Michael Francis, Ronald Wittmann, October 2017
Spherical wave coefficients are chosen to minimize a cost function that is a norm of the residual of the fit. For example, in standard orthogonality-based processing algorithms [1], the cost function is an integral (over 4 p steradians) of the squared amplitude of the difference between actual measurements and predicted values. Some recent work [2,3] at NIST has led to the use of discrete norms where the integral is replaced by a weighted sum. We explore issues regarding the choice of these weights, the relative performance of different weighting schemes, and the relation between the continuous and discrete cases.
These norms are mathematically equivalent if there is a solution with zero residual. In practice, we have observed noticeable variation due to the presence of measurement errors, including multiple reflections, room reflections… Also, different weighting schemes are associated with widely varying condition numbers. When the condition number is large, small measurement errors can lead to large errors in the result. Additionally, we show that the integral cost function mentioned above can be reduced to a discrete quadrature.
M.H. Francis and R.C. Wittmann, Chp. 19, “Near-Field Scanning Measurements: Theory and Practice” in Modern Antenna Handbook, ed. C.A. Balanis, John Wiley & Sons, 2008.
R.C. Wittmann, B.K. Alpert, M.H. Francis, “Near-field, spherical-scanning antenna measurements with nonideal probe locations,”IEEE Antennas and Propagat., vol. 52, pp. 2184 – 2186, August 2004.
R.C. Wittmann, B.K. Alpert, M.H. Francis, “Near-field antenna measurements with nonideal measurement locations,”IEEE Antennas and Propagat., vol. 46, pp. 716 – 722, May 1998.