Multilateration and Signal Matching with Unknown Emission Times

TL;DR

This paper introduces algebraic methods for solving multilateration and signal matching problems in 3D space, enabling source localization and wall reconstruction from sensor data, even with multiple sources and echoes.

Contribution

It presents a novel algebraic approach for matching signals and solving multilateration, including echo scenarios, applicable in all dimensions.

Findings

Algorithm correctly matches signals for almost all sensor positions.

Unique source and emission time determination from reception times.

Reconstruction of reflecting wall positions from echo signals.

Abstract

Assume that a source emits a signal in $3$-dimensional space at an unknown time, which is received by at least~$5$ sensors. In almost all cases the emission time and source position can be worked out uniquely from the knowledge of the times when the sensors receive the signal. The task to do so is the multilateration problem. But when there are several emission events originating from several sources, the received signals must first be matched in order to find the emission times and source positions. In this paper, we propose to use algebraic relations between reception times to achieve this matching. A special case occurs when the signals are actually echoes from a single emission event. In this case, solving the signal matching problem allows one to reconstruct the positions of the reflecting walls. We show that, no matter where the walls are situated, our matching algorithm works correctly for almost all positions of the sensors. In the first section of this paper we consider the multilateration problem, which is equivalent to the GPS-problem, and give a simple algebraic solution that applies in all dimensions.