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# Linear Estimation

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

### 1.2 Problem formulation

#### Process model

Process model of a system could be defined in discrete state space with the help of a difference equation described with $$X_{k+1}=\phi_{k}X_{k}+w_{k},$$ where, state vector $X_{k}\in \mathbb{R}^{n}$ state transition matrix $\phi_{k}\in\mathbb{R}^{n\times n}$, and process noise $w_{k}\in \mathbb{R}^{n}$. Process noise, $w_{k}$, is assumed as white Gaussian with zero mean, $Q_{k}$ covariance i.e. $$w_{k} \sim \aleph(0,Q_{k}),$$ where, $$Q_{k}=E[w_{k}w_{k}^{T}],$$ where, $E[.]$ is an expectation operator, $T$ denotes the transpose of a matrix.

#### Measurement model

The measurement vector of the state at step $k$ is $Y_{k}$ and it is defined as, $$Y_{k}=H_{k}X_{k}+v_{k},$$ where, measurement vector $Y_{k}\in\mathbb{R}^{m}$, Measurement matrix $H_{k}\in\mathbb{R}^{m\times n}$, and measurement noise $v_{k}\in\mathbb{R}^{m}$. Measurement noise, $v_{k}$, is white Gaussian with zero mean, $R_{k}$ covariance i.e. $$v_{k} \sim \aleph(0,R_{k}),$$ where, $$R_{k}=E[v_{k}v_{k}^{T}].$$ We also assume the process noise $w_{k}$ and measurement noise $v_{k}$ are independent and uncorrelated to each other. $$\label{3c} E[w_{i}w_{j}^{T}]=Q_{k}\delta(i,j),$$ $$E[v_{i}v_{j}^{T}]=R_{k}\delta(i,j)$$ $$E[w_{i}v_{j}^{T}]=0,$$ where, $\delta(i,j)$ is the unit impulse signal, which is defined as, \begin{equation*} \delta(i, j)=\begin{cases} 1 & \text{for $i=j$}\\ 0 & \text{for $i\neq j$} \end{cases} \end{equation*}