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# Linear Estimation

## 1. Kalman Filter Tutorial (Cont'd...)

### 1.5 Simulation (cont'd...)

#### Noise covariance

Now, we want to measure only position and velocity of a particle, where the applied acceleration excitation is unknown. We assume that the acceleration excitation $g(t)$ as a white Gaussian random process with zero mean. If we don't know any information about the maneuverability of the particle, then we choose any value for variance, where, we assume the continuous random process $g(t)$ has standard deviation $\sigma$, and its covariance can be defined as, $$\label{5f} E[g(\lambda)g(\mu)]=\sigma^{2}\delta(\lambda-\mu).$$ Now, we define the process noise covariance, $Q_{k}$, i.e. $$\label{5g} Q_{k}=E[w_{k,\tau}w_{k,\tau}^{T}]=E[\int_{\lambda=k-\tau}^{\tau}\int_{\mu=k-\tau}^{\tau}\phi_{k-\lambda}G_{\lambda}G_{\mu}^{T}\phi_{k-\mu}^{T}d\lambda d\mu]$$ or, $$\label{5h} \begin{split} &Q_{k}=\int_{\lambda=k-\tau}^{\tau}\int_{\mu=k-\tau}^{\tau}\phi_{k-\lambda}E[G_{\lambda}G_{\lambda}^{T}]\phi_{k-\mu}^{T}d\lambda d\mu\\ &=\sigma^{2}\int_{\lambda=k-\tau}^{\tau}\phi_{k-\lambda}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} \phi_{k-\lambda}^{T}d\lambda \end{split}$$ Now, substituting $k-\lambda=\gamma$ and the value of $\phi_{k,k-\tau}$, we obtain $$\label{5i} Q_{k}=\sigma^{2}\int_{\gamma=0}^{\tau}\phi_{\gamma}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} \phi_{\gamma}^{T}d\gamma= \sigma^{2}\int_{\gamma=0}^{\tau}\begin{bmatrix} \dfrac{1}{2}\gamma^{2} \\ \gamma \\ 1 \end{bmatrix} \begin{bmatrix} \dfrac{1}{2}\gamma^{2} & \gamma & 1 \end{bmatrix} d\gamma.$$ Now, expanding and integrating the above expression, we obtain $$\label{5j} Q_{k}=\sigma^{2}\tau\begin{bmatrix} \dfrac{1}{20}\tau^{4} & \dfrac{1}{8}\tau^{3} & \dfrac{1}{6}\tau^{2} \\ \dfrac{1}{8}\tau^{3} & \dfrac{1}{3}\tau^{2} & \dfrac{1}{2}\tau \\ \dfrac{1}{6}\tau^{2} & \dfrac{1}{2}\tau & 1 \end{bmatrix}$$ This type of noise covariances is used only when acceleration excitation of the particle is unknown.