# Detection Estimation and Grid matching of Multiple Targets with Single Snapshot Measurements

## TL;DR

This paper presents a method using sparse signal recovery and group lasso to detect and estimate the directions of multiple targets from a single snapshot, addressing grid mismatch issues in DoA estimation.

## Contribution

It introduces asymptotic and finite sample test statistics for target detection and proposes a joint estimation approach to improve accuracy in single snapshot scenarios.

## Key findings

- Effective detection at moderate to high SNRs
- Mitigates grid mismatch problem in DoA estimation
- Accurate estimation of target directions and number

## Abstract

In this work, we explore the problems of detecting the number of narrow-band, far-field targets and estimating their corresponding directions from single snapshot measurements. The principles of sparse signal recovery (SSR) are used for the single snapshot detection and estimation of multiple targets. In the SSR framework, the DoA estimation problem is grid based and can be posed as the lasso optimization problem. However, the SSR framework for DoA estimation gives rise to the grid mismatch problem, when the unknown targets (sources) are not matched with the estimation grid chosen for the construction of the array steering matrix at the receiver. The block sparse recovery framework is known to mitigate the grid mismatch problem by jointly estimating the targets and their corresponding offsets from the estimation grid using the group lasso estimator. The corresponding detection problem reduces to estimating the optimal regularization parameter ($\tau$) of the lasso (in case of perfect grid-matching) or group-lasso estimation problem for achieving the required probability of correct detection ($P_c$). We propose asymptotic and finite sample test statistics for detecting the number of sources with the required $P_c$ at moderate to high signal to noise ratios. Once the number of sources are detected, or equivalently the optimal $\hat{\tau}$ is estimated, the corresponding estimation and grid matching of the DoAs can be performed by solving the lasso or group-lasso problem at $\hat{\tau}$