Complexity Of Sphere Decoding Research Articles

Dynamic branch pruning aided low switching fixed complexity sphere decoding for small scale and massive MIMO detection

AbstractIn this article, we have proposed a dynamic branch pruning aided low switching fixed complexity sphere decoding (LSFSD) algorithm that can reduce the number of node visits in conventional fixed‐complexity sphere decoder (FSD) or imbalanced fixed‐complexity sphere decoder (IFSD) detector dynamically based on the channel condition. Further, we propose a low complexity QR‐based FSD ordering scheme in the real domain that requires half of the number of real arithmetic operations in the conventional case. As a case study, we have simulated the proposed algorithm for and MIMO system with 64‐QAM modulation. The simulation results show that the proposed algorithm can reduce approximately 50% of node visits to achieve a BER of . Additionally, a low complexity solution of the soft bit information generation method is also proposed. The concept is further extended to massive MIMO applications where the number of receiving antenna to transmit antenna ratio is close to one. A smaller node extension parameter is defined for the proposed detector to reduce the complexity in massive MIMO detection. Simulation results demonstrate that the proposed detection algorithm has significantly lower computational complexity than other recent state‐of‐the‐art detection algorithms for massive MIMO and large MIMO systems.

LR-aided sphere decoding of generalized spatial modulation signals

This paper presents a new low complexity sphere decoding (SD) for the generalized spatial modulation (GSM). We introduce a pre-processing stage using the lattice reduction (LR) aided minimum mean squared error (MMSE) equalization in the GSM systems. This stage speeds up the search in the decoding tree and provides a lattice dependent (LD) initial choice of the radius. Moreover, we derive a lattice independent (LI) initial radius that guarantees the optimal performance at a high signal-to-noise ratio (SNR). We also propose an iterative method to increase the radius in order to achieve the maximum likelihood (ML) performance at all SNRs. We show that the proposed algorithm achieves the ML performance while requiring prominently less computational complexity (CC) than an exhaustive search. In addition, we analyze the CC of the resulting algorithm at high SNRs and we derive an analytical expression for the complexity. The simulation results demonstrate a noticeable decrease in the CC of the proposed sphere decoding algorithm in comparison with its counterparts, particularly at low SNRs.

Reduced Complexity Detection Schemes for Golden Code Systems

The Golden code has full-rate and full-diversity. However, its applications are limited because of the very high detection complexity. The complexity of sphere decoding depends on the size of signal set, $M^{2}$ , and the depth of search. Meanwhile, the complexity of fast essentially maximum likelihood (ML) detection is still $\mathcal {O}(M^{2})$ for $M$ -ary quadrature amplitude modulation ( $M$ QAM). In this paper, we propose two reduced complexity detection schemes, fast essentially ML with detection subset and sphere decoding with detection subset. Two theoretical bounds on the average bit error probability for the Golden code with $M$ QAM are also formulated in this paper. Simulation results demonstrate that both the fast essentially ML with detection subset and sphere decoding with detection subset agree well with the formulated theoretical bounds and can achieve the error performance of the conventional fast essentially ML detector and sphere decoding.

Investigation of Sphere Decoder and Channel Tracking Algorithms for Media-Based Modulation over Time-Selective Channels

The performance of media-based modulation (MBM) systems, where additional information can be conveyed by the indices of the channel states created by RF mirrors, over time-selective channels is investigated. By transforming the MBM system model into a traditional MIMO system model, we first propose a reduced complexity sphere decoder algorithm. Then two channel tracking algorithms, which are based on least mean square adaptive filter and recursive least-squares adaptive filter, are employed in order to combat the performance loss caused by the time-varying channels. Numerical results show that the proposed sphere decoder and these two channel tracking algorithms perform well in MBM systems.

Reduced complexity sphere decoding using probabilistic threshold based Schnorr–Euchner enumeration

The use of multiple antennas at both the transmitter and the receiver is the key technology for future wireless communications since it achieves higher spectral efficiency. Symbol decoding at the receiver for such a system is still a challenge. We propose a new enumeration technique, termed as probabilistic threshold-based Schnorr–Euchner (PSE) enumeration, to reduce the complexity of sphere decoding (SD). The conventional SE-SD algorithm visits all nodes satisfying the sphere constraint in ascending order of their branch metric values. However, this algorithm is computationally expensive in real-time scenarios. Therefore, we use a threshold limit for the branch metric values in the Schnorr–Euchner (SE) enumeration which is based on the noise statistics, to reduce the overall complexity. In each layer of the decoding process, the proposed PSE-SD technique performs a threshold test before visiting the next candidate in an SE ordered list. As a result, unlikely nodes, which do not satisfy the threshold test, are pruned earlier than in the conventional SE-SD algorithm. From the simulation results, we examine the influence of the proposed probabilistic threshold for various MIMO configurations and witness that the PSE-SD technique achieves a significant complexity reduction over existing methods with only a negligible performance loss.

Energy-Efficient Soft-Input Soft-Output Signal Detector for Iterative MIMO Receivers

This paper presents the VLSI design of an energy-efficient, high-throughput soft-input soft-output signal detector for iterative multiple-input multiple-output (MIMO) receiver. The detector is evolved from our previously developed imbalanced fixed complexity sphere decoder and adopts several new algorithm-level techniques to exploit the available a priori information of transmitted bits. More specifically, an adaptive tree-travel control scheme, a reliability-dependent log-likelihood ratio correction method and an iteration-based hybrid node enumeration technique are proposed to provide near-optimal detection performance with much reduced computational complexity. A multi-stage parallel VLSI architecture is developed to implement the proposed algorithm with high detection throughput. Furthermore, the block-level clock gating is deployed to save power when the tree-search space is reduced, while still preserving the constant-throughput feature. As a proof of concept, we designed the iterative detector using a 65-nm CMOS technology and conducted post-layout simulation. The core area is 0.64 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> with 198.2 k gates. Working at 240-MHz clock frequency with 1.0-V voltage supply, the detector achieves a maximum 1.44-Gbps throughput. Under frequency-selective channels, the detector core consumes 98.5-, 127.9-, and 149.5-pJ energy per bit detection in open-loop, 2-iteration, and 4-iteration modes, respectively.

Reduced K-best sphere decoding algorithm based on minimum route distance and noise variance

This paper focuses on reducing the complexity of K-best sphere decoding (SD) algorithm for the detection of uncoded multiple input multiple output (MIMO) systems. The proposed algorithm utilizes the threshold-pruning method to cut nodes with partial Euclidean distances (PEDs) larger than the threshold. Both the known noise value and the unknown noise value are considered to generate the threshold, which is the sum of the two values. The known noise value is the smallest PED of signals in the detected layers. The unknown noise value is generated by the noise power, the quality of service (QoS) and the signal-to-noise ratio (SNR) bound. Simulation results show that by considering both two noise values, the proposed algorithm makes an efficient reduction while the performance drops little.

A Low Complex Modified Golden Code in a Cooperative Relaying System

In recent years, cooperative communication is a hot topic of research and it is a powerful physical layer technique to combat fading in wireless relaying scenario. Concerning with the physical layer issues, in this paper it is focussed on with providing a better space time block coding (STBC) scheme and incorporating it in the cooperative relaying nodes to upgrade the system performance. Recently, the golden codes have proven to exhibit a superior performance in a wireless MIMO (Multiple Input Multiple Output) scenario than any other code. However, a serious limitation associated with it is its increased decoding complexity. This paper attempts to resolve this challenge through suitable modification of golden code such that a less complex sphere decoder could be used without much compromising the error rates. The decoder complexity is analyzed through simulation and it proves to exhibit less complexity compared to the conventional (Maximum likelihood) ML decoder. The single relay cooperative STBC consisting of source, relay and destination are considered. The cooperative protocol strategy considered in the relay node is Decode and forward (DF) protocol. The proposed modified golden code with less complex sphere decoder is implemented in the nodes of the cooperative relaying system to achieve better performance in the system. The simulation results have validated the effectiveness of the proposed scheme by offering better BER performance, minimum outage probability and increased spectral efficiency compared to the non cooperative transmission method.

A Semifixed Complexity Sphere Decoder for Uncoded Symbols for Wireless Communications

The development of high data rate wireless communications systems using Multiple Input — Multiple Output (MIMO) antenna techniques requires detectors with reduced complexity and good Bit Error Rate (BER) performance. In this paper, we present the Semi-fixed Complexity Sphere Decoder (SCSD), which executes the process of detection in MIMO systems with a significantly lower computation cost than the high-performance/reduced-complexity detectors: Sphere Decoder (SD), K-best, Fixed Complexity Sphere Decoder (FSD) and Adaptive Set Partitioning (ASP). Simulation results show that when the Signal-to-Noise Ratio (SNR) is less than 15dB, the SCSD reduces the complexity by up to 90% with respect to SD, up to 60% with respect to K-best or ASP and by up to 90% with respect to FSD. In the proposed algorithm, the BER performance does not show significant degradation and therefore, can be considered as a complexity reduction scheme suitable for implementing in MIMO detectors.

Minimizing the Complexity of Fast Sphere Decoding of STBCs

Decoding of linear space-time block codes (STBCs) with sphere-decoding (SD) is well known. A fast-version of the SD known as fast sphere decoding (FSD) was introduced by Biglieri, Hong and Viterbo. Viewing a linear STBC as a vector space spanned by its defining weight matrices over the real number field, we define a quadratic form (QF), called the Hurwitz-Radon QF (HRQF), on this vector space and give a QF interpretation of the FSD complexity of a linear STBC. It is shown that the FSD complexity is only a function of the weight matrices defining the code and their ordering, and not of the channel realization (even though the equivalent channel when SD is used depends on the channel realization) or the number of receive antennas. It is also shown that the FSD complexity is completely captured into a single matrix obtained from the HRQF. Moreover, for a given set of weight matrices, an algorithm to obtain an optimal ordering of them leading to the least FSD complexity is presented. The well known classes of low FSD complexity codes (multi-group decodable codes, fast decodable codes and fast group decodable codes) are presented in the framework of HRQF.

An Improved Complex Decoding Scheme for Diagonal Algebraic STBC over Fading Channels

We present a new decoding scheme of complex lattice for diagonal algebraic Space-time (ST) block codes in multiple-antennas systems. With multidimensional complex constellation symbols rotation and Hadamard transform, the constructed code is capable of achieving full diversity over a fast fading channels. At the receiver the received signals have structure similar to diagonal matrix, which is easy to decode. The Maximum-likelihood (ML) decoding of the proposed codes can be implemented by the improved complex sphere decoder, without doubling the channel matrix. It is shown that the complex lattice decoding scheme for proposed codes in a multiple-antennas system yields excellent performance and low decoding complexity.

On the Sphere Decoding Complexity of High-Rate Multigroup Decodable STBCs in Asymmetric MIMO Systems

A space-time block code (STBC) is said to be multigroup decodable if the information symbols encoded by it can be partitioned into two or more groups such that each group of symbols can be maximum-likelihood (ML) decoded independently of the other symbol groups. In this paper, we show that the upper triangular matrix R encountered during the sphere decoding of a linear dispersion STBC can be rank-deficient even when the rate of the code is less than the minimum of the number of transmit and receive antennas. We then show that all known families of high-rate (rate greater than 1) multigroup decodable codes have rank-deficient R matrix even when the rate is less than the number of transmit and receive antennas, and this rank-deficiency problem arises only in asymmetric MIMO systems when the number of receive antennas is strictly less than the number of transmit antennas. Unlike the codes with full-rank R matrix, the complexity of the sphere decoding-based ML decoder for STBCs with rank-deficient R matrix is polynomial in the constellation size, and hence is high. We derive the ML sphere decoding complexity of most of the known high-rate multigroup decodable codes, and show that for each code, the complexity is a decreasing function of the number of receive antennas.

Effects of the LLL Reduction on the Success Probability of the Babai Point and on the Complexity of Sphere Decoding

A common method to estimate an unknown integer parameter vector in a linear model is to solve an integer least squares (ILS) problem. A typical approach to solving an ILS problem is sphere decoding. To make a sphere decoder faster, the well-known LLL reduction is often used as preprocessing. The Babai point produced by the Babai nearest plane algorithm is a suboptimal solution of the ILS problem. First, we prove that the success probability of the Babai point as a lower bound on the success probability of the ILS estimator is sharper than the lower bound given by Hassibi and Boyd [1]. Then, we show rigorously that applying the LLL reduction algorithm will increase the success probability of the Babai point and give some theoretical and numerical test results. We give examples to show that unlike LLL's column permutation strategy, two often used column permutation strategies SQRD and V-BLAST may decrease the success probability of the Babai point. Finally, we show rigorously that applying the LLL reduction algorithm will also reduce the computational complexity of sphere decoders, which is measured approximately by the number of nodes in the search tree in the literature.

MIMO Detector for Uncoded Symbols based on Complex Sphere Decoder and Manhattan Metric

For communication standards with high transfer rates (WiMAX, WiFi, LTE), which use MIMO (Multiple Input - Mul- tiple Output) systems, detectors with reduced computational complexity that achieve a good Bit Error Rate (BER) performance are of great interest. In the literature, it has been recognized the maximum likelihood (ML) detector as the optimum, but this algorithm experiences an exponential complexity making it an impractical alternative for implemen- tation. However, there are also alternatives such as the K-best and sphere decoder (SD) which can reach a quasi-ML performance with lower computational complexity. This paper presents a variation of the SD algorithm based on the Complex Sphere Decoder and K-best algorithm, named as KLSD, which limits the number of searching points during the predetection and has a performance similar to that given by the algorithm Fixed Complexity Sphere Decoder (FSD) without channel ordering. Furthermore, for the calculation of the weights of each candidate node is proposed to replace the use of the Euclidean distance by the Manhattan Metric to reduce the number calculation performed. When compar- ing performance and complexity against others algorithms, it can be seen that a similar performance without increasing its complexity. Additionally, the results show that the change of metric, does not affect the performance of the proposed algorithm, so it is considered a feasible complexity reduction scheme.

On the average complexity of sphere decoding in lattice space–time coded multiple‐input multiple‐output channel

AbstractThe exact average complexity analysis of the basic sphere decoder for general space–time codes applied to multiple‐input multiple‐output (MIMO) wireless channel is known to be difficult. In this work, we shed the light on the computational complexity of sphere decoding for the quasi‐static, lattice space–time (LAST) coded MIMO channel. Specifically, we drive an upper bound of the tail distribution of the decoder's computational complexity. We show that when the computational complexity exceeds a certain limit, this upper bound becomes dominated by the outage probability achieved by LAST coding and sphere decoding schemes. We then calculate the minimum average computational complexity that is required by the decoder to achieve near optimal performance in terms of the system parameters. Our results indicate that there exists a cut‐off rate (multiplexing gain) for which the average complexity remains bounded.Copyright © 2012 John Wiley &amp; Sons, Ltd.

Approximate MIMO Iterative Processing With Adjustable Complexity Requirements

Targeting only the best achievable bit error rate (BER) performance in iterative receivers operating over multiple-input-multiple-output (MIMO) channels may result in a significant waste of resources, particularly when the achievable BER is considerably better than the target performance [e.g., under good channel conditions and at high signal-to-noise ratio (SNR)]. In contrast to the typical iterative schemes, a practical iterative decoding framework that approximates the soft information exchange is proposed, allowing reduced complexity sphere and channel decoding that is adjustable to the transmission conditions and the required BER. With the proposed approximate soft information exchange, the performance of the exact soft information can still be reached with significant complexity reduction.

Reduced complexity sphere decoding

ABSTRACTIn multiple‐input multiple‐output (MIMO) systems, sphere decoding (SD) can achieve a performance equivalent to a full‐search maximum likelihood decoding, with reduced complexity. Several researchers reported techniques that reduce the complexity of SD further. In this paper, a new technique that decreases the computational complexity of SD substantially, without sacrificing performance, is introduced. The reduction is accomplished by deconstructing the decoding metric to decrease the number of computations and by exploiting the structure of a lattice representation. Furthermore, an application of SD employing a proposed smart implementation with very low computational complexity is introduced. This application calculates the soft bit metrics of a bit‐interleaved convolutional‐coded MIMO system in an efficient manner. On the basis of the reduced complexity SD, the proposed smart implementation employs the initial radius acquired by zero‐forcing decision feedback equalization, which ensures no empty spheres. Other than that, a technique of a particular data structure is also incorporated to efficiently reduce the number of executions carried out by SD. Simulation results show that these approaches achieve substantial gains in terms of the computational complexity for both uncoded and coded MIMO systems. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Parallel High Throughput Soft-Output Sphere Decoding Algorithm

Multiple-Input-Multiple-Output communication systems demand fast sphere decoding with high performance. To speed up the computation, we propose a scheme with multiple fixed complexity sphere decoders to construct a parallel soft-output fixed complexity sphere decoder (PFSD). The proposed decoder is highly parallel and has performance comparable to soft-output list fixed complexity sphere decoder (LFSD) and K-best sphere decoder. In addition, we propose a parallel QR decomposition algorithm to lower the preprocessing overhead, and a low complexity LLR algorithm to allow parallel update of LLR values. We demonstrate that the PFSD algorithm can increase the throughput and reduce bit error rate of a soft-output solution in a 4?×?4 16-QAM system, and has superior performance compared to other soft decoders with comparable throughput and computation complexity. The PFSD algorithm has been mapped onto Xilinx XC4VLX160 FPGA. The resulting PFSD decoder can achieve up to 75 Mbps throughput for 4?×?4 64-QAM configuration at 100MHz with low control overhead.

A Fast Generalized Complex Sphere Decoder for MIMO Systems

摘要: 该文提出了一种广义复球形解码算法。它能处理多输入多输出系统(MIMO)中发送天线多于接收天线的情形,并能同时检测具有格型结构和不具有格型结构的二维空间星座信号。该算法对信号矢量的超定部分进行优化搜索,从而避免了穷尽搜索的高复杂度。仿真结果表明该广义复球形解码算法的复杂度明显低于采用穷尽搜索策略的复杂度。 关键词: 无线通信; 广义复球形解码算法; 多输入多输出系统

On Two-User Gaussian Multiple Access Channels With Finite Input Constellations

Constellation Constrained (CC) capacity regions of two-user Single-Input Single-Output (SISO) Gaussian Multiple Access Channels (GMAC) are computed for several Non-Orthogonal Multiple Access schemes (NO-MA) and Orthogonal Multiple Access schemes (O-MA). For NO-MA schemes, a metric is proposed to compute the angle(s) of rotation between the input constellations such that the CC capacity regions are maximally enlarged. Further, code pairs based on Trellis Coded Modulation (TCM) are designed with PSK constellation pairs and PAM constellation pairs such that any rate pair within the CC capacity region can be approached. Such a NO-MA scheme which employs CC capacity approaching trellis codes is referred to as Trellis Coded Multiple Access (TCMA). Then, CC capacity regions of O-MA schemes such as Frequency Division Multiple Access (FDMA) and Time Division Multiple Access (TDMA) are also computed and it is shown that, unlike the Gaussian distributed continuous constellations case, the CC capacity regions with FDMA are strictly contained inside the CC capacity regions with TCMA. Hence, for finite constellations, a NO-MA scheme such as TCMA is better than FDMA and TDMA which makes NO-MA schemes worth pursuing in practice for two-user GMAC. Then, the idea of introducing rotations between the input constellations is used to construct Space-Time Block Code (STBC) pairs for two-user Multiple-Input Single-Output (MISO) fading MAC. The proposed STBCs are shown to have reduced Maximum Likelihood (ML) decoding complexity and information-losslessness property. Finally, STBC pairs with reduced sphere decoding complexity are proposed for two-user Multiple-Input Multiple-Output (MIMO) fading MAC.