This paper characterizes the performance limits of optimal array designs using orthogonal and coherent waveforms for both linear and planar arrays. For orthogonal waveforms, we show that the single-target Cramér-Rao Bound (CRB) depends on the sum of the so-called spatial variances of the transmit (Tx) and receive (Rx) arrays, or equivalently, the spatial variance of the sum co-array weighted by the multiplicities of the virtual sensors. This reveals that CRB-optimal geometries are inherently redundant, highlighting a fundamental trade-off between mean squared error (MSE) and identifiability in parameter estimation. Moreover, we derive optimal Tx-Rx sensor allocations given a total sensor budget and show that unequal allocation (favoring the Rx) is optimal even for nonredundant arrays, questioning conventional designs. We extend our results to planar arrays, providing a new general condition that the spatial covariances of the Tx and Rx arrays should satisfy for the optimal waveforms to direct power in the target direction. Additionally, we establish a connection between Diophantine equations and array geometries with equal CRB, along with a constructive method for designing such arrays. Our work provides new guidelines for and insights into optimal array and waveform design with relevance in emerging active sensing multiple-input multiple-output systems.

翻译：本文刻画了使用正交和相干波形进行线阵与平面阵优化的性能极限。对于正交波形，我们证明单目标的克拉美-罗下界（CRB）取决于发射（Tx）和接收（Rx）阵列的所谓空间方差之和，或等价地取决于由虚拟传感器重数加权的和协阵列的空间方差。这一发现揭示了CRB最优几何结构本质上是冗余的，凸显了参数估计中均方误差（MSE）与可辨识性之间的根本性权衡。此外，我们推导了给定总传感器预算下的最优Tx-Rx传感器分配方案，并表明即使对于非冗余阵列，非均衡分配（偏向Rx）也是最优的，这对传统设计提出了质疑。我们将结果推广至平面阵，给出了Tx和Rx阵列空间协方差应满足的新一般条件，以实现最优波形将功率导向目标方向。同时，我们在丢番图方程与具有相等CRB的阵列几何之间建立了联系，并提供了设计这类阵列的构造方法。我们的工作为新兴主动传感多输入多输出系统中的最优阵列与波形设计提供了新准则与新见解。