This paper studies a multiple-input multiple-output (MIMO) radar system for sensing the unknown and random angular location (angle) of a point target, based on the target-reflected echo signals and known prior distribution information about the target's angle specified by a probability density function (PDF). We consider a challenging yet practical scenario where the knowledge of such PDF is imperfect, due to the inaccuracy in PDF acquisition or unpredicted change of target appearance pattern; while the real (actual) PDF is modeled as an unknown perturbed version of the imperfect known PDF bounded by a given uncertainty radius. Such PDF imperfection motivates us to study the robust transmit beamforming design to optimize the worst-case sensing performance among all possible real PDFs. Since the sensing mean-squared error (MSE) is difficult to be characterized explicitly, we adopt the worst-case posterior Cramér-Rao bound (PCRB) as the performance metric. We formulate the beamforming optimization problem to minimize the maximum PCRB among all possible real PDFs, which is highly non-trivial since the PCRB has a complex intractable expression over the real PDF, and there are infinite constraints corresponding to the continuous set of real PDFs bounded by the uncertainty radius. To address these challenges, we derive a tractable quadratic approximation of the PCRB via second-order Taylor expansion, and leverage the S-procedure to equivalently transform the infinite constraints into a linear matrix inequality, based on which the problem is reformulated into a convex optimization problem solvable with polynomial time complexity. The obtained solution approaches the globally optimal robust beamforming solution as the uncertainty radius decreases. Numerical results validate the effectiveness of our proposed robust beamforming design.

翻译：本文研究基于目标反射回波信号及由概率密度函数（PDF）描述的目标角度先验分布信息，对未知随机方向（角度）点目标进行感知的多输入多输出（MIMO）雷达系统。我们考虑一个具有挑战性的实际场景：由于PDF获取不准确或目标出现模式不可预测的变化，该PDF的知识并不完美；而真实（实际）PDF被建模为已知非完美PDF的未知扰动版本，其范围受给定不确定半径约束。这种PDF不完美性促使我们研究鲁棒发射波束赋形设计，以优化所有可能真实PDF中的最差感知性能。由于感知均方误差（MSE）难以显式表征，我们采用最差后验克拉美-罗界（PCRB）作为性能指标。我们建立波束赋形优化问题，旨在最小化所有可能真实PDF中的最大PCRB。该问题高度复杂，因为PCRB对真实PDF的表达式复杂且难以处理，同时存在对应于由不确定半径约束的连续真实PDF集的无限约束。为应对这些挑战，我们通过二阶泰勒展开推导PCRB的易处理二次近似，并利用S-过程将无限约束等价转化为线性矩阵不等式，据此将问题重构为具有多项式时间复杂度的凸优化问题。当不确定半径减小时，所求解趋近于全局最优鲁棒波束赋形解。数值结果验证了所提鲁棒波束赋形设计的有效性。