Jointly optimal array geometries and waveforms in active sensing: New insights into array design via the Cram\'er-Rao bound

TL;DR

This paper explores the joint design of array geometries and waveforms in active sensing to minimize the Cramér-Rao bound, revealing unique array configurations and proposing a sparse array design with improved properties.

Contribution

It introduces a joint optimization framework for array geometry and waveforms, identifying conditions for CRB equivalence and designing a novel sparse array with advantageous co-array properties.

Findings

Multiple array-waveform pairs can achieve the same CRB due to Diophantine solutions.

The receive array geometry minimizing CRB is unique under physical constraints.

A new sparse array design with a nonredundant, contiguous co-array is proposed.

Abstract

This paper investigates jointly optimal array geometry and waveform designs for active sensing. Specifically, we focus on minimizing the Cram\'er-Rao lower bound (CRB) of the angle of a single target in white Gaussian noise. We first find that several array-waveform pairs can yield the same CRB by virtue of sequences with equal sums of squares, i.e., solutions to certain Diophantine equations. Furthermore, we show that under physical aperture and sensor number constraints, the CRB-minimizing receive array geometry is unique, whereas the transmit array can be chosen flexibly. We leverage this freedom to design a novel sparse array geometry that not only minimizes the single-target CRB given an optimal waveform, but also has a nonredundant and contiguous sum co-array, a desirable property when launching independent waveforms, with relevance also to the multi-target case.