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Error Rates of the MaximumLikelihood 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 frequencyflat slowfading 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 frequencyflat slowfading 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 twodimensional 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 unitaryprecoded OFDM system. For example, the error rate bounds of a unitaryprecoded 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 VBLAST
 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 mfold SNR gain. As m increases to infinity, the BLER decreases to zero, which is a manifestation of the spacetime autocoding effect in the VBLAST. While the suboptimal ordering (based on the beforeprojection 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 VBLAST and can be an attractive solution for lowcomplexity/lowenergy systems. Uncoded DBLAST exhibits the same outage and error rate performance as that of the VBLAST. An SNR penalty of the linear receiver interfaces compared to the BLAST is also analytically evaluated. Index Terms — Multiantenna (MIMO) system, VBLAST, performance analysis, autocoding effect I.
Optimum Power and Rate Allocation for Coded VBLAST
, 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 VBLAST 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 VBLAST 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 VBLAST, 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 VBLAST with the average rate allocation (when perstream rates are set to match the perstream capacity). The results reported also apply to multiuser detection and channel equalization relying on successive interference cancelation. Index Terms—Multiantenna (MIMO) system, spatial multiplexing, coded VBLAST, power/rate allocation, waterfilling, performance analysis I.
Error Rates of CapacityAchieving Codes Are Convex
"... Abstract — Motivated by a widespread 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 maximumlikelihood decoding ..."
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Abstract — Motivated by a widespread 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 maximumlikelihood 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 capacityachieving 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 capacityachieving 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 maximumlikelihood/mindistance 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 maximumlikelihood/mindistance 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 signaltonoise ratio (SNR) in the highSNR regime with an explicitly determined threshold, which depends only on the constellation dimensionality and minimum distance, thus enabling an application of the powerful tools of convex optimization to such digital communication 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/lowSNR bounds of the convexity/concavity regions are tightened and no further improvement is shown to be possible in general. The highSNR bound fits closely into the channel coding theorem: all codes, including capacityachieving ones, whose decision regions include the hardened noise spheres (from the noise sphere hardening argument in the channel coding theorem), satisfy this highSNR requirement and thus has convex error rates in both SNR and noise power. We conjecture that all capacityachieving codes have convex error rates. Convexity properties in signal amplitude and noise power are also investigated. Some applications of the results are discussed. In particular, it is shown that fading is convexitypreserving and is never good in low dimensions under spherically invariant noise, which may also include any linear diversity combining.
Performance Analysis of VBLAST with Optimum Power Allocation
"... Abstract—Comprehensive performance analysis of the unordered VBLAST algorithm with various power allocation strategies is presented, which makes use of analytical tools and resorts to MonteCarlo simulations for validation purposes only. HighSNR approximations for the optimized average block and t ..."
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Abstract—Comprehensive performance analysis of the unordered VBLAST algorithm with various power allocation strategies is presented, which makes use of analytical tools and resorts to MonteCarlo simulations for validation purposes only. HighSNR 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 closedform approximations. The optimized algorithm is shown to be robust to perturbations in individual and total transmit powers. Based on the algorithm robustness, a preset 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 VBLAST with Optimum Power and Rate Allocation
"... Abstract—Several optimization strategies for instantaneous rate and/or power allocation in the coded VBLAST 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 VBLAST 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 VBLAST, a recentlyproposed ”fractional waterfilling ” algorithm is studied, which simultaneously maximizes the system capacity and minimizes the outage probability. A comparative, closedform performance analysis of this and other algorithms is presented, including bounds on the outage probability and its lowoutage 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 widespread 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 maximumlikelihood decod ..."
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Abstract — Motivated by a widespread 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 maximumlikelihood 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 explicitlydetermined 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 VBLAST: Average Optimization
, 1010
"... Abstract—An analytical framework for performance analysis and optimization of coded VBLAST is developed. Average power and/or rate allocations to minimize the outage probability as well as their robustness and dual problems are investigated. Compact, closedform expressions for the optimum allocati ..."
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Abstract—An analytical framework for performance analysis and optimization of coded VBLAST is developed. Average power and/or rate allocations to minimize the outage probability as well as their robustness and dual problems are investigated. Compact, closedform 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—Multiantenna (MIMO) system, spatial multiplexing, coded VBLAST, power/rate allocation, performance analysis I.
Optimal Detection Ordering for Coded VBLAST
"... Abstract—Optimum ordering strategies for the coded Vertical Bell Labs Layered SpaceTime (VBLAST) architecture with capacity achieving temporal codes on each stream are analytically studied, including 4 different power/rate allocation strategies among data streams. Compact closedform solutions are ..."
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Abstract—Optimum ordering strategies for the coded Vertical Bell Labs Layered SpaceTime (VBLAST) architecture with capacity achieving temporal codes on each stream are analytically studied, including 4 different power/rate allocation strategies among data streams. Compact closedform solutions are obtained for the case of zeroforcing (ZF) VBLAST 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 VBLAST 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 nearlyoptimal in many cases. An SNR gain of ordering is introduced and studied, including closedform expressions as well as lower and upper bounds and the conditions for their achievability. For the minimum mean square error (MMSE) VBLAST under an optimal rate allocation, any ordering is shown to deliver the same system capacity. All the results also apply to a multipleaccess channel with the successive interference cancelation receiver. Index Terms—MIMO, VBLAST, optimal ordering, successive interference cancellation. I.