List Sphere Decoder Research Articles

A Capacity Achieving MIMO Detector Based on Stochastic Sampling

Spatial-multiplexing multiple-input multiple-output (MIMO) systems have been developed and enhanced over the past two decades. In particular, a great amount of effort has gone towards development of capacity achieving detectors with affordable computational complexity. The developed detectors may be broadly divided into two classes: (i) deterministic sampling, such as list sphere decoding detector; and (ii) stocastic sampling, such as those based on Markov chain Monte Carlo (MCMC) search schemes. This paper proposes a novel detection scheme that is based on stochastic sampling, but is fundamentally different from the MCMC detectors. While MCMC follows a set of sequential sampling steps, hence, the sample sets obtained are highly correlated, the method proposed in this paper takes stochastic samples that are completely independent. This new approach of stochastic sampling leads to a detector with significantly reduced complexity. It also allows reduction in the detector latency.

An Efficient Software Stack Sphere Decoder for Polar Codes

Proved to achieve the symmetric capacity of the binary-input discrete memoryless channels, polar codes have been chosen for the eMBB control channels in the 5th generation mobile communication systems. Besides the main decoding algorithms like successive cancellation (SC) decoding and CRC-aid SC list (CA-SCL) decoding, sphere decoder (SD) and list SD (LSD) are the alternatives for short codes with less required memory bits. Existing SD and LSD attain high calculation complexity, for SD requires a back-tracking process and LSD needs a large list size $L$ to achieve satisfying performance. To reduce complexity, an efficient software stack sphere decoder (ESSD) based on the synchronous determination is firstly proposed in this article. With the dynamic set-by-set decoding in the stack structure, it achieves the lowest complexity in SD-based decoders (SD/LSD/ESSD) while sharing the same performance on low-rate codes and high-rate codes. Compared with the CA-SCL decoder, the complexity and latency of the proposed ESSD are also competitive at high signal-to-noise-ratio on the displayed codes. Implemented on C++ , the proposed ESSD reduces $\text{44.77}\%$ latency compared with CA-SCL-32 for $\mathcal {P}(128,120)$ at the BER of $10^{-5}$ with $E_b/N_0 = \text{7}$ dB.

A Linear-Complexity Channel-Independent Code Construction Method for List Sphere Polar Decoder

Besides the main decoding methods for polar codes such as successive cancellation (SC) decoding, SC list decoding (SCL), and belief propagation (BP) decoding, list sphere decoding (LSD) is an alternative for short codes regarding its lower complexity. Though the improved LSD (ILSD) can achieve better performance than LSD by a synchronous determination scheme, its performance is still not satisfactory with an inappropriate code construction. Good polar code construction is to pick the K most reliable bits from N bits. However, existing polar construction methods are designed for SC or BP decoders. These construction methods might not be suitable for ILSD, which searches based on the Euclidean distance. In this paper, a linear-complexity channel-independent polar code construction method for ILSD is proposed to improve the performance. Considering the Hamming distance as the reliability for ILSD, the proposed method arranges more frozen bits in the early synchro sets and improves the evaluation of the paths at each decoding level. With linear complexity, this method is channel-independent, which is efficient for offline constructions. Numerical results show the proposed construction with low code rates outperforms the state-of-the-art by 2 dB with ILSD.

An Efficient Software List Sphere Decoder for Polar Codes

Polar codes, first achieving the capacity of symmetric binary-input discrete memoryless channels (B-DMCs), have been standardized for eMBB control channels. Since 5G cellular requires flexible architecture which is realized by the software defined networking paradigm, efficient polar decoder is anticipated. Though successive cancellation list (SCL) decoder achieves satisfactory performance, it requires a large amount of memory. For short control channel codes, sphere decoder (SD) is an alternative, but costs unbearable time complexity at low signal-to-noise ratio. List sphere decoder (LSD) abandons the radius and keeps a list of best paths to gain a fixed complexity. However, LSD needs a large list size L for satisfactory performance. In this paper, an efficient software LSD with path pruning and efficient sorting is proposed. We recall the radius as the bound to delete the paths out of the sphere at very early levels. Since L is dynamic, efficient sorting is proposed to reduce the copy operations. Implemented with C++, the proposed decoder can reduce up to 65.3% latency compared with the original LSD, with the same performance and lower complexity.

Bridging the Gap Between MMSE-DFE and Optimal Detection of MIMO Systems

In this paper, we propose a novel low-complexity, near-optimal soft-input soft-output detector for $\text {N}\times \text {M}$ multiple-input multiple-output (MIMO) systems. Our algorithm is based on the combination of minimum mean square error decision feedback equalization (MMSE-DFE) and conditional optimization. In a round-robin fashion, one symbol is detected using exhaustive search in such a way that all $\text {N}\times (\text {M}-1)$ submatrices of the baseband channel matrix are considered and the one with the best metric is chosen. This search over all columns of the channel matrix, which can be performed in parallel, has the advantages of improving the performance of the hard-output version of the detector, and refining the list of candidates for efficient implementation of the soft-output detector for MIMO systems with error correcting codes. In particular, it is shown that the error performance of the soft-output system is comparable to that of the list sphere decoder (LSD) but with much smaller list size, and hence smaller complexity than the latter. We also analyze the performance and complexity of the proposed algorithm and discuss different techniques to further reduce its complexity without affecting the performance. Finally, the obtained theoretical results are validated via simulations.

An Improved Software List Sphere Polar Decoder With Synchronous Determination

Being the first codes achieving the capacity of symmetric binary-input discrete memoryless channels, polar codes have been favorable and standardized for enhanced mobile broadband (eMBB) control channels. In future mobile communications, the entire system is required to be flexible and adaptive to varying scenarios. Therefore, efficient software implementation of polar decoder is expected. Existing software successive cancellation list polar decoder can achieve good performance but with a good memory consumption. For short control channel codes, a more memory-efficient alternative is sphere decoder, but its latency gives rise to another issue. Equipped with a breadth-first search to keep a fixed complexity, list sphere decoder (LSD) is also considered. To improve the performance of LSD, a software LSD with synchronous determination is proposed in this paper. We precalculate the synchro sets to improve the decoding, which leads to a performance gain. Meanwhile, the decoding process is simplified due to the precalculation, which attains even lower complexity. Implemented with C++ , the proposed decoder can achieve up to 2 dB performance gain compared with the existing LSD.

Relaying strategy based on estimated information for multi-antenna cooperative networks

A sphere-based list forwarding scheme for multiple-input multiple-output (MIMO) relay networks is proposed and analyzed. Firstly, an estimate forwarding (EF) method is proposed, which forwards the minimum mean squared error (MMSE) estimate of the source data to the destination. Since it performs like amplify-and-forward (AF) and decode-and-forward (DF) for the low and high signal-to-noise ratio (SNR) regions, respectively, the EF relay thus outperforms conventional AF and DF across all SNRs without the need for switching algorithms for different SNRs. Because computational complexity is however high for relays with a large number of antennas (large MIMO) and/or high order constellations, list EF for large MIMO relay networks is proposed. It computes a list sphere decoder based MMSE estimate and retains the advantages of the exact EF relay at a negligible performance loss. The proposed list EF could offer a flexible trade-off between the performance and computational complexity.

Low Complexity Iterative Receiver Design for Sparse Code Multiple Access

Sparse code multiple access (SCMA) is one of the most promising methods among all the non-orthogonal multiple access techniques in the future 5G communication. Compared with some other non-orthogonal multiple access techniques such as low density signature (LDS), SCMA can achieve better performance due to the shaping gain of the SCMA codewords. However, despite of the sparsity of the codewords, the decoding complexity of the current message passing algorithm (MPA) utilized by SCMA is still prohibitively high. In this paper, by exploring the lattice structure of SCMA codewords, we propose a low complexity decoding algorithm based on list sphere decoding (LSD). The LSD avoids the exhaustive search for all possible hypotheses and only considers signal within a hypersphere. As LSD can be viewed a depth-first tree search algorithm, we further propose several methods to prune the redundancy visited nodes in order to reduce the size of the search tree. Simulation results show that the proposed algorithm can reduce the decoding complexity substantially while the performance loss compared with the existing algorithm is negligible.

ML Symbol Detection for MIMO Systems in the Presence of Channel Estimation Errors

In wireless communication, the multiple-input multiple-output (MIMO) system is a well-known approach to improve the reliability as well as the data rate. In MIMO systems, channel state information (CSI) is typically required at the receiver to detect transmitted signals; however, in practical systems, the CSI is imperfect and contains errors, which affect the overall system performance. In this paper, we propose a novel maximum likelihood (ML) scheme for MIMO systems that is robust to the CSI errors. We apply an optimization method to estimate an instantaneous covariance matrix of the CSI errors in order to improve the detection performance. Furthermore, we propose the employment of the list sphere decoding (LSD) scheme to reduce the computational complexity, which is capable of efficiently finding a reduced set of the candidate symbol vectors for the computation of the covariance matrix of the CSI errors. An iterative detection scheme is also proposed to further improve the detection performance.

Joint maximum likelihood detection and link selection for cooperative MIMO relay systems

In this study, the authors propose a cross layer design strategy that consists of a cooperative maximum likelihood detector operating in conjunction with link selection for cooperative multiple-input multiple-output relay networks. The system considered is a two phase relaying model, with amplify-and forward relays. The model considers the effect of relay placement and of large scale shadowing. The authors develop a cooperative maximum likelihood detector for an arbitrary number of relays, and link selection schemes are devised for two scenarios relating to the available knowledge at the destination node, which considers relay combination, with complexity analysis. The authors also propose a cooperative list sphere decoder that processes soft information and implements an iterative detection and decoding scheme. The results of simulations of the systems are presented, with comparisons to system models used in previous literature, and show how the cooperative detector and the proposed link selection schemes affect the bit error rate performance at the destination node, for both the hard decision non-iterative case, and the soft information iterative processing case.

Two-Stage List Sphere Decoding for Under-Determined Multiple-Input Multiple-Output Systems

A two-stage list sphere decoding (LSD) algorithm is proposed for under-determined multiple-input multiple-output (UD-MIMO) systems that employ N transmit antennas and M<;N receive antennas. The two-stage LSD algorithm exploits the unique structure of UD-MIMO systems by dividing the N detection layers into two groups. Group 1 contains layers 1 to M that have similar structures as a symmetric MIMO system; while Group 2 contains layers M+1 to N that contribute to the rank deficiency of the channel Gram matrix. Tree search algorithms are used for both groups, but with different search radii. A new method is proposed to adaptively adjust the tree search radius of Group 2 based on the statistical properties of the received signals. The employment of the adaptive tree search can significantly reduce the computation complexity. We also propose a modified channel Gram matrix to combat the rank deficiency problem, and it provides better performance than the generalized Gram matrix used in the Generalized Sphere-Decoding (GSD) algorithm. Simulation results show that the proposed two-stage LSD algorithm can reduce the complexity by one to two orders of magnitude with less than 0.1 dB degradation in the Bit-Error-Rate (BER) performance.

반복 MIMO 검출을 위한 저 복잡도 LSD 기법

논문에서는 부호화된 MIMO 시스템에서 연판정 반복 검출 방식과 결합하여 효율적으로 사용될 수 있는 복잡도가 감소된 리스트 구 복호 (list sphere decoding; LSD) 방식을 제안한다. 기존의 LSD 방식은 고정된 값의 반지름을 이용하여 후보 신호들을 검색하였다. 그러나, 정해진 수 만큼의 후보 리스트들이 모두 채워지면, 이들에 대한 최대 또는 평균 반지름은 줄어들 가능성이 매우 크다. 반지름을 줄이게 되면 복잡도 또한 줄일 수 있게 된다. 본 논문에서는 LSD 기반의 MIMO 검출 시스템에서 효율적으로 사용될 수 있는 매우 간단한 반지름 갱신 방식 및 이를 이용한 연판정 반복 MIMO 검출 기법을 제안한다. 또한 터보 부호를 이용하여, 연판정 입출력을 이용하는 반복 MIMO 검출 방식에 대한 성능 평가 결과를 제시한다. 본 논문에서 제시된 시뮬레이션 결과에 따르면, 제안된 방식은 기존의 LSD 방식을 사용하는 것에 비해 거의 유사한 비트오류율 성능을 유지하면서도 복잡도를 크게 감소시킬 수 있다. This paper proposes a complexity reduced list sphere decoding (LSD) scheme for joint iterative soft detection scheme for coded MIMO system. The conventional LSD scheme is based on searching the candidates with a fixed radius. However, once the candidate list is full, it is highly probable that the radius can be reduced. By reducing the radius, the complexity can be also reduced. We propose a simple and efficient radius update method for complexity reduction of list version sphere decoding and its application to iterative soft MIMO detection. We evaluate the performance of the proposed scheme with a joint soft-input-soft-ouput iterative MIMO detection in combination with turbo codes. Simulation results show that the proposed methods provide substantial complexity reduction while achieving similar bit error rate (BER) performance as the conventional LSD scheme.

Reduced complexity list sphere decoding for MIMO systems

Depth-first sphere decoding of MIMO systems has near maximum likelihood performance with reasonable computational complexity. In this paper, lower complexity depth-first sphere decoding and list sphere decoding algorithms are proposed. Several criteria for re-ordering the search dimensions are proposed. The proposed sphere decoders are shown to have a significantly reduced decoding complexity at low SNRs. To further reduce the complexity at high SNRs, the point search-space at each ordered dimension is adaptively reduced. Further reductions in the decoding complexity are achieved by inter-layer interference cancellation. It is shown that the proposed sphere decoding algorithms maintain their near-optimal performance, concurrently with a significant complexity reduction, over a wide SNR range.

Decoding by Sampling — Part II: Derandomization and Soft-Output Decoding

In this paper, a derandomized algorithm for sampling decoding is proposed to achieve near-optimal performance in lattice decoding. By setting a probability threshold to sample candidates, the whole sampling procedure becomes deterministic, which brings considerable performance improvement and complexity reduction over to the randomized sampling. Moreover, the upper bound on the sample size K, which corresponds to near-maximum likelihood (ML) performance, is derived. We also find that the proposed algorithm can be used as an efficient tool to implement soft-output decoding in multiple-input multiple-output (MIMO) systems. An upper bound of the sphere radius R in list sphere decoding (LSD) is derived. Based on it, we demonstrate that the derandomized sampling algorithm is capable of achieving near-maximum a posteriori (MAP) performance. Simulation results show that near-optimum performance can be achieved by a moderate size K in both lattice decoding and soft-output decoding.

MIMO-OFDM系统中基于变分Bayes EM算法的联合符号检测与鲁棒Kalman信道跟踪

A new joint symbol detection and channel estimation algorithm is proposed for MIMO-OFDM systems over fast fading channels based on the Variational Bayes Expectation-Maximization (VBEM) algorithm and turbo principle. By applying VBEM algorithm,a modified List Sphere Decoder (LSD) and soft-input Kalman Filter (KF) are derived for symbol detection and channel estimation, respectively. The covariance matrices of channel estimation are considered in the symbol detector, and the covariances of symbols are taken into account in channel estimator. In fast time-varying channels, soft symbols with low reliability may introduce outliers in soft-input kalman filter. Since the KF lacks robustness to outliers, filter divergence may occur and gives rise to propagation and error floor phenomenon. In order to mitigate the outliers effect, a robust soft-input kalman filter under the VBEM framework is proposed based on the theory of robust statistics, which can keep robust channel-tracking ability in fast fading channels. Simulation results show that the proposed VBEM algorithm has better performance over the conventional VBEM algorithm and EM algorithm in fast time-varying multipath channels.

Reducing the Complexity of Quasi-ML Detectors for MIMO Systems Through Simplified Branch Metric and Accumulated Branch Metric Based Detection

A detection scheme built on a combination of a simplified branch metric based search and refined calculation is proposed in this work to reduce the computational complexity of the quasi-maximum-likelihood (QML) detectors, such as the QR-decomposition combined with M-algorithm (QRD-M), list sequential sphere decoder (LISS) and list sphere decoding (LSD), for multiple-input multiple-output (MIMO) multiplexing systems, while maintaining similar performance to the original methods. More specifically, the proposed method utilizes a multiplication-free accumulated branch metric to first identify the potential candidates with significantly reduced search complexity. The optimal solution is then determined by the standard squared Euclidean distance metric from the initial candidates. The key idea of the proposed method lies in the fact that the best candidate identified by the simplified branch metric coincides with that of which is based on the conventional one with a high probability for a properly designed parameter. Numerical results demonstrates that the QRD-M detector using the proposed method achieves significantly better performance than the conventional one with comparable (or even less) computational complexity.

Improved K-Best Sphere Detection for Uncoded and Coded MIMO Systems

The conventional K-best sphere decoder (KSD) keeps the best K nodes at each level of the search tree. In addition to retaining the best K nodes, we also consider all the nodes whose costs are within a certain margin of the cost of the Kth best node. The resulting algorithm is called improved K-best sphere decoder (IKSD). Three IKSD variants are considered in this letter, which are fixed threshold, normalized threshold and adaptive threshold IKSD. The proposed IKSD requires a smaller K (indicating lower complexity) while still achieving a better and near optimal performance compared to the conventional KSD. These gains are confirmed by the simulation results. For example, for the fixed threshold IKSD in a 4 x 4 16-QAM multiple-input multiple-output (MIMO) system, with K=2, it achieves the same performance as the conventional KSD (K=16), yielding about 80% complexity savings. For coded MIMO systems, the IKSD is also extended as a list sphere decoder for joint iterative detection and decoding.

Reducing the Complexity of Quasi-Maximum-Likelihood Detectors Through Companding for Coded MIMO Systems

A companding-based transformation method is introduced to quasi-maximum-likelihood (ML) detectors, such as the QR-decomposition-based M-algorithm (QRD-M) and list sphere decoding, for coded multiple-input-multiple-output (MIMO) systems in this paper. The key idea of the proposed companding technique is to compress (i.e., down-weight) the dubious observation of the accumulated branch metric by taking into account its statistical characteristics so that, after companding, the estimation error of the unreliable detected information bits due to insufficient candidate size and/or channel estimation error is significantly mitigated without disproportionate compromise of the reliable information bits. By employing the proposed companding method, the original leptokurtically distributed log-likelihood ratio of the detected information bits becomes more Gaussian distributed. As an illustrative example, the QRD-M detector is employed in this paper. Numerical results show that the QRD-M detector based on the proposed companding paradigm achieves significant performance gain over the conventional method and approaches the performance of the ML detector for a 16-ary quadrature-amplitude-modulated (16-QAM) 4×4 MIMO multiplexing system with lower-than-linear-detector computational complexity.

Low-Complexity Detection of Full-Rate SFBC in BICM-OFDM Systems

Last-generation and future wireless communication standards, such as DVB-T2 or DVB-NGH, are including multi-antenna transmission and reception in order to increase bandwidth efficiency and receiver robustness. The main goal is to combine diversity and spatial multiplexing in order to fully exploit the multiple-input multiple-output (MIMO) channel capacity. Full-rate full-diversity (FRFD) space-time codes (STC) such as the Golden code are studied for that purpose. However, despite their larger achievable capacity, most of them present high complexity for soft detection, which hinders their combination with soft-input decoders in bit-interleaved coded modulation (BICM) schemes. This article presents a novel low-complexity soft detection algorithm for the reception of FRFD space-frequency block codes in BICM orthogonal frequency-division multiplexing (OFDM) systems. The proposed detector maintains a reduced and fixed complexity, avoiding the variable nature of the list sphere decoder (LSD) due to its dependence on the noise and channel conditions. Complexity and simulation-based performance results are provided which show that the proposed detector performs close to the optimal log-maximum a posteriori (MAP) detection in a variety of DVB-T2 broadcasting scenarios.

Efficient Soft-Output Demodulation of MIMO QPSK via Semidefinite Relaxation

Two efficient list-based “soft”-output demodulators are developed for iterative receivers in multiple-input multiple-output (MIMO) communication systems with QPSK signaling. The proposed demodulators are based on the semidefinite relaxation (SDR) technique, and hence their computational costs are bounded by a low-order polynomial of the number of bits transmitted per channel use. The first demodulator applies the SDR technique once per demodulation-decoding iteration, and generates list members via the randomization procedure that is inherent in the SDR technique. The second demodulator is based on an approximation of that randomization procedure by a set of independent Bernoulli trials, and this approximation reduces the number of semidefinite programs that need to be solved to just one per channel use. List-free implementations that reduce the memory requirements of list demodulators with moderate to long lists are also developed. Analysis suggests that the proposed “Single-SDR” demodulator should offer good performance at moderate computational cost, especially for larger systems. This is quantified using simulations of a richly scattered environment, in which the performance of the Single-SDR demodulator is similar to that of the list sphere decoder with moderate sized lists and better than that of the minimum mean square error soft interference canceler. The average computational cost of a straightforward implementation of the Single-SDR demodulator is competitive with that of the list sphere decoder with moderate sized lists, and the distribution of its computational cost is quite concentrated around the average.