CRB-Optimal Arrays and Waveforms in Active Sensing: Role of Redundancy and Spatial Covariance of Array Geometry

2026-05-29

AI Summary

This work investigates the joint optimization of array geometry and waveform design in active sensing systems to approach the Cramér-Rao bound (CRB) performance limit for parameter estimation, balancing mean squared error and identifiability. By analyzing the CRB under both orthogonal and coherent waveforms for linear and planar arrays, it reveals that single-target estimation performance is governed by the sum of the spatial variances of the transmit and receive arrays. Building on this insight, the study proposes an asymmetric allocation strategy of transmit and receive sensors, departing from conventional symmetric designs. It further establishes a connection between Diophantine equations and CRB-equivalent array constructions, leveraging weighted virtual array multiplicity and beam steering optimization to derive a general optimality criterion. This yields constructible high-performance array configurations, offering a new paradigm for MIMO active sensing systems.

Abstract

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.

Problem

• Cramér-Rao Bound
• array geometry
• spatial covariance
• active sensing
• redundancy

Innovation

• Cramér-Rao Bound
• array redundancy
• spatial covariance
• optimal waveform design
• Diophantine equations