Optimization of the Transmit Weighting Matrix for MIMO Radar Based on the Uniform Elemental Power Constraint
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摘要: 为提高多输入多输出(MIMO)雷达的目标角度估计性能，同时兼顾雷达发射功率利用率的需求，该文以合成发射导向矢量与期望导向矢量二范数误差为目标函数，研究了阵元等功率约束下的发射加权矩阵优化问题。推导了发射加权矩阵向量化条件下的等价优化模型，并基于循环优化方法和改进PDR算法对其进行了求解。在每次迭代过程中，所提方法均能获得子优化问题的闭式解，因而其计算复杂度非常低。在此基础之上，从理论上证明了所提方法的收敛性。由于该方法实现了MIMO雷达发射功率在期望空域的聚焦，在同等条件下相比传统MIMO雷达能够有效提高目标的角度估计性能。最后，仿真实验表明了所提方法的有效性。Abstract: To improve the angle estimation performance of the MIMO (Multiple-Input-Multiple-Output) radar, the optimization of the transmit weighting matrix by setting the 2-norm error between the actual synthesized transmitting steering vector and the desired one as the objective function is studied in this paper. The maximization of the transmit power utilization can be enforced via imposing the uniform elemental power constraint on the formulation. Furthermore, a method based on the cyclic algorithm and improved Projection Descent and Retraction (PDR) is provided to settle the equivalent problem under the vectorization of the transmit weighting matrix. The closed solution can be achieved at each iteration, then the computational complexity is low of the proposed method. And the convergence can also be proved. The proposed method obtains the superior performance in angle estimation of MIMO radar based on focusing the transmit power into the desired spatial sector. Finally, simulation results are presented to verify the efficiency of the proposed method.
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表 1 所提算法的具体流程
输入：发射阵元数目$ {M_{\text{T}}} $，正交基波形数目$ K $，加权系数${{\bar {\boldsymbol{w}}} }$，期望导向矢量${ {\tilde {\boldsymbol{A} } } } = [{ {\tilde {\boldsymbol{a} } } }_1^{\text{T} },{ {\tilde {\boldsymbol{a} } } }_2^{\text{T} },\cdots,{ {\tilde {\boldsymbol{a} } } }_N^{\text{T} }]$。 输出：最优发射加权矩阵(向量)$ {{\boldsymbol{w}}^ * } $。 初始化：$ {{\boldsymbol{w}}^{(l)}} $，且$ l = 0 $。 (1) 计算$ {\boldsymbol{P}} = {\boldsymbol{A\bar w}}{{\boldsymbol{A}}^{\text{H}}} $，$ {\boldsymbol{q}} = {\boldsymbol{A\bar w}}{{{\tilde {\boldsymbol{A}}}}^{\text{H}}} $，$ {{\tilde {\boldsymbol{A}}\bar {\boldsymbol{w}}}}{{{\tilde {\boldsymbol{A}}}}^{\text{H}}} $。 (2) 外循环操作： (a) 令$ l = l + 1 $； (b) 根据式(10)更新$ {\alpha ^{(l)}} $； (c) 令$ q = 0 $，$ {{\boldsymbol{w}}^{(l,q)}} = {{\boldsymbol{w}}^{(l - 1)}} $，计算参数$ {{\boldsymbol{\hat P}}^{(l)}} $； (d) 内循环操作： ① 根据式(13)计算$ {\nabla ^{(l,q)}} $； ② 根据式(15)计算$ {\hat \nabla ^{(l,q)}} $； ③ 根据式(16)更新$ {{{\tilde {\boldsymbol{w}}}}^{(l,q)}} $； ④ 根据式(17)进行缩放操作获得$ {{\boldsymbol{w}}^{(l,q + 1)}} $； ⑤ 令$ q = q + 1 $； ⑥ 判断是否满足收敛条件，如果是：内循环结束，令$ {{\boldsymbol{w}}^{(l)}} = {{\boldsymbol{w}}^{(l,q)}} $并转入步骤(e)；如果否：转入步骤①。 (e) 判断是否满足收敛条件，如果是：外循环结束，转入步骤(3)；如果否：转入步骤(a)。 (3) $ {{\boldsymbol{w}}^ * } = {{\boldsymbol{w}}^{(l)}} $。 
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