Single channel (univariate) signal - Practical Biomedical Signal Analysis Using MATLAB - page 33

x 0 , P 0

Update :
K k

P k M T MP k M T R − 1

Predict :
x k =

= +

A x k − 1 + B u k

x k + M x k ) x k = K k ( z k −

P k = AP k − 1 A T + Q

P k P k =( I − K k M )

FIGURE 2.3: The computation of successive system states with Kalman filter.

Kalman filter may be found in [Welch et al., 1960].
MATLAB toolbox by Kevin Murphy avaliable at http://people.cs.ubc.ca/
˜ murphyk/Software/Kalman/kalman.html provides Kalman filter formalism for
filtering, smoothing, and parameter estimation for linear dynamical systems.
Many real dynamical systems do not exactly fit the model assumed by Kalman
filter design. The not-modeled dynamics contributes to the component w k in equation
2.19. In practice we cannot distinguish between the uncertainty of the measurement
and the not modeled dynamics. We have to assume a certain value of R . Too big R
results in the slow adaptation of the filter and too small R leads to instability of the
results. R controls the speed of adaptation and it has to be chosen in an optimal way.

2.3 Stationary signals

2.3.1 Analytic tools in the time domain

2.3.1.1 Mean value, amplitude distributions

In Sect. 1.1 formulas for calculation of moments such as mean, variance, standard
deviation, correlation, etc., were based on ensemble averaging paradigm. Quite often
we have only one realization of a process. Under the assumption of ergodicity one
can calculate the estimators by averaging over time instead of averaging over an
ensemble.
We have to bear in mind that in case of application of ergodicity assumption, the
strict value of the estimator is obtained for an infinitely long time. In practice the
shorter the time epochs, the higher the error of estimate will be.