# Atomic Norm Minimization for Modal Analysis from Random and Compressed Samples

## TL;DR

This paper formulates modal analysis as an atomic norm minimization problem, enabling efficient recovery of modal parameters from limited and compressed sensor data, with theoretical bounds on sample complexity and extensions to noisy and multiple measurement scenarios.

## Contribution

It introduces a novel atomic norm minimization approach for modal analysis from compressed samples and establishes sample complexity bounds, including for random temporal compression and MMV settings.

## Key findings

- Atomic norm minimization can recover modal parameters efficiently.
- Sample complexity decreases as the number of sensors increases.
- Extensions to noisy data and multiple measurement vectors are provided.

## Abstract

Modal analysis is the process of estimating a system's modal parameters such as its natural frequencies and mode shapes. One application of modal analysis is in structural health monitoring (SHM), where a network of sensors may be used to collect vibration data from a physical structure such as a building or bridge. There is a growing interest in developing automated techniques for SHM based on data collected in a wireless sensor network. In order to conserve power and extend battery life, however, it is desirable to minimize the amount of data that must be collected and transmitted in such a sensor network. In this paper, we highlight the fact that modal analysis can be formulated as an atomic norm minimization (ANM) problem, which can be solved efficiently and in some cases recover perfectly a structure's mode shapes and frequencies. We survey a broad class of sampling and compression strategies that one might consider in a physical sensor network, and we provide bounds on the sample complexity of these compressive schemes in order to recover a structure's mode shapes and frequencies via ANM. A main contribution of our paper is to establish a bound on the sample complexity of modal analysis with random temporal compression, and in this scenario we prove that the samples per sensor can actually decrease as the number of sensors increases. We also extend an atomic norm denoising problem to the multiple measurement vector (MMV) setting in the case of uniform sampling.