Subspace Coding for Spatial Sensing

A subspace code is defined as a collection of subspaces of an ambient vector space, where each information-encoding codeword is a subspace. This paper studies a class of spatial sensing problems, notably direction of arrival (DoA) estimation using multisensor arrays, from a novel subspace coding perspective. Specifically, we demonstrate how a canonical (passive) sensing model can be mapped into a subspace coding problem, with the sensing operation defining a unique structure for the subspace codewords. We introduce the concept of sensing subspace codes following this structure, and show how these codes can be controlled by judiciously designing the sensor array geometry. We further present a construction of sensing subspace codes leveraging a certain class of Golomb rulers that achieve near-optimal minimum codeword distance. These designs inspire novel noise-robust sparse array geometries achieving high angular resolution. We also prove that codes corresponding to conventional uniform linear arrays are suboptimal in this regard. This work is the first to establish connections between subspace coding and spatial sensing, with the aim of leveraging insights and methodologies in one field to tackle challenging problems in the other.

Sparse Spatial Smoothing: Reduced Complexity and Improved Beamforming Gain via Sparse Sub-Arrays

This paper addresses the problem of single snapshot Direction-of-Arrival (DOA) estimation, which is of great importance in a wide-range of applications including automotive radar. A popular approach to achieving high angular resolution when only one temporal snapshot is available is via subspace methods using spatial smoothing. This involves leveraging spatial shift-invariance in the antenna array geometry, typically a uniform linear array (ULA), to rearrange the single snapshot measurement vector into a spatially smoothed matrix that reveals the signal subspace of interest. However, conventional approaches using spatially shifted ULA sub-arrays can lead to a prohibitively high computational complexity due to the large dimensions of the resulting spatially smoothed matrix. Hence, we propose to instead employ judiciously designed sparse sub-arrays, such as nested arrays, to reduce the computational complexity of spatial smoothing while retaining the aperture and identifiability of conventional ULA-based approaches. Interestingly, this idea also suggests a novel beamforming method which linearly combines multiple spatially smoothed matrices corresponding to different sets of shifts of the sparse (nested) sub-array. This so-called shift-domain beamforming method is demonstrated to boost the effective SNR, and thereby resolution, in a desired angular region of interest, enabling single snapshot low-complexity DOA estimation with identifiability guarantees.

Importance of array redundancy pattern in active sensing

This paper further investigates the role of the array geometry and redundancy in active sensing. We are interested in the fundamental question of how many point scatterers can be identified (in the angular domain) by a given array geometry using a certain number of linearly independent transmit waveforms. We consider redundant array configurations (with repeated virtual transmit-receive sensors), which we have recently shown to be able to achieve their maximal identifiability while transmitting fewer independent waveforms than transmitters. Reducing waveform rank in this manner can be beneficial in various ways. For example, it may free up spatial resources for transmit beamforming. In this paper, we show that two array geometries with identical sum co-arrays, and the same number of physical and virtual sensors, need not achieve equal identifiability, regardless of the choice of waveform of a fixed reduced rank. This surprising result establishes the important role the pattern (not just the number) of repeated virtual sensors has in governing identifiability, and reveals the limits of compensating for unfavorable array geometries via waveform design.

Effect of Beampattern on Matrix Completion with Sparse Arrays

We study the problem of noisy sparse array interpolation, where a large virtual array is synthetically generated by interpolating missing sensors using matrix completion techniques that promote low rank. The current understanding is quite limited regarding the effect of the (sparse) array geometry on the angle estimation error (post interpolation) of these methods. In this paper, we make advances towards solidifying this understanding by revealing the role of the physical beampattern of the sparse array on the performance of low rank matrix completion techniques. When the beampattern is analytically tractable (such as for uniform linear arrays and nested arrays), our analysis provides concrete and interpretable bounds on the scaling of the angular error as a function of the number of sensors, and demonstrates the effectiveness of nested arrays in presence of noise and a single temporal snapshot.

On array geometry and self-interference in full-duplex massive MIMO communications

This paper studies the role of the joint transmit-receive antenna array geometry in shaping the self-interference (SI) channel in full-duplex communications. We consider a simple spherical wave SI model and two prototypical linear array geometries with uniformly spaced transmit and receive antennas. We show that the resulting SI channel matrix has a regular (Toeplitz) structure in both of these cases. However, the number of significant singular values of these matrices - an indication of the severity of SI - can be markedly different. We demonstrate that both reduced SI and high angular resolution can be obtained by employing suitable sparse array configurations that fully leverage the available joint transmit-receive array aperture without suffering from angular ambiguities. Numerical electromagnetic simulations also suggest that the worst-case SI of such sparse arrays need not increase - but can actually decrease - with the number of antennas. Our findings provide preliminary insight into the extent to which the array geometry alone can mitigate SI in full-duplex massive MIMO communications systems employing a large number of antennas.

Harnessing Holes for Spatial Smoothing with Applications in Automotive Radar

This paper studies spatial smoothing using sparse arrays in single-snapshot Direction of Arrival (DOA) estimation. We consider the application of automotive MIMO radar, which traditionally synthesizes a large uniform virtual array by appropriate waveform and physical array design. We explore deliberately introducing holes into this virtual array to leverage resolution gains provided by the increased aperture. The presence of these holes requires re-thinking DOA estimation, as conventional algorithms may no longer be easily applicable and alternative techniques, such as array interpolation, may be computationally expensive. Consequently, we study sparse array geometries that permit the direct application of spatial smoothing. We show that a sparse array geometry is amenable to spatial smoothing if it can be decomposed into the sum set of two subsets of suitable cardinality. Furthermore, we demonstrate that many such decompositions may exist - not all of them yielding equal identifiability or aperture. We derive necessary and sufficient conditions to guarantee identifiability of a given number of targets, which gives insight into choosing desirable decompositions for spatial smoothing. This provides uniform recovery guarantees and enables estimating DOAs at increased resolution and reduced computational complexity.

Array-Informed Waveform Design for Active Sensing: Diversity, Redundancy, and Identifiability

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.