MIMO雷达波形设计
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摘要
MIMO雷达是一种新兴的有源探测技术,现已成为雷达技术领域的一个研究热点。它的发射天线和接收天线根据系统要求可以进行灵活布置并且每个辐射单元可以发射不同的信号波形。正是这种波形分集能力使得MIMO雷达,与传统的雷达相比,具有更多的优势。分布式MIMO雷达利用空间分集增益克服目标的闪烁效应,提高对起伏目标的探测性能。集中式MIMO雷达具有更高的空间分辨率、更好的参数辨别能力和更加灵活地设计发射方向图等优点。
波形分集是MIMO雷达的重要特征,波形设计是实现波形分集优点的重要手段。目前,MIMO雷达的波形设计主要包含正交波形优化设计、发射方向图匹配设计和发射信号波形合成等问题。论文围绕这些问题,先回顾了已有的方法,然后借助最优化方法,提出了新的方法,并展开了深入的比较和讨论。
论文主要内容可概括如下六部分：
第一部分,介绍了最优化方法的基本理论和算法,并举例说明了最优化方法在波形设计和信号处理中的一些应用。首先,介绍了无约束优化和约束优化的一阶和二阶最优性条件以及最优化方法的基本结构。接着,介绍了凸优化理论,总结了一些常用的判断凸性的方法,用最优峰值旁瓣电平失配滤波器的设计说明了凸优化在信号处理中的简单应用。然后,讨论了最速下降法、牛顿法、共轭梯度法和拟牛顿法等无约束优化方法,以及这些优化方法的联系和区别。最后,介绍了序列二次规划和信赖域算法等约束优化方法。在介绍完各种优化方法后,以设计脉冲压缩信号为例说明了它们的应用,并比较了各种优化方法的性能,最后选用序列二次规划法作为解决MIMO波形设计问题的主要方法。
第二部分,研究了正交波形的设计问题。首先,回顾了已有的正交波形设计方法—模拟退火、遗传算法和改进的Flethcher-Reeves算法,并用遗传算法研究了主瓣展宽的正交连续相位编码信号,通过仿真实验发现自相关峰值旁瓣电平和峰值互相关电平得到了极大改善。接着,提出了基于软件Lingo和基于序列二次规划的正交连续相位编码信号设计方法,实验结果表明,与改进的Flethcher-Reeves算法相比,优化得到的正交波形,其自相关峰值旁瓣电平和峰值互相关电平更低。最后,基于序列二次规划研究了自相关峰值旁瓣电平和峰值互相关电平与码长N、序列数L和加权系数λ的数值关系,得出了适当调整加权系数λ,损失一些峰值互相关电平的性能,可以极大改善自相关峰值旁瓣电平。
第三部分,研究了最优峰值旁瓣电平失配滤波器组的设计问题,即如何通过失配滤波的方法改善正交波形的性能。首先,回顾了已有的迭代加权最小二乘方法。接着,提出了基于凸优化的设计方法。然后,通过仿真实验表明,采用增加失配滤波器系数长度和牺牲信噪比的措施,以上两种方法都可以进一步地改善正交波形的自相关旁瓣电平和互相关电平。最后,通过比较得出了凸优化方法性能更加优越,不仅可以控制信噪比损失,而且可以得到更低的自相关峰值旁瓣电平和更低的峰值互相关电平的结论。
第四部分,研究了发射方向图匹配设计的问题,即如何从给定的理想发射方向图得到信号协方差矩阵。首先,回顾了已有的基于半正定规划设计发射方向图的方法。接着,松弛阵元等功率的约束条件,即对阵列进行固定的幅度加权,提出了任意发射方向图是由一组基波束合成的思想,并采用线性规划快速求解出合成给定发射方向图的基波束和其发射比例。最后,通过仿真实验表明,MIMO雷达凭借波形分集能力确实可以灵活地设计发射方向图,将雷达系统的电磁能量最大化地辐射到感兴趣的区域和目标上,在一定程度上抑制回波之间的相互干扰；比较半正定规划和线性规划这两种方法,后者不仅能快速求解得到信号协方差矩阵,并且所合成的发射方向图以及互相关方向图具有更低的空域峰值旁瓣电平。
第五部分,研究了发射信号波形合成的问题,即如何从已知的信号协方差矩阵合成恒模发射信号波形。首先,回顾了已有的循环算法。然后,提出了基于序列二次规划的发射信号波形合成方法。最后,通过仿真实验比较了这两种方法,得出用序列二次规划方法优化得到的恒模发射信号波形,其回波信号具有更好的自相关和互相关特性的结论。
第六部分,研究了在线设计发射方向图和在线合成发射信号波形的问题,即如何快速合成发射信号波形,既匹配给定的发射方向图,又使得回波信号具有良好的相关特性。首先,提出了用基波束和基波形快速合成发射信号波形的思想。然后,采用序列二次规划算法设计基波形。最后,通过仿真实验表明,用序列二次规划算法得到的基波形具有良好的相关特性；用基波束和基波形快速合成的发射信号波形,其回波信号具有良好的自相关和互相关特性。
Multi-input multi-output (MIMO) radar is a new technology of active radar detection, and has become a hot research topic. Transmit antennas and receive antennas can be flexibly configured according to system requirements and each transmit antenna can freely choose signal waveform. Compared with traditional radar, MIMO radar has waveform diversity with more advantages. Distributed MIMO radar makes full use of spatial diversity to overcome the fade of targets and to improve target detection performance. Centralized MIMO radar has higher spatial resolution, better parameter identification capacity and greater flexibility of transmit beam-pattern design.
Waveform diversity is an important feature of MIMO radar, and waveform design is a mean to achieve the advantages of waveform diversity. Recently, MIMO radar waveform design consists mainly of orthogonal waveform design, transmit beam-pattern matching design, transmit signal waveform synthesis and so on. This dissertation focuses on these issues. Firstly, we review the existing methods, and then propose new approaches based on optimization methods. Finally, a thorough comparison and discussion are provided.
The main content of this dissertation is summarized as follows.
The first part introduces the basic theory of optimization methods, and takes examples of waveform design and signal processing. First of all, the first order and second order optimality conditions of unconstrained optimization and constrained optimization as well as the basic structure of optimization methods are introduced. Then, the convex optimization theory is described, and some common methods of judging convexity are summarized. An example of the optimal peak sidelobe mismatch filter design applies the convex optimization to signal processing. Next, the steepest descent method, Newton method, conjugate gradient method and quasi-Newton method for unconstrained optimization are explained. The links and differences of these optimization methods are pointed out. Finally, the sequential quadratic programming and trust region method for constrained optimization are described. After the introduction to a variety of optimization methods, we take the design of pulse compression signal as an example to illustrate their applications, and compare the performance of various optimization methods. Sequential quadratic programming method is ultimately chosen as the solution to the problem with MIMO waveform design method.
The second part is contributed to the orthogonal waveform design problem. Firstly, we review the existing orthogonal waveform design methods-simulated annealing, genetic algorithms and improved Flethcher-Reeves algorithm.Orthogonal polyphase codes with broaden main-lobe are designed via gentic algorithms. Simulations show that the peak sidelobe level has been greatly improved. Next, a software-based Lingo of constrained nonlinear programming and sequential quadratic programming method for the continuous phase encoding orthogonal signal are proposed. Experimental results show that, the optimized orthogonal waveforms have flat sidelobes with lower peak sidelobe level, compared with the improved Flethcher-Reeves algorithm. Finally, based on sequential quadratic programming method, we study the relationship among autocorrelation peak sidelobe level, peak cross-correlation level, the code length N, the signal number L, and the weighted coefficientλ. The results show that, a appropriate; adjustment ofλcan contribute a great improvement of autocorrelation peak sidelobe: level with little loss of peak cross-correlation level
The third part focuses on the optimal peak sidelobe level mismatched filter design problem, namely, how to design mismatched filters to improve the performance of orthogonal waveforms. Firstly, we review the existing iterative weighted least squares method. Then, we propose a mismatched filter design method based on convex optimization. Next, the simulation results show that, increasing the coefficient length of mismatched filter and with a little loss of signal to noise ratio, both methods can further improve the peak sibelobe level. Finally, by comparison, the convex optimization method has superior performance. It can not only control the SNR loss, but also get a lower peak sidelobe level.
The fourth part is contributed to transmit beam-pattern matching design issue, namely, how to obtaine signal covariance matrix from a given ideal beampattern. At first, we review the existing method based on semi-definite programming. Next, we replace the uniform array constraint with a fix-weighted and nonuniform one. We propose an idea that an arbitrary beam-pattern can be synthesized from a set of basic beams. The basic beams and their weight coefficients can be fast obtained via linear programming. Finally, simulation results show that, MIMO radar can freely choose its signal waveforms not only to maximize the electromagnetic energy onto the targets of interest, but also to minimize cross-correlation of echo signals; compared with the semidefinite programming method, the linear programming method can not only fastly get the signal covariance matrix, but also make the synthesized pattern and the correlation pattern have a lower spatial peak sidelobe level.
The fifth part focuses on the transmit signal waveform synthesis issue, namely, how to get constant modulus signal waveforms from the known covariance matrix. Firstly, we review the existing cyclic algorithm. Then, we propose the sequential quadratic programming method. Finally, the simulation results show that, the constant modulus signal waveforms, which are optimized via the sequential quadratic programming, make their echo signals have better autocorrelation and cross correlation.
The sixth part is contributed to the online design issue on transmit beam-pattern and signal waveforms, namely, how to rapidly synthesize signal waveforms, not only to match the given beam-pattern, but also to make the echo signals have good correlation properties. Firstly, we propose an idea that transmitted signal waveforms can be rapidly synthesized with basic beams and basic signals. Then, we use sequential quadratic programming to design basic signals. Finally, the simulation results show that the basic signals obtained via sequential quadratic programming method have good correlation properties; the transmit signal waveforms rapidly synthesized from basic beams and basic signals, make their echo signals have good autocorrelation and cross correlation properties.
波形分集是MIMO雷达的重要特征,波形设计是实现波形分集优点的重要手段。目前,MIMO雷达的波形设计主要包含正交波形优化设计、发射方向图匹配设计和发射信号波形合成等问题。论文围绕这些问题,先回顾了已有的方法,然后借助最优化方法,提出了新的方法,并展开了深入的比较和讨论。
论文主要内容可概括如下六部分：
第一部分,介绍了最优化方法的基本理论和算法,并举例说明了最优化方法在波形设计和信号处理中的一些应用。首先,介绍了无约束优化和约束优化的一阶和二阶最优性条件以及最优化方法的基本结构。接着,介绍了凸优化理论,总结了一些常用的判断凸性的方法,用最优峰值旁瓣电平失配滤波器的设计说明了凸优化在信号处理中的简单应用。然后,讨论了最速下降法、牛顿法、共轭梯度法和拟牛顿法等无约束优化方法,以及这些优化方法的联系和区别。最后,介绍了序列二次规划和信赖域算法等约束优化方法。在介绍完各种优化方法后,以设计脉冲压缩信号为例说明了它们的应用,并比较了各种优化方法的性能,最后选用序列二次规划法作为解决MIMO波形设计问题的主要方法。
第二部分,研究了正交波形的设计问题。首先,回顾了已有的正交波形设计方法—模拟退火、遗传算法和改进的Flethcher-Reeves算法,并用遗传算法研究了主瓣展宽的正交连续相位编码信号,通过仿真实验发现自相关峰值旁瓣电平和峰值互相关电平得到了极大改善。接着,提出了基于软件Lingo和基于序列二次规划的正交连续相位编码信号设计方法,实验结果表明,与改进的Flethcher-Reeves算法相比,优化得到的正交波形,其自相关峰值旁瓣电平和峰值互相关电平更低。最后,基于序列二次规划研究了自相关峰值旁瓣电平和峰值互相关电平与码长N、序列数L和加权系数λ的数值关系,得出了适当调整加权系数λ,损失一些峰值互相关电平的性能,可以极大改善自相关峰值旁瓣电平。
第三部分,研究了最优峰值旁瓣电平失配滤波器组的设计问题,即如何通过失配滤波的方法改善正交波形的性能。首先,回顾了已有的迭代加权最小二乘方法。接着,提出了基于凸优化的设计方法。然后,通过仿真实验表明,采用增加失配滤波器系数长度和牺牲信噪比的措施,以上两种方法都可以进一步地改善正交波形的自相关旁瓣电平和互相关电平。最后,通过比较得出了凸优化方法性能更加优越,不仅可以控制信噪比损失,而且可以得到更低的自相关峰值旁瓣电平和更低的峰值互相关电平的结论。
第四部分,研究了发射方向图匹配设计的问题,即如何从给定的理想发射方向图得到信号协方差矩阵。首先,回顾了已有的基于半正定规划设计发射方向图的方法。接着,松弛阵元等功率的约束条件,即对阵列进行固定的幅度加权,提出了任意发射方向图是由一组基波束合成的思想,并采用线性规划快速求解出合成给定发射方向图的基波束和其发射比例。最后,通过仿真实验表明,MIMO雷达凭借波形分集能力确实可以灵活地设计发射方向图,将雷达系统的电磁能量最大化地辐射到感兴趣的区域和目标上,在一定程度上抑制回波之间的相互干扰；比较半正定规划和线性规划这两种方法,后者不仅能快速求解得到信号协方差矩阵,并且所合成的发射方向图以及互相关方向图具有更低的空域峰值旁瓣电平。
第五部分,研究了发射信号波形合成的问题,即如何从已知的信号协方差矩阵合成恒模发射信号波形。首先,回顾了已有的循环算法。然后,提出了基于序列二次规划的发射信号波形合成方法。最后,通过仿真实验比较了这两种方法,得出用序列二次规划方法优化得到的恒模发射信号波形,其回波信号具有更好的自相关和互相关特性的结论。
第六部分,研究了在线设计发射方向图和在线合成发射信号波形的问题,即如何快速合成发射信号波形,既匹配给定的发射方向图,又使得回波信号具有良好的相关特性。首先,提出了用基波束和基波形快速合成发射信号波形的思想。然后,采用序列二次规划算法设计基波形。最后,通过仿真实验表明,用序列二次规划算法得到的基波形具有良好的相关特性；用基波束和基波形快速合成的发射信号波形,其回波信号具有良好的自相关和互相关特性。
Multi-input multi-output (MIMO) radar is a new technology of active radar detection, and has become a hot research topic. Transmit antennas and receive antennas can be flexibly configured according to system requirements and each transmit antenna can freely choose signal waveform. Compared with traditional radar, MIMO radar has waveform diversity with more advantages. Distributed MIMO radar makes full use of spatial diversity to overcome the fade of targets and to improve target detection performance. Centralized MIMO radar has higher spatial resolution, better parameter identification capacity and greater flexibility of transmit beam-pattern design.
Waveform diversity is an important feature of MIMO radar, and waveform design is a mean to achieve the advantages of waveform diversity. Recently, MIMO radar waveform design consists mainly of orthogonal waveform design, transmit beam-pattern matching design, transmit signal waveform synthesis and so on. This dissertation focuses on these issues. Firstly, we review the existing methods, and then propose new approaches based on optimization methods. Finally, a thorough comparison and discussion are provided.
The main content of this dissertation is summarized as follows.
The first part introduces the basic theory of optimization methods, and takes examples of waveform design and signal processing. First of all, the first order and second order optimality conditions of unconstrained optimization and constrained optimization as well as the basic structure of optimization methods are introduced. Then, the convex optimization theory is described, and some common methods of judging convexity are summarized. An example of the optimal peak sidelobe mismatch filter design applies the convex optimization to signal processing. Next, the steepest descent method, Newton method, conjugate gradient method and quasi-Newton method for unconstrained optimization are explained. The links and differences of these optimization methods are pointed out. Finally, the sequential quadratic programming and trust region method for constrained optimization are described. After the introduction to a variety of optimization methods, we take the design of pulse compression signal as an example to illustrate their applications, and compare the performance of various optimization methods. Sequential quadratic programming method is ultimately chosen as the solution to the problem with MIMO waveform design method.
The second part is contributed to the orthogonal waveform design problem. Firstly, we review the existing orthogonal waveform design methods-simulated annealing, genetic algorithms and improved Flethcher-Reeves algorithm.Orthogonal polyphase codes with broaden main-lobe are designed via gentic algorithms. Simulations show that the peak sidelobe level has been greatly improved. Next, a software-based Lingo of constrained nonlinear programming and sequential quadratic programming method for the continuous phase encoding orthogonal signal are proposed. Experimental results show that, the optimized orthogonal waveforms have flat sidelobes with lower peak sidelobe level, compared with the improved Flethcher-Reeves algorithm. Finally, based on sequential quadratic programming method, we study the relationship among autocorrelation peak sidelobe level, peak cross-correlation level, the code length N, the signal number L, and the weighted coefficientλ. The results show that, a appropriate; adjustment ofλcan contribute a great improvement of autocorrelation peak sidelobe: level with little loss of peak cross-correlation level
The third part focuses on the optimal peak sidelobe level mismatched filter design problem, namely, how to design mismatched filters to improve the performance of orthogonal waveforms. Firstly, we review the existing iterative weighted least squares method. Then, we propose a mismatched filter design method based on convex optimization. Next, the simulation results show that, increasing the coefficient length of mismatched filter and with a little loss of signal to noise ratio, both methods can further improve the peak sibelobe level. Finally, by comparison, the convex optimization method has superior performance. It can not only control the SNR loss, but also get a lower peak sidelobe level.
The fourth part is contributed to transmit beam-pattern matching design issue, namely, how to obtaine signal covariance matrix from a given ideal beampattern. At first, we review the existing method based on semi-definite programming. Next, we replace the uniform array constraint with a fix-weighted and nonuniform one. We propose an idea that an arbitrary beam-pattern can be synthesized from a set of basic beams. The basic beams and their weight coefficients can be fast obtained via linear programming. Finally, simulation results show that, MIMO radar can freely choose its signal waveforms not only to maximize the electromagnetic energy onto the targets of interest, but also to minimize cross-correlation of echo signals; compared with the semidefinite programming method, the linear programming method can not only fastly get the signal covariance matrix, but also make the synthesized pattern and the correlation pattern have a lower spatial peak sidelobe level.
The fifth part focuses on the transmit signal waveform synthesis issue, namely, how to get constant modulus signal waveforms from the known covariance matrix. Firstly, we review the existing cyclic algorithm. Then, we propose the sequential quadratic programming method. Finally, the simulation results show that, the constant modulus signal waveforms, which are optimized via the sequential quadratic programming, make their echo signals have better autocorrelation and cross correlation.
The sixth part is contributed to the online design issue on transmit beam-pattern and signal waveforms, namely, how to rapidly synthesize signal waveforms, not only to match the given beam-pattern, but also to make the echo signals have good correlation properties. Firstly, we propose an idea that transmitted signal waveforms can be rapidly synthesized with basic beams and basic signals. Then, we use sequential quadratic programming to design basic signals. Finally, the simulation results show that the basic signals obtained via sequential quadratic programming method have good correlation properties; the transmit signal waveforms rapidly synthesized from basic beams and basic signals, make their echo signals have good autocorrelation and cross correlation properties.
引文
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[24]Deng H.. Polyphase Code Design for Orthogonal Netted Radar Systems. IEEE Trans. on Signal Processing.2004, Vol.52(11).3126-3135.
[25]Deng H.. Discrete frequency-coding waveform design for netted radar systems.IEEE Signal Processing Letters.2004, Vol.11(2).179-182.
[26]刘波MIMO雷达正交波形设计及信号处理研究.成都：电子科技大学博士学位论文.2008.4.
[27]Liu B., He Z.-S., Zeng J.-K., et al. Polyphase Orthogonal Code Design for MIMO Radar Systems. International Conference on Radar.2006.1-4.
[28]Liu B., He Z.-S., He Q. Optimization of Orthogonal Discrete Frequency-Coding Waveform Based on Modified Genetic Algorithm for MIMO Radar. International Conference on Communications, Circuits and Systems.2007.966-970.
[29]Khan H. A., Edwards D. J.. Doppler problems in orthogonal MIMO radars.2006 IEEE Conference on Radar. Vol.1.24-27.
[30]Kevin J. Gartz. Generation of Uniform Amplitude Complex Code Sets with Low Correlation Sidelobes. IEEE Trans.on Signal Processing.1992, Vol.40(2).343-351.
[31]Stoica P., He H., Li J.. New Algorithms for Designing Unimodular Sequences With Good Correlation Properties. IEEE Trans. on Signal Processing.2009, Vol.57(4). 1415-1425.
[32]Fuhrmann D. R. and Antonio G. S.. Transmit beamforming for MIMO radar systems using partial signal correlation. Proc.38th Asilomar Conference on Signals, System and Computers. Nov.2004, Vol.1.295-299.
[33]Tuomas Aittomaki and Visa Koivunen. Low-complexity method for transmit beamforming in MIMO radars. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing.2007, Vol.2.305-308.
[34]Stoica P., Li J., Xie Y.. On probing signal design for MIMO radar. IEEE Trans, on Signal Processing.2007, Vol.55(8).4151-4161.
[35]Luca A.. Experimental results on SIAR digital beamforming radar. Proceedings of the IEEE International Radar Conference.Brighton, UK.1992.505-510.
[36]Antonio G. S., Fuhrmann D. R., Beampattem synthesis for wideband MIMO radar systems.2005 Proc.1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive. Vol.1.105-108.
[37]李军MIMO雷达中的数字波束形成与信号处理技术研究.成都：电子科技大学博士学位论文.2009.6.
[38]Stoica P., Li J., Zhu X.. Waveform Synthesis for Diversity-Based Transmit Beampattern Design. IEEE Trans. on Signal Processing.2008, Vol.56(6). 2593-2598.
[1]袁亚湘,孙文瑜.最优化理论与方法.北京：科学出版社.2003.
[2]Jorge Nocedal, Stephen J. Wright. Numerical Optimization. Springer.2006.
[3]Andreas Antoniou, Lu Wu-Sheng. Practical optimization algorithms and engineering applications. Springer.2007.
[4]Fletcher R.. Practical Methods of Optimization,2nd ed.. New York:Wiley.1987.
[5]Stephen Boyd. Convex Optimization. UK:Cambridge University Press.2004.
[6]Michael Grant, Stephen Boyd. Cvx users'guide for cvx version 1.2.2009.
[7]Alexandre D'aspremont, Laurent El Ghaoui, Michael I. Jordan, et al.. A Direct Formulation for Sparse PCA Using Semidefinite Programming. Proceeding of the Neural Information Processing Systems.2004.
[8]F. Alizadeh. Interior point methods in semidefinite programming with applications to combi-natorial optimization, SIAM Journal on Optimization.1995, Vol.5.13-51.
[9]胡亮兵,刘宏伟,吴顺君.基于凸优化的最优峰值旁瓣电平失配滤波器设计.已投电子学报.2010.
[10]Ackroyd H. M., Ghani F.. Optimum mismatched filters for sidelobe suppression. IEEE Transactions on Aerospace and Electronic.1973.
[11]Baden J. M. and Cohen M. N.. Optimal Peak Sidelobe Filters for Biphase Pulse Compression. IEEE International Radar Conference.1990.
[12]Nunn C. Constrained optimization applied to pulse compression codes and filters. IEEE International Radar Conference.2005,190-194
[13]Hu C.-H., Liu R.-B., Zhou Q.-F.. Mismatched-Filter Design for Biphase-coded Pulse for High Frequency Untrasound Imaging. IEEE Ultrasonics Symposium. 2006.
[14]Griep R. K., Ritcey A. J., Burlingame J. J.. Poly-Phase Codes and Optimal Filters for Multiple User Ranging. IEEE Transactions on Aerospace and Electronic Systems. 1995, Vol.31(2).752-767.
[15]Sturm J. F.. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones.Optim. Meth. Softw..1999,11-12.625-653.
[16]Grant M and Boyd S. CVX:Matlab software for disciplined convex programming. http://stanford.edu/～boyd/cvx,2008, Dec..
[17]Schittkowski. A Comparative Performance Evaluation of 27 Nonlinear Programming Codes, Computing.1983, Vol.30.335.
[1]Deng H. Polyphase Code Design for Orthogonal Netted Radar Systems. IEEE Trans, on Signal Processing.2004, Vol.52(11).3126-3135.
[2]Deng H.. Discrete frequency-coding waveform design for netted radar systems.IEEE Signal Processing Letters.2004, Vol.11 (2).179-182.
[3]Liu B., He Z.-S., Zeng J.-K., et al. Polyphase Orthogonal Code Design for MIMO Radar Systems. International Conference on Radar.2006,1-4.
[4]Liu B., He Z.-S., He Q. Optimization of Orthogonal Discrete Frequency-Coding Waveform Based on Modified Genetic Algorithm for MIMO Radar. International Conference on Communications, Circuits and Systems.2007,966-970.
[5]Khan H. A., Zhang Y., Ji C.. Optimizing Polyphase Sequences Optimizing Polyphase Sequences. IEEE Trans. on Signal Processing Letters.2006, Vol.13(10). 589-592.
[6]Kevin J. Gartz. Generation of Uniform Amplitude Complex Code Sets with Low Correlation Sidelobes. IEEE Trans. on Signal Processing.1992, Vol.40(2). 343-351.
[7]Stoica P., He H., Li J.. New Algorithms for Designing Unimodular Sequences With Good Correlation Properties. IEEE Trans. on Signal Processing.2009, Vol.57(4). 1415-1425.
[8]Richard O. Duda, Peter E. Hart, David G. Stork李宏东,姚天翔译.模式识别.北京：机械工业出版社.2003.
[9]王小平,曹立明.遗传算法—理论、应用与软件实现.西安：西安交通大学出版社.2002.
[10]Lingo.http://www.lindo.com.
[11]胡亮兵,刘宏伟,吴顺君.基于约束非线性规划的MIMO雷达正交波形设计.系统工程与电子技术学报.已录用.2010.
[1]Ackroyd H. M., Ghani F.. Optimum mismatched filters for sidelobe suppression. IEEE Transactions on Aerospace and Electronic.1973.
[2]Baden J. M. and Cohen M. N.. Optimal Peak Sidelobe Filters for Biphase Pulse Compression. IEEE International Radar Conference.1990.
[3]Nunn C.. Constrained optimization applied to pulse compression codes, and filters. IEEE International Radar Conference.2005,190-194.
[4]Hu C.-H., Liu R.-B., Zhou Q.-F.. Mismatched-Filter Design for Biphase-coded Pulse for High Frequency Untrasound Imaging. IEEE Ultrasonics Symposium.2006.
[5]Griep R. K., Ritcey A. J., Burlingame J. J.. Poly-Phase Codes and Optimal Filters for Multiple User Ranging. IEEE Transactions on Aerospace and Electronic Systems. 1995, Vol.31(2).752-767.
[6]Hu Liang-Bing, Liu Hong-Wei., Feng Da-Zheng. Optimal Mismatched Filter Bank Design for MIMO Radar via Convex Optimization. Submitted to 5 Waveform Diversity Conference, Canada.2010.
[7]Stephen Boyd. Convex Optimization. UK:Cambridge University Press.2004.
[8]Sturm J. F.. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones.Optim. Meth. Softw..1999,11-12.625-653.
[9]Grant M and Boyd S. CVX:Matlab software for disciplined convex programming. http://stanford.edu/-boyd/cvx,2008,Dec.
[10]Deng H.. Polyphase Code Design for Orthogonal Netted Radar Systems. IEEE Trans. on Signal Processing.2004, Vol.52(11).3126-3135.
[1]Stoica P., Li J., Xie Y.. On probing signal design for MIMO radar. IEEE Transactions on Signal Processing. Aug.2007, Vol.55(8).4151-4161.
[2]Fuhrmann D. R. and Antonio G. S.. Transmit beamforming for MIMO radar systems using partial signal correlation. Proc.38th Asilomar Conference on Signals, System and Computers. Nov.2004, Vol.1.295-299.
[3]Tuomas Aittomaki and Visa Koivunen. Low-complexity method for transmit beamforming in MIMO radars. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing.2007, Vol.2.305-308.
[4]Luca A.. Experimental results on SIAR digital beamforming radar. Proceedings of the IEEE International Radar Conference.Brighton,UK.1992,505-510.
[5]Antonio G. S., Fuhrmann D. R., Beampattem synthesis for wideband MIMO radar systems.2005 Proc.1st IEEE International Workshop on Computational Advances in Multi—Sensor Adaptive. Vol.1.105-108.
[6]李军MIMO雷达中的数字波束形成与信号处理技术研究.成都：电子科技大学博士学位论文.2009.6.
[7]胡亮兵,刘宏伟,杨晓超,吴顺君.集中式MIMO雷达快速发射方向图设计方法.电子与信息学报.2010,Vo1.32(2).481-484.
[8]Grant M. and Boyd S.. CVX:Matlab software for disciplined convex programming. http://stanford.edu/～boyd/cvx,Dec.2008.
[1]Stoica P., Li J., Xie Y.. On probing signal design for MIMO radar. IEEE Transactions on Signal Processing. Aug.2007, Vol.55(8).4151-4161.
[2]Fuhrmann D. R. and Antonio G. S.. Transmit beamforming for MIMO radar systems using partial signal correlation. Proc.38th Asilomar Conference on Signals, System and Computers. Nov.2004, Vol.1.295-299.
[3]Tuomas Aittomaki and Visa Koivunen. Low-complexity method for transmit beamforming in MIMO radars. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing.2007, Vol.2.305-308.
[4]Luca A.. Experimental results on SIAR digital beamforming radar. Proceedings of the IEEE International Radar Conference.Brighton,UK.1992,505-510.
[5]Antonio G. S., Fuhrmann D. R., Beampattem synthesis for wideband MIMO radar systems.2005 Proc.1st IEEE International Workshop on Computational Advances in Multi—Sensor Adaptive. Vol.1.105-108.
[6]李军MIMO雷达中的数字波束形成与信号处理技术研究.成都：电子科技大学博士学位论文.2009.6.
[7]胡亮兵,刘宏伟,杨晓超,吴顺君.集中式MIMO雷达快速发射方向图设计方法.电子与信息学报.2010,Vo1.32(2).481-484..
[8]Stoica P., Li J., Zhu X.. Waveform Synthesis for Diversity-Based Transmit Beampattern Design. IEEE Trans. on Signal Processing.2008, Vol.56(6). 2593-2598.
[1]胡亮兵,刘宏伟,杨晓超,吴顺君.集中式MIMO雷达快速发射方向图设计方法.电子与信息学报.2010,Vol.32(2).481-484.
[2]Stoica P., Li J., Xie Y.. On probing signal design for MIMO radar. IEEE Transactions on Signal Processing. Aug.2007, Vol.55(8).4151-4161.
[3]Fuhrmann D. R. and Antonio G. S.. Transmit beamforming for MIMO radar systems using partial signal correlation. Proc.38th Asilomar Conference on Signals, System and Computers. Nov.2004, Vol.1.295-299.
[4]Tuomas Aittomaki and Visa Koivunen. Low-complexity method for transmit beamforming in MIMO radars. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing.2007, Vol.2.305-308.
[5]Stoica P., Li J., Zhu X.. Waveform Synthesis for Diversity-Based Transmit Beampattern Design. IEEE Trans. on Signal Processing.2008, Vol.56(6). 2593-2598.
[2]Bliss D.W. and Forsythe K.W.. Multiple-input multiple-output (MIMO) radar and imaging:Degrees of freedom and resolution. Proc.37th Asilomar Conference Signals System Computer. Pacific Grove, CA, Nov.2003, Vol.1.54-59.
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[21]Khan H. A., Zhang Y., Ji C.. Optimizing Polyphase Sequences Optimizing Polyphase Sequences. IEEE Trans. on Signal Processing Letters.2006, Vol.13(10). 589-592.
[22]Forsythe K W., Bliss D. W. Waveform Correlation and Optimization Issues for MIMO Radar.2006 Conference Record of the Thirty-Ninth Asilomar Conference on Signals, Systems and Computers.2006, Vol.1.1306-1310.
[23]Bekkerman I., Tabrikian J.. Spatially coded transmission for active arrays.2004 Proc. Int.Conf.Acoustics, Speech, Signal Processing(ICASSP). Vol.2.209-212.
[24]Deng H.. Polyphase Code Design for Orthogonal Netted Radar Systems. IEEE Trans. on Signal Processing.2004, Vol.52(11).3126-3135.
[25]Deng H.. Discrete frequency-coding waveform design for netted radar systems.IEEE Signal Processing Letters.2004, Vol.11(2).179-182.
[26]刘波MIMO雷达正交波形设计及信号处理研究.成都：电子科技大学博士学位论文.2008.4.
[27]Liu B., He Z.-S., Zeng J.-K., et al. Polyphase Orthogonal Code Design for MIMO Radar Systems. International Conference on Radar.2006.1-4.
[28]Liu B., He Z.-S., He Q. Optimization of Orthogonal Discrete Frequency-Coding Waveform Based on Modified Genetic Algorithm for MIMO Radar. International Conference on Communications, Circuits and Systems.2007.966-970.
[29]Khan H. A., Edwards D. J.. Doppler problems in orthogonal MIMO radars.2006 IEEE Conference on Radar. Vol.1.24-27.
[30]Kevin J. Gartz. Generation of Uniform Amplitude Complex Code Sets with Low Correlation Sidelobes. IEEE Trans.on Signal Processing.1992, Vol.40(2).343-351.
[31]Stoica P., He H., Li J.. New Algorithms for Designing Unimodular Sequences With Good Correlation Properties. IEEE Trans. on Signal Processing.2009, Vol.57(4). 1415-1425.
[32]Fuhrmann D. R. and Antonio G. S.. Transmit beamforming for MIMO radar systems using partial signal correlation. Proc.38th Asilomar Conference on Signals, System and Computers. Nov.2004, Vol.1.295-299.
[33]Tuomas Aittomaki and Visa Koivunen. Low-complexity method for transmit beamforming in MIMO radars. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing.2007, Vol.2.305-308.
[34]Stoica P., Li J., Xie Y.. On probing signal design for MIMO radar. IEEE Trans, on Signal Processing.2007, Vol.55(8).4151-4161.
[35]Luca A.. Experimental results on SIAR digital beamforming radar. Proceedings of the IEEE International Radar Conference.Brighton, UK.1992.505-510.
[36]Antonio G. S., Fuhrmann D. R., Beampattem synthesis for wideband MIMO radar systems.2005 Proc.1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive. Vol.1.105-108.
[37]李军MIMO雷达中的数字波束形成与信号处理技术研究.成都：电子科技大学博士学位论文.2009.6.
[38]Stoica P., Li J., Zhu X.. Waveform Synthesis for Diversity-Based Transmit Beampattern Design. IEEE Trans. on Signal Processing.2008, Vol.56(6). 2593-2598.
[1]袁亚湘,孙文瑜.最优化理论与方法.北京：科学出版社.2003.
[2]Jorge Nocedal, Stephen J. Wright. Numerical Optimization. Springer.2006.
[3]Andreas Antoniou, Lu Wu-Sheng. Practical optimization algorithms and engineering applications. Springer.2007.
[4]Fletcher R.. Practical Methods of Optimization,2nd ed.. New York:Wiley.1987.
[5]Stephen Boyd. Convex Optimization. UK:Cambridge University Press.2004.
[6]Michael Grant, Stephen Boyd. Cvx users'guide for cvx version 1.2.2009.
[7]Alexandre D'aspremont, Laurent El Ghaoui, Michael I. Jordan, et al.. A Direct Formulation for Sparse PCA Using Semidefinite Programming. Proceeding of the Neural Information Processing Systems.2004.
[8]F. Alizadeh. Interior point methods in semidefinite programming with applications to combi-natorial optimization, SIAM Journal on Optimization.1995, Vol.5.13-51.
[9]胡亮兵,刘宏伟,吴顺君.基于凸优化的最优峰值旁瓣电平失配滤波器设计.已投电子学报.2010.
[10]Ackroyd H. M., Ghani F.. Optimum mismatched filters for sidelobe suppression. IEEE Transactions on Aerospace and Electronic.1973.
[11]Baden J. M. and Cohen M. N.. Optimal Peak Sidelobe Filters for Biphase Pulse Compression. IEEE International Radar Conference.1990.
[12]Nunn C. Constrained optimization applied to pulse compression codes and filters. IEEE International Radar Conference.2005,190-194
[13]Hu C.-H., Liu R.-B., Zhou Q.-F.. Mismatched-Filter Design for Biphase-coded Pulse for High Frequency Untrasound Imaging. IEEE Ultrasonics Symposium. 2006.
[14]Griep R. K., Ritcey A. J., Burlingame J. J.. Poly-Phase Codes and Optimal Filters for Multiple User Ranging. IEEE Transactions on Aerospace and Electronic Systems. 1995, Vol.31(2).752-767.
[15]Sturm J. F.. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones.Optim. Meth. Softw..1999,11-12.625-653.
[16]Grant M and Boyd S. CVX:Matlab software for disciplined convex programming. http://stanford.edu/～boyd/cvx,2008, Dec..
[17]Schittkowski. A Comparative Performance Evaluation of 27 Nonlinear Programming Codes, Computing.1983, Vol.30.335.
[1]Deng H. Polyphase Code Design for Orthogonal Netted Radar Systems. IEEE Trans, on Signal Processing.2004, Vol.52(11).3126-3135.
[2]Deng H.. Discrete frequency-coding waveform design for netted radar systems.IEEE Signal Processing Letters.2004, Vol.11 (2).179-182.
[3]Liu B., He Z.-S., Zeng J.-K., et al. Polyphase Orthogonal Code Design for MIMO Radar Systems. International Conference on Radar.2006,1-4.
[4]Liu B., He Z.-S., He Q. Optimization of Orthogonal Discrete Frequency-Coding Waveform Based on Modified Genetic Algorithm for MIMO Radar. International Conference on Communications, Circuits and Systems.2007,966-970.
[5]Khan H. A., Zhang Y., Ji C.. Optimizing Polyphase Sequences Optimizing Polyphase Sequences. IEEE Trans. on Signal Processing Letters.2006, Vol.13(10). 589-592.
[6]Kevin J. Gartz. Generation of Uniform Amplitude Complex Code Sets with Low Correlation Sidelobes. IEEE Trans. on Signal Processing.1992, Vol.40(2). 343-351.
[7]Stoica P., He H., Li J.. New Algorithms for Designing Unimodular Sequences With Good Correlation Properties. IEEE Trans. on Signal Processing.2009, Vol.57(4). 1415-1425.
[8]Richard O. Duda, Peter E. Hart, David G. Stork李宏东,姚天翔译.模式识别.北京：机械工业出版社.2003.
[9]王小平,曹立明.遗传算法—理论、应用与软件实现.西安：西安交通大学出版社.2002.
[10]Lingo.http://www.lindo.com.
[11]胡亮兵,刘宏伟,吴顺君.基于约束非线性规划的MIMO雷达正交波形设计.系统工程与电子技术学报.已录用.2010.
[1]Ackroyd H. M., Ghani F.. Optimum mismatched filters for sidelobe suppression. IEEE Transactions on Aerospace and Electronic.1973.
[2]Baden J. M. and Cohen M. N.. Optimal Peak Sidelobe Filters for Biphase Pulse Compression. IEEE International Radar Conference.1990.
[3]Nunn C.. Constrained optimization applied to pulse compression codes, and filters. IEEE International Radar Conference.2005,190-194.
[4]Hu C.-H., Liu R.-B., Zhou Q.-F.. Mismatched-Filter Design for Biphase-coded Pulse for High Frequency Untrasound Imaging. IEEE Ultrasonics Symposium.2006.
[5]Griep R. K., Ritcey A. J., Burlingame J. J.. Poly-Phase Codes and Optimal Filters for Multiple User Ranging. IEEE Transactions on Aerospace and Electronic Systems. 1995, Vol.31(2).752-767.
[6]Hu Liang-Bing, Liu Hong-Wei., Feng Da-Zheng. Optimal Mismatched Filter Bank Design for MIMO Radar via Convex Optimization. Submitted to 5 Waveform Diversity Conference, Canada.2010.
[7]Stephen Boyd. Convex Optimization. UK:Cambridge University Press.2004.
[8]Sturm J. F.. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones.Optim. Meth. Softw..1999,11-12.625-653.
[9]Grant M and Boyd S. CVX:Matlab software for disciplined convex programming. http://stanford.edu/-boyd/cvx,2008,Dec.
[10]Deng H.. Polyphase Code Design for Orthogonal Netted Radar Systems. IEEE Trans. on Signal Processing.2004, Vol.52(11).3126-3135.
[1]Stoica P., Li J., Xie Y.. On probing signal design for MIMO radar. IEEE Transactions on Signal Processing. Aug.2007, Vol.55(8).4151-4161.
[2]Fuhrmann D. R. and Antonio G. S.. Transmit beamforming for MIMO radar systems using partial signal correlation. Proc.38th Asilomar Conference on Signals, System and Computers. Nov.2004, Vol.1.295-299.
[3]Tuomas Aittomaki and Visa Koivunen. Low-complexity method for transmit beamforming in MIMO radars. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing.2007, Vol.2.305-308.
[4]Luca A.. Experimental results on SIAR digital beamforming radar. Proceedings of the IEEE International Radar Conference.Brighton,UK.1992,505-510.
[5]Antonio G. S., Fuhrmann D. R., Beampattem synthesis for wideband MIMO radar systems.2005 Proc.1st IEEE International Workshop on Computational Advances in Multi—Sensor Adaptive. Vol.1.105-108.
[6]李军MIMO雷达中的数字波束形成与信号处理技术研究.成都：电子科技大学博士学位论文.2009.6.
[7]胡亮兵,刘宏伟,杨晓超,吴顺君.集中式MIMO雷达快速发射方向图设计方法.电子与信息学报.2010,Vo1.32(2).481-484.
[8]Grant M. and Boyd S.. CVX:Matlab software for disciplined convex programming. http://stanford.edu/～boyd/cvx,Dec.2008.
[1]Stoica P., Li J., Xie Y.. On probing signal design for MIMO radar. IEEE Transactions on Signal Processing. Aug.2007, Vol.55(8).4151-4161.
[2]Fuhrmann D. R. and Antonio G. S.. Transmit beamforming for MIMO radar systems using partial signal correlation. Proc.38th Asilomar Conference on Signals, System and Computers. Nov.2004, Vol.1.295-299.
[3]Tuomas Aittomaki and Visa Koivunen. Low-complexity method for transmit beamforming in MIMO radars. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing.2007, Vol.2.305-308.
[4]Luca A.. Experimental results on SIAR digital beamforming radar. Proceedings of the IEEE International Radar Conference.Brighton,UK.1992,505-510.
[5]Antonio G. S., Fuhrmann D. R., Beampattem synthesis for wideband MIMO radar systems.2005 Proc.1st IEEE International Workshop on Computational Advances in Multi—Sensor Adaptive. Vol.1.105-108.
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