# Optimal Number of Measurements in a Linear System with Quadratically Decreasing SNR

## TL;DR

This paper analyzes the optimal number of measurements in a linear sensing system with energy constraints, showing that the SNR decreases quadratically with more measurements, leading to an optimal measurement count.

## Contribution

It introduces a model accounting for quadratically decreasing SNR with measurements, providing a novel analysis of optimal measurement number under energy constraints.

## Key findings

- There exists an optimal number of measurements for given energy constraints.
- The SNR per measurement decreases quadratically as the number of measurements increases.
- Numerical results validate the theoretical analysis across various distributions.

## Abstract

We consider the design of a linear sensing system with a fixed energy budget assuming that the sampling noise is the dominant noise source. The energy constraint implies that the signal energy per measurement decreases linearly with the number of measurements. When the maximum sampling rate of the sampling circuit is chosen to match the designed sampling rate, the assumption on the noise implies that its variance increases approximately linearly with the sampling rate (number of measurements). Therefore, the overall SNR per measurement decreases quadratically in the number of measurements. Our analysis shows that, in this setting there is an optimal number of measurements. This is in contrast to the standard case, where noise variance remains unchanged with sampling rate, in which case more measurements imply better performance. Our results are based on a state evolution technique of the well-known approximate message passing algorithm. We consider both the sparse (e.g. Bernoulli-Gaussian and least-favorable distributions) and the non-sparse (e.g. Gaussian distribution) settings in both the real and complex domains. Numerical results corroborate our analysis.