Ae. Jones et Rjp. Bain, A GENERALIZATION OF PLANAR MAGNETIC GRADIOMETER DESIGN VIA ORTHOGONALPOLYNOMIALS, Journal of computational physics, 118(2), 1995, pp. 191-199
We describe a problem in magnetic field detection involving a form of
spatial filtering to detect weak signal sources in the presence of noi
se. Conventionally N-th order magnetic field gradiometers of fixed geo
metry are used in this situation. The pre-defined geometry completely
determines the spatial sensitivity of such gradiometers. We demonstrat
e a method of making such devices much more flexible in that the near-
source response can be modified while maintaining gradiometric order.
The problem is described by the solution of N equations in sums and di
fferences of powers, up to order N-r of m variables, with m greater th
an or equal to N. The values of (m - N) variables are chosen on physic
al considerations. We show that when values of the m variables are a s
olution set, they may be expressed as the roots of two polynomial equa
tions, whose order is no greater than (m + 1)/2 when m is odd, or m/2
when m is even. These polynomial equations can be expressed as a linea
r combination of Chebyshev polynomials of the first and second kinds i
n the case of m odd, and a related pair,fully described, in the case o
f m even. Existence of, and bounds on, solution sets are discussed and
examples given. (c) 1995 Academic Press.