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158
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 153 (3 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.
On the Complexity of Sphere Decoding in Digital Communications
 IN DIGITAL COMMUNICATIONS,” IEEE TRANSACTIONS ON SIGNAL PROCESSING, TO APPEAR
, 2005
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A unified framework for tree search decoding: rediscovering the sequential decoder
 IEEE Trans. Inform. Theory
, 2006
"... Abstract—We consider receiver design for coded transmission over linear Gaussian channels. We restrict ourselves to the class of lattice codes and formulate the joint detection and decoding problem as a closest lattice point search (CLPS). Here, a tree search framework for solving the CLPS is adopte ..."
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Cited by 50 (2 self)
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Abstract—We consider receiver design for coded transmission over linear Gaussian channels. We restrict ourselves to the class of lattice codes and formulate the joint detection and decoding problem as a closest lattice point search (CLPS). Here, a tree search framework for solving the CLPS is adopted. In our framework, the CLPS algorithm is decomposed into the preprocessing and tree search stages. The role of the preprocessing stage is to expose the tree structure in a form matched to the search stage. We argue that the forward and feedback (matrix) filters of the minimum meansquare error decision feedback equalizer (MMSEDFE) are instrumental for solving the joint detection and decoding problem in a single search stage. It is further shown that MMSEDFE filtering allows for solving underdetermined linear systems and using lattice reduction methods to diminish complexity, at the expense of a marginal performance loss. For the search stage, we present a generic method, based on the branch and bound (BB) algorithm, and show that it encompasses all existing sphere decoders as special cases. The proposed generic algorithm further allows for an interesting classification of tree search decoders, sheds more light on the structural properties of all known sphere decoders, and inspires the design of more efficient decoders. In particular, an efficient decoding algorithm that resembles the wellknown Fano sequential decoder is identified. The excellent performance–complexity tradeoff achieved by the proposed MMSEDFE Fano decoder is established via simulation results and analytical arguments in several multipleinput multipleoutput (MIMO) and intersymbol interference (ISI) scenarios. Index Terms—Closest lattice point search (CLPS), Fano decoder, lattice codes, sequential decoding, sphere decoding, tree search. I.
LowComplexity NearMaximumLikelihood Detection and Precoding for MIMO Systems using Lattice Reduction
 IEEE Information Theory Workshop 2003
, 2003
"... Abstract — We consider the latticereductionaided detection scheme for 2×2 channels recently proposed by Yao and Wornell [11]. Using an equivalent realvalued substitute MIMO channel model their lattice reduction algorithm can be replaced by the wellknown LLL algorithm, which enables the applicati ..."
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Cited by 41 (4 self)
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Abstract — We consider the latticereductionaided detection scheme for 2×2 channels recently proposed by Yao and Wornell [11]. Using an equivalent realvalued substitute MIMO channel model their lattice reduction algorithm can be replaced by the wellknown LLL algorithm, which enables the application to MIMO systems with arbitrary numbers of dimensions. We show how lattice reduction can also be favourably applied in systems that use precoding and give simulation results that underline the usefulness of this approach. I.
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 31 (2 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.
A Deterministic Single Exponential Time Algorithm for Most Lattice Problems based on Voronoi Cell Computations (Extended Abstract)
, 2009
"... We give deterministic 2O(n)time algorithms to solve all the most important computational problems on point lattices in NP, including the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), and Shortest Independent Vectors Problem (SIVP). This improves the nO(n) running time of the best pre ..."
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Cited by 31 (2 self)
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We give deterministic 2O(n)time algorithms to solve all the most important computational problems on point lattices in NP, including the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), and Shortest Independent Vectors Problem (SIVP). This improves the nO(n) running time of the best previously known algorithms for CVP (Kannan, Math. Operation Research 12(3):415440, 1987) and SIVP (Micciancio, Proc. of SODA, 2008), and gives a deterministic alternative to the 2 O(n)time (and space) randomized algorithm for SVP of (Ajtai, Kumar and Sivakumar, STOC 2001). The core of our algorithm is a new method to solve the closest vector problem with preprocessing (CVPP) that uses the Voronoi cell of the lattice (described as intersection of halfspaces) as the result of the preprocessing function. In the process, we also give algorithms for several other lattice problems, including computing the kissing number of a lattice, and computing the set of all Voronoi relevant vectors. All our algorithms are deterministic, and have 2 O(n) time and space complexity 1 1
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 30 (2 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.
SoftInput SoftOutput Lattice Sphere Decoder for Linear Channels
 Proc. of the IEEE GLOBECOM’03
, 2003
"... Soft output detection for signals transmitted on linear channels is investigated. A particular emphasis is made for signal detection on multiple antenna channels. The a posteriori information at the detector output is evaluated from a shifted spherical list of point candidates. The spherical list is ..."
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Cited by 29 (7 self)
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Soft output detection for signals transmitted on linear channels is investigated. A particular emphasis is made for signal detection on multiple antenna channels. The a posteriori information at the detector output is evaluated from a shifted spherical list of point candidates. The spherical list is centered on the maximum likelihood point, which has the great advantage of stabilizing the list size. Thus, the sphere radius is selected in order to control the list size and to cope with the boundaries of the finite multiple antenna constellation. Our new soft output sphere decoder is then applied to the computation of constrained channel capacity and to the iterative detection of a coded transmission. For example, we achieved a signaltonoise ratio at 1.25dB from capacity limit on a 44 MIMO channel with 16QAM modulation and a 4state rate 1/2 parallel turbo code.
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 26 (4 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.
On the sphere decoding algorithm II. Generalizations, secondorder statistics, and applications to communications
 IEEE Trans. Signal Processing
, 2005
"... In Part I, we found a closedform expression for the expected complexity of the sphere decoding algorithm, both for the infinite and finite lattice. We continue the discussion in this paper by generalizing the results to the complex version of the problem and using the expected complexity expression ..."
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Cited by 24 (3 self)
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In Part I, we found a closedform expression for the expected complexity of the sphere decoding algorithm, both for the infinite and finite lattice. We continue the discussion in this paper by generalizing the results to the complex version of the problem and using the expected complexity expressions to determine situations where sphere decoding is practically feasible. In particular, we consider applications of sphere decoding to detection in multiantenna systems. We show that, for a wide range of signaltonoise ratios, rates, and numbers of antennas, the expected complexity is polynomial, in fact often roughly cubic. Since many communications systems operate at noise levels for which the expected complexity turns out to be polynomial, this suggests that maximumlikelihood decoding, which was hitherto thought to be computationally intractable, can in fact be implemented in realtime—a result with many practical implications. To provide complexity information beyond the mean, we derive a closedform expression for the variance of the complexity of sphere decoding algorithm in a finite lattice. Furthermore, we consider the expected complexity of sphere decoding for channels with memory, where the latticegenerating matrix has a special Toeplitz structure. Results indicate that the expected complexity in this case is, too, polynomial over a wide range of SNRs, rates, data blocks, and channel impulse response lengths. Index Terms—Sphere decoding, wireless communications, multipleantenna systems, frequencyselective channels, expected complexity, polynomialtime complexity. 1