Single channel (univariate) signal - Practical Biomedical Signal Analysis Using MATLAB - page 32

perturbation v k . Expressing it as an equation we get:

x k = A k x k − 1 + B k u k + v k (2.20)

where A k is the state transition matrix, B k is the matrix transforming the control
input, v k is the process noise coming from the zero mean normal distribution with
covariance matrix Q k . The matrixes M k , R k , A k , B k ,and Q k mayvaryintime.The
initial state, and the noise vectors at each step

{ x 0 , w 1 ,..., w k , v 1 ,..., v k } are all as-

sumed to be mutually independent.
In the estimation of the current state of the system all the above mentioned ma-
trixes are assumed to be known. Two types of error can be defined: apriori estimate
error:

x k (2.21)
where x k is the estimate of state x k based only on the knowledge of previous state of
the system, with covariance matrix given by:

e k = x k −

T

P k = e k e k E [ ] (2.22)

and a posteriori estimate error, with the result z k of observation known:

e k = x k − x k (2.23)

with covariance matrix given by:

e k e k ] P k = E [ (2.24)

Posterior estimate of the system state in current step can be obtained as a linear
mixture of the apriori estimate of the system state and the error between the actual
and estimated result of the observation:

x k + M x k ) x k = K ( z k − (2.25)

The matrix K is chosen such that the Frobenius norm of the a posteriori covariance
matrix is minimized. This minimization can be accomplished by first substituting
(2.25) into the above definition for e k (2.23), substituting that into (2.24), performing
the indicated expectations, taking the derivative of the trace of the result with respect
to K , setting that result equal to zero, and then solving for K . The result can be
written as:

P k M T MP k M T R − 1

K k = + (2.26)

The Kalman filter estimates the state of LDS by using a form of feedback control:
the filter estimates the process state at some time and then obtains feedback in the
form of (noisy) measurements. Two phases of the computation may be distinguished:
predict and update. The predict phase uses the state estimate from the previous time
step to produce an estimate of the current time step. In the update phase the current a
priori prediction is combined with the current observation to refine the a posteriori
estimate. This algorithm is presented in Figure 2.3 . Excellent introduction to the