13_Low Complexity Motion Estimation for H.264 AVC Based Depth Encoding in Free Viewpoint Video

第 22卷第 7期2023年 7月Vol.22 No.7Jul.2023软件导刊Software Guide基于提前终止策略改进的运动估计算法朱鑫磊，汪伟（上海理工大学光电信息与计算机工程学院，上海 200093）摘要：针对HM-16.14中TZSearch标准算法存在的计算复杂度高、耗时相对较长等问题，提出一种基于提前终止策略的改进TZSearch算法。

首先，根据编码产生的率失真代价对编码单元、变换单元和预测单元的深度进行划分，有效避免了额外的划分深度；然后，在TZSearch初始网格搜索过程中，采用钻石搜索和六边形搜索两种搜索方式，根据运动矢量分布位置选择一种更为有效的方式，精确找出最佳匹配点；最后，使用OARP栅格搜索和精细搜索完成运动估计。

由实验结果可知，该方法与标准算法相比，平均降低了60％以上的TZSearch运动估计耗时，且基本不影响视频质量。

关键词：TZSearch算法；提前终止策略；栅格搜索；精细搜索；运动估计DOI：10.11907/rjdk.221887开放科学（资源服务）标识码（OSID）：中图分类号：TP391.1 文献标识码：A文章编号：1672-7800（2023）007-0051-08A Modified Motion Estimation Algorithm Based on Early Termination StrategyZHU Xinlei， WANG Wei（School of Optical-Electrical and Computer Engineering， University of Shanghai for Science and Technology，Shanghai 200093， China）Abstract：Considering the high computational complexity and relatively long time consumption of the TZSearch standard algorithm within HM-16.14， an improved TZSearch algorithm based on early termination strategy is proposed to improve the efficiency of video coding. Firstly，the depth sorting of the coding unit， transform unit and prediction unit is calculated according to the performance of rate distortion， which can effectively decrease additional division depths. Secondly， two search methods， i.e. diamond search and hexagonal search， are employed within the initial grid search step of TZSearch in order to precisely find the best matching point according to the motion vector distribution. Finally，OARP raster search and fine search are used to acquire the motion estimation results. Compared with the standard algorithm， experimental re‐sults show that the proposed method reduces more than 60% motion estimation time consumption on average， yet keeps the similar video quali‐ty .Key Words：TZSearch algorithm； early termination strategy； raster search； fine search； motion estimation0 引言随着视频技术的快速发展，依靠视频传递信息变得越来越普及，这使得视频流数据在互联网传输中的占比越来越大。

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Algorithm to Reduce PAPRof an OFDM Signal Using Partial Transmit Sequences Technique》原文链接:https:///academic-journal-foreign_broadcasting-ieee-transactions_thesis/020*********.html128.《Improved Decoding Algorithm of Bit-Interleaved Coded Modulation for LDPC Code》原文链接:https:///academic-journal-foreign_broadcasting-ieee-transactions_thesis/020*********.html129.《Precoding for PAPR Reduction of OFDM Signals With Minimum Error Probability》原文链接:https:///academic-journal-foreign_broadcasting-ieee-transactions_thesis/020*********.html130.《Network Design and Field Application of ATSC Distributed Translators》原文链接:https:///academic-journal-foreign_broadcasting-ieee-transactions_thesis/020*********.html131.《On the Channel and Signal Cross Correlation of Downlink and Uplink Mobile UHF DTV Channels With Antenna Diversity》原文链接:https:///academic-journal-foreign_broadcasting-ieee-transactions_thesis/020*********.html132.《Performance Evaluation of TV Over Broadband Wireless Access 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Behaviors》原文链接:https:///academic-journal-foreign_broadcasting-ieee-transactions_thesis/020*********.html144.《On the Methodology for Calculating SFN Gain in Digital Broadcast Systems》原文链接:https:///academic-journal-foreign_broadcasting-ieee-transactions_thesis/020*********.html145.《Statistical Multiplexing of Upstream Transmissions in DOCSIS Cable Networks》原文链接:https:///academic-journal-foreign_broadcasting-ieee-transactions_thesis/020*********.html146.《Bit-Rate Allocation for Broadcasting of Scalable Video Over Wireless Networks》原文链接:https:///academic-journal-foreign_broadcasting-ieee-transactions_thesis/020*********.html147.《Full-Reference Video Quality Metric for Fully Scalable and Mobile SVC Content》。

基于导频受限短突发通信的高精度快速频偏估计袁静珍【摘要】针对导频符号辅助调制(PSAM)的导频受限短突发通信载波同步,提出了一种高精度快速的频偏估计算法.其基本原理是:利用自相关算子得到具有较大估计范围和较低信噪比门限的联合自相关算法,再利用互相关算子获得具有高精度的简化互相关算法.理论分析和仿真结果表明,提出的算法能够消除大频偏,而且还具有较低的复杂度和良好的解调性能.【期刊名称】《电讯技术》【年(卷),期】2018(058)010【总页数】5页(P1201-1205)【关键词】短突发通信;载波同步;导频符号辅助调制;频偏估计【作者】袁静珍【作者单位】韩山师范学院物理与电子工程学院,广东潮州521041【正文语种】中文【中图分类】TN911.231 引言近年来，短突发通信已经广泛应用于卫星遥感、深空通信等前沿领域，同时还将应用到第五代(5G)移动通信中[1-2]。

在这些通信领域中，通信双方的相对移动会产生多普勒效应，大多普勒频移会造成同步接收机无法实现相干解调，从而导致通信质量的急剧下降。

为了对抗大载波频偏，传统的估计算法可以分为数据辅助、非数据辅助两大类[3-4]，其中，非数据辅助这一类估计算法的信噪比门限和复杂度较数据辅助估计算法高，因此，在短突发通信中，普遍采用基于已知的数据符号的数据辅助估计算法。

用于第二代数字视频广播(Digital Video Broadcasting-Second Generation，DVB-S2)的数据帧结构将90个已知的数据符号作为帧头，再以1 440个数据符号附加36个导频符号为单元周期地构成DVB-S2数据帧结构。

文献[5-6]提出了一种基于导频符号辅助调制(Pilot-Symbol-Assisted-Modulation，PSAM)的数据帧结构，即将一定长度的导频符号分成两部分，含有若干个连续符号的部分作为帧头，细分成单个离散符号的部分插至帧中和帧尾。

MeGUI参数设置1 Main图1 Main选项卡1.1 Encoding ModeABR：一次成形。

码率的分配来自即时验算。

通常不应该使用这个模式。

ConstQuantizer：固定量化值输出。

每一类帧采用相同的量化值来量化，使得全片的质量大体相近。

这里指定的应该是P帧的qr，I和B帧的qr由第2页的比例计算来得到。

CQ模式适用与追求质量而不计全片码率和文件大小的片子使用。

2pass - 1st/2nd：手动进行2pass压制。

1st时可以用turbo以加快压制速度。

在1st pass时MeGUI不会输出任何的视频信息，也不会生成一个空的文件。

Automated 2pass：自动2pass压制。

一次设置就能生成2个任务，加快了处理速度。

3pass：高级功能，供高级用户使用。

Const Quality：恒定画质，对于体积没有要求的压制。

Quality中的数字越大表明质量越差文件越小，越小表明质量越好文件越大，0代表无损。

1.2 TuningsTunings下拉菜单中有多种片源预置参数可供选择，一般可以直接选择Default默认，然后在后面的选项卡中再进行具体的设置）。

1.3 A VC ProfilesHigh Profiles：主要用于电脑播放和高清视频的制作。

Main Profiles：主要用于移动设备视频的制作，如PSP等设备。

Baseline Profiles：主要用于低清视频的制作。

1.4A VC Level移动视频选择Level 3.0即可。

电脑视频可选择Level 4.1以上的选项。

默认值：Unrestricted/Autoguess建议值：Unrestricted/Autoguess2 Frame-Type2.1 H.264 FeaturesDeblocking：开启环路滤波，除马赛克。

默认值：勾选（当不勾选该项时该参数显示为：--no-deblock）建议值：勾选Deblocking Strength：--deblock 0:0：设置环路滤波的AlphaC的参数，范围-6－6。

专利名称：Low complexity cost function for sub-pixelmotion estimation发明人：Richard John Allen申请号：US13867788申请日：20130422公开号：US09667960B1公开日：20170530专利内容由知识产权出版社提供专利附图：摘要：Methods and circuitry are provided for finding an optimum vector for motion-compensated prediction in video coding. Processing circuitry is operable to define a block in an image, the defined block having a plurality of pixels. A first set of residuals iscalculated using a first motion vector. Each residual of the first set of residuals corresponds to a respective pixel within the plurality of pixels. Additionally, a first plurality of absolute differences is calculated. Each absolute difference in the first plurality of absolute differences is between a pair of residuals in the first set of residuals. The first plurality of absolute differences is summed, and is compared to sums of absolute differences of residuals calculated using other motion vectors. The motion vector corresponding to the lowest sum may be identified as an optimum motion vector.申请人：Altera Corporation地址：San Jose CA US国籍：US代理人：Michael H. Lyons更多信息请下载全文后查看。

Research on Super-resolution Reconstruction Algorithm Based on L1 normA Thesis Submitted to Chongqing Universityin Partial Fulfillment of the Requirement for theMaster’s Degree of EngineeringByWang BinSupervised by Prof. He Zhong ShiSpecialty: Computer Software and TheoryCollege of Computer Science of ChongqingUniversity, Chongqing, ChinaApril, 201摘要在道路病害检测中，由于摄像机多处于野外环境，通常摄像设备较为简陋，获取的道路图片分辨率较低。

同时由于硬件成本较高，且成像系统本身的技术瓶颈，使得在很多领域高分辨率图像通常难以获取。

图像超分辨率重建技术的基本原理是通过软件手段从多幅具有非冗余互补信息的低分辨率图像重构出包含更多信息的高分辨率图像的过程。

超分辨率重建技术不需要在硬件上改变成像系统便能提高图像的空间分辨率，这样能极大降低高分辨率图像的获取成本。

超分辨率重建技术广泛应用的同时也存在着许多亟待解决的问题，如超分辨率重建过程是病态的，解不唯一的。

另外超分辨率重建的全过程涉及多个环节，如图像配准中的运动估计、插值放大、能量函数选择、算法求解、去模糊，如何保证高效的实现重建过程需要各个环节的优化协调。

本文主要研究内容及成果如下：①提出基于高斯金字塔的小十字形搜索算法：针对图像配准中求取运动位移估计计算量大的缺点，采用基于高斯金字塔思想的运动估计算法，通过使用高斯金字塔将图像分层，先在金字塔上层搜索运动偏移量，然后以此偏移量为初始偏移量，再在金字塔下层进行搜索，由于金字塔上层搜索范围远小于原图，因此能大大提高搜索速度。

中国科学技术大学硕士学位论文第三章新一代视频编码标准——Ｈ２６４／ＡＶＣ得多，从而实现更高效的数据压缩。

量化是一种有损压缩过程，实际上就是将变换的系数除以相应的量化步长后四舍五入取整数。

ｔｔ．２６４／ＡＶＣ采用的一个重要变换方法是类似ＤＣＴ变换的４Ｘ４整数变换“”：”…，是一种只需加法和移位运算即可实现的低复杂度的精确变换。

同时，Ｈ．２６４／ＡＶＣ将变换过程的系数缩放运算并入量化过程，并将缩放系数与量化步长联合通过查表实现，既简化了核心变换运算，又避免了量化过程的除法运算。

３．３．１变换的类型Ｈ．２６４／Ａｖｃ采用各种优化技术提高了预测的效果，但同时对预测误差的编码提出更高的要求，希望预测误差编码无漂移（Ｄｒｉｆｔ—Ｆｒｅｅ）。

另外，为了适应更广泛的应用环境，Ｈ．２６４／ＡＶＣ希望变换只需１６一ｂｉｔ运算。

而４Ｘ４整数变换很好地满足了以上要求，是Ｈ．２６４／ＡＶＣ最重要、晟基本的一种变换。

一当宏块采用Ｉｎｔｒａ＿１６Ｘ１６预测方式时，由于图象相对平滑，交换后大部分能量集中在Ｄｃ系数上。

为了进一步减小Ｄｃ系数之间的相关性，将１６个（图３．３．１中编号Ｏ一１５）４Ｘ４块的Ｄｃ系数构成一个４Ｘ４的ＤＣ系数块（图３．３．１中编号一１），再进行Ｈａｄａｍａｒｄ变换处理，以进一步减，ｊ，ＤＣ系数之间的相关性。

对于色度分量也作类似处理，得Ｎ２个２×２５ｊＤＣ系数块（图３．３．１中编号１６—１７），然后再进行Ｈａｄａｍａｒｄ变换。

采用这种分级变换结构进一步提高了压缩效率。

图３．３．１４４块的扫描顺序以及Ｄｃ块的构成”１Ｈ．２６４／ＡＶＣ共支持３种变换方式，即针对４Ｘ４差值信号块的整数变换、针对４Ｘ４。

440C-CR30COMLOCK 00 01 02 03 04 05 06 07 08 0 10 1112 13 14 15 16 17 18 1 20 21+DC -DC 24 24Note: Configuration is what we tested,not the max capacityMicro800 PTO Motion PopularConfiguration Drawing• Optimized for compatibility with MicroLogix™ , SLC™, and Micro800 controllers.• Features include unicode language switching, alarm messages and history, and basic recipe capability• User Interface through Web browser•Panelview Component Configuration Publication CCSIMP-QR002A-EN-P• Supports communication via RS-232 (DH-485), RS-232 (DF1), RS-485 and Ethernet• Two USB ports for transferring files or updating firmware , Supports SD memory cardsPanelview Component Capacity• 4.3, 5.7 ,10.7 inch Display Sizes • Monochrome Transmissive FSTN Options/Color Transmissive TFT, Analog Touch• Automatic motor recognition and gain settings • Cost-effective motion control solution for smaller, low-axis count applications• Flexible command interfaces including digital I/O, analog, pulse train and Modbus-RTU• Configurable with Connected Component Workbench SoftwareKinetix 3 Configuration• Input volt range of 170…264V AC single and three-phase• Continuous power output of 0.05 W…1.5 kW• Modbus-RTU serial communications for changing drive parameters during runtimeKinetix 3 Capacity• CCAT contains Building blocks for indexing up to 3 axisKinetix 3 PerformanceExternal Power Supply• Embedded I/O 28 In, 20 Out,Base Analog I/O channels via plug in or expansion. Maximum Digital I/O :132, Max Expansion I/O Modules:4, 6 High Speed Counters, 3 PTOMicro850 Configuration• Designed for larger standalone machine applications that require more I/O or higher performance analog I/O• Number of I/O points embedded in the base 24,48 points (2085 expansion I/O’s)Micro850 PerformanceMicro 850 Overview 2080-LC50-48xxx Embedded communication USB, Serial, Ethernet Basic I/O size 48 points Max number of I/OUp to 132Max number of Expansion I/O 4Motion axis (Only available in model with transistor output)Up to 3 axis HSCUp to 6 embeddedTo Plant Wide Ethernet network To Computer for ProgrammingStratix 2000PanelView ComponentMicro850Kinetix 3External Encoder Differential / Open collector typeStratix 2000 Unmanaged Switch - 5 Copper PortsPanelview Component Performance 3• Compact DIN rail mount safety relay• 22 embedded Safety I/O • Support for 16 standardI/O via plug-ins440C-CR30 Safety Relay Micro850 Capacity• LC50-48xxx has 10Ksteps of instructions and 20KB of data memory and supports 5 plug in I/O modules & 4 expansion modules.• Supports Ladder Programming,Function Block Diagram, Structured TextC400Micro850 with2080-SERIALISOL (slot 1) & 2080-MOT-HSC (slot 2-4) plug-in modules.• Rotary Motors • TL-Series • Linear Motors• LDC-Series Iron Core • LCL-Series IronlessServo motorsSafety Diagnostics to Micro850 and to PVcSafety I/OConnectionsPulse Train Outputs (24V)High Speed feedback signals from Kinetix 3 Servo drive 5V Differential3High Speed feedback signals from External Encoder . (Up to 5V/24V, Up to 100kHz open collector 250kHz differential)Kinetix 3Kinetix 3Protocol RS-232Micro850: Modbus RTU Master 440C-CR30: SlaveDaisy ChainRS-485 to three Kinetix 3About this ConfigurationThis Micro 850 Controller based low cost system demonstrates the power and scalability of the Component class on an Ethernet I/P based network. This system utilizes standard Ethernet I/P technology to connect to plant wide network. This configuration also highlights Micro850 Capabilities in Connecting Safety relay, Motion control & HMl .A key advantage of this architecture is the ability to use Connected Components Workbench a common integrated environment for programming, configuration, commissioning, and motion tools for the Kinetix 3 ,Panelview Component , and Micro 800 Controller family of products.About the ProductsMicro 850 Controllers• The Micro850 controller is equipped with the same form factor, plug-in support, instruction/data size and motion capabilities as the 24-pt and 48-pt Micro830 controllers• Support up to four Micro850 Expansion I/O modules, Up to a maximum of 132 I/O points (with 48-pt model)• Designed for larger standalone machine applications that require more I/O or higher performance analog I/O than supported by Micro830.• Structured Text, Ladder Diagram and Function Block editors that support • Symbolic addressing• Connected Components Workbench software is used for PLC programming, HMI, drives and safety relay configurationMicro 850 Plugin Module2080-MOT-HSC• The new Allen-Bradley Micro800 Motion Feedback Axis plug-in enhances high speed counter function to Micro800 controllers.• With input frequency to 250 kHz, it supports both 5V differential line driver input and 24VDC input with 1 physical digital output and 15 virtual digital outputs.• Its programmable limit switch function makes it easy for user to turn ON and OFF output based on the counter value.• Through its motion feedback axis mode, virtual axis mode and touch probe function, the new plug-in addition provides an easy and intuitive way for users to obtain motion positioning information..440C-CR30 Safety Relay• Programmable Safety Relay designed for flexibility in a simple safety system• Predefined Safety Functions provide an easy to use system that is programed and configured using CCW software• Compact DIN rail mount designed with 22 embedded Safety I/O and support for 16 standard I/O via plug-ins in a 110mm wide package • Built in communications for programming and HMI diagnostic displayPanelView Component• A full line of displays ranging from 2” to 10”,• Communicate to MicroLogix, Micro800 and SLC controllers via serial (RS232 or RS422/RS485) networks on all terminals, and Ethernet on the C400, C600 and C1000 displays.• Features include unicode language switching, alarm messages and history, and basic recipe capability• PanelView Component software (DesignStation) is an offline programming software offering better user experience with significant DesignTime performance improvementKinetix 3 Servo DrivesSingle-axis solution for low-complexity motion applications, with or without a PLC • Digital I/O, analog, preset velocity, and pulse-train command interfaces• Performs indexing on up to 64 points through serial communication or over digital I/O • 170…264V AC, (200V-class) single-phase or three-phase1783 Stratix 2000™ Unmanaged Switches• Multiple port count and fiber options are available • Copper ports are 10/100 Mbps, full- or half-duplex• Fiber ports are 100 Mbps, full-duplex, with an LC fiber optic connector • Ideal Ethernet switch for small, isolated networks • Auto negotiates speed and duplex settings • Operates on 20V AC or 24V DC power • Automatic cable cross-over detectionFor More Information and HelpFor more information contact your local distributor or Rockwell Automation sales representative.• • Publication Library • My Support• A – Z Product DirectoryBill of MaterialsQty Catalog #DescriptionSystem: Controller Hardware 12080-LC50-48QBB Micro 850 Controller12080- SERIALISOL RS232/485 isolated serial port Plug in Module 32080-MOT-HSC Micro800 Motion Feedback Axis plug-in ModuleHMI Hardware12711C T4T PanelView C400, touch terminalSafety Relay Module1440C-CR30-22BBB Compact DIN rail mount safety relay designed with 22 embedded Safety I/O and support for 16 standard I/O via plug-ins 11761-CBL-HM02Communication cableMotion Hardware22071-AP1Kinetix 3 , 2.38Amps 2TL-A130P-BJ32AA Servo Motor22090-CFBM6DD-CCAA03Cable,Feedback,M-Circ.Plastic.D-DB15,INCR CC ,Std,3M 22090-CPWM6DF-16AA03Cable,Power,M-Circ.Plastic.D-Flying Lead,16AWG ,Std,3M12090-CCMDSDS-48AA01Serial Communication cable between K3 drive & 2080-SERIALISOL plug-in 1Serial Communication cable between K3 drivesNote: Catalog numbers consist of characters, each of which identifies a specific version or option for that component. Reference Publication GMC-SG001-EN-E Kinetix Motion Control Selection Guide for additional informationSystem: Communication Hardware11783-US05T Stratix 2000 Unmanaged Switch - 5 Copper Ports 41585J-M8TBJM-1M Shielded Ethernet CableConfiguration Tool Required19328-SO001-EN-CConnected Components Workbench software。

ｄｏｉ：１０．３９６９／ｊ．ｉｓｓｎ．１００３－３１１４．２０２２．０２．０１７引用格式：施育鑫，鲁信金，孙艺夫，等．ＭＩＭＯ通信模型下的相关矩阵组低复杂度设计［Ｊ］．无线电通信技术，２０２２，４８（２）：３２７－３３５．［ＳＨＩＹｕｘｉｎ，ＬＵＸｉｎｊｉｎ，ＳＵＮＹｉｆｕ，ｅｔａｌ．Ｌｏｗ⁃ｃｏｍｐｌｅｘｉｔｙＤｅｓｉｇｎｏｆＣｏｒｒｅｌａｔｉｏｎＭａｔｒｉｘＧｒｏｕｐｕｎｄｅｒＭＩＭＯＣｏｍｍｕｎｉｃａｔｉｏｎＭｏｄｅｌ［Ｊ］．ＲａｄｉｏＣｏｍｍｕｎｉｃａｔｉｏｎｓＴｅｃｈｎｏｌｏｇｙ，２０２２，４８（２）：３２７－３３５．］ＭＩＭＯ通信模型下的相关矩阵组低复杂度设计施育鑫１，鲁信金２，孙艺夫２，雷㊀菁２，李玉生１（１．国防科技大学第六十三研究所，江苏南京２１００００；２．国防科技大学电子科学学院，湖南长沙４１００００）摘㊀要：矩阵组常用于无线通信中的数据表示㊂在多输入多输出（Ｍｕｌｔｉｐｌｅ⁃ＩｎｐｕｔＭｕｌｔｉｐｌｅ⁃Ｏｕｔｐｕｔ，ＭＩＭＯ）通信模型中，基站利用信道数据设计适应信道的最小均方误差（ＭｉｎｉｍｕｍＭｅａｎＳｑｕａｒｅＥｒｒｏｒ，ＭＭＳＥ）均衡器接收矩阵组，以低复杂度地处理来自多用户的上行数据㊂首先分析了矩阵数据的关联性，通过时谱图确定矩阵组在时间维所具有的强相关性；其次采用插值算法进行低复杂度的矩阵估计，并提出最大插值比搜索算法计算各类插值算法的性能及其复杂度；接着利用一种改进的Ｓｔｒａｓｓｅｎ矩阵求逆算法来降低ＭＭＳＥ求逆过程的复杂度㊂相比传统的接收矩阵组，显著降低了计算复杂度㊂关键词：ＭＩＭＯ通信模型；奇异值分解；最小均方误差；计算复杂度中图分类号：ＴＮ９２９．５㊀㊀㊀文献标志码：Ａ㊀㊀㊀开放科学（资源服务）标识码（ＯＳＩＤ）：文章编号：１００３－３１１４（２０２２）０２－０３２７－０９Ｌｏｗ⁃ｃｏｍｐｌｅｘｉｔｙＤｅｓｉｇｎｏｆＣｏｒｒｅｌａｔｉｏｎＭａｔｒｉｘＧｒｏｕｐｕｎｄｅｒＭＩＭＯＣｏｍｍｕｎｉｃａｔｉｏｎＭｏｄｅｌＳＨＩＹｕｘｉｎ１，ＬＵＸｉｎｊｉｎ２，ＳＵＮＹｉｆｕ２，ＬＥＩＪｉｎｇ２，ＬＩＹｕｓｈｅｎｇ１（１．６３ｒｄＲｅｓｅａｒｃｈＩｎｓｔｉｔｕｔｅ，ＮａｔｉｏｎａｌＵｎｉｖｅｒｓｉｔｙｏｆＤｅｆｅｎｓｅＴｅｃｈｎｏｌｏｇｙ，Ｎａｎｊｉｎｇ２１００００，Ｃｈｉｎａ；２．ＳｃｈｏｏｌｏｆＥｌｅｃｔｒｏｎｉｃＳｃｉｅｎｃｅ，ＮａｔｉｏｎａｌＵｎｉｖｅｒｓｉｔｙｏｆＤｅｆｅｎｓｅＴｅｃｈｎｏｌｏｇｙ，Ｃｈａｎｇｓｈａ４１００００，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｍａｔｒｉｘｇｒｏｕｐｓａｒｅｏｆｔｅｎｕｓｅｄｔｏｒｅｐｒｅｓｅｎｔｄａｔａｉｎｗｉｒｅｌｅｓｓｃｏｍｍｕｎｉｃａｔｉｏｎ．ＩｎｔｈｅＭＩＭＯ（Ｍｕｌｔｉｐｌｅ⁃ＩｎｐｕｔＭｕｌｔｉｐｌｅ⁃Ｏｕｔｐｕｔ）ｃｏｍｍｕｎｉｃａｔｉｏｎｍｏｄｅｌ，ｔｈｅｂａｓｅｓｔａｔｉｏｎｏｆｔｅｎｕｓｅｓｔｈｅｃｈａｎｎｅｌｄａｔａｔｏｄｅｓｉｇｎａｃｈａｎｎｅｌ⁃ａｄａｐｔｅｄＭＭＳＥ（ＭｉｎｉｍｕｍＭｅａｎＳｑｕａｒｅＥｒｒｏｒ）ｅｑｕａｌｉｚｅｒｒｅｃｅｉｖｉｎｇｍａｔｒｉｘ，ｉｎｏｒｄｅｒｔｏｐｒｏｃｅｓｓｔｈｅｕｐｌｉｎｋｄａｔａｆｒｏｍｍｕｌｔｉｐｌｅｕｓｅｒｓｗｉｔｈｌｏｗｃｏｍｐｌｅｘｉｔｙ．Ｆｉｒｓｔ，ｔｈｅｃｏｒｒｅｌａｔｉｏｎｏｆｍａｔｒｉｘｄａｔａｉｓａｎａｌｙｚｅｄ，ａｎｄｔｈｅｓｔｒｏｎｇｃｏｒｒｅｌａｔｉｏｎｏｆｔｈｅｍａｔｒｉｘｇｒｏｕｐｉｎｔｈｅｔｉｍｅｄｉｍｅｎｓｉｏｎｉｓｄｅｔｅｒｍｉｎｅｄｔｈｒｏｕｇｈｔｉｍｅ⁃ｓｐｅｃｔｒｏｇｒａｍ．Ｓｅｃｏｎｄｌｙ，ａｎｉｎｔｅｒｐｏｌａｔｉｏｎａｌｇｏｒｉｔｈｍｉｓｕｓｅｄｆｏｒｌｏｗ⁃ｃｏｍｐｌｅｘｉｔｙｍａｔｒｉｘｅｓｔｉｍａｔｉｏｎ，ａｎｄａｍａｘｉｍｕｍｉｎｔｅｒｐｏｌａｔｉｏｎｒａｔｉｏｓｅａｒｃｈａｌｇｏｒｉｔｈｍｉｓｐｒｏｐｏｓｅｄｔｏｃａｌｃｕｌａｔｅｔｈｅｐｅｒｆｏｒｍａｎｃｅａｎｄｃｏｍｐｌｅｘｉｔｙｏｆｖａｒｉｏｕｓｉｎｔｅｒｐｏｌａｔｉｏｎａｌｇｏｒｉｔｈｍｓ．ＴｈｅｎａｎｉｍｐｒｏｖｅｄＳｔｒａｓｓｅｎｍａｔｒｉｘｉｎｖｅｒｓｉｏｎａｌ⁃ｇｏｒｉｔｈｍｉｓｕｓｅｄｔｏｒｅｄｕｃｅｔｈｅｃｏｍｐｌｅｘｉｔｙｏｆｔｈｅＭＭＳＥｉｎｖｅｒｓｉｏｎｐｒｏｃｅｓｓ．Ｃｏｍｐａｒｅｄｗｉｔｈｔｈｅｔｒａｄｉｔｉｏｎａｌｒｅｃｅｉｖｉｎｇｍａｔｒｉｘ，ｔｈｅｃｏｍｐｕｔａ⁃ｔｉｏｎａｌｃｏｍｐｌｅｘｉｔｙｉｓｓｉｇｎｉｆｉｃａｎｔｌｙｒｅｄｕｃｅｄ．Ｋｅｙｗｏｒｄｓ：ＭＩＭＯｃｏｍｍｕｎｉｃａｔｉｏｎｍｏｄｅｌ；ｓｉｎｇｕｌａｒｖａｌｕｅｄｅｃｏｍｐｏｓｉｔｉｏｎ（ＳＶＤ）；ＭＭＳＥ；ｃｏｍｐｕｔａｔｉｏｎａｌｃｏｍｐｌｅｘｉｔｙ收稿日期：２０２１－１２－１６论文来源：基于２０２１年 华为杯 第十八届中国研究生数学建模竞赛Ａ题建模改写，参赛团队获得竞赛一等奖（获奖率约１．１％）及华为专项奖（共１０项）㊂０㊀引言矩阵常常被用于表示无线通信㊁图像视频处理㊁计算机视觉㊁相控阵雷达的数据表示㊂随着用户需求的不断增加，数据规模㊁通信阵列的持续扩大，矩阵的大小和维度也随之快速增加，这给矩阵的数据存储㊁算法计算带来了很大的困难㊂矩阵的关联性是指矩阵数据在某些维度上的相关特性，例如视频中时间相邻的帧具有很强的矩阵关联性㊂因此，充分挖掘矩阵间关联性，以实现低复杂度的计算具有十分重要的价值和意义㊂１㊀问题描述对于所给定复数矩阵Ｈ＝Ｈｊ，ｋ{}，Ｈｊ，ｋɪＭˑＮ，ｊ＝１，２， ，Ｊ；ｋ＝１，２， ，Ｋ㊂其中，矩阵之间以及同一矩阵的元素之间有一定的相关性，包括：相同ｊ下标㊁不同ｋ下标的矩阵间存在一定的关联，即Ｈｊ，１，Ｈｊ，２，Ｈｊ，３， ，Ｈｊ，Ｋ{}间存在关联；且矩阵的各个元素间ｈ（ｊ，ｋ）ｍ，ｎ{}，ｍ＝１，２， ，Ｍ；ｎ＝１，２， ，Ｎ也存在关联，矩阵Ｈｊ，ｋɪＭˑＮ，ｊ＝１，２， ，Ｊ；ｋ＝１，２， ，Ｋ可表示为：Ｈｊ，ｋ＝ｈ（ｊ，ｋ）１，１ｈ（ｊ，ｋ）１，２ｈ（ｊ，ｋ）１，３ｈ（ｊ，ｋ）１，Ｎｈ（ｊ，ｋ）２，１ｈ（ｊ，ｋ）２，２ｈ（ｊ，ｋ）２，３ ｈ（ｊ，ｋ）２，Ｎ︙︙︙︙ｈ（ｊ，ｋ）Ｍ，１ｈ（ｊ，ｋ）Ｍ，２ｈ（ｊ，ｋ）Ｍ，３ ｈ（ｊ，ｋ）Ｍ，Ｎéëêêêêêùûúúúúú㊂（１）此外定义矩阵组Ｈ＝Ｈｊ，ｋ{}的一组数学运算，其中间结果Ｖ＝Ｖｊ，ｋ{}由式（２）给出：Ｖｊ，ｋ＝ｓｖｄ（Ｈｊ，ｋ）Ｈｊ，ｋ＝Ｕｊ，ｋＳｊ，ｋＶ Ｈｊ，ｋＶｊ，ｋ＝Ｖ Ｈｊ，ｋ（：，１：Ｌ）ìîíïïïï，（２）式中，ｊ＝１，２， ，Ｊ；ｋ＝１，２， ，Ｋ，ｓｖｄ（㊃）为矩阵奇异值分解（ＳｉｎｇｕｌａｒＶａｌｕｅＤｅｃｏｍｐｏｓｉｔｉｏｎ，ＳＶＤ）中求解右奇异向量的过程；Ｖｊ，ｋ是由Ｈｊ，ｋ的前Ｌ个右奇异向量构成的矩阵，维度为ＮˑＬ㊂为得到最终输出结果Ｗ＝Ｗｊ，ｋ{}，先将不同ｊ下标㊁相同ｋ下标的Ｖｊ，ｋ进行横向的拼接，得到维度ＮˑＬＪ的Ｖｋ＝［Ｖ１，ｋ， Ｖｊ，ｋ， ＶＪ，ｋ］，然后根据式（３）获取Ｗｋ：Ｗｋ＝Ｖｋ（ＶＨｋＶｋ＋σ２Ｉ）－１，（３）式中，σ２为固定常数；Ｗｋ维度同Ｖｋ；Ｉ为单位矩阵，维度为ＬＪˑＬＪ㊂最后将各Ｗｋ按如式（４）进行拆解：Ｗｋ＝［Ｗ１，ｋ， Ｗｊ，ｋ， ＷＪ，ｋ］，（４）式中，Ｗｊ，ｋ为Ｗｋ中顺序排列的子矩阵，维度为ＮˑＬ㊂为了降低计算和储存的复杂度，分析相关矩阵组的关联性，通过建模对输出结果进行估计，建模过程可用式（５）表示：Ｗ＾＝ｆ（Ｈ），（５）式中，Ｗ＾即为对输出结果Ｗ的建模估计㊂该建模过程可拆分为如式（６）的两个步骤㊂Ｖ＾＝ｆ１（Ｈ）Ｗ＾＝ｆ２（Ｖ＾）{，（６）式中，ｆ１（㊃）表示从输入矩阵组Ｈ到中间结果Ｖ的建模过程，Ｖ＾表示中间结果Ｖ的建模估计，ｆ２（㊃）表示从中间结果Ｖ到最终结果Ｗ的建模过程，Ｗ＾表示最终结果Ｗ的建模估计㊂定义Ｗ的建模估计精度为：ρｌ，ｊ，ｋ（Ｗ）＝Ｗ＾Ｈｌ，ｊ，ｋＷｌ，ｊ，ｋ２Ｗ＾ｌ，ｊ，ｋ２ Ｗｌ，ｊ，ｋ ２，ｌ＝１，２， ，Ｌ，（７）式中， ㊃ ２表示矢量的欧几里得范数（即２范数，对于列矢量ａ， ａ ２＝ａＨａ），Ｗｌ，ｊ，ｋ表示Ｗｊ，ｋ的第ｌ列㊂上式中，Ｗ＾Ｈｌ，ｊ，ｋＷｌ，ｊ，ｋ为复数标量，此处取其欧几里得范数即获取其模值㊂为描述方便，额外定义Ｗ的最低建模精度为ρｍｉｎ（Ｗ）：ρｍｉｎ（Ｗ）ｍｉｎｌɪ｛１，２， ，Ｌ｝ｊɪ｛１，２， ，Ｊ｝ｋɪ｛１，２， ，Ｋ｝ρｌ，ｊ，ｋ（Ｗ），（８）式中，ｍｉｎｌ，ｊ，ｋ（㊃）表示在ｌ，ｊ，ｋ三个维度上取最小值㊂另外，中间结果Ｖ的建模估计精度ρｌ，ｊ，ｋ（Ｗ）的定义及最低建模精度ρｍｉｎ（Ｗ）的定义与此相同㊂计算复杂度定义为由矩阵组Ｈ计算得到结果矩阵组Ｗ所需要的总计算复杂度㊂复数矩阵运算可拆解为基本的复数运算，而基本的复数运算又可进一步拆解为基本的实数运算㊂例如，复数乘法（ａ＋ｂｊ）（ｃ＋ｄｊ）＝（ａｃ－ｂｄ）＋（ａｄ＋ｂｃ）ｊ的复杂度为４次实数乘法和２次实数加（减）法㊂实数基本运算的复杂度按照表１计算㊂表１㊀实数基本运算的计算复杂度Ｔａｂ．１㊀Ｃｏｍｐｕｔａｔｉｏｎａｌｃｏｍｐｌｅｘｉｔｙｏｆｂａｓｉｃｏｐｅｒａｔｉｏｎｓｏｎｒｅａｌｎｕｍｂｅｒｓ运算类型计算复杂度加（减）法１乘法３倒数２５平方根２５自然指数２５自然对数２５正弦２５余弦２５其他１００２０２１研究生数学建模Ａ题提供的数据集附件（Ｄａｔａ１ Ｄａｔａ６）给出的详细数据，包括输入矩阵组㊁标准中间矩阵组和标准输出矩阵组的数据及其维度，其中Ｍ＝４，Ｎ＝６４，Ｌ＝２，Ｊ＝４，Ｋ＝３８４，σ２＝０．０１，数据为十进制格式㊂根据所给数据Ｄａｔａ１ Ｄａｔａ６中的Ｍ＝４，Ｎ＝６４，Ｊ＝４，可对应通信模型中共有Ｊ＝４个用户，每个用户的发射天线数为Ｍ＝４，基站的接收天线数为Ｎ＝６４㊂Ｌ＝２表示取信道衰落程度最低的２个信道向量，即对矩阵进行压缩㊂Ｋ表示信道探测时隙数㊂σ２表示信道中高斯白噪声的噪声方差㊂基于给定的所有矩阵数据，本文通过分析数据间的关联性，解决相关矩阵组的低复杂度计算问题，即以减少计算复杂度为目标进行模型优化㊂设计相应的近似分析模型Ｗ＾＝ｆ（Ｈ），在满足ρｍｉｎ（Ｖ）ȡρｔｈ＝０．９９的情况下，使根据表格计算的总计算复杂度最低㊂２　通信模型建立建立基于矩阵的多输入多输出（Ｍｕｌｔｉｐｌｅ⁃ＩｎｐｕｔＭｕｌｔｉｐｌｅ⁃Ｏｕｔｐｕｔ，ＭＩＭＯ）通信模型［１］如图１所示，Ｊ个用户发送信息，信号经过信道Ｈ到达基站，基站有Ｎ根接收天线㊂图１㊀矩阵关系的通信模型建立示意图Ｆｉｇ．１㊀Ｓｃｈｅｍａｔｉｃｄｉａｇｒａｍｏｆｅｓｔａｂｌｉｓｈｉｎｇｃｏｍｍｕｎｉｃａｔｉｏｎｍｏｄｅｌｏｆｍａｔｒｉｘｒｅｌａｔｉｏｎｓｈｉｐ如图２所示，对于某个用户，拥有Ｍ个天线，各个天线均可与基站天线进行通信㊂其通信信道矩阵Ｈｊ，ｋ＝Ｕｊ，ｋＳｊ，ｋＶＨｊ，ｋ，通过ＳＶＤ分解，将信道矩阵分解成方向酉矩阵和信道随机衰落矩阵的乘积，其中，Ｓｊ，ｋ为随机矩阵，代表波束随机衰落主信道矩阵，Ｕｊ，ｋ和Ｖ Ｈｊ，ｋ分别是用户和基站特征向量矩阵的相关矩阵㊂由于ＶＨｊ，ｋ与基站和用户位置相关且各个节点的位置相对固定，可以取信道衰落程度最低的Ｌ个信道向量压缩Ｖ Ｈｊ，ｋ矩阵，即Ｖｊ，ｋ＝ＶＨｊ，ｋ（：，１：Ｌ）㊂以上模型建立与式（２）一致㊂图２㊀单个用户和基站的通信示意图Ｆｉｇ．２㊀Ｓｃｈｅｍａｔｉｃｄｉａｇｒａｍｏｆｃｏｍｍｕｎｉｃａｔｉｏｎｂｅｔｗｅｅｎａｓｉｎｇｌｅｕｓｅｒａｎｄａｂａｓｅｓｔａｔｉｏｎ此时，可使用压缩矩阵Ｖｊ，ｋ来表示Ｈ矩阵㊂进一步，引入最小均方误差（ＭｉｎｉｍｕｍＭｅａｎＳｑｕａｒｅＥｒｒｏｒ，ＭＭＳＥ）的概念来解释题设条件㊂如图３所示的信道模型中，信号ｘ经过信道Ｖ＝Ｖｊ，ｋ，由于白噪声ｎ的影响，接收信号ｙ可表示为：ｙ＝Ｖｘ＋ｎ㊂（９）图３㊀信号传输模型（求解ＭＭＳＥ流程）Ｆｉｇ．３㊀Ｓｉｇｎａｌｔｒａｎｓｍｉｓｓｉｏｎｍｏｄｅｌ（ｐｒｏｃｅｓｓｏｆｓｏｌｖｉｎｇＭＭＳＥ）ＭＭＳＥ的目的是找到一个矩阵Ｗ＝Ｗｊ，ｋ{}，使得Ｗｙ更加接近ｘ㊂得到ｘ ＝Ｗｙ与原始发送信号ｘ的差值为：ｅ＝ｘ －ｘ＝Ｗｙ－ｘ㊂（１０）此时的ＭＭＳＥ为：ＭＭＳＥ＝ＥｅＨｅ{}㊂（１１）假设接收到的数据ｙ和误差ｅ是不相关的，即Ｅｅ㊃ｙＨ{}＝０㊂（１２）将式（１０）代入式（１２）可得：Ｅ（Ｗｙ－ｘ）㊃ｙＨ{}＝０㊂（１３）将式（１３）左边进一步展开可得：㊀㊀Ｅ（Ｗｙ－ｘ）㊃ｙＨ{}＝ＥＷｙｙＨ{}－ＥｘｙＨ{}＝ＷＥｙｙＨ{}－ＥｘｙＨ{}㊂（１４）由式（１３）和式（１４）可得：Ｗ＝ＥｘｙＨ{}ＥｙｙＨ{}－１㊂（１５）接下来对ＥｙｙＨ{}和ＥｘｙＨ{}进行处理，首先对于ＥｙｙＨ{}，将其进一步展开：㊀㊀ＥｙｙＨ{}＝Ｅ（Ｖｘ＋ｎ）（Ｖｘ＋ｎ）Ｈ{}＝ＥＶｘｘＨＶＨ＋ＶｘｎＨ＋ｎｘＨＶＨ＋ｎｎＨ{}㊂（１６）此处假设输入信号ｘ和噪声ｎ不相关，则ｎｘＨ与ｎｘＨ值为０，可得：㊀㊀㊀ＥｙｙＨ{}＝ＥＶｘｘＨＶＨ＋ｎｎＨ{}＝ＶＥｘｘＨ{}ＶＨ＋ＥｎｎＨ{}＝Ｖ（Ｐ㊃Ｉ）ＶＨ＋σ２㊃Ｉ，（１７）式中，Ｐ为发送信号ｘ的能量，σ２为噪声ｎ的方差㊂其次对于ＥｘｙＨ{}，展开如下：㊀㊀㊀ＥｘｙＨ{}＝Ｅｘ（Ｖｘ＋ｎ）Ｈ{}＝ＥｘｘＨＶＨ＋ｘｎＨ{}＝ＥｘｘＨＶＨ{}＝ＥｘｘＨ{}ＶＨ＝（Ｐ㊃Ｉ）ＶＨ㊂（１８）得到ＥｙｙＨ{}和ＥｘｙＨ{}后，将其代入式（１５）可得到Ｗ的表达式：㊀㊀Ｗ＝ＥｘｙＨ{}ＥｙｙＨ{}－１＝（Ｐ㊃Ｉ）ＶＨ（Ｖ（Ｐ㊃Ｉ）ＶＨ＋σ２㊃Ｉ）－１＝Ｐ㊃ＶＨ（ＰＶＶＨ＋σ２㊃Ｉ）－１＝ＶＨ（ＶＶＨ＋㊃Ｉ）－１㊂（１９）当发送信号ｘ的能量Ｐ为１时，则可得：Ｗ＝ＶＨ（ＶＶＨ＋σ２㊃Ｉ）－１㊂（２０）综合上述分析，ＭＩＭＯ模型中利用信道数据计算信道的ＭＭＳＥ均衡器接收矩阵的复杂度主要来源于ＳＶＤ分解与式（２０）中的矩阵求逆㊂３㊀相关矩阵组的低复杂度计算由前文可知，Ｈ㊁Ｖ和Ｗ之间的关系可以由图４表示㊂图４㊀Ｈ㊁Ｖ和Ｗ的关系示意图Ｆｉｇ．４㊀ＲｅｌａｔｉｏｎｓｈｉｐｏｆＨ，ＶａｎｄＷ３．１㊀利用相关矩阵组的关联性降低计算复杂度利用相关矩阵组的关联性以降低计算复杂度，其具体分析及操作如下㊂３．１．１㊀矩阵数据的关联性对于信道系数复数矩阵Ｈ＝Ｈｊ，ｋ{}，其中Ｈｊ，ｋɪＭˑＮ，ｊ＝１，２， ，Ｊ；ｋ＝１，２， ，Ｋ㊂因此，该矩阵是一个ＭˑＮˑＪˑＫ维的信道矩阵，其中Ｍ表示接收天线的数量，Ｎ表示发射天线的数量，Ｊ表示用户数量，Ｋ表示时隙个数㊂由前文建立的通信模型，对Ｈ矩阵内在的关联性进行分析㊂首先，Ｈｊ，ｋ内部的关系可表示为不同天线构建出的不同信道之间的相关性㊂在一般高斯白噪声信道条件下，天线阵列之间固定的距离和入射角关系将带来一定的规律，但数据集中未能发现Ｈｊ，ｋ内部可靠的相关特性㊂这可能是由于天线之间的方向性㊁距离之间的差异较大，使得信道在空间上的相关特性不再明显㊂考虑不同ｋ时隙，同一用户ｊ的信道系数情况，即ＭＮ个信道的时间相关性㊂图５给出了ＭＮ个信道在时隙ｋ＝１，２，３情况下的幅度响应和相位响应㊂可以看出，在不同ｋ下的幅度和角度的变化程度不大，这可以理解为在相关时间内，信道的变化很小，这进一步验证通信建模的可行性㊂（ａ）不同信道系数的幅度响应（ｂ）不同信道系数的相位响应图５㊀同ｊ不同ｋ的Ｈ块之间的幅度和相位响应关系Ｆｉｇ．５㊀ＡｍｐｌｉｔｕｄｅａｎｄｐｈａｓｅｒｅｓｐｏｎｓｅｒｅｌａｔｉｏｎｓｈｉｐｂｅｔｗｅｅｎｂｌｏｃｋｓＨｗｉｔｈｔｈｅｓａｍｅｊａｎｄｄｉｆｆｅｒｅｎｔｋ基于上述两层分析，得出Ｈｊ，ｋ块在时间维上的相关性㊂而在时间维上利用ＳＶＤ与ＭＭＳＥ求Ｗ矩阵是独立的，无法直接用于算法简化，为此，本文利用时间相关性，并运用插值算法直接估计Ｗ矩阵㊂具体的，由于同ｊ不同ｋ的块在时间维上的相关性，在经过函数Ｗ＾＝ｆ（Ｈ）后，具有相关性的输入Ｈ，与得到的Ｗ之间将保持相关性㊂因此，可以利用同ｊ不同ｋ的Ｗ的相关性，通过插值算法获取某些ｋ值上的Ｗ矩阵㊂这将直接减少ＳＶＤ与ＭＭＳＥ求逆过程的计算数量㊂为了便于理解，将Ｌ维与Ｊ维（用户数）进行合并，因此Ｗ矩阵可以改写为三维矩阵㊂将矩阵按照Ｋ维展开，不同ｋ下标的矩阵可以由二维平面示意，其插值过程如图６所示㊂图６㊀Ｗ矩阵的插值示意图Ｆｉｇ．６㊀ＳｃｈｅｍａｔｉｃｄｉａｇｒａｍｏｆｉｎｔｅｒｐｏｌａｔｉｏｎｏｆＷｍａｔｒｉｘ３．１．２矩阵数据Ｗ的插值算法对于矩阵数据Ｗ的插值算法，采用ｌｉｎｅａｒ插值法㊁ｓｐｌｉｎｅ插值法与Ｐｃｈｉｐ插值法进行建模插值［２－４］㊂此处，引入 最大插值比 作为评估方法来评价插值性能，参数寻优的过程可以表示为：㊀㊀Ｒａｔｅ＝ｍａｘ１Ｒ{}ｓ．ｔ．ρｍｉｎ（Ｗ）ｍｉｎｌɪ｛１，２， ，Ｌ｝ｊɪ｛１，２， ，Ｊ｝ｋɪ｛１，２， ，Ｋ｝ρｌ，ｊ，ｋ（Ｗ）＞０．９９，（２１）式中，Ｒ表示每隔Ｒ个点进行一次插值运算㊂因此，式（２１）表示插值结果满足最小建模精度的约束下，使得插值数量越多的寻优目标㊂因此，为满足题设对于拟合后Ｗ矩阵对最小建模精度的要求，需选择合适的插值方法并计算最大插值比，为此进一步提出了最大插值比搜索方法，以评估在不同信道条件数据集下可用的插值参数，最大插值比搜索算法的详细过程在算法１中给出，其基本思想为通过给定的初始插值比Ｒａｔｅ，和选定的插值类型，不断迭代和逼近给定插值类型下的满足要求的最大插值比㊂当取得的插值比越大时，意味着Ｗ矩阵的更多部分可以通过在Ｋ维度上的相关性插值得到，不需要通过ＳＶＤ与ＭＭＳＥ求逆过程，这将大大减少计算的复杂度㊂此外，Ｄａｔａ集最大插值比的计算过程可以理解为适应信道的训练过程㊂在后续过程中，在外部信道条件未剧烈改变的情况下，不需要再次执行，因此最大插值比搜索可以在线下执行，不会影响算法的复杂度㊂算法１所提出的最大插值比搜索算法输入：训练数据集Ｄａｔａ，包含通过ＭＭＳＥ计算的标准Ｗ矩阵；初始化：初始插值比Ｒａｔｅ＝１／Ｒ，插值间隔Ｒ的初始值可以设定为Ｒ＝２㊂Ｒ的中止值可以设定为Ｒｍａｘ＝２００选定的插值类型：Ｌｉｎｅａｒ，Ｓｐｌｉｎｅ，Ｐｃｈｉｐ㊂执行过程：ＦｏｒＲ＜Ｒｍａｘ＝２００㊀ＦｏｒｉＮ＝１：Ｎ㊀㊀ＦｏｒｉＬＪ＝１：ＬＪ１．根据选定的插值比Ｒａｔｅ，确定插值点所在的横坐标序列ｘ，其中ｘ＝［１，Ｒ，２Ｒ， ，ｐＲ］Ｔ，ｐＲɤＫ＝３８４２．合并Ｊ个用户的Ｗ矩阵，使其降维为ＮˑＬＪˑＫ３．将Ｗ矩阵的第三维度中与ｘ重合的部分置零，新建为Ｗ＾，在Ｍａｔｌａｂ中可采用ｓｅｔｄｉｆｆ函数㊂置零部分准备后续进行插值填充４．执行插值操作Ｗ＾（ｉＮ，ｉＬＪ，１：Ｋ）＝ｉｎｔｅｒｐ１（ｚ，ｙ，１：Ｋ，插值类型）其中ｉｎｔｅｒｐ１表示插值函数，具体使用方法可参考Ｍａｔｌａｂ中对应函数５．评估插值结果ρｌ，ｊ，ｋ（Ｗ）＝Ｗ＾Ｈｌ，ｊ，ｋＷｌ，ｊ，ｋ ２Ｗ＾ｌ，ｊ，ｋ ２ Ｗｌ，ｊ，ｋ ２，ｌ＝１，２， ，Ｌ若ρｍｉｎ（Ｗ） ｍｉｎｌɪ｛１，２， ，Ｌ｝ｊɪ｛１，２， ，Ｊ｝ｋɪ｛１，２， ，Ｋ｝ρｌ，ｊ，ｋ（Ｗ）＞０．９９，则跳出循环（ｂｒｅａｋ）㊀㊀Ｅｎｄｉｆ㊀ＥｎｄｉｆＥｎｄｉｆ输出：插值类型，最大插值比Ｒａｔｅ表２给出了３种常见插值方法在６个Ｄａｔａ集的最大插值比㊂由于Ｄａｔａ１ Ｄａｔａ６来自不同的信道条件，因此同一插值方法在插值过程中计算出的最大插值比有较大区别㊂例如Ｄａｔａ３数据集的最大插值比显著较小，这可以理解为信道的时变性强或受到干扰噪声较大，插值算法在此时难以满足要求，需要降低最大插值比㊂进一步，横向对比３种插值方法，可见在多数的数据集下，Ｌｉｎｅａｒ插值的最大插值比最小，性能最差，这是由于简单的Ｌｉｎｅａｒ插值精度较低㊂Ｓｐｌｉｎｅ插值与Ｐｃｈｉｐ插值的最大插值比性能相近，Ｓｐｌｉｎｅ插值在Ｄａｔａ１与Ｄａｔａ３上表现比Ｐｃｈｉｐ插值较好㊂从原理上分析，可以理解为当基础函数振荡时，Ｓｐｌｉｎｅ比Ｐｃｈｉｐ更好地捕获点之间的移动，后者会在局部极值附近急剧扁平化，这在该场景下带来了更好的插值性能［５－６］㊂表２㊀３种插值方法在６个Ｄａｔａ集的最大插值比Ｔａｂ．２㊀Ｍａｘｉｍｕｍｉｎｔｅｒｐｏｌａｔｉｏｎｒａｔｉｏｏｆｔｈｅｔｈｒｅｅｉｎｔｅｒｐｏｌａｔｉｏｎｍｅｔｈｏｄｓｉｎ６ｄａｔａｓｅｔｓ最大插值比Ｌｉｎｅａｒ插值Ｓｐｌｉｎｅ插值Ｐｃｈｉｐ插值Ｄａｔａ１１／２０１／１５１／２０Ｄａｔａ２１／３０１／３０１／３０Ｄａｔａ３１／１３０１／１１１１／１１８Ｄａｔａ４１／３１／２１／２Ｄａｔａ５１／９１／９１／９Ｄａｔａ６１／１７１／１０１／１０复数矩阵运算可拆解为基本的复数运算，而基本的复数运算又可进一步拆解为基本的实数运算，实数基本运算的复杂度按照表１计算㊂对于Ｌｉｎｅａｒ插值法，根据上述描述，可得需要加减法６次㊁乘法２次㊁倒数１次，且对于复数，幅度和相位要分别插值，总复杂度是实数插值的两倍㊂然而，特殊的是这里插值点的横坐标是均匀的，因此计算的复杂度可以大大简化，仅需要３次加法㊁２次乘法和１次倒数，因此总复杂度为６８㊂每个插值时刻ｋ，共需要ＮＪＬ次插值，因此需要复杂度６８ＮＪＬ㊂本文的参数取值为Ｎ＝６４，Ｊ＝４，Ｌ＝２，因此计算复杂度为１７４０８㊂对于Ｓｐｌｉｎｅ插值法，其计算步骤为：求三次函数的系数，然后将插值点横坐标代入三次函数，计算又需要６次加法㊁６次乘法㊂且对于复数，幅度和相位要分别插值，总复杂度是实数插值的２倍，因此总复杂度为１２４，每个插值时刻ｋ，计算复杂度１２４ＮＪＬ㊂取本文参数，计算复杂度为３１７４４㊂对于Ｐｃｈｉｐ插值，由于其性能不如Ｓｐｌｉｎｅ且计算复杂度相似，因此不在此处进行考虑㊂３．２㊀降低矩阵求逆的计算复杂度对于求解逆矩阵Ｖｋ（ＶＨｋＶｋ＋σ２Ｉ）－１过程中的计算复杂度，当使用高斯消元法时［７］，求解维度ＬＪˑＬＪ的矩阵的逆矩阵的复杂度近似为Ｏ（（ＬＪ）３）；当矩阵求逆过程中使用的矩阵乘法使用文献［８］中的Ｓｔｒａｓｓｅｎᶄｓ方法时，其提出的矩阵相乘公式将常规的矩阵相乘的运算量减少很多，可以将上述复杂度降低到Ｏ（（ＬＪ）２．８０７）㊂为此，进一步研究矩阵求逆降低复杂度算法，明显看出Ｖｋ（ＶＨｋＶｋ＋σ２Ｉ）－１为Ｈｅｒｍｉｔｅ正定阵［９］，采用了一种改进的Ｓｔｒａｓｓｅｎ矩阵求逆算法［１０］，该算法结合Ｓｔｒａｓｓｅｎ矩阵求逆的高效性以及Ｈｅｒｍｉｔｅ正定阵的共轭对称性特点，使得算法运算量小，结构也简化许多㊂首先，对于矩阵分块直接求逆，假设一个Ｎ阶（这里的Ｎ被重新定义）的方阵Ａ，分块如下：Ａ＝［ａ１１］ｎˑｎ［ａ１２］ｎˑｍ［ａ２１］ｍˑｎ［ａ２２］ｍˑｎæèçöø÷ＮˑＮ㊂（２２）设Ａ的逆矩阵分块如下：Ａ＝［ｃ１１］ｎˑｎ［ｃ１２］ｎˑｍ［ｃ２１］ｍˑｎ［ｃ２２］ｍˑｎæèçöø÷ＮˑＮ，（２３）则根据矩阵分块求逆的原理有：ｃ１１＝（ａ１１－ａ１２ˑａ１２１ˑａ２１）－１ｃ１２＝－ｃ１１ˑａ１２ˑａ－１２２ｃ２１＝－ａ－１２２ˑａ２１ˑｃ１１ìîíïïï，（２４）式中，对于ＮˑＮ阶矩阵，需要的矩阵相乘的运算量级为Ｎ３㊂对于ｎ＞１，ｍ＞１，直接分块求逆算法在具体实现中需要利用递归实现，具体的运算量按照复数乘加次数统计，对于上述的直接求逆算法，以乘加次数统计理论运算量，式（２４）各部分运算量：ｃ１１的运算量为４ｍ２ｎ＋４ｍｎ２－ｍｎ，ｃ１２和ｃ２１的运算量相等，均为４ｍ２ｎ＋４ｍｎ２－６ｍｎ，同理，ｃ２２的运算量为４ｍ２ｎ＋４ｍｎ２－ｍ２，为此，矩阵Ａ利用一次分块求逆的总的运算量为：㊀㊀㊀Ｔ（１）（Ｎ）＝Ｔ（ｍ）＋Ｔ（ｎ）＋１６ｍ２ｎ＋１２ｍｎ２－１３ｍｎ－ｍ２＋４ｍ３，（２５）式中，Ｔ（１）（Ｎ）表示利用一次矩阵分块求逆算法计算矩阵求逆的总计算量，Ｔ（ｍ）和Ｔ（ｎ）分别表示对ｍ和ｎ阶复矩阵求逆所需的运算量㊂由式（２４）可知，经过一次分块求逆之后的运算量依然很高，即Ｔ（１）（Ｎ） Ｏ［（ｍａｘ（ｍ，ｎ））３］，同样可知，式（２４）中的（ａ１１－ａ１２ˑａ－１２２ˑａ２１）－１和ａ－１２２可以继续作为需要求逆的复矩阵，利用式（２２） （２４）可以继续分块求逆，所需运算量即式（２５）中Ｔ（ｍ）和Ｔ（ｎ）部分㊂采用改进的Ｓｔｒａｓｓｅｎ矩阵求逆算法，结合式（２２） （２３），Ｓｔｒａｓｓｅｎ算法应用到求逆运算有如下公式：Ｒ１＝ａ－１１１Ｒ２＝ａ２１ˑＲ１Ｒ３＝Ｒ１ˑａ１２Ｒ４＝ａ２１ˑＲ３Ｒ５＝Ｒ４－ａ２２Ｒ６＝Ｒ－１５ｃ１２＝Ｒ３ˑＲ６ｃ２１＝Ｒ６ˑＲ２Ｒ７＝Ｒ３ˑｃ２１ｃ１１＝Ｒ１－Ｒ７ｃ２２＝－Ｒ６ìîíïïïïïïïïïïïïïïïï㊀㊀㊂（２６）式中，对于ＮˑＮ阶矩阵，需要的矩阵相乘的运算量级为Ｎｌｏｇ６２＝Ｎ２．５８５，相比于矩阵分块直接求逆，运算量随着矩阵维数将有显著降低㊂按照前面相同的运算量统计方法，根据式（２５），矩阵利用一次分块求逆的总运算量为：㊀㊀㊀㊀Ｔ（１）＝Ｔ（ｍ）＋Ｔ（Ｎ）＋１３ｍ２ｎ＋１１ｍｎ２－４ｍｎ＋３ｍ２㊂（２７）对于ｎ＞１，ｍ＞１，Ｓｔｒａｓｓｅｎ矩阵求逆算法也是利用递归实现的，但因为Ｓｔｒａｓｓｅｎ算法减少了矩阵复乘次数，所以相比直接分块的常规算法运算量有明显降低㊂又由于ａ１１，ａ２２，ａ－１２２均为Ｈｅｒｍｉｔｅ矩阵，且ａＨ１２＝ａ２１，代入到式（２４）中得Ｒ３＝ＲＨ２，ｃ２１＝ｃＨ１２，根据Ｈｅｒ⁃ｍｉｔｅ矩阵的共轭对称性，式（２６）可进一步改写为：Ｒ１＝ａ－１１１Ｒ２＝ａ２１ˑＲ１Ｒ３＝ＲＨ２Ｒ４＝ａ２１ˑＲ３Ｒ５＝Ｒ４－ａ２２Ｒ６＝Ｒ－１５ｃ１２＝Ｒ３ˑＲ６ｃ２１＝ｃＨ１２Ｒ７＝ＲＨ２ˑｃ２１ｃ１１＝Ｒ１－Ｒ７ｃ２２＝－Ｒ６ìîíïïïïïïïïïïïïïïïï，（２８）式中，对于ＮˑＮ阶矩阵，需要的矩阵相乘的运算量级为Ｎ２㊂为了便于比较新求逆算法的性能改善，按照前面相同的运算量统计方法，根据式（２８），矩阵Ａ利用一次分块求逆的总的运算量为：Ｔ（１）（Ｎ）＝Ｔ（ｍ）＋Ｔ（Ｎ）＋８ｍ２ｎ＋８ｍｎ２＋ｍｎ，（２９）式中，Ｔ（１）（Ｎ）表示利用改进算法计算一次矩阵求逆的运算量，Ｔ（ｍ）和Ｔ（ｎ）部分表示对ｍ和ｎ阶复矩阵求逆所需的运算量㊂由式（２８）可知，经过一次分块求逆之后的运算量Ｔ（１）（Ｎ） Ｏ［（ｍａｘ（ｍ，ｎ））３］㊂同样可以得出，式（２８）中的Ｒ－１５可以继续作为需要求逆的复矩阵，利用式（２８）可以继续分块求逆，所需运算量即式（２９）中Ｔ（ｍ）和Ｔ（ｎ）部分㊂和矩阵直接分块求逆算法相比，新的求逆算法虽然增加了一些加减运算，但复乘次数降低㊂对于维数较高的矩阵，其中有大量的复矩阵运算，复乘消耗的运算量将远大于加减法，而这个运算量随着矩阵维数增加将有显著增大，因此新算法对于复乘次数的降低将显著改善求逆运算的实时性能㊂和常规Ｓｔｒａｓｓｅｎ矩阵求逆算法相比，改进的算法由于利用了求逆矩阵的特点，即对Ｈｅｒｍｉｔｅ矩阵进行求逆运算，所以在运算量和算法复杂度上都有明显的降低㊂常规算法计算一次矩阵求逆的计算复杂度如表３所示，改进算法计算一次矩阵求逆的计算复杂度如表４所示㊂表３㊀常规算法计算一次矩阵求逆的计算复杂度Ｔａｂ．３㊀Ｃｏｍｐｕｔａｔｉｏｎａｌｃｏｍｐｌｅｘｉｔｙｏｆｔｈｅｃｏｎｖｅｎｔｉｏｎａｌａｌｇｏｒｉｔｈｍｔｏｃａｌｃｕｌａｔｅｔｈｅｉｎｖｅｒｓｉｏｎｏｆａｍａｔｒｉｘ单项乘法次数加法次数其他Ｒ１Ｔ（ｎ）求逆Ｒ２４ｍｎ２４ｍｎ２－２ｍｎＲ３４ｍｎ２４ｍｎ２－２ｍｎＲ４４ｍ２ｎ４ｍ２ｎ－２ｍ２Ｒ５２ｍ２Ｒ６Ｔ（ｍ）求逆Ｒ７４ｍｎ２４ｍｎ２－２ｎ２ｃ１１２ｎ２ｃ２１４ｍ２ｎ４ｍ２ｎ－２ｍｎｃ１２４ｍ２ｎ４ｍ２ｎ－２ｍｎｃ２２ｍ２复杂度合计９ｍｎ（４ｍ＋４ｎ－１）＋３ｍ２＋Ｔ（ｎ）＋Ｔ（ｍ）表４㊀改进算法计算一次矩阵求逆的计算复杂度Ｔａｂ．４㊀Ｉｍｐｒｏｖｅｄａｌｇｏｒｉｔｈｍｔｏｃａｌｃｕｌａｔｅｔｈｅｃｏｍｐｕｔａｔｉｏｎａｌｃｏｍｐｌｅｘｉｔｙｏｆａｍａｔｒｉｘｉｎｖｅｒｓｉｏｎ单项乘法次数加法次数其他Ｒ１Ｔ（ｎ）求逆Ｒ２４ｍｎ２４ｍｎ２－２ｍｎＲ３Ｒ４４ｍ２ｎ４ｍ２ｎ－２ｍ２Ｒ５２ｍ２Ｒ６Ｔ（ｍ）求逆Ｒ７４ｍｎ２４ｍｎ２－２ｎ２ｃ１１２ｎ２ｃ２１ｃ１２４ｍ２ｎ４ｍ２ｎ－２ｍｎｃ２２ｍ２复杂度合计８ｍｎ４ｍ＋４ｎ－１２()＋３ｍ２＋Ｔ（ｎ）＋Ｔ（ｍ）在本文中，当矩阵维度为８（Ｊ＝４）时，改进算法总共复杂度为４７７６，常规算法总共５２５５，复杂度降低１０．０３％㊂进一步，图７给出了不同用户数量Ｊ时，改进的Ｓｔｒａｓｓｅｎ算法与常规Ｓｔｒａｓｓｅｎ算法的复杂度比较㊂可以看出，随着用户数量增加，矩阵求逆时的维度增加，采用改进的Ｓｔｒａｓｓｅｎ算法对复杂度的降低更加明显㊂图７㊀在不同用户个数Ｊ下，改进的Ｓｔｒａｓｓｅｎ算法与常规Ｓｔｒａｓｓｅｎ算法的复杂度比较Ｆｉｇ．７㊀ＵｎｄｅｒｄｉｆｆｅｒｅｎｔｎｕｍｂｅｒｏｆｕｓｅｒｓＪ，ｔｈｅｃｏｍｐｌｅｘｉｔｙｃｏｍｐａｒｉｓｏｎｏｆｔｈｅｉｍｐｒｏｖｅｄＳｔｒａｓｓｅｎａｌｇｏｒｉｔｈｍａｎｄｔｈｅｃｏｎｖｅｎｔｉｏｎａｌＳｔｒａｓｓｅｎａｌｇｏｒｉｔｈｍ３．３㊀所提算法对最小建模精度的影响利用改进的Ｓｔｒａｓｓｅｎ矩阵求逆算法求得的Ｗ＾Ｈｌ，ｊ，ｋ与原来的矩阵求逆算法求得的Ｗｌ，ｊ，ｋ进行建模估计精度计算，对于所给的数据集Ｄａｔａ１ Ｄａｔａ６，仿真不同数据集的最小建模精度，发现各数据集最小建模精度均为１，如图８所示㊂可见所提改进Ｓｔｒａｓｓｅｎ矩阵求逆算法不会对建模精度带来影响，这是因为该算法采用了分块迭代方法在变换的过程中不会带来计算误差㊂图８㊀采用改进的矩阵求逆算法后对各数据集最小建模精度的影响Ｆｉｇ．８㊀Ｉｎｆｌｕｅｎｃｅｏｆｔｈｅｉｍｐｒｏｖｅｄｍａｔｒｉｘｉｎｖｅｒｓｉｏｎａｌｇｏｒｉｔｈｍｏｎｔｈｅｍｉｎｉｍｕｍｍｏｄｅｌｉｎｇａｃｃｕｒａｃｙｏｆｅａｃｈｄａｔａｓｅｔ３．４㊀综合复杂度分析本节分别对插值算法和改进的矩阵求逆的综合复杂度进行分析㊂利用相关矩阵组的关联度降低计算复杂度，即通过Ｓｐｌｉｎｅ插值操作降低ＳＶＤ的总复杂度㊂其中，所需的乘法次数（５ＭＮ２－ＭＮ）㊁加法次数（３ＭＮ２－ＭＮ）㊁除法次数（０．５Ｎ（Ｎ－１）＋２ＭＮ）以及平方根次数（ＭＮ２）㊂最终的复杂度合计为Ｎｉｔｅ（４３ＭＮ２＋７５２Ｎ（Ｎ－１）＋１４８ＭＮ）㊂另外，通过改进的矩阵求逆，即采用改进的Ｓｔｒａｓｓｅｎᶄｓ矩阵求逆进一步降低矩阵求逆Ｖｋ（ＶＨｋＶｋ＋σ２Ｉ）－１的复杂度，得出了改进后矩阵求逆算法后所需的乘法次数（４ＭＬ２＋４Ｎ２ＬＪ）㊁加法次数（４ＭＬ２＋４Ｎ２ＬＪ－２ＭＬ－２Ｎ２）以及求逆复杂度（４７７６）㊂最终的复杂度合计为２Ｎ２（８ＬＪ－１）＋２ＭＬ（８Ｌ－１）＋４７７６㊂对于不采用插值算法的情况，每个插值时刻ｋ，由Ｈ矩阵到Ｗ矩阵，需要进行Ｊ（Ｊ＝４）次ＳＶＤ和１次ＭＭＳＥ均衡（主要复杂度在于求逆）的计算㊂其中４次ＳＶＤ需要３５７４４００次计算㊂ＭＭＳＥ均衡需要５２１１１２次计算，因此共需要计算复杂度Ｃ１＝４０９５５１２㊂采用Ｓｐｌｉｎｅ插值时，每个插值时刻的复杂度为３１７４４㊂可以计算出每次插值的复杂度收益为Ｃ２＝３５７４４００＋５２１１１２－３１７４４＝４０６３７６８㊂因此采用插值的最终复杂度收益为：ΔＣ＝（Ｃ１－Ｃ２）ˑＫˑＲａｔｅ㊂（３０）假设Ｒａｔｅ＝１／３时，ΔＣ＝５２４２２５５３６㊂可见，插值对于计算复杂度的降低比较明显㊂同样，计算复杂度可以降低为：ΔＲＣ＝Ｃ１（１－Ｒａｔｅ）＋Ｃ２ˑＲａｔｅＣ１㊂（３１）当Ｒａｔｅ＝１／３时，计算复杂度降低为原来的６６．９３％㊂当Ｒａｔｅ＝１／１０时，计算复杂度降低为原来的９０．０８％㊂综上，当采用改进的Ｓｔｒａｓｓｅｎ矩阵求逆算法时，复杂度降低了１０．０３％㊂进一步采用插值算法后，计算复杂度能够在上述的基础上再降低９．９２％（插值比为１／１０）㊁３３．０７％（插值比为１／３）㊂４㊀结论本论文主要解决ＭＩＭＯ场景下的相关矩阵组的低复杂度计算问题，首先利用Ｈ矩阵在时间相关性推导了Ｗ矩阵的相关性，通过对已有Ｗ矩阵的相关性直接插值出部分缺失的Ｗ㊂这使得在接收Ｈ矩阵时，在求取部分Ｗ矩阵后通过相关性重建完整的Ｗ矩阵；也避免了一部分Ｈ矩阵的存储以及这部分Ｈ矩阵计算ＳＶＤ与求逆获得Ｗ矩阵的过程㊂相比ＳＶＤ与求逆的复杂度，插值的复杂度要低得多㊂此外，采用了一种改进的Ｓｔｒａｓｓｅｎ矩阵求逆算法来降低求逆过程的复杂度㊂该算法结合了Ｓｔｒａｓｓｅｎ矩阵求逆的高效性以及Ｈｅｒｍｉｔｅ正定阵的共轭对称性特点，结构更简化㊂参考文献［１］㊀ＲＵＳＥＫＦ，ＰＥＲＳＳＯＮＤ，ＬＡＵＢＫ，ｅｔａｌ．ＳｃａｌｉｎｇＵｐＭＩＭＯ：ＯｐｐｏｒｔｕｎｉｔｉｅｓａｎｄＣｈａｌｌｅｎｇｅｓｗｉｔｈＶｅｒｙＬａｒｇｅＡｒｒａｙｓ［Ｊ］．ＩＥＥＥＳｉｇｎａｌＰｒｏｃｅｓｓｉｎｇＭａｇａｚｉｎｅ，２０１３（３０）１：４０－６０．［２］㊀蔡锁章，杨明，雷英杰．数值计算方法［Ｍ］．２版．北京：国防工业出版社，２０１６．［３］㊀许小勇，钟太勇．三次样条插值函数的构造与Ｍａｔｌａｂ实现［Ｊ］．兵工自动化，２００６（１１）：７６－７８．［４］㊀陈帅，岳迎春，徐巍，等．小波时间序列对非平稳信号中突变点的辨识与处理［Ｊ］．测绘科学，２０１３，３８（５）：１１－１２．［５］㊀ＤＥＢＯＯＲＣ．ＡＰｒａｃｔｉｃａｌＧｕｉｄｅｔｏＳｐｌｉｎｅｓ［Ｊ］．ＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃａｌＳｃｉｅｎｃｅｓＮｅｗＹｏｒｋＳｐｒｉｎｇｅｒ，１９７８，２７（１４９）：１５７－１５７．［６］㊀ＦＲＩＴＳＣＨＦＮ，ＣＡＲＬＳＯＮＲＥ．ＭｏｎｏｔｏｎｅＰｉｅｃｅｗｉｓｅＣｕｂｉｃＩｎｔｅｒｐｏｌａｔｉｏｎ［Ｊ］．ＳｉａｍＪｏｕｒｎａｌｏｎＮｕｍｅｒｉｃａｌＡｎａｌｙｓｉｓ，１９８０，１７（２）：２３８－２４６．［７］㊀颜志升，郑昱．基于高斯消元的自适应信号处理的实现方法：ＣＮ１１１４２７０１４Ａ［Ｐ］．２０２０－０７－１７．［８］㊀ＳＴＲＡＳＳＥＮＶ．ＧａｕｓｓｉａｎＥｌｉｍｉｎａｔｉｏｎｉｓＮｏｔＯｐｔｉｍａｌ［Ｊ］．ＮｕｍｅｒｉｓｃｈｅＭａｔｈｅｍａｔｉｋ，１９６９，１３（４）：３５４－３５６．［９］㊀杨忠鹏，林志兴．关于Ｈｅｒｍｉｔｉａｎ矩阵的特征的注记［Ｊ］．大学数学，２００３，１９（５）：５２－５３．［１０］李瑞，李晓明，董晔．ＳＴＡＰ中的协方差矩阵求逆快速算法研究［Ｊ］．计算机仿真，２０１１，２８（２）：２５－２８．作者简介：㊀㊀施育鑫㊀国防科技大学第六十三研究所博士研究生㊂主要研究方向：通信抗干扰㊁ＯＦＤＭ㊂㊀㊀鲁信金㊀国防科技大学电子科学学院博士研究生㊂主要研究方向：信息论㊁索引调制㊁ｐｏｌａｒ码㊁物理层安全㊁无线通信技术等㊂在各类期刊和会议论文集上发表论文多篇㊂㊀㊀孙艺夫㊀国防科技大学电子科学学院博士研究生㊂主要研究方向：可重构信息超表面㊁通信抗干扰㊁物理层安全㊂㊀㊀雷㊀菁㊀国防科技大学电子科学学院教授，博士生导师㊂主要研究方向：信息论㊁ＬＤＰＣ码㊁物理层安全㊁ｐｏｌａｒ码㊁无线通信等㊂㊀㊀李玉生㊀国防科技大学第六十三研究所正高级工程师，硕士生导师㊂主要研究方向：通信抗干扰㊂。

第十一届全国研究生数学建模竞赛C题无线通信中的快时变信道建模一、背景介绍1.基本模型宽带移动通信传输正在改变着人们的生活，更为快速和准确的传递信息是其基本需求。

据预测，到2020年，数以千亿的“物”，包括汽车、计量表、医疗设备和家电等都将连入移动通信网络，人们的移动数字生活也将更加美好。

由于移动通信网络连接环境复杂多变，对实现高速宽带数据传递提出了更高的要求和挑战。

例如，高速铁路和高速公路的开通和应用，使未来移动通信系统面临高速移动环境，而在高速移动环境下，无线通信信道会发生快速变化，若不能适应这种变化，通信系统性能将会受到严重影响，极大降低信息传输的速度和质量。

分析现有通信模型的不足，建立新的数学模型，对提升信道容量、增加信息传输速率和降低误码率会有很好的促进作用。

在通信系统中，发送端通过信道传输信号到接收端，在传输过程中，不可避免地要引入干扰噪声。

接收端对包含噪声的信号进行合理解码，得到正确的信息，完成信息传输过程，原理用图1表示。

图1 通信基本模型示意图通信过程的数学模型可以表示为：WXHY+⋅=(1)从式（1）可以看出，在已知接收端信号Y的情况下，要得知发送端的信号X，还需要知道信道变量H和噪声W的统计特征。

W可视为加性高斯白噪声AWGN（Additive White Gaussian Noise），因此问题的关键就是对H规律的探索。

在无线信道中，发送和接收之间通常存在多于一条的信号传播路径。

多径的存在是因为发射机和接收机之间建筑物和其他物体的反射、绕射、散射等引起的，其传播特征如图2所示。

图2 无线信道传播特征图中LOS（line of sight）是信号直接到达的传播路径。

可以看出，由于环境的复杂性，信号传播途径也复杂多变，需要对其进行简化和抽象，建立描述、估计信道传播的数学模型。

发端信号传输信道H噪声WYX收端信号当信号在无线信道传播时，多径反射和衰减的变化将使信号经历随机波动。

无线多径传输系统的时间离散形式的数学表达式为[1]：∑-=-=+-=101,...,0],[][][][L l l K n n w l n x n h n y (2)式中L 为信道的多径数，K 为传输信号的长度，)(n w 可视为AWGN ，[]l h n 就是信道参数。