A Lower Bound and Other Properties of the Mean-Square Error of Linear Measurement Systems

Abstract

The central purpose of this paper is to establish a lower bound on the mean-square error of linear, noisy, dissipative measurement systems designed to measure a constant quantity. A lower bound on mean-square error is determined to be (2kTA)/t, where k is Boltzmann's constant, T is the absolute system temperature, t is the time consumed by the measurement, and A is a parameter of the measurement system closely related to its driving point immittance. This lower bound shows, among other things, that a nonzero measurement interval is required to obtain a useful estimate of the quantity being measured. The lower bound is arrived at by considering a logical structure for linear noisy measurement systems and mathematical expressions derived for the mean-square error of such systems. The key step in establishing the lower bound is the consideration of the unavoidable random perturbation at the system input due to the generalized Nyquist noise discussed earlier by Callen and Welton. Use of the general lower bound is illustrated by establishing more specific lower bounds for two classes of systems, namely, current measuring galvanometers and thermal detectors. In these cases the lower bounds are consistent with minimum values for the mean-square error of such systems as previously reported in the literature by McCombie.