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# Nonlinear Estimation

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

### Generation of deterministic sampling points and weights

#### Gauss Hermite filter (GHF)

$\bullet$ Consider a symmetric tridiagonal matrix with zero diagonal element $J_{i,i+1}=\sqrt{i/2}; 1\leq i\leq (N-1)$.

$\bullet$ The quadrature points are at $q_i=\sqrt{2}\psi_i$, where $\psi_i$ is the $i_{th}$ eigenvalue of the matrix J.

$\bullet$ The $i_{th}$ weight $w_{i}$ is chosen as $w_i=\kappa_{i1}^2$, where $\kappa_{i1}$ is the first element of the $i_{th}$ normalized eigenvector of J.

$\bullet$ The extension of quadrature rules to multi dimensional problem could be done using product rule.

$\bullet$ The $n$ dimensional integral, \begin{equation*} I_N =\int_{R_n} f(s)\dfrac{1}{(2\pi)^{n/2}}e^{-\dfrac{1}{2}|s|^{2}}ds \end{equation*} can be approximately evaluated as \begin{equation*} I_N=\sum_{i_1=1}^{N}...\sum_{i_n=1}^{N}f(q_{i_1},q_{i_2},q_{i_3},..,q_{i_n})w_{i_1}w_{i_2}...w_{i_n} \end{equation*} As an example, for a second order system and three point GHF, nine quadrature points and weights will be ${(q_i,q_j)}$ and ${w_iw_j}$ respectively for $i=1,2,3$ and $j=1,2,3$.