Sphere Decoding Algorithm Research Articles

Look-ahead sphere decoding: algorithm and VLSI architecture

Multiple-input-multiple-output (MIMO) systems are recognised as a key enabling technology in high-performance wireless communications; however, the implementation of high-throughput MIMO detectors still is a critical task. Among known MIMO detectors, sphere decoder algorithm (SDA) is capable of optimal performance with acceptable processing complexity. The study presents an improved SDA, which enables significant throughput increase at a very limited additional complexity and with no degradation in terms of bit error rate performance. The modified detection method, called LASDA (look-ahead SDA), is based on formal algorithm transformations, namely look-ahead, retiming and pipelining, and requires a modified tree search strategy. The VLSI design of LASDA detector supporting a 4 × 4 MIMO channel with 16 QAM modulation is detailed in the study for a 130 nm CMOS standard cell technology: synthesis results show that the proposed solution achieves an average throughput of 380 Mbps at a signal-to-noise ratio of 22 dB (Rayleigh fading channel), with an occupied silicon area of 0.18 mm 2 . Comparisons with a number of previous implementations are also provided.

Multiple-symbol M-sphere Detection based on Linear Radius for Differential Unitary Apace-time Modulation Systems

Abstract In view of the problems of high computational complexity and difficulties in pipelining and parallel processing existed in sphere decoding (SD) algorithm for multiple-symbol differential detection (MSDD) system, the M-SD detection which combined M algorithm with SD is presented.. In M-SD, a new spherical radius scheme called linear radius is proposed to reduce the retained branches. It largely reduces the times of computing Frobenius norm square by a gradually decreasing radius with signal-to-noise ratio (SNR) increases. Based on the linear radius, M algorithm is combined with SD at low SNR to maintain a stable retained branch size and further reduce complexity. Simulation results show that M-SD detection with the constraint of linear radius significantly reduces the overall complexity compared with regular SD and its complexity is more stable. , Especially the complexity of M-SD can get more than 90% reduction when SNR £ 5dB, while the bit error rate (BER) performance is very close to SD within 0.1dB loss.

Efficiency Improvement of the Fixed-complexity Sphere Decoder

In this paper, we propose two schemes to reduce the complexity of fixed-complexity sphere decoder (FSD) algorithm in the ordering and tree-search stages, respectively, while achieving quasi-ML performance. In the ordering stage, we propose a QR-decomposition-based FSD signal ordering based on the zero-forcing criterion (FSD-ZF-SQRD) that requires only a few number of additional complex flops compared to the unsorted QRD. Also, the proposed ordering algorithm is extended using the minimum mean square error (MMSE) criterion to achieve better performance. In the tree-search stage, we introduce a threshold-based complexity reduction approach for the FSD depending on the reliability of the signal with the largest noise amplification. Numerical results show that in 8 ? 8 MIMO system, the proposed FSD-ZF-SQRD and FSD-MMSE-SQRD only require 19.5% and 26.3% of the computational efforts required by Hassibi’s scheme, respectively. Moreover, a third threshold vector is outlined which can be used for high order modulation schemes. In 4 ? 4 MIMO system using 16-QAM and 64-QAM, simulation results show that when the proposed threshold-based approach is employed, FSD requires only 62.86% and 53.67% of its full complexity, respectively.

A Near-ML Complex K-Best Decoder with Efficient Search Design for MIMO Systems

A low-complexity near-ML K-Best sphere decoder is proposed. The development of the proposed K-Best sphere decoding algorithm (SDA) involves two stages. First, a new candidate sequence generator (CSG) is proposed. The CSG directly operates in the complex plane and efficiently generates sorted candidate sequences with precise path weights. Using the CSG and an associated parallel comparator, the proposed K-Best SDA can avoid performing a large amount of path weight evaluations and sorting. Next, a new search strategy based on a derived cumulative distribution function (cdf), and an associated efficient procedure is proposed. This search procedure can be directly manipulated in the complex plane and performs ML search in a few preceding layers. It is shown that incorporating detection ordering into the proposed SDA offers a systematic method for determining the numbers of required ML search layers. With the above features, the proposed SDA is shown to provide near ML performance with a lower complexity requirement than conventional K-Best SDAs.

Soft–Input Soft–Output Single Tree-Search Sphere Decoding

Soft-input soft-output (SISO) detection algorithms form the basis for iterative decoding. The computational complexity of SISO detection often poses significant challenges for practical receiver implementations, in particular in the context of multiple-input multiple-output (MIMO) wireless communication systems. In this paper, we present a low-complexity SISO sphere-decoding algorithm, based on the single tree-search paradigm proposed originally for soft-output MIMO detection in Studer (“Soft-output sphere decoding: Algorithms and VLSI implementation,” IEEE J. Sel. Areas Commun., vol. 26, no. 2, pp. 290-300, Feb. 2008). The new algorithm incorporates clipping of the extrinsic log-likelihood ratios (LLRs) into the tree-search, which results in significant complexity savings and allows to cover a large performance/complexity tradeoff region by adjusting a single parameter. Furthermore, we propose a new method for correcting approximate LLRs - resulting from sub-optimal detectors - which (often significantly) improves detection performance at low additional computational complexity.

Trellis Coded Spatial Modulation

Trellis coded modulation (TCM) is a well known scheme that reduces power requirements without any bandwidth expansion. In TCM, only certain sequences of successive constellation points are allowed (mapping by set partitioning). The novel idea in this paper is to apply the TCM concept to the antenna constellation points of spatial modulation (SM). The aim is to enhance SM performance in correlated channel conditions. SM considers the multiple transmit antennas as additional constellation points and maps a first part of a block of information bits to the transmit antenna indices. Therefore, spatial multiplexing gains are retained and spectral efficiency is boosted. The second part of the block of information bits is mapped to a complex symbol using conventional digital modulation schemes. At any particular time instant, only one antenna is active. The receiver estimates the transmitted symbol and the active antenna index and uses the two estimates to retrieve the original block of data bits. In this paper, TCM partitions the entire set of transmit antennas into sub-sets such that the spacing between antennas within a particular sub-set is maximized. The scheme is called trellis coded spatial modulation (TCSM). Tight analytical performance bounds over correlated fading channels are proposed in this paper. In addition, the performance and complexity of TCSM is compared to the performance of SM, coded V-BLAST (vertical Bell Labs layered space-time) applying near optimum sphere decoder algorithm, and Alamouti scheme combined with TCM. Also, the performance of all schemes with turbo coded modulation is presented. It is shown that under the same spectral efficiency, TCSM exhibits significant performance enhancements in the presence of realistic channel conditions such as Rician fading and spatial correlation (SC). In addition, the complexity of the proposed scheme is shown to be 80% less than the V-BLAST complexity.

Performance Comparison of Sphere Decoding and Global Search Algorithm based Multiuser Detectors for MC-CDMA system under Clipping Noise

Performance Comparison of Sphere Decoding and Global Search Algorithm based Multiuser Detectors for MC-CDMA system under Clipping Noise

A List Sphere Decoding Algorithm with Improved Radius Setting Strategies

Complexity of different sphere-decoding (SD) algorithms is heavily influenced by their own distinctive computational features. Sphere radius setting for initialization and reduction during the sear...

Implementation aspects of list sphere decoder algorithms for MIMO-OFDM systems

A list sphere decoder (LSD) can be used to approximate the optimal maximum a posteriori (MAP) detector for the detection of multiple-input multiple-output (MIMO) signals. In this paper, we consider two LSD algorithms with different search methods and study some algorithm design choices which relate to the performance and computational complexity of the algorithm. We show that by limiting the dynamic range of log-likelihood ratio, the required LSD list size can be lowered, and, thus, the complexity of the LSD algorithm is decreased. We compare the real and the complex-valued signal models and their impact on the complexity of the algorithms. We show that the real-valued signal model is clearly the less complex choice and a better alternative for implementation. We also show the complexity of the sequential search LSD algorithm can be reduced by limiting the maximum number of checked nodes without sacrificing the performance of the system. Finally, we study the complexity and performance of an iterative receiver, analyze the tradeoff choices between complexity and performance, and show that the additional computational cost in LSD is justified to get better soft-output approximation.

On further reduction of complexity in tree pruning based sphere search

In this letter, we propose an extension of the probabilistic tree pruning sphere decoding (PTP-SD) algorithm that provides further improvement of the computational complexity with minimal extra cost and negligible performance penalty. In contrast to the PTP-SD that considers the tightening of necessary conditions in the sphere search using per-layer radius adjustment, the proposed method focuses on the sphere radius control strategy when a candidate lattice point is found. For this purpose, the dynamic radius update strategy depending on the lattice point found as well as the lattice independent radius selection scheme are jointly exploited. As a result, while maintaining the effectiveness of the PTP-SD, further reduction of the computational complexity, in particular for high SNR regime, can be achieved. From simulations in multiple-input and multiple-output (MIMO) channels, it is shown that the proposed method provides a considerable improvement in complexity with near-ML performance.

Selection of Design Parameters for Generalized Sphere Decoding Algorithms

Various efficient generalized sphere decoding (GSD) algorithms have been proposed to approach optimal ML performance for underdetermined linear systems, by transforming the original problem into the full-column-rank one so that standard SD can be fully applied. However, their design parameters are heuristically set based on observation or the possibility of an ill-conditioned transformed matrix can affect their searching efficiency. This paper presents a better transformation to alleviate the ill-conditioned structure and provides a systematic approach to select design parameters for various GSD algorithms in order to high efficiency. Simulation results on the searching performance confirm that the proposed techniques can provide significant improvement.

Memory-Constrained Tree Search Detection and New Ordering Schemes

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Hardware implementations of tree search-based multiple-input multiple-output (MIMO) detection often have limited performance due to large memory requirement or high computational complexity of sophisticated MIMO detection algorithms. In this paper, we propose new tree search-based detection algorithms that achieve maximum-likelihood (ML) performance under any given memory constraints and with reduced computational complexity. To this end, we make two main contributions. First, we propose a memory-constrained tree search (MCTS) algorithm that bridges the gap between the sphere decoding (SD) and stack algorithms. Our MCTS algorithm dynamically adapts to any pre-specified memory constraint and offers a graceful tradeoff between computational complexity and memory requirement while maintaining the ML performance. When the memory size is set as the minimum, our MCTS algorithm is similar to the SD algorithm. As the memory size increases, the average computational complexity of our MCTS algorithm decreases. When the memory size becomes large, our MCTS algorithm is similar to the stack algorithm, having similar average computational complexity but requiring significantly less memory. To further reduce the computational complexity of tree search-based ML detection algorithms, we propose novel ordering schemes that can be easily embedded in the QR decomposition and take into account both the channel matrix and the received signal (noise); simulation results show that our ordering schemes lead to reduced average computational complexity for the SD and MCTS algorithms, and the reduction is significant at low to medium signal-to-noise ratio region. </para>

A low-complexity generalized sphere decoding approach for underdetermined linear communication systems: performance and complexity evaluation

For underdetermined linear systems, original sphere decoding (SD) algorithms fail due to zero diagonal elements in the upper-triangular matrix of the QR or Cholesky factorization of the underdetermined matrix. To solve this problem, this paper presents a low-complexity generalized sphere decoding (GSD) approach by transforming the original underdetermined problem into the full-column-rank one so that standard SD can be directly applied on the transformed problem. Since the introduced transformation maintains the dimension of the original problem for all M-QAM's, the proposed GSD approach provides significant reduction in complexity as compared to other GSD schemes, especially for M-QAM with large signaling constellation. Both performance and expected complexity are analyzed to provide the comprehensive relationships between the performance and complexity of the proposed GSD and its parameters. Illustrative simulation and analytical results are in good agreement in terms of both the performance and complexity and indicate that with the properly selected design parameters, the proposed GSD scheme can approach the optimum maximumlikelihood decoding (MLD) performance with low complexity for underdetermined linear communication systems including underdetermined MIMO systems, and the proposed expected complexity analysis can be used as reliable complexity estimation for practical implementation of the proposed algorithm and serve as reference for other GSD algorithms.

Multiple symbol differential detection based on sphere decoding for unitary space-time modulation

Recently, a multiple symbol differential (MSD) sphere decoding (SD) algorithm for unitary space-time modulation over quasi-static channel has been proved to achieve the performance of maximum-likelihood (ML) detection with relatively low complexity. However, an error floor occurs if the algorithm is applied over rapid-fading channels. Based on the assumption of continuous fading, a multiple symbol differential automatic sphere decoding (MSDASD) algorithm is developed by incorporating a recursive form of an ML metric into automatic SD (ASD) algorithm. Furthermore, two algorithms, termed as MSD approximate ASD (MSDAASD) and MSD pruning ASD (MSDPASD), are proposed to reduce computational complexity and the number of comparisons, respectively. Compared with the existing typical algorithms, i.e., multiple symbol differential feedback detection (MS-DFD) and noncoherent sequence detection (NSD), the performance of the proposed algorithms is much superior to that of MS-DFD and a little inferior to that of NSD, while the complexity is lower than that of MS-DFD in most cases and significantly lower than that of NSD.

Adaptive Group Detection Based on the Sort-Descending QR Decomposition for V-BLAST Architectures

Combining the sphere decoding (SD) algorithm and the sequential detection method, we propose an adaptive group detection (AGD) scheme based on the sort-descending QRD (S-D-QRD) for V-BLAST architectures over an i.i.d. Rayleigh flat fading channel. Simulation results show that the proposed scheme, which encompasses the SD algorithm and the sequential detection method as two extreme cases in a probability sense, can achieve a very flexible tradeoff between the detection performance and computational complexity by adjusting the group parameter.

An Efficient Searching Algorithm for Receive Minimum Distance in MIMO Systems with ML Receiver

A modified Schnorr-Euchner sphere decoding (SE-SD) algorithm to search for the receive minimum distance is presented. In the proposed algorithm, the visit to negative symmetric vectors of already spanned vectors is avoided by using a biased spanning, and the redundant processes to visit the all-zero vector are also eliminated. A numerical experiment shows that the modified SE-SD algorithm is much more efficient than the conventional algorithm in terms of average computational complexity.

Partial regularisation approach for detection problems in underdetermined linear systems

The maximum likelihood detection problem in many underdetermined linear communications systems can be described as an underdetermined integer least squares (ILS) problem. To solve it efficiently, a partial regularisation approach is proposed. The original underdetermined ILS problem is first transformed to an equivalent overdetermined ILS problem by using part of the transmit vector to do the regularisation. Then the overdetermined ILS problem is solved by conventional sphere decoding algorithms. Simulation results indicate that this approach can be much more efficient than other approaches for any square constellation higher than 4QAM.

Decoding OvTDM with sphere-decoding algorithm

Overlapped time division multiplexing (OvTDM) is a new type of transmission scheme with high spectrum efficiency and low threshold signal-to-noise ratio (SNR). In this article, the structure of OvTDM is introduced and the sphere-decoding algorithm of complex domain is proposed for OvTDM. Simulations demonstrate that the proposed algorithm can achieve maximum likelihood (ML) decoding with lower complexity as compared to traditional maximum likelihood sequence demodulation (MLSD) or viterbi algorithm (VA).

New List Sphere Decoding (LSD) and Iterative Synchronization Algorithms for MIMO-OFDM Detection With LDPC FEC

New sphere decoding and synchronization algorithms for multiple-input-multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems are proposed in this paper. In particular, an iterative list branch-and-bound (BB) algorithm based on the basic BB algorithm is described to obtain a candidate list to compute soft information that is used in the iterative detector. Furthermore, an improved algorithm that uses prior information from the preceding iteration to calculate the lower bound is proposed, and the candidate list is updated every iteration. To obtain a complete modem architecture, we propose an efficient expectation-maximization (EM)-based iterative algorithm for synchronization and channel estimation to interface with the proposed list-sphere-decoding detector, and we investigate the performance of the designed MIMO-OFDM modem on a realistic fading channel. The obtained performance results show that it is possible to practically design a performing MIMO-OFDM modem with high spectral efficiency, i.e., 8 bit/s/Hz with a 4times4 16-QAM MIMO-OFDM system.

On the ML Decoding of Quasi-Orthogonal Space-Time Block Codes via Sphere Decoding and Exhaustive Search

For the large family of quasi-orthogonal space-time block codes, it is not clear when is best to decode using an exhaustive search, and when the sphere decoding algorithm (SDA) should be applied. In investigating this practical issue, a generic maximum-likelihood metric expression is utilized which is more conducive to lowering the decoding complexity. It is then shown that for smaller constellations and/or number of transmit antenna, the complexity advantage of smart exhaustive search in comparison with sphere decoding can be significant.