Kalman Filter for Signal Tracking

The Kalman gain matrix K determines how much weight is given to measurement correction versus model prediction. Large Kalman gain trusts measurements more (and prediction less), while small Kalman gain trusts the model more. The Kalman gain is automatically computed as K = P_pred·H'/(H·P_pred·H' + R), where P_pred is predicted error covariance, H is measurement matrix, and R is measurement noise covariance. This optimal weighting minimizes mean-square error.

In theory, Q (process noise covariance) and R (measurement noise covariance) should match true noise statistics. In practice, estimate Q and R from data: R from pure measurement noise when signal is known constant, and Q by observing how model predictions diverge from reality. Often these require empirical tuning—monitor innovation sequence and residuals to verify filter performance and adjust Q and R if needed.

Both minimize mean-square error for Gaussian noise, but apply to different scenarios. Wiener filter assumes stationary signals and provides a batch solution with fixed coefficients. Kalman filter handles time-varying systems with recursive updates as new measurements arrive, making it suitable for real-time tracking. Kalman filter also naturally incorporates process models, while Wiener filter is limited to fixed linear relationships.

Use EKF when your system is nonlinear (either process model or measurement model). EKF linearizes the nonlinear functions around current state estimate and applies standard Kalman filtering to the linearized system. This approximation works well when nonlinearity is mild. For strongly nonlinear systems, Unscented Kalman Filter (UKF) or particle filters provide better performance at higher computational cost.