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The Quadratic Data-Fitting
Problem: Estimation of
Linear Model Parameters
from Observations in
Multiplicative Noise

Daniel R. Fuhrmann
Washington University
Department of Electrical Engineering
St. Louis, MO 63130
email: danf@saturn.wustl.edu

Abstract This presentation describes a parameter estimation problem in which a finite number of linear model parameters or coefficients are to be determined from observations with finite resolution in multiplicative noise. Our motivation for studying this problem comes from an airborne radar calibration application.

We presume the existence of a linear constant-coefficient model for some continuous-parameter function. We assume that the data consists of values found by multiplying the function by a stochastic process with known statistics, and integrating.

Such a problem arises in determining the azimuthal response of an airborne antenna system using ground clutter, an idea proposed by Robey et al. 1 Suppose one transmits a radar pulse of known beam shape toward a ground patch with known statistics. The data point is the complex (demodulated) receiver output for one range bin, and the statistics of the multiplicative stochastic process are determined by the transmit beam shape and the clutter distribution. The complex antenna pattern could be modeled using a spatial Fourier series. 2

The computational problem becomes one of fitting the magnitude-squared data points to a quadratic function of the unknown coefficients. We refer to this as the quadratic data-fitting problem. Under a Gaussian assumption, maximum-likelihood methodology yields the cost function for which we have determined analytic expressions for the gradient and Hessian; from these a Newton algorithm could be crafted.

1F. Robey, D. Fuhrmann, and S. Krich, "Array calibration utilizing clutter scattering,"
Proc. IEEE 7th SP Workshop on Statistical Signal and Array Processing (Quebec, Canada),
pp. 317­320, June 1994.

2M. Koerber and D. Fuhrmann, "Array calibration by Fourier series parameterization: Scaled principal components method," Proc. ICASSP '93 (Minneapolis, Minnesota), vol. 4, pp. 340­343, April 1993.

This work was supported in part by the Department of the Air Force under contract F19628-95-C-0002.



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