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Subspace Coding for Spatial Sensing

Hessam Mahdavifar, Robin Rajamäki, Piya Pal

TL;DR

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.

Abstract

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.

Subspace Coding for Spatial Sensing

TL;DR

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.

Abstract

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.
Paper Structure (9 sections, 29 equations, 2 figures)

This paper contains 9 sections, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Subspace Coding Meets Spatial Sensing
  3. Receiver System Model for Spatial Sensing
  4. A Subspace Coding Perspective of Spatial Sensing
  5. Sensing Subspace Codes: New insights and (near) Optimal Designs for Single Target
  6. Near Optimal Code Constructions
  7. Sub-optimality of uniform sensing
  8. Numerical Results
  9. Discussion and Future Directions

Figures (2)

  • Figure 1: Minimum distance of sensing subspace codes. As the code length $M$ (number of antennas) increases, $d^{(s)}_{\min}$ approaches its maximum value $1$ in case of the Bose-Chowla ruler, and minimum value $0$ in case of ULA ($N\!=\!M^2\!-1$).
  • Figure 2: Probability of error of minimum distance decoder. The Bose-Chowla ruler achieves a low probability of error $P_e$ both when SNR (left) or $M$ (right) increases. The ULA is significantly less robust to noise due to the smaller minimum distance of the associated sensing subspace code---indeed, when the SNR is fixed, $P_e$ approaches $1$ even as $M$ grows ($N=M^2-1$).

Theorems & Definitions (3)

  • proof
  • proof
  • proof