##Purpose and Usage
 Eliminate noise in measurements
 Generate nonobservable states(e.g., Velocity from position signals)
 For prediction of future state
 Optimal filtering
##Framework and Model
###Given:

A discrete stochastic linear controlled dynamical system:
Each current signal value x^k is a combination of previous signal value $x_{k1}$ times a constant, a control signal $u_{k}$ and a process noise and a process noise signal $w_{k1}$(which usually considered as zero).

A measurement function, where $v_{k}$ is the measurement noise.

Assume the process noise and the measurement noise are both considered to be normal distribution that
In reality, covariance matrix Q and R may change in every iteration. We assume they are constant here however.
###Goal:
Find the best (recursive) estimate of the state x of the system.
###Computational Origins
Define $e_{k}^{}$ to be a priori state estimate at step k given knowledge of the process prior to step $k$, and define $e_{k}$ to be a posteriori state estimate at step $k$ given measurement $z_{k}$. Then a priori and a posteriori estimate errors can be defined as:
The a priori estimate error covariance is then
The a posteriori estimate error covariance is then
Then How can we optimally (linearly) combine the estimate and measurement to obtain the best reconstruction of ￼the true x? The answer given in “The Probabilistic Origins of the Filter” found.
Where residual is $z_k  H \hat{x}_{k}^{}$. It also can be called as measurement innovation.
The Kalman filter gains are derived by minimizing the posterior error covariance, resulting in
If the a priori estimate of the process noise is zero, Then
If the measurement noise is zero