Digital Signal Processing Reference
In-Depth Information
Process equation
Measurement equation
˜
u
(
n
)
x
(
n
)
y
(
n
)
χ
(
n
)
G
(
n
)
˜
n
(
n
)
x
(
n
-1)
z
-1
I
F
(
n
)
FIGURE B.1
Representation of a linear, discrete-time dynamical system.
⎧
⎨
⎧
⎨
Q
(
n
)
,
n
=
k
R
(
n
)
,
n
=
k
E
u
H
and E
H
(
n
)
u
(
k
)
=
n
˜
(
n
)
˜
n
(
n
)
=
0
,
n
=
k
0
,
n
=
k
⎩
⎩
(B.3)
Hence, given the state-space model of a system, the objective of the
Kalman filter can be formally stated as follows.
DEFINITION B.1
(Discrete-Time Kalman Filtering Problem) Consider the
linear, finite-dimensional, discrete-time system represented by (B.1) and
(B.2), defined for
n
be inde-
pendent, zero-mean, Gaussian white processes with covariance matrices
given by (B.3). Using the entire observed data
≥
0. Let the noise sequences
{
u
(
n
)
}
and
{ ˜
n
(
n
)
}
y
˜
(
1
)
,
y
˜
(
2
)
,
...
,
y
˜
(
n
)
,findthe
minimum mean-square estimate of the state
x
(
i
)
.
The above definition encompasses three slightly different problems that
can be solved by the Kalman filter. If
i
n
, i.e., we want to estimate the
current state based on observations up to time index
n
, we have a filtering
problem. On the other hand, if
i
=
>
n
it means we are facing a prediction prob-
lem. Finally, if 1
n
we have a so-called smoothing problem, in which we
observe a longer sequence of observations to estimate the state.
≤
i
<
B.2 Deriving the Kalman Filter
A well known result from the estimation theory is that the minimum mean-
squared error (MMSE) estimator is given by the conditional mean
E
x
|
˜
y
ˆ
=
x
MMSE
(B.4)
