The effect of redundancy on measurement

Abstract

The parameters of a mechanical or electrical system are usually determined by making as many measurements as there are degrees of freedom. This article demonstrates that increasing the number of measurements made on a system beyond this minimum can reduce the effect of measurement errors. The possible systems to which this technique may be applied are too diverse to be captured by a simple and compact analytical formulation. Thus, we discuss techniques for three different measurement models: scalar difference measurements, complex product measurements, and scalar additive measurements. All have both a compact mathematical formalization and wide practical applicability. A simple example of a measurement problem discussed is the generation of a binary ruler using a compass and a straight edge. Error is introduced whenever the mid-point of a line is found. The straight-forward approach to the ruler's construction is iterative bisection of the distance between two adjacent points till the required resolution is obtained. The maximum error introduced using this procedure scales logarithmically with the number of points. The accuracy may be improved by bisecting the distance from one of the end points to an already obtained point iteratively. Using this technique the error of any constructed point can be bounded by a constant, i.e., it does not scale with the number of points. We show, for three broad classes of measurements, that, as the measurement redundancy increases, the residual error falls to a small constant value. The overall effect is analogous to the improvements in communications reliability demonstrated by the coding theorem. Only scalar difference measurements are considered.