Biomedical Engineering Reference
In-Depth Information
by a discrete time delay operator, denoted by
z
−1
in the
Z
-domain. Let
x
j
(
n
) be an input sample for
the
j
th channel at time instance
n
. The temporal pattern of the
j
th channel is represented in the
embedded vector space as
x
j
(
n
)
L
∈
, where
L
is the number of taps in the delay line. The linear
approximation of the time-varying input-output mapping between all input channels and desired
output signals, denoted by
y
(
n
), is given by
M
å
w
y n
( )
=
( )
n
T
x
( )
n
+
b n
( )
(4.42)
j
j
j
=
1
∈
is the coefficient vector for the
j
th channel at time instance
n
,
b
(
n
) is bias, and
M
is the number of channels. Note that the operation of
w
j
(
n
)
T
x
j
(
n
) can be considered as filtering the
j
th channel input signal by a linear time-variant FIR filter. Therefore, ∑
j
w
j
(
n
)
T
x
j
(
n
) can be the sum
of the FIR filter outputs from every channel. We can also remove the bias term
b
(
n
) by normalizing
input and output, forcing them to have zero-mean.
The coefficient vector
w
j
(
n
) can be adapted by a number of methods, among which the LMS
algorithm plays a central role because of its computational simplicity and tracking capability. With
the LMS algorithm, the coefficient vector is updated as
L
where
w
j
(
n
)
w
(
n
+ =
1
)
w
( )
n
+
h
e n
( )
x
( )
n
(4.43)
j
j
j
for
j
= 1,… ,
M
. η is the learning rate, controlling convergence speed and the misadjustment, and
e
(
n
) is the instantaneous error such that
M
∑
w
e n
( )
= − = −
y n
( )
y n
ˆ( )
y n
( )
( )
n
T
x
( )
n
.
(4.44)
j
j
j
=
1
See [
31
] for the review of the tracking linear time-variant systems using LMS.
Assuming linear independence between input channels in MIMO systems, we can apply on-
line variable selection to channels instead of each tap output. To do so, we need to identify a variable
that can represent the temporal patterns of input time series at each channel. If we consider learning
a linear MIMO system, the estimation of desired signal is simply the sum of the FIR filters outputs
from each channel. These individual filtered outputs indicate the relationship between desired out-
puts and the filtered channel input. Because online variable selection operates based on correlation,
the filter output is hypothesized to be a sufficient variable to provide the correlation information
between desired outputs and input temporal patterns
Hence, we choose the filter outputs as input to the online variable selection procedure.
Then, the remaining question is how to learn each channel filter in real time, and here we choose
to utilize LMS because of its simplicity and reasonable tracking performance in nonstationary en-
vironments [
14
].
