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1. The difference and connection between Wiener filter, Kalman filter and adaptive filter
Wiener filtering:
(1) Suppose the signal s(k) and the observation process x(k)) are generalized stationary, and its power spectrum or autocorrelation function is known, then based on the observation process x(k), according to the linear minimum average The square error estimation criterion, the optimal estimation of the signal s(k) is called Wiener filtering. (Definition)
(2) Wiener filter parameters are fixed, suitable for optimal filtering under stationary random conditions and simple to implement;
(3) Wiener filter is to estimate the current value of the signal based on past and current observation data. Its solution is given in the form of the system's transfer function or unit impulse response. (Estimated process perspective)
Kalman filter:
(1) Set the dynamic model measurement equation of the known signal, then based on the process x(k) and initial conditions, according to the linear minimum mean square error recursive estimation criterion, the optimal estimation of the state s(k) It is called Kalman filter. (Definition)
(2) Kalman filter parameters are time-varying, which is suitable for optimal filtering under non-stationary random conditions and has superior performance;
(3) Kalman filter estimates the current value of the signal based on the previous estimated value and the latest observation data; It is estimated using state equations and recursive methods, and its solution is given in the form of estimated values of state variables. (Estimated process perspective)
Adaptive filter: The
adaptive filter can also provide excellent filtering performance when the signal and noise statistical characteristics are unknown a priori.
The characteristics of the filter are as follows: it
can automatically adjust its own parameters to meet the requirements of the system characteristics; it
requires little or no prior knowledge about the signal and noise; the
implementation is almost as simple as Wiener filtering, and the performance is close to Kalman Filter
2. Two kinds of optimal linear filtering (Wiener filtering/Kalman filtering) are summarized, including applicable conditions, optimal criteria, limitations, and estimation process.
Applicable conditions:
Wiener filtering: Let the signal s(k) and the observation process x(k)) be generalized stationary, and its power spectrum or autocorrelation function is known.
Kalman filter: Set up the dynamic model measurement equation of the known signal
Limitations:
Only when the statistical characteristics of the signal and noise are known a priori, the two filters
can obtain the optimal filtering. In practical applications, it is often impossible to obtain
the prior knowledge of these statistical characteristics, or the statistical characteristics change with time, and then it is
impossible to use these two filters to achieve optimal filtering.
Optimal criterion:
Wiener filtering: linear minimum mean square error estimation criterion
Kalman: linear minimum mean square error recursive estimation criterion
Estimation process:
Wiener filtering is to estimate the current value of the signal based on the past and current observation data.
Kalman filter estimates the current signal value based on the previous estimate and the current observation.