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34
Lattice Reduction  A survey with applications in wireless communications
, 2011
"... Lattice reduction is a powerful concept for solving diverse problems involving point lattices. Signal processing applications where lattice reduction has been successfully used include global positioning system (GPS), frequency estimation, color space estimation in JPEG pictures, and particularly da ..."
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Lattice reduction is a powerful concept for solving diverse problems involving point lattices. Signal processing applications where lattice reduction has been successfully used include global positioning system (GPS), frequency estimation, color space estimation in JPEG pictures, and particularly data detection and precoding in wireless communication systems. In this article, we first provide some background on point lattices and then give a tutorialstyle introduction to the theoretical and practical aspects of lattice reduction. We describe the most important lattice reduction algorithms and comment on their performance and computational complexity. Finally, we discuss the application of lattice reduction in wireless communications and statistical signal processing. Throughout the article, we point out open problems and interesting questions for future research.
Decoding by Embedding: Correct Decoding Radius and DMT Optimality
, 2013
"... Abstract—The closest vector problem (CVP) and shortest (nonzero) vector problem (SVP) are the core algorithmic problems on Euclidean lattices. They are central to the applications of lattices in many problems of communications and cryptography. Kannan’s embedding technique is a powerful technique fo ..."
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Abstract—The closest vector problem (CVP) and shortest (nonzero) vector problem (SVP) are the core algorithmic problems on Euclidean lattices. They are central to the applications of lattices in many problems of communications and cryptography. Kannan’s embedding technique is a powerful technique for solving the approximate CVP, yet its remarkable practical performance is not well understood. In this paper, the embedding technique is analyzed from a bounded distance decoding (BDD) viewpoint. We present two complementary analyses of the embedding technique: We establish a reduction from BDD to Hermite SVP (via unique SVP), which can be used along with any Hermite SVP solver (including, among others, the Lenstra, Lenstra and Lovász (LLL) algorithm), and show that, in the special case of LLL, it performs at least as well as Babai’s nearest plane algorithm (LLLaided SIC). The former analysis helps to explain the folklore practical observation that unique SVP is easier than standard approximate SVP. It is proven that when the LLL algorithm is employed, the embedding technique can solve the CVP provided that the noise norm is smaller than a decoding radius λ1/(2γ), where λ1 is the minimum distance of the lattice, and γ ≈ O(2 n/4). This substantially improves the previously best known correct decoding bound γ ≈ O(2 n). Focusing on the applications of BDD to decoding of multipleinput multipleoutput (MIMO) systems, we also prove that BDD of the regularized lattice is optimal in terms of the diversitymultiplexing gain tradeoff (DMT), and propose practical variants of embedding decoding which require no knowledge of the minimum distance of the lattice and/or further improve the error performance. Index Terms—closest vector problem, lattice decoding, lattice reduction, MIMO systems, shortest vector problem I.
Precoded IntegerForcing Universally Achieves the MIMO Capacity to Within a Constant Gap
, 2013
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Augmented lattice reduction for MIMO decoding
 IEEE Trans. Wireless Commun
, 2010
"... Abstract—Lattice reduction algorithms, such as the LenstraLenstraLovasz (LLL) algorithm, have been proposed as preprocessing tools in order to enhance the performance of suboptimal receivers in multipleinput multipleoutput (MIMO) communications. A different approach, introduced by Kim and Park ..."
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Abstract—Lattice reduction algorithms, such as the LenstraLenstraLovasz (LLL) algorithm, have been proposed as preprocessing tools in order to enhance the performance of suboptimal receivers in multipleinput multipleoutput (MIMO) communications. A different approach, introduced by Kim and Park, allows to combine right preprocessing and detection in a single step by performing lattice reduction on an “augmented channel matrix”. In this paper we propose an improvement of the augmented matrix approach which guarantees a better performance. We prove that our method attains the maximum receive diversity order of the channel. Simulation results evidence that it significantly outperforms LLL reduction followed by successive interference cancellation (SIC) while requiring a moderate increase in complexity. A theoretical bound on the complexity is also derived. Index Terms—Lattice reductionaided decoding, LLL algorithm, right preprocessing. I.
HKZ and Minkowski Reduction Algorithms for LatticeReductionAided MIMO Detection
"... Abstract—Recently, lattice reduction has been widely used for signal detection in multiinput multioutput (MIMO) communications. In this paper, we present three novel lattice reduction algorithms. First, using a unimodular transformation, a significant improvement on an existing HermiteKorkineZolot ..."
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Abstract—Recently, lattice reduction has been widely used for signal detection in multiinput multioutput (MIMO) communications. In this paper, we present three novel lattice reduction algorithms. First, using a unimodular transformation, a significant improvement on an existing HermiteKorkineZolotareffreduction algorithm is proposed. Then, we present two practical algorithms for constructing Minkowskireduced bases. To assess the output quality, we compare the orthogonality defect of the reduced bases produced by LLL algorithm and our new algorithms, and find that in practice Minkowskireduced basis vectors are the closest to being orthogonal. An errorrate analysis of suboptimal decoding algorithms aided by different reduction notions is also presented. To this aim, the proximity factor is employed as a measurement. We improve some existing results and derive upper bounds for the proximity factors of Minkowskireductionaided decoding (MRAD) to show that MRAD can achieve the same diversity order with infinite lattice decoding (ILD). Index Terms—HKZ, lattice reduction, LLL, MIMO detection, Minkowski, proximity factors.
Channel state feedback over the MIMOMAC
 in Proc. IEEE Int. Symp. Information Theory
, 2009
"... We consider the problem of designing low latency and low complexity schemes for channel state feedback over the MIMOMAC (multipleinput multipleoutput multiple access channel). We develop a framework for analyzing this problem in terms of minimizing the MSE distortion, and come up with separated s ..."
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We consider the problem of designing low latency and low complexity schemes for channel state feedback over the MIMOMAC (multipleinput multipleoutput multiple access channel). We develop a framework for analyzing this problem in terms of minimizing the MSE distortion, and come up with separated sourcechannel schemes and joint sourcechannel schemes that perform better than analog feedback. We also develop a strikingly simple code design based on scalar quantization and uncoded QAM modulation that achieves the theoretical asymptotic performance limit of the separated approach with very low complexity and latency, in the case of singleantenna users.
IntegerForcing Architectures for MIMO: Distributed Implementation and SIC
"... Abstract—Linear receivers are often used in multipleantenna systems due to ease of implementation. However, traditional linear receivers such as the Decorrelator and the linear minimummean squared error (MMSE) receiver often have a significant performance loss compared to the optimal joint maximum ..."
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Abstract—Linear receivers are often used in multipleantenna systems due to ease of implementation. However, traditional linear receivers such as the Decorrelator and the linear minimummean squared error (MMSE) receiver often have a significant performance loss compared to the optimal joint maximum likelihood (ML) receiver. In previous work, we proposed the IntegerForcing linear receiver, which bridges the rate gap between traditional linear receivers and the joint ML receiver at the cost of some additional signal processing. In this paper, we examine a distributed implementation of the IntegerForcing architecture where the frontend linear receiver is eliminated. This reduces the signal processing complexity at the receiver side and allows for distribution in the MIMO system. Our results show that although this distributedness does come at a price in performance, the IntegerForcing architecture still achieves both rate and diversity gains over traditional linear architectures. We also propose the use of Successive Interference Cancellation (SIC) in the IntegerForcing Linear Receiver. I.
Achieving a vanishing SNRgap to exact lattice decoding at a subexponential complexity
, 2012
"... Abstract—The work identifies the first lattice decoding solution that achieves, in the general outagelimited MIMO setting and in the highrate and highSNR limit, both a vanishing gap to the errorperformance of the (DMT optimal) exact solution of preprocessed lattice decoding, as well as a computa ..."
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Abstract—The work identifies the first lattice decoding solution that achieves, in the general outagelimited MIMO setting and in the highrate and highSNR limit, both a vanishing gap to the errorperformance of the (DMT optimal) exact solution of preprocessed lattice decoding, as well as a computational complexity that is subexponential in the number of codeword bits. The proposed solution employs lattice reduction (LR)aided regularized (lattice) sphere decoding and proper timeout policies. These performance and complexity guarantees hold for most MIMO scenarios, all reasonable fading statistics, all channel dimensions and all fullrate lattice codes. In sharp contrast to the above very manageable complexity, the complexity of other standard preprocessed lattice decoding solutions is revealed here to be extremely high. Specifically the work is first to quantify the complexity of these lattice (sphere) decoding solutions and to prove the surprising result that the complexity required to achieve a certain ratereliability performance, is exponential in the lattice dimensionality and in the number of codeword bits, and it in fact matches, in common scenarios, the complexity of MLbased solutions. Through this sharp contrast, the work was able to, for the first time, rigorously demonstrate and quantify the pivotal role of lattice reduction as a special complexity reducing ingredient. Finally the work analytically refines transceiver DMT analysis which generally fails to address potentially massive gaps between theory and practice. Instead the adopted vanishing gap condition guarantees that the decoder’s error curve is arbitrarily close, given a sufficiently high SNR, to the optimal error curve of exact solutions, which is a much stronger condition than DMT optimality which only guarantees an error gap that is subpolynomial in SNR, and can thus be unbounded and generally unacceptable for practical implementations. I.
An Improved LRaided KBest Algorithm for MIMO Detection
 Proceeding IEEE International Conference on Wireless Communication and Signal Processing (WCSP
, 2012
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