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# Nonlinear Estimation

## 2 Literature review on quadrature based nonlinear estimation (Cont'd...)

### 2.4 Gauss-Hermite filter (GHF)

Gauss-Hermite Filters make use of the Gauss-Hermite Quadrature rule for integrating the intractable integrals encountered in non linear Bayesian Filtering. This rule for integration was already available in literature [Hilderbrand 2008] [Krylov 2005] , but it was used in estimation problems by the work of Ito and Xiong [Ito 2000] . In this filtering technique, the unknown probability density function is approximated as Gaussian. This is done by defining a set of weights and their corresponding Gauss-Hermite quadrature points.

Consider an integral of any function $f(x)$ $$I=\int_{-\infty}^{\infty}f(x)e^{-x^{2}}dx$$ It can be evaluated numerically with N quadrature points $$I\approx\sum_{j=1}^{N}f(q_{j})w_{j}$$ where $q_{j}$ and $w_{j}$ are the $j_{th}$ quadrature point and it's corresponding weight. The quadrature points and its weights can be evaluated using a symmetric tridiagonal matrix $J$ with $J_{j,j+1}=\sqrt{j/2};1\leq j \leq(N-1)$.The quadrature points are obtained as $q_{j}=\sqrt{2}\Psi_{j}$, where $\Psi_{j}$ is the $j_{th}$ eigenvalue of matrix $J$. The $j_{th}$ weight $w_{j}$ can be defined as $w_{j}=k_{j1}^{2}$, where $k_{j1}$ is the first element of the $j_{th}$ normalized eigenvector of $J$ [Arasaratnam 2009] [Chalasani 2012] . This method was actually proposed by Golub et. al. [Golun 1969] and was later used by Arasaratnam et.al. [Arasaratnam 2009] for filtering applications.