A GENERALIZATION OF PLANAR MAGNETIC GRADIOMETER DESIGN VIA ORTHOGONALPOLYNOMIALS

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.