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Sparse Sensor Arrays for Active Sensing: Models, Configurations and Applications

Robin Rajamäki, Visa Koivunen

TL;DR

The chapter addresses how sparse active sensor arrays can extend the effective aperture via the sum co-array, enabling higher angular resolution with fewer physical sensors. It develops a practical framework for low-redundancy, symmetric array designs, and links the Tx and Rx sensing legs through a sum-coarray perspective to enable joint beamforming and image addition. Key contributions include formalizing redundancy for active sensing, detailing symmetric-array constructions (notably CNAs and NA-based schemes), and introducing image addition as a means to efficiently realize desired Tx-Rx beampatterns with a controllable number of component images. The discussion highlights concrete applications in imaging, MIMO radar, and wireless communications, and underscores the trade-offs between array sparsity, waveform rank, and beampattern synthesis time, with implications for real-world sensing systems.

Abstract

This chapter focuses on active sensing using sparse arrays. In active sensing applications, such as radar, sonar, wireless communications, and medical ultrasound, a collection of sensors probes the environment by emitting self-generated energy. A key benefit of such active multi-sensor arrays is their ability to focus and steer energy in desired directions by beamforming on transmit. Sparse sensor arrays offer several advantages over conventional uniform arrays, including improved resolution using fewer physical sensors and the capability to identify more scatterers than sensors. This is facilitated by the effective transmit-receive virtual array known as the sum co-array, which can have many more virtual sensors than the number of physical transmit or receive sensors. Herein, we focus on the design of low-redundancy sparse array configurations and on employing transmit-receive (Tx-Rx) beamforming using sparse arrays. We discuss the optimal, but computationally intractable Minimum-redundancy array, and a scalable symmetric array framework, which extends many well-known passive sparse array geometries to the active case. We also examine mitigating side lobes arising from spatial undersampling by a synthetic beamforming method known as image addition. We briefly present approaches for finding the physical beamforming weights synthesizing a desired Tx-Rx beampattern, and consider related spatio-temporal trade-offs. We conclude by discussing selected applications of sparse arrays in active sensing.

Sparse Sensor Arrays for Active Sensing: Models, Configurations and Applications

TL;DR

The chapter addresses how sparse active sensor arrays can extend the effective aperture via the sum co-array, enabling higher angular resolution with fewer physical sensors. It develops a practical framework for low-redundancy, symmetric array designs, and links the Tx and Rx sensing legs through a sum-coarray perspective to enable joint beamforming and image addition. Key contributions include formalizing redundancy for active sensing, detailing symmetric-array constructions (notably CNAs and NA-based schemes), and introducing image addition as a means to efficiently realize desired Tx-Rx beampatterns with a controllable number of component images. The discussion highlights concrete applications in imaging, MIMO radar, and wireless communications, and underscores the trade-offs between array sparsity, waveform rank, and beampattern synthesis time, with implications for real-world sensing systems.

Abstract

This chapter focuses on active sensing using sparse arrays. In active sensing applications, such as radar, sonar, wireless communications, and medical ultrasound, a collection of sensors probes the environment by emitting self-generated energy. A key benefit of such active multi-sensor arrays is their ability to focus and steer energy in desired directions by beamforming on transmit. Sparse sensor arrays offer several advantages over conventional uniform arrays, including improved resolution using fewer physical sensors and the capability to identify more scatterers than sensors. This is facilitated by the effective transmit-receive virtual array known as the sum co-array, which can have many more virtual sensors than the number of physical transmit or receive sensors. Herein, we focus on the design of low-redundancy sparse array configurations and on employing transmit-receive (Tx-Rx) beamforming using sparse arrays. We discuss the optimal, but computationally intractable Minimum-redundancy array, and a scalable symmetric array framework, which extends many well-known passive sparse array geometries to the active case. We also examine mitigating side lobes arising from spatial undersampling by a synthetic beamforming method known as image addition. We briefly present approaches for finding the physical beamforming weights synthesizing a desired Tx-Rx beampattern, and consider related spatio-temporal trade-offs. We conclude by discussing selected applications of sparse arrays in active sensing.
Paper Structure (42 sections, 1 theorem, 42 equations, 9 figures)

This paper contains 42 sections, 1 theorem, 42 equations, 9 figures.

Table of Contents

  1. Sparse Sensor Arrays for Active Sensing: Models, Configurations and Applications
  2. Introduction
  3. Goals, scope, and organization
  4. Notation
  5. Active sensing signal model
  6. Physical array model
  7. Angle-delay-Doppler model
  8. Simplified angle-only model
  9. Waveform matrix
  10. Waveform rank
  11. Practical waveform constraints
  12. Virtual array model
  13. Sparse array configurations
  14. Categorization of array configurations based on overlap between Tx and Rx arrays
  15. Minimum-redundancy array (MRA)
  16. ...and 27 more sections

Key Result

Theorem 9.1

Consider the symmetric array $\mathbb{S}$ in eq:symm_gen with generator $\mathbb{G}$ and offset $\ell$. If $\mathbb{G}-\mathbb{G}\supseteq [0\mathop{\mathrm{:}}\nolimits \max\mathbb{G}]$ and $\mathbb{G}+\mathbb{G}\supseteq[0\mathop{\mathrm{:}}\nolimits \ell-1]$, then $\mathbb{S}+\mathbb{S}=[0\mathop

Figures (9)

  • Figure 1: Active sensing signal model. A linear Tx array with $N_{\mathop{\mathrm{t}}\nolimits}$ sensors illuminates $K$ far field point scatterers using $N_s\leq N_{\mathop{\mathrm{t}}\nolimits}$ linearly independent waveforms. A linear Rx array with $N_{\mathop{\mathrm{r}}\nolimits}$ sensors, co-located with the Tx array, measures a superposition of the backscattered planar wavefronts.
  • Figure 2: Categorization of (aligned) active array configurations based on the overlap between Tx and Rx sensors. The fully overlapping geometry in \ref{['fig:array_fo']} achieves a contiguous sum co-array of exactly twice the physical Tx/Rx array aperture. The non-overlapping configuration in \ref{['fig:array_no']} attains the same co-array using fewer physical sensors at the expense of a larger physical array aperture.
  • Figure 3: Active MRAs with $N=11$ physical (transceiving) sensors hoctor1996arrayredundancy. Most of the configurations are symmetric. In fact, there exists a symmetric MRA configuration for (at least) each $N\leq48$kohonen2014meetkohonen2015early.
  • Figure 4: The \ref{['fig:G_NA']} NA generator $\mathbb{G}=\mathbb{D}_\text{\normalfont NA}$ and shift parameter $\ell$ define the \ref{['fig:S-NA']} symmetric NA, which reduces to the configuration in \ref{['fig:CNA']}, when $\ell=N_1$. The minimum-redundancy symmetric NA solving \ref{['p:SA']} has the structure in \ref{['fig:CNA']}. Adapted from rajamaki2021sparsesymmetric © 2021 IEEE
  • Figure 5: NA and symmetric NA configurations using minimum-redundancy parameters, with corresponding sum co-array $\mathbb{D}_\Sigma$ and difference co-array $\mathbb{D}_\Delta$ (shifted to non-negative half plane). The \ref{['fig:sna_N']} symmetric NA attains a larger contiguous sum co-array than the NA, both for the same \ref{['fig:na_L']} physical array aperture and \ref{['fig:na_N']} number of physical sensors. The difference co-array of the symmetric NA is also contiguous.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Definition 9.1: Sum co-array
  • Definition 9.2: Redundancy pattern
  • Example 9.1: Redundancy pattern of ULA
  • Theorem 9.1: Sufficient condition for contiguous co-array
  • proof