[Paper Review] Array-Informed Waveform Design for Active Sensing: Diversity, Redundancy, and Identifiability

Robin Rajamäki, Piya Pal|arXiv (Cornell University)|May 10, 2023

Sparse and Compressive Sensing Techniques4 citations

TL;DR

This paper proposes a unified framework for designing transmit waveforms in MIMO active sensing systems by jointly optimizing array geometry and waveform diversity to maximize identifiability of sparse scatterers. It establishes that redundant array configurations can achieve maximum identifiability using fewer linearly independent waveforms than transmitters, reducing hardware cost and transmission time while preserving performance through Kruskal rank maximization of the structured sensing matrix.

ABSTRACT

This paper investigates the combined role of transmit waveforms and (sparse) sensor array geometries in active sensing multiple-input multiple-output (MIMO) systems. Specifically, we consider the fundamental identifiability problem of uniquely recovering the unknown scatterer angles and coefficients from noiseless spatio-temporal measurements. Assuming a sparse scene, identifiability is determined by the Kruskal rank of a highly structured sensing matrix, which depends on both the transmitted waveforms and the array configuration. We derive necessary and sufficient conditions that the array geometry and transmit waveforms need to satisfy for the Kruskal rank -- and hence identifiability -- to be maximized. Moreover, we propose waveform designs that maximize identifiability for common array configurations. We also provide novel insights on the interaction between the waveforms and array geometry. A key observation is that waveforms should be matched to the pattern of redundant transmit-receive sensor pairs. Redundant array configurations are commonly employed to increase noise resilience, robustify against sensor failures, and improve beamforming capabilities. Our analysis also clearly shows that a redundant array is capable of achieving its maximum identifiability using fewer linearly independent waveforms than transmitters. This has the benefit of lowering hardware costs and transmission time. We illustrate our findings using multiple examples with unit-modulus waveforms, which are often preferred in practice.

Motivation & Objective

• To address the fundamental identifiability problem in MIMO active sensing systems where unknown scatterer angles and coefficients must be uniquely recovered from noiseless spatio-temporal measurements.
• To determine the necessary and sufficient conditions on array geometry and transmit waveforms for maximizing Kruskal rank of the sensing matrix, which governs identifiability.
• To design waveforms that maximize identifiability for common redundant array configurations, especially under reduced waveform rank (WR).
• To reveal the interaction between waveform design and array redundancy, showing that waveforms should be matched to redundant sensor pairs to improve performance.
• To demonstrate that redundant arrays can achieve maximum identifiability with fewer linearly independent waveforms than the number of transmitters, enabling cost and time savings.

Proposed method

• The paper models the sensing process using a highly structured sensing matrix whose Kruskal rank determines identifiability of sparse scatterers in the scene.
• It derives necessary and sufficient conditions on the transmit waveforms and array geometry to maximize the Kruskal rank, ensuring unique recovery of scatterer parameters.
• The method involves constructing a matrix Q via tensor products and permutation matrices to analyze the rank properties of the composite waveform-array system.
• It uses Vandermonde matrix structure and Kruskal rank theory to prove that full identifiability (Kruskal rank = NΣ) is achievable even with reduced WR when array redundancy is exploited.
• The approach leverages the sum co-array concept, where virtual array elements are formed by pairwise sums of transmit and receive sensor positions, enabling enhanced resolution and identifiability.
• Theoretical analysis is validated using unit-modulus waveforms, which are common in practical systems, to illustrate the design principles and performance gains.

Experimental results

Research questions

• RQ1 What conditions on array geometry and transmit waveforms are necessary and sufficient to maximize identifiability in MIMO active sensing systems?
• RQ2 How does waveform rank (WR) interact with array redundancy to affect the Kruskal rank of the sensing matrix and thus identifiability?
• RQ3 Can redundant array configurations achieve maximum identifiability with fewer linearly independent waveforms than the number of transmitters?
• RQ4 How should waveforms be designed to match the pattern of redundant transmit-receive sensor pairs for optimal performance?
• RQ5 What is the theoretical trade-off between waveform diversity, array redundancy, and the number of required transmissions in MIMO sensing?

Key findings

• Redundant array configurations can achieve maximum identifiability using fewer linearly independent waveforms than the number of transmitters, reducing hardware and transmission costs.
• The Kruskal rank of the sensing matrix—critical for identifiability—is maximized when waveforms are matched to the pattern of redundant transmit-receive pairs.
• For any array geometry, identifiability is maximized when the Kruskal rank of the composite waveform-array matrix reaches NΣ, the total number of virtual array elements.
• Even with reduced waveform rank (WR), full identifiability is achievable if the waveform design aligns with the array's redundancy structure, as proven via rank analysis of structured matrices.
• The proposed waveform designs ensure that the sensing matrix achieves full Kruskal rank NΣ, enabling unique recovery of up to NΣ sparse scatterers.
• Unit-modulus waveforms are shown to be effective in achieving maximum identifiability, making the results applicable to practical systems like automotive radar and JCS.