Measurement time dependency of asymptotic Cramér-Rao bound for an unknown constant in stationary Gaussian noise

In metrology, the knowledge of uncertainty principles helps to improve the performance of measurement systems and to understand measurement limits. Especially an uncertainty principle that explains the relation between the measurement uncertainty and the measurement time of a measurand would be beneficial, because repeating and averaging measurements is a common technique for reducing random errors. Although many research covered the analysis of the Cramér-Rao bound (CRB) as minimum achievable measurement uncertainty, its relation to the measurement time of the measurand was only studied for specific measurands yet. For this reason, the dependency of the CRB on the measurement time is studied for an unknown constant in stationary Gaussian noise, while the unknown constant can be any physical quantity as measurand. Here, the correlations of the samples of the Gaussian process are assumed to be independent of the measurand. As a result, the CRB can be asymptotically linked with the measurement time by the noise power spectral density at the frequency zero and the sensitivity. A comparison between the proposed asymptotic and the exact CRB solution shows a good agreement for common types of colored Gaussian noise. Hence, the analytic expression of the asymptotic solution is proven to be applicable for estimating the measurement uncertainty limits of an unknown constant in stationary Gaussian noise for a desired measurement time and vice versa.