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Bayesian Estimation and Tracking: A Practical Guide by Anton J. Haug

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Chapter 11

The Sigma Point Class: The Gauss–Hermite Kalman Filter

Practical methods for one-dimensional Gauss–Hermite integration has its origins in the work of Wilf [1], as noted in the Numerical Recipes book [2], and the short but excellent paper by Ball [3]. Unlike the methods addressed in the preceding chapters, Gauss–Hermite integration is a quadrature method that uses orthogonal polynomials instead of simple polynomials.

Consider the general multidimensional integral (5.60), rewritten here in more explicit form

(11.1) equation

Changing variables to remove the factor of 1/2 from the exponent, let

(11.2) equation

or

(11.3) equation

Thus, (11.1) becomes

(11.4) equation

with

(11.5) equation

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