Maximum signal-to-noise ratio enhancement by averaging under a limited measurement time

Averaging through repetitive measurement is a ubiquitous strategy for improving signal-to-noise ratio (SNR) and is commonly assumed to yield a $\sqrt{N}$ enhancement with the number of repetitions $N$. This assumption, however, implicitly requires the signal amplitude to be independent of measurement duration. This condition does not generally hold in dynamical sensing systems with finite response time and a fixed measurement time. We derive a closed-form expression for the SNR enhancement factor by analytically accounting for the competition between statistical noise reduction and dynamical signal attenuation, and demonstrate the existence of a strict upper bound on the SNR enhancement. The enhancement factor is a non-monotonic function of $N$ with a well-defined maximum at an optimal repetition number, beyond which further averaging degrades the SNR. Moreover, below a threshold set by the ratio of measurement time to response time, averaging yields no enhancement at all. These two regimes delimit where the conventional $\sqrt{N}$ law breaks down. Experimental validation using nanomechanical gas sensing, with two receptor-analyte systems deliberately chosen to bracket this enhancement transition, confirms the theoretical predictions. Our results show that measurement time is a finite resource to be optimally partitioned between signal accumulation and averaging, and provide a quantitative guideline for selecting the repetition number in time-constrained sensing such as real-time and repetitive gas or odor detection.