摘要
本文研究了基于半正定规划(SDP)的发射方向图设计方法和稳健波束形成，SDP优化模型具有凸优化特性，具有SDP特性的波束形成方法优化模型可以求得全局最优解。传统的向量加权发射方向图设计方法和稳健波束形成方法并不能很好控制主瓣形状，而MIMO雷达发射方向图设计方法的优化变量是发射信号协方差矩阵，有效提高了方向图设计方法中可利用的自由度，自由度的提高为设计具有期望主瓣形状和低旁瓣特性的方向图提供了一个很好的先决条件。可以将发射信号协方差矩阵看作是加权向量的协方差矩阵，将加权向量的协方差矩阵作为阵列信号处理中波束形成方法的优化变量，设计具有期望特性的波束方向图。
本文具体的研究内容：低旁瓣MIMO雷达发射方向图设计方法，基于阵列结构划分的MIMO雷达发射方向图设计方法，基于SDP的优化布阵方法，基于矩阵加权和半正定秩松弛(SDR)方法的稳健波束形成，基于时变向量加权的稳健波束形成。
1.低旁瓣MIMO雷达发射方向图可以在保证主瓣发射能量的前提下，抑制旁瓣杂波和虚假目标能量，从而达到提升回波信噪比的目的。本文提出了两种低旁瓣MIMO雷达发射方向图设计方法。(1)通过对已有发射信号协方差矩阵的非对角线元素进行修正来实现低旁瓣方向图的优化设计。首先，利用方向图匹配设计方法得到具有期望主瓣形状的发射方向图，并以其作为初始值；其次，以最小化峰值旁瓣或积分旁瓣为目标函数，在约束修正后的信号协方差矩阵为半正定矩阵的前提下，利用修正矩阵的Frobenius范数为量度约束主瓣形状失真程度。(2)以最小化峰值旁瓣或积分旁瓣为目标函数，约束主瓣幅度波动范围、零点深度以及主波束间的互相关性（多波束方向图）建立优化模型；或者以最小化幅度波动范围为目标函数，约束峰值旁瓣电平、零点深度以及主波束间的互相关性（多波束方向图）建立优化模型。这两个方法都可以得到具有期望主瓣形状和低旁瓣特性的方向图，第二种方法得到的方向图还可有效设置零点深度和主波束间的互相关性。上述优化问题都是SDP凸优化问题，可以求得全局最优解。
2.针对现有MIMO雷达发射方向图设计方法无法直接推广到大规模阵列MIMO雷达的问题，本文提出了基于阵列结构划分方法的MIMO雷达发射方向图设计方法。(1)将阵列划分为阵列结构相同的子阵，各个子阵的发射信号不同，子阵内阵元发射信号相同，以每个子阵的第一个阵元组成新的稀疏阵列MIMO雷达，子阵划分以后的发射方向图为稀疏阵列MIMO雷达的方向图与子阵方向图的乘积。该方法可以降低发射信号矩阵的维度以及方向图设计中的角度范围，这就有效降低了发射方向图设计方法的运算量。(2)基于平面阵方向图可以由水平和垂直方向的线阵方向图合成的思想，本文提出了应用基波束和概率选择方法设计平面阵MIMO雷达发射方向图的方法。该方法首先将期望方向图沿方位角累加，形成一维俯仰角方向图，建立垂直方向线阵的俯仰角基波束集合和该集合元素的概率选择优化模型；其次针对每个俯仰角对应的一维方位角期望方向图，建立水平方向线阵的方位角基波束集合和该集合元素的概率选择优化模型；最后合成两维基波束集合和集合中元素的选择概率，并求得平面阵MIMO雷达的发射方向图和发射信号。以上方法中的优化模型都是SDP优化模型，可以求得全局最优解。
3.现有优化布阵方法的优化模型是以阵元位置为优化变量，通常以指数函数出现，该优化问题非凸。针对该问题，本文中提出了基于SDP的优化布阵方法，该方法的实现步骤：首先，将需要稀疏布阵的区域划分为很小的栅格，每个栅格点上有一个待选阵元；其次，以每个待选阵元的选择概率为优化变量，设计具有强指向性的发射方向图；最后，在最小阵元间距的约束下，以选择概率的大小以及重心准则将待选阵元进行聚合，得到满足最小阵元间距要求的稀疏布阵。该方法可以看作是基于概率选择加权向量的发射方向图设计方法，优化模型是一个SDP优化模型，可以求得全局最优解。相比传统的智能优化算法的优化布阵方法，本文方法只需进行单次求解，并且可以从单次求解结果中选择不同阵元数的稀疏布阵。
4.本文提出了基于矩阵加权和半正定秩松弛(SDR)方法的稳健波束形成。(1)约束主瓣幅度波动范围的矩阵加权稳健波束形成方法，与已有方法相比该方法可以有效控制方向图的主瓣形状、旁瓣电平以及零点深度。存在噪声和系统误差时，该方法对于信号功率的估计具有更好的稳健性。通过约束主瓣幅度波动范围波束形成方法求得加权矩阵的协方差矩阵，对该协方差矩阵做特征值分解求得加权矩阵，通过划分特征值大小来确定最小维度的加权矩阵，该加权矩阵可以在不损失方向图形状和信号功率估计性能的条件下有效降低系统实现复杂度。(2)与向量加权稳健波束形成方法相比，矩阵加权稳健波束形成方法系统实现复杂度较大。针对该问题，本文中给出了基于SDR方法的稳健波束形成，该方法优化模型与矩阵加权方法优化模型的不同只是多了协方差矩阵的秩为1的约束。应用SDR方法求得加权向量的协方差矩阵，将该矩阵中的每一行(列)转化为加权向量，然后选择加权向量使方向图主瓣与0dB之间失真最大值最小。该方法的系统实现复杂度与传统向量加权方法一致，对信号功率的估计性能与矩阵加权方法相当。
5.尽管矩阵加权稳健波束形成方法对于目标信号功率估计较为准确，对于误差也更稳健，但是该方法的系统实现复杂度较大。针对这个问题，我们提出了基于时变向量加权的稳健波束形成方法。已有的波束形成方法是对每次快拍信号加相同的权向量或权矩阵，这里我们借鉴MIMO雷达的思想，对不同时刻的快拍信号加不同的权向量，这些加权向量组成一个加权向量组，以加权向量组的协方差矩阵为优化变量，建立与矩阵加权稳健波束形成方法相同的优化模型。该方法的信号功率估计性能与矩阵加权方法一致，但是系统的匹配滤波输出会有信干噪比(SINR)损失。通过约束每次快拍的阵列向量加权幅度响应，可以有效降低SINR损失，并且可以显式求解加权向量组。
Transmit pattern Design and robust beamforming via semi-definite programming(SDP) were studied in this dissertation. SDP optimization model possesses convexoptimization features, and the optimization model of beamforming with SDP can obtainglobal optimal solution. The traditional vector weighted transmit pattern design methodand robust beamforming didn’t controlle the mainlobe shape effectively. The transmitsignal covariance matrix is the optimization variable of MIMO radar transmit patterndesign, and the more available degree of freedom can be used than traditional methods,the desired mainlobe shape and low sidelobe pattern can be designed under thecondition of more available degree of freedom. The transmit signal covariance matrixcan be regarded as a weighted vector covariance matrix, which is the optimizationvariable in beamforming methods, and the desired beam pattern can be designed byoptimizing the weighted vector covariance matrix.
Specific contents of this dissertation: low sidelobe transmit pattern design forMIMO radar, MIMO radar transmit pattern design via array structure divided,optimization of array elements position via SDP, robust beamforming via matrixweighted and semi-definite rank relaxation (SDR), robust beamforming via time variableweighted vector.
1. Low sidelobe transmit pattern of MIMO radar can concentrate majority energyin the mainlobe region to improve the target detection, and reduce the energy of clutterand false targets from sidelobe, so as to enhance the signal-to-noise ratio(SNR) of echo.(1) Two low sidelobe transmit pattern optimization algorithms for MIMO radar areproposed. Based on a modified non-diagonal elements of transmit signal covariancematrix, and a low sidelobe transmit pattern optimization algorithm for MIMO radar ispresented. Firstly, the pattern matching design method is implemented to obtain thedesired mainlobe shape of transmit pattern, and the results are set as the initial value.Secondly, the cost function can be formulated as the minimization of peak-sidelobe orintegrated-sideloble; the mainlobe shape distortion can be constrained by the Frobeniusnorm of modification matrix, and the modified transmit signal covariance matrix needsto be positive semi-definite.(2) The cost function can be formulated as theminimization of peak-sidelobe or integrated-sideloble, the amplitude ripple of mainlabe,null depth and cross correlation of mainlobes (multiple beam pattern) can be constrained;The cost function can be formulated as the minimization of the amplitude ripple ofmainlabe, the peak-sidelobe level, null depth and cross correlation of mainlobes (multiple beam pattern) can be constrained. The desired mainlobe shape and lowsidelobe pattern can be obtaibed with the proposed methods, the null depth and crosscorrelation of mainlobes can be controlled effectively in second method. The aboveoptimization problem is SDP convex optimization problem, we can get the globaloptimal solution.
2. The transmit pattern methods available for the MIMO radar can not be extendedinto the large array of the MIMO radar. MIMO radar transmit pattern design viadividing array structure is presented on this dissertation.(1) The large array is dividedinto the same subarray, each subarray transmitted different signal, the same signal wastransmitted by each element in subarray, the first subarray element formed a new sparsearray of MIMO radar, and the sum transmit pattern is the pattern product of sparse arrayMIMO radar and subarray. The size of transmit signal covariance matrix and anglerange with pattern design, the amount of calculation of transmit pattern design methodcan be reduced effectively.(2) With the application of the base-beam and probabilityselection methods, an approach to design the transmit pattern for the planer arrayMIMO radar is presented in this dissertation, which is based on the idea that thepattern of planer array can be synthetised by the pattern of a horizontal and vertical linearray. First, the desired pattern is accumulated along the azimuth and consequently the1-D pattern along the elevation can be formed; the elevation base-beam collection of thevertical line array and the corresponding probability selecting optimization model areformulated. Then, the azimuth base-beam collection of the horizontal line array and thecorresponding probability selecting optimization model can also be formulated for thedesired azimuth pattern of a candidate elevation. Finally, the2-D base-beam collectionis synthetised and the corresponding selected probability of the collection elements canbe calculated. The above optimization problem is SDP convex optimization problem,we can obtain the global optimal solution.
3. The elements position is the optimization variable in the proposed methods,usually expressed in exponential function, thus the optimization problem isn’t convex.To solve this problem, optimization of array elements position via SDP is proposed inthis dissertation, implementation steps of the proposed method are described asfollows: Firstly, the region of array elements position can be divided in to many smallgrid, there is an array element to be selected on each grid point; Secondly, set selectionprobability of each element as optimization variables, and the transmit pattern with thefeature of strong directivity can be designed; Finally, the array element position isaggregated with the selection probability and gravity method under the minimum array element distance constraint, the sparse array can be obtained by this method. Thismethod can be regarded as transmit pattern design method that optimization variable isthe weighted vector of selection probability. The optimization problem is SDP convexoptimization problem, we can obtain the global optimal solution, and the different sparsearray can be selected in the single result.
4. The robust beamforming via matrix weighted and semi-definite rank relaxation(SDR) methods are proposed in this dissertation.(1) Compared with the availablemethods, matrix weighted beamforming method which constrains the magnitude ripplerange of the mainlobe can control the shape of the mainlobe, the sidelobe level and thenull depth of pattern more effectively, the estimation of signal power is robust to thenoise and systematic errors. The covariance matrix of the weighting matrix can beobtained by the matrix weighted beamforming method, and the weighting matrix can beobtained by the eigen-decomposition of the covariance matrix. And the minimum scaleof the weighting matrix dimension can be determined by the dominant eigenvalues. Withthe beampattern shape controlled, this method can be able to maintain the performanceof the signal power estimation and the system complexity can be reduced efficiently.(2)Because The system implementation complexity of matrix weighted beamformingmethod is more complex than vector weighted beamforming method. In order to solvethese problems, this dissertation presents new robust beamforming approach based onsemi-definite rank relaxation (SDR). Detailed description of the proposed method aregiven as: the optimal model has the same objective as that of the Capon algorithm; theoptimization variable is the covariance matrix of weighting vector with constraints posedon the ripple of mainlobe amplitude and sidelobe level, and the rank of covariancematrix is1; the covariance matrix of the weighting vector can be obtained by the SDRmethod, and each row or column of the matrix is translated into weight vector, then aweighting vector is chosen which allows it become minimal one in the maximumdistortions between the mainlobe of pattern and0dB. The system implementationcomplexity of the proposed method is the same as the vector weighted methods, and thesignal power estimation performance is similar to the matrix weighted method.
5. Although matrix weighted robust beamforming method can estimate signalpower accurately, and more robust to errors, this method has a disadvantage that thesystem implementation complexity is more complex than vector weighted beamforming.In order to solve this problem, the time-varing vector weighted robust beamforming isproposed in this dissertation. Each snapshot signal multiplies the same weighted vectoror weighted matrix in the available beamforming methods, we use the idea of MIMO radar that the different moment snapshot signal multiplies different weighted vector,weighted vectors formed weighted vector group, the same optimization model can beformulated which is the same as matrix weighted robust beamforming method, and thecovariance matrix of weighted vector group is the optimization variable. The signalpower estimation performance is equal to the matrix weighted method, but the matchedfilter output of system would arise signal to interference noise ratio (SINR) loss. TheSINR loss can be reduced by constraining magnitude response of weighted vector group,which can be solved by explicitly.
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