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Table of Integrals, Series, and Products

About: The article was published on 2007-01-01 and is currently open access. It has received 4923 citations till now. The article focuses on the topics: Table (landform).
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Journal ArticleDOI
TL;DR: In this paper, the tradeoff between the energy efficiency and spectral efficiency of a single-antenna system is quantified for a channel model that includes small-scale fading but not large scale fading, and it is shown that the use of moderately large antenna arrays can improve the spectral and energy efficiency with orders of magnitude compared to a single antenna system.
Abstract: A multiplicity of autonomous terminals simultaneously transmits data streams to a compact array of antennas. The array uses imperfect channel-state information derived from transmitted pilots to extract the individual data streams. The power radiated by the terminals can be made inversely proportional to the square-root of the number of base station antennas with no reduction in performance. In contrast if perfect channel-state information were available the power could be made inversely proportional to the number of antennas. Lower capacity bounds for maximum-ratio combining (MRC), zero-forcing (ZF) and minimum mean-square error (MMSE) detection are derived. An MRC receiver normally performs worse than ZF and MMSE. However as power levels are reduced, the cross-talk introduced by the inferior maximum-ratio receiver eventually falls below the noise level and this simple receiver becomes a viable option. The tradeoff between the energy efficiency (as measured in bits/J) and spectral efficiency (as measured in bits/channel use/terminal) is quantified for a channel model that includes small-scale fading but not large-scale fading. It is shown that the use of moderately large antenna arrays can improve the spectral and energy efficiency with orders of magnitude compared to a single-antenna system.

2,770 citations

Journal ArticleDOI
TL;DR: In this article, the authors proposed a new approach to sparsity called the horseshoe estimator, which is a member of the same family of multivariate scale mixtures of normals.
Abstract: This paper proposes a new approach to sparsity called the horseshoe estimator. The horseshoe is a close cousin of other widely used Bayes rules arising from, for example, double-exponential and Cauchy priors, in that it is a member of the same family of multivariate scale mixtures of normals. But the horseshoe enjoys a number of advantages over existing approaches, including its robustness, its adaptivity to dierent sparsity patterns, and its analytical tractability. We prove two theorems that formally characterize both the horseshoe’s adeptness at large outlying signals, and its super-ecient rate of convergence to the correct estimate of the sampling density in sparse situations. Finally, using a combination of real and simulated data, we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full Bayesian model-averaging approach using a discrete mixture prior to model signals and noise.

1,260 citations

Journal ArticleDOI
TL;DR: The idea of space-time coding devised for multiple-antenna systems to the problem of communications over a wireless relay network with Rayleigh fading channels is applied and it is shown that for high SNR, the pairwise error probability (PEP) behaves as (logP/P)min{TH}, with T the coherence interval.
Abstract: We apply the idea of space-time coding devised for multiple-antenna systems to the problem of communications over a wireless relay network with Rayleigh fading channels. We use a two-stage protocol, where in one stage the transmitter sends information and in the other, the relays encode their received signals into a "distributed" linear dispersion (LD) code, and then transmit the coded signals to the receive node. We show that for high SNR, the pairwise error probability (PEP) behaves as (logP/P)min{TH}, with T the coherence interval, that is, the number of symbol periods during which the channels keep constant, R the number of relay nodes, and P the total transmit power. Thus, apart from the log P factor, the system has the same diversity as a multiple-antenna system with R transmit antennas, which is the same as assuming that the R relays can fully cooperate and have full knowledge of the transmitted signal. We further show that for a network with a large number of relays and a fixed total transmit power across the entire network, the optimal power allocation is for the transmitter to expend half the power and for the relays to collectively expend the other half. We also show that at low and high SNR, the coding gain is the same as that of a multiple-antenna system with R antennas. However, at intermediate SNR, it can be quite different, which has implications for the design of distributed space-time codes

1,050 citations

Journal ArticleDOI
TL;DR: The optimal detector for the so-called spatial modulation (SM) system introduced by Mesleh et al. in 2006 is derived, and it is shown that SM with the optimal detector achieves performance gains over popular multiple antenna systems, making it an excellent candidate for future wireless communication standards.
Abstract: In this letter, we derive the optimal detector for the so-called spatial modulation (SM) system introduced by Mesleh et al. in (Mesleh, 2006). The new detector performs significantly better than the original (~ 4 dB gain), and we support our results by deriving a closed form expression for the average bit error probability. As well, we show that SM with the optimal detector achieves performance gains (~ 1.5 - 3 dB) over popular multiple antenna systems, making it an excellent candidate for future wireless communication standards.

875 citations

Journal ArticleDOI
TL;DR: Several SNR-suboptimal multiple relay selection schemes are proposed, whose complexity is linear in the number of relays and are proved to achieve full diversity.
Abstract: This paper is on relay selection schemes for wireless relay networks. First, we derive the diversity of many single-relay selection schemes in the literature. Then, we generalize the idea of relay selection by allowing more than one relay to cooperate. The SNR-optimal multiple relay selection scheme can be achieved by exhaustive search, whose complexity increases exponentially in the network size. To reduce the complexity, several SNR-suboptimal multiple relay selection schemes are proposed, whose complexity is linear in the number of relays. They are proved to achieve full diversity. Simulation shows that they perform much better than the corresponding single relay selection methods and very close to the SNR-optimal multiple relay selection scheme. In addition, for large networks, these multiple relay selection schemes require the same amount of feedback bits from the receiver as single relay selection schemes.

739 citations