Closest Lattice Point Research Articles

An Analysis of Valid Nodes Distribution for Sphere Decoding in the MIMO Wireless Communication System

Sphere Decoding (SD) algorithms can achieve a quasi-maximum likelihood (ML) decoder performance over Gaussian multiple input-multiple output (MIMO) channels with much lower complexity compared to the exhaustive search method. The SD algorithm is based on a closest lattice point search over a limited search space (hypersphere). On top of that, QR-decomposition simplifies the SD linear system's matrix to be an upper triangle matrix. The solution solver then is done by searching in the exponentially expanding search tree, started from the top with only a single node then increases by M times every level (in $N_T\times N_R$ MIMO system). Fortunately, the SD algorithm shrinks its hypersphere at every level (once the level node is determined) and phases out a vast number of the candidates, remaining only specific valid nodes in the current considered level. In this work, we proposed the statistical approach for evaluating the adequate number of valid search nodes at every level of the search tree, aiming to optimize the overall computational workload. We use a massive number of inputs patterns and extensive simulation to project the number of remaining valid nodes during the searching process. The simulations have been conducted for $4\times 4$ and $8\times 8$ MIMO systems. Our results indicate that for a particular targeted BER, choosing an appropriate sphere radius is essentially important and the number of necessary calculations increases only at the middle layer and can be generically quantified regardless of the system characteristics. This finding is beneficial for the hardware implementation of the SD, where the number of computational units has to be fixed in advance.

Basis Construction for Range Estimation by Phase Unwrapping

We consider the problem of estimating the distance, or range, between two locations by measuring the phase of a sinusoidal signal transmitted between the locations. This method is only capable of unambiguously measuring range within an interval of length equal to the wavelength of the signal. To address this problem signals of multiple different wavelengths can be transmitted. The range can then be measured within an interval of length equal to the least common multiple of these wavelengths. Estimation of the range requires solution of a problem from computational number theory called the closest lattice point problem. Algorithms to solve this problem require a basis for this lattice. Constructing a basis is non-trivial and an explicit construction has only been given in the case that the wavelengths can be scaled to pairwise relatively prime integers. In this paper we present an explicit construction of a basis without this assumption on the wavelengths. This is important because the accuracy of the range estimator depends upon the wavelengths. Simulations indicate that significant improvement in accuracy can be achieved by using wavelengths that cannot be scaled to pairwise relatively prime integers.

Fast GNSS ambiguity resolution by ant colony optimisation

Global Navigation Satellite System (GNSS) carrier phase ambiguity resolution (AR) is the key technique to high precision positioning and navigation. Ant colony optimisation (ACO) as a stochastic meta-heuristic method solves combinatorial optimisation problems by construsting solutions iteratively using a colony of ants guided by pheromone trails and heuristic information. This paper seeks to explore the effectiveness of ACO to deal with the AR problem and closest lattice point problem. The performance of this new method is evaluated considering several simulated examples with different dimensions. The results show that the proposed algorithm can compete efficiently with other promising approaches to the problem and provide integer optimal solutions in often simulated scenarios. We hope that this paper provides a starting point for researches in applying ACO algorithm and other stochastic methods in the AR problem and other GNSS problems due to the simplicities involved in algebraic manipulation.

Towards the Performance of ML and the Complexity of MMSE: A Hybrid Approach for Multiuser Detection

In this paper, we consider a low-complexity multiuser detection for downlink of high speed packet access (HSPA) of universal mobile telecommunications system (UMTS). Instead of attempting to perform maximum likelihood (ML) detection of all users in multiple cells, which is impractical for battery powered mobile receiver, we utilize interference cancelled chips obtained from iterative linear minimum mean square error (LMMSE) estimation to perform a near ML detection in a reduced dimensional system. As a result, the worst case complexity of the detection process, achieved by the closest lattice point search (CLPS), is bounded to a controllable level irrespective of multipath spans. Furthermore, by exploiting the LMMSE estimate in tightening the hypersphere condition of the CLPS algorithm so called sphere decoding, we achieve significant improvement in search complexity. From simulations on realistic downlink communication scenario in HSPA systems, we show that the proposed method offers substantial performance gain over conventional receiver algorithms with reasonable complexity.

Efficient soft decoding of Reed–Solomon codes based on sphere decoding

A novel soft-decision decoding method motivated by the idea of sphere decoding is proposed for Reed–Solomon (RS) codes. Sphere decoding reduces the complexity of finding the closest lattice point to a given point by confining the search to points that fall inside a sphere around the given point. In the authors’ proposed scheme, in order to reduce the search even further, the search effort is concentrated on the most probable lattice points. To do so, they first find the most reliable positions of the codeword. Then a sphere decoder is used to select symbol values for these positions. The proposed sphere decoder chooses the acceptable symbol values for each position from a pre-determined ordered set of most probable transmitted symbols. Each time the most reliable code symbols are selected, they are used to find the rest of RS symbols. If the resulting codeword is within the search radius, it is saved as a candidate transmitted codeword. The ordering used in the algorithm helps finding the candidate codewords quickly resulting in an efficient decoding method. Simulation results indicate considerable coding gains over hard decision decoding with a feasible complexity. The performance is also superior to the soft decision Koetter–Vardy method.

Decision-Feedback Closest Lattice Point Search for UMTS HSPA System

This letter considers a low-complexity multiuser detection based on the closest lattice point search (CLPS) for high speed packet access (HSPA) system. Instead of attempting to solve the ML detection problem in the presence of intersymbol and inter-cell interference, we utilize interference cancelled chips obtained from a bidirectional decision feedback operation to detect symbols. As a result, the worst case complexity of the CLPS is bounded to a controllable level irrespective of multipath spans. From the simulation on single and multi cell downlink communications in HSPA systems, we show that the proposed method offers substantial performance gain over conventional RAKE and MMSE equalizer.

Lattice quantizers: Entropy reduction by proper tie-handling (Corresp.)

The entropy of a lattice quantizer can be lowered by breaking ties in rounding off in favor of the closest lattice point of highest probability. For a differential pulse-code modulation (DPCM) signal, this means rounding off toward the component of smallest norm. This variant of the algorithm of Conway and Sloane reduces entropy without sacrificing mean-square error.

Soft decoding techniques for codes and lattices, including the Golay code and the Leech lattice

Two kinds of algorithms are considered. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1)</tex> If *** is a binary code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> , a "soft decision" decoding algorithm for *** changes an arbitrary point of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> into a nearest codeword (nearest in Euclidean distance). <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2)</tex> Similarly, a decoding algorithm for a lattice <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Lambda</tex> in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> changes an arbitrary point of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> into a closest lattice point. Some general methods are given for constructing such algorithms, ami are used to obtain new and faster decoding algorithms for the Gosset lattice <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E_{8}</tex> , the Golay code the Leech lattice.