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82
On MaximumLikelihood Detection and the Search for the Closest Lattice Point
 IEEE TRANS. INFORM. THEORY
, 2003
"... Maximumlikelihood (ML) decoding algorithms for Gaussian multipleinput multipleoutput (MIMO) linear channels are considered. Linearity over the field of real numbers facilitates the design of ML decoders using numbertheoretic tools for searching the closest lattice point. These decoders are colle ..."
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Cited by 273 (9 self)
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Maximumlikelihood (ML) decoding algorithms for Gaussian multipleinput multipleoutput (MIMO) linear channels are considered. Linearity over the field of real numbers facilitates the design of ML decoders using numbertheoretic tools for searching the closest lattice point. These decoders are collectively referred to as sphere decoders in the literature. In this paper, a fresh look at this class of decoding algorithms is taken. In particular, two novel algorithms are developed. The first algorithm is inspired by the Pohst enumeration strategy and is shown to offer a significant reduction in complexity compared to the ViterboBoutros sphere decoder. The connection between the proposed algorithm and the stack sequential decoding algorithm is then established. This connection is utilized to construct the second algorithm which can also be viewed as an application of the SchnorrEuchner strategy to ML decoding. Aided with a detailed study of preprocessing algorithms, a variant of the second algorithm is developed and shown to offer significant reductions in the computational complexity compared to all previously proposed sphere decoders with a nearML detection performance. This claim is supported by intuitive arguments and simulation results in many relevant scenarios.
A unified framework for tree search decoding: rediscovering the sequential decoder,”
 IEEE Transactions on Information Theory,
, 2006
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Nearmaximumlikelihood detection of MIMO systems using MMSEbased latticereduction
 IEEE Conf. on Commun
, 2004
"... Abstract — In recent publications the use of latticereduction for signal detection in multiple antenna systems has been proposed. In this paper, we adopt these latticereductionaided schemes to the MMSE criterion. We show that an obvious way to do this is infeasible and propose an alternative meth ..."
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Cited by 69 (3 self)
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Abstract — In recent publications the use of latticereduction for signal detection in multiple antenna systems has been proposed. In this paper, we adopt these latticereductionaided schemes to the MMSE criterion. We show that an obvious way to do this is infeasible and propose an alternative method based on an extended system model, which in conjunction with simple successive interference cancellation nearly reaches the performance of maximumlikelihood detection. Furthermore, we demonstrate that a sorted QR decomposition can significantly reduce the computational effort associated with latticereduction. Thus, the new algorithm clearly outperforms existing methods with comparable complexity.
Complex lattice reduction algorithms for lowcomplexity MIMO detection
 IN IEEE GLOBAL TELECOMMN. CONF. (GLOBECOM
, 2006
"... Recently, latticereductionaided detectors have been proposed for multipleinput multipleoutput (MIMO) systems to give performance with full diversity like maximum likelihood receiver, and yet with complexity similar to linear receivers. However, these latticereductionaided detectors are based ..."
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Cited by 59 (7 self)
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Recently, latticereductionaided detectors have been proposed for multipleinput multipleoutput (MIMO) systems to give performance with full diversity like maximum likelihood receiver, and yet with complexity similar to linear receivers. However, these latticereductionaided detectors are based on the traditional LLL reduction algorithm that was originally introduced for reducing real lattice bases, in spite of the fact that the channel matrices are inherently complexvalued. In this paper, we introduce the complex LLL algorithm for direct application to reduce the basis of a complex lattice which is naturally defined by a complexvalued channel matrix. We prove that complex LLL reductionaided detection can also achieve full diversity. Our analysis reveals that the new complex LLL algorithm can achieve a reduction in complexity of nearly 50 % over the traditional LLL algorithm, and this is confirmed by simulation. It is noteworthy that the complex LLL algorithm aforementioned has nearly the same biterrorrate performance as the traditional LLL algorithm.
Latticereductionaided broadcast precoding
 IEEE Trans. Commun
, 2004
"... Abstract—A precoding scheme for multiuser broadcast communications is described, which fills the gap between the lowcomplexity Tomlinson–Harashima precoding and the sphere decoderbased system of Peel et al. Simulation results show that, replacing the closestpoint search with the Babai approximatio ..."
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Cited by 55 (4 self)
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Abstract—A precoding scheme for multiuser broadcast communications is described, which fills the gap between the lowcomplexity Tomlinson–Harashima precoding and the sphere decoderbased system of Peel et al. Simulation results show that, replacing the closestpoint search with the Babai approximation, the full diversity order supported by the channel is available to each user, as in the system of Peel et al., and unlike Tomlinson–Harashima precoding, which suffers some diversity penalty. The complexity of the scheme is similar to that of Tomlinson–Harashima precoding. Index Terms—Lattice reduction, multipleinput multipleoutput (MIMO) broadcast channels, MIMO precoding.
LLL reduction achieves the receive diversity in MIMO decoding
 IEEE TRANS. INFORM. THEORY
, 2007
"... Diversity order is an important measure for the performance of communication systems over MIMO fading channels. In this paper, we prove that in MIMO multiple access systems (or MIMO pointtopoint systems with VBLAST transmission), latticereductionaided decoding achieves the maximum receive diver ..."
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Cited by 42 (1 self)
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Diversity order is an important measure for the performance of communication systems over MIMO fading channels. In this paper, we prove that in MIMO multiple access systems (or MIMO pointtopoint systems with VBLAST transmission), latticereductionaided decoding achieves the maximum receive diversity (which is equal to the number of receive antennas). Also, we prove that the naive lattice decoding (which discards the outofregion decoded points) achieves the maximum diversity.
DMT optimality of LRaided linear decoders for a general class of channels, lattice designs, and system models
 IEEE TRANS. INFOM. THEORY
, 2010
"... The work identifies the first general, explicit, and nonrandom MIMO encoderdecoder structures that guarantee optimality with respect to the diversitymultiplexing tradeoff (DMT), without employing a computationally expensive maximumlikelihood (ML) receiver. Specifically, the work establishes the ..."
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Cited by 33 (4 self)
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The work identifies the first general, explicit, and nonrandom MIMO encoderdecoder structures that guarantee optimality with respect to the diversitymultiplexing tradeoff (DMT), without employing a computationally expensive maximumlikelihood (ML) receiver. Specifically, the work establishes the DMT optimality of a class of regularized lattice decoders, and more importantly the DMT optimality of their latticereduction (LR)aided linear counterparts. The results hold for all channel statistics, for all channel dimensions, and most interestingly, irrespective of the particular latticecode applied. As a special case, it is established that the LLLbased LRaided linear implementation of the MMSEGDFE lattice decoder facilitates DMT optimal decoding of any lattice code at a worstcase complexity that grows at most linearly in the data rate. This represents a fundamental reduction in the decoding complexity when compared to ML decoding whose complexity is generally exponential in rate. The results’ generality lends them applicable to a plethora of pertinent communication scenarios such as quasistatic MIMO, MIMOOFDM, ISI, cooperativerelaying, and MIMOARQ channels, in all of which the DMT optimality of the LRaided linear decoder is guaranteed. The adopted approach yields insight, and motivates further study, into joint transceiver designs with an improved SNR gap to ML decoding.
A Near Maximum Likelihood Decoding Algorithm for MIMO Systems Based on SemiDefinite Programming
, 2005
"... In MultiInput MultiOutput (MIMO) systems, MaximumLikelihood (ML) decoding is equivalent to finding the closest lattice point in an Ndimensional complex space. In general, this problem is known to be NP hard. In this paper, we propose a quasimaximum likelihood algorithm based on SemiDefinite Pr ..."
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Cited by 28 (4 self)
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In MultiInput MultiOutput (MIMO) systems, MaximumLikelihood (ML) decoding is equivalent to finding the closest lattice point in an Ndimensional complex space. In general, this problem is known to be NP hard. In this paper, we propose a quasimaximum likelihood algorithm based on SemiDefinite Programming (SDP). We introduce several SDP relaxation models for MIMO systems, with increasing complexity. We use interiorpoint methods for solving the models and obtain a nearML performance with polynomial computational complexity. Lattice basis reduction is applied to further reduce the computational complexity of solving these models. The proposed relaxation models are also used for soft output decoding in MIMO systems.
Communication Over MIMO Broadcast Channels Using LatticeBasis Reduction
, 2006
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On the Proximity Factors of Lattice ReductionAided Decoding
"... Lattice reductionaided decoding enables significant complexity saving and nearoptimum performance in multiinput multioutput (MIMO) communications. However, its remarkable performance largely remains a mystery to date. In this paper, a first step is taken towards a quantitative understanding of ..."
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Cited by 22 (7 self)
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Lattice reductionaided decoding enables significant complexity saving and nearoptimum performance in multiinput multioutput (MIMO) communications. However, its remarkable performance largely remains a mystery to date. In this paper, a first step is taken towards a quantitative understanding of its performance limit. To this aim, the proximity factors are defined to measure the worstcase gap to maximumlikelihood (ML) decoding in terms of the signaltonoise ratio (SNR) for given error rate. The proximity factors are derived analytically and found to be bounded above by a function of the dimension of the lattice alone. As a direct consequence, it follows that lattice reductionaided decoding can always achieve full receive diversity of MIMO fading channels. The study is then extended to the dualbasis reduction. It is found that in some cases reducing the dual can result in smaller proximity factors than reducing the primal basis. The theoretic bounds on the proximity factors are further compared with numerical results.