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Estimating the magnitude of the sum of two magnetic fields with uncertain spatial orientations, polarizations, and/or relative phase
Author(s) -
Kaune W.T.
Publication year - 2002
Publication title -
bioelectromagnetics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.435
H-Index - 81
eISSN - 1521-186X
pISSN - 0197-8462
DOI - 10.1002/bem.98
Subject(s) - estimator , random error , magnitude (astronomy) , physics , root mean square , phase (matter) , phase difference , statistics , square (algebra) , mathematics , square root , combinatorics , quantum mechanics , geometry , astrophysics
Abstract A problem frequently encountered when modeling the power frequency magnetic fields, B and A , produced by two sources is the necessity of estimating the root mean square (rms) magnitude of their sum, i.e., T = | B + A |, when the rms magnitudes, B and A , of the fields are specified by the model, but not necessarily their spatial directions, polarizations, and/or relative phase. The estimator $Q=\sqrt {B^2+A^2}$ was proposed many years ago for this purpose. The accuracy of this estimator is characterized in this paper. If it is known that B and A are approximately linearly polarized and in phase, the maximum bias (i.e., systematic) and random errors for Q used to estimate T are 6.1 and 35%, respectively, when B = A . These errors decrease as the difference between B and A increases. The bias and random errors are, respectively, 3.2 and 26% when B = 2 A or A /2 and 0.2 and 5.8% when B = 10 A or A /10. If the directions, relative phase, and polarizations of the two fields are unknown, Q has maximum bias and random errors of ≈ 2.6 and ≈ 23%, respectively, when B = A . These errors decrease to ≈ 1.5 and ≈ 18% when B =2 A or A /2 and ≈ 0.08 and ≈ 4.0% when B = 10 A or A /10. If B and A are known to be linearly polarized and collinear, but with unknown phase between them, the maximum bias and random errors are 11 and 48%, respectively, when B = A . The errors are 5.1 and 32% when B = 2 A or A /2 and 0.2 and 7.0% when B = 10 A or A /10. Estimators for T with zero bias can be derived, but they are more complicated and increase overall accuracy very little. Bioelectromagnetics 23:59–67, 2002. © 2002 Wiley‐Liss, Inc.