The traditional filtering method can only be realized when the useful signal and noise have different frequency bands. In the 1940s, N. Wiener and A.H. Kolmogorov introduced the statistical properties of signal and noise into the filtering theory, and under the assumption that the signal and noise are stationary processes, the true value of the signal is estimated by the optimization method to achieve the purpose of filtering, which is conceptually linked with the traditional filtering method and is called Wiener filtering. This method requires that both signal and noise must be based on stationary process. In the early 1960s, R.E.Kalman and R.S. Bucy published an important paper "New Achievements of Linear Filtering and Prediction Theory", and put forward a new linear filtering and prediction theory, called Kalman filtering. It is characterized by processing the noisy input and observation signals on the basis of linear state space representation to obtain the system state or real signal.

This theory is expressed in time domain, and the basic concept is: based on the state space representation of linear system, the optimal estimation of system state is obtained from the output and input observation data. The system state mentioned here is a set of minimum parameters, which summarizes the influence of all past inputs and disturbances on the system. Knowing the system state can determine the overall behavior of the system and the future input and interference.

Kalman filtering does not require that both signal and noise are stationary processes. For the system disturbance and observation error (i.e. noise) at each moment, as long as some appropriate assumptions are made on their statistical properties, the estimated value of the real signal with the smallest error can be obtained by processing the observation signal containing noise. Therefore, since the advent of Kalman filter theory, it has been applied in many departments such as communication system, power system, aerospace, environmental pollution control, industrial control, radar signal processing and so on, and has achieved many successful results. For example, in image processing, Kalman filter is applied to restore blurred images caused by some noises. After assuming some statistical properties of noise, we can use Kalman algorithm to recursively get the real image with the smallest mean square error from the blurred image, so as to restore the blurred image.