Discrete-Time Minimum-Variance Prediction and Filtering
Garry Einicke · InTech eBooks · 2012
Discrete-Time Minimum-Variance Prediction and Filtering 75 4 Discrete-Time Minimum-Variance Prediction and Filtering , where the predictor gain, K k , is a function of the noise statistics and the model parameters.The above formula was reported by Rudolf E. Kalman in the 1960s [1], [2].He has since received many awards and prizes, including the National Medal of Science, which was presented to him by President Barack Obama in 2009.The Kalman filter calculations are simple and well-established.A possibly troublesome obstacle is expressing problems at hand within a state-space framework.This chapter derives the main discrete-time results to provide familiarity with state-space techniques and filter application.The continuous-time and discrete-time minimum-square-error Wiener filters were derived using a completing-the-square approach in Chapters 1 and 2, respectively.Similarly for time-varying continuous-time signal models, the derivation of the minimum-variance Kalman filter, presented in Chapter 3, relied on a least-mean-square (or conditional-mean) formula.This formula is used again in the solution of the discrete-time prediction and filtering problems.Predictions can be used when the measurements are irregularly spaced or missing at the cost of increased mean-square-error.This chapter develops the prediction and filtering results for the case where the problem is nonstationary or time-varying.It is routinely assumed that the process and measurement noises are zero mean and uncorrelated.Nonzero mean cases can be accommodated by including deterministic inputs within the state prediction and filter output updates.Correlated noises can be handled by adding a term within the predictor gain and the underlying Riccati equation.The same approach is employed when the signal model "Man will occasionally stumble over the truth, but most of the time he will pick himself up and continue on."Winston Leonard Spencer-Churchill www.intechopen.