-form approximations. 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 lowcomplexity alternative to the other optimization strategies, which exhibits only a minor loss in performance over the practical SNR range. I.
Performance Analysis of Coded V-BLAST with Optimum Power and Rate Allocation
"... Abstract—Several optimization strategies for instantaneous rate and/or power allocation in the coded V-BLAST are studied analytically. Outage probabilities and system capacities of these strategies in a spatial multiplexing system are compared under generic settings. Since the conventional waterfill ..."
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Abstract—Several optimization strategies for instantaneous rate and/or power allocation in the coded V-BLAST are studied analytically. Outage probabilities and system capacities of these strategies in a spatial multiplexing system are compared under generic settings. Since the conventional waterfilling algorithm is suboptimal for the coded V-BLAST, a recently-proposed ”fractional waterfilling ” algorithm is studied, which simultaneously maximizes the system capacity and minimizes the outage probability. A comparative, closed-form performance analysis of this and other algorithms is presented, including bounds on the outage probability and its low-outage approximations. The fractional waterfilling algorithm attains the full MIMO channel diversity and outperforms the other algorithms by a wide margin. I.
Bit Error Rate is Convex at High SNR
"... Abstract — Motivated by a wide-spread use of convex optimization techniques, convexity properties of bit error rate of the maximum likelihood detector operating in the AWGN channel are studied for arbitrary constellations and bit mappings, which may also include coding under maximum-likelihood decod ..."
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Abstract — Motivated by a wide-spread use of convex optimization techniques, convexity properties of bit error rate of the maximum likelihood detector operating in the AWGN channel are studied for arbitrary constellations and bit mappings, which may also include coding under maximum-likelihood decoding. Under this generic setting, the pairwise probability of error and bit error rate are shown to be convex functions of the SNR in the high SNR regime with explicitly-determined boundary. The bit error rate is also shown to be a convex function of the noise power in the low noise/high SNR regime. I.
1 Optimum Power and Rate Allocation for Coded V-BLAST: Average Optimization
, 1010
"... Abstract—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 allocati ..."
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Abstract—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 cancelation. Index Terms—Multi-antenna (MIMO) system, spatial multiplexing, coded V-BLAST, power/rate allocation, performance analysis I.
Optimal Detection Ordering for Coded V-BLAST
"... Abstract—Optimum ordering strategies for the coded Vertical Bell Labs Layered Space-Time (V-BLAST) architecture with capacity achieving temporal codes on each stream are analytically studied, including 4 different power/rate allocation strategies among data streams. Compact closed-form solutions are ..."
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Abstract—Optimum ordering strategies for the coded Vertical Bell Labs Layered Space-Time (V-BLAST) architecture with capacity achieving temporal codes on each stream are analytically studied, including 4 different power/rate allocation strategies among data streams. Compact closed-form solutions are obtained for the case of zero-forcing (ZF) V-BLAST with two transmit antennas and necessary optimality conditions are found for the general case. The optimal rate allocation is shown to have a major impact (stronger streams are detected last) while the optimal power allocation does not alter the original Foschini ordering (stronger streams are detected first). Sufficient conditions for the optimality of the greedy ordering are established: it is optimal for the ZF V-BLAST under an optimal rate allocation with two transmit antennas at any SNR and with any number of antennas in the low and high SNR regimes. It satisfies the necessary optimality conditions for larger systems at any SNR and is nearly-optimal in many cases. An SNR gain of ordering is introduced and studied, including closed-form expressions as well as lower and upper bounds and the conditions for their achievability. For the minimum mean square error (MMSE) V-BLAST under an optimal rate allocation, any ordering is shown to deliver the same system capacity. All the results also apply to a multiple-access channel with the successive interference cancelation receiver. Index Terms—MIMO, V-BLAST, optimal ordering, successive interference cancellation. I.
