Kevin Buckley  2007
1
ECE 8770 Topics in Digital Communications  Sp. 2007
Lecture 10a
4
Channel Equalization
4.4
Adaptive Equalization
4.5
Alternative Adaptation Schemes
(continued)
The Kalman Filter as an Adaptive Equalizer:
In this Subsection we develop the Kalman filtering algorithm for adaptation of the
coefficient vector of a linear equalizer or DFE. We start with the general Kalman fil
tering problem formulation and solution, and we then discuss its application to channel
equalization.
1.
Kalman Filtering:
The Kalman filter is popular as an effective estimator of the
state of a random process
because it is the
minimum meansquared state estimator
.
It is also the
conditional mean estimator of the state
. We will see that it can also
be used as an effective adaptive equalizer.
The StateSpace Model or a Random Process:
Consider the following general
discretetime linear statespace model
w
k
+1
=
A
k
w
k
+
G
k
v
k
(1)
z
k
=
H
k
w
k
+
η
k
(2)
where
–
z
k
is the
L
×
1 dimensional observation (i.e. data) vector at time
k
,
–
w
k
is the
M
×
1 dimensional state vector,
–
H
k
is the
L
×
L
dimensional transition matrix from the state to the output
at time
k
,
–
η
k
is the
L
×
1 measurement noise vector,
–
A
k
is the
M
×
M
dimensional state transition matrix at time
k
,
–
v
k
is the
P
×
1 dimensional process noise vector, and
–
G
k
is the
M
×
P
dimensional transition matrix at time
k
from the state
transition noise to the next state.
We assume that
η
k
and
v
k
are zeromean vector sequences, uncorrelated across
time, with covariance matrices
S
k
and
Q
k
at time
k
, respectively.
η
k
and
v
n
are
assumed uncorrelated with one another for all
k
and
n
.
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Kevin Buckley  2007
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Figure 1 illustrates this general discretetime linear statespace model. This can
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 Spring '09
 Signal Processing, Estimation theory, Kalman filter, Kalman

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