Biomedical Engineering Reference
In-Depth Information
A
j
}
M
A
components or
sources
originating in the atria,
{s
(
t
)
j
=1
, and the ventricles,
}
M
V
}
M
N
{s
V
j
(
t
)
{s
N
j
(
t
)
j
=1
, as well as other sources of noise, interference and artifacts,
j
=1
,
[
20
,
21
]:
x
i
(
t
)=
M
A
(
t
)+
M
V
(
t
)+
M
N
A
A
j
V
V
j
N
N
j
j
=1
h
ij
s
j
=1
h
ij
s
j
=1
h
ij
s
(
t
)
i
=1
,
2
,...,L.
(3.16)
With the aid of some additional notations, this model accepts a convenient matrix
formulation. Let vectors
A
A
A
M
A
(
t
)]
T
s
A
(
t
)=[
s
1
(
t
)
,s
2
(
t
)
,...,s
,
1
(
t
)
,s
2
(
t
)
,...,s
V
M
V
(
t
)]
T
s
V
(
t
)=[
s
,
N
1
(
t
)
,s
N
2
(
t
)
,...,s
N
M
N
(
t
)]
T
s
N
(
t
)=[
s
contain, respectively, the atrial, ventricular and noise sources. Let the linear
superposition coefficients be stored in matrices
[
H
A
]
ij
A
ij
=
h
,
1
≤ i ≤ L
,
V
ij
N
ij
1
≤ j ≤ M
A
,
[
H
V
]
ij
=
h
,
1
≤ i ≤ L
,
1
≤ j ≤ M
V
,and
[
H
N
]
ij
=
h
,
1
≤ j ≤ M
N
. Finally, let the lead outputs be stacked in vector
x
(
t
)=[
x
1
(
t
)
,x
2
(
t
)
,...,x
L
(
t
)]
T
. According to these notations, model (
3.16
) can
be compactly expressed as
≤ i ≤ L
,
1
⎡
⎤
A
(
t
)
s
⎣
⎦
=
Hs
(
t
)
,
x
(
t
)=[
H
A
,
H
V
,
H
N
]
V
(
t
)
(3.17)
s
N
(
t
)
s
∈R
L×M
and
s
(
t
)=[
s
A
(
t
)
T
,
s
V
(
t
)
T
,
s
N
(
t
)
T
]
T
where
H
=[
H
A
,
H
V
,
H
N
]
∈
R
M
, with
M
=
M
A
+
M
V
+
M
N
. Since most often vertical offsets do not convey
any physiological information, the source signals are assumed to have zero mean.
The mixing coefficients are determined by the relative location between sources and
electrodes, and the propagation characteristics of the body as a conductive medium,
which can be considered as purely resistive in the frequency range of interest [
14
].
Each mixing-matrix column represents the contribution of the corresponding source
to the different electrodes and can be associated with the electric potential spatial
distribution of that source on the body surface; hence, a mixing-matrix column is
also known as the
spatial topography
of its respective source [
3
].
Now, if the sources
s
(
t
)
and the mixing matrix
H
were available in model (
3.17
),
the atrial contributions to the recordings could easily be computed free from
ventricular activity and other disturbances by isolating
H
A
and
s
A
(
t
)
:
x
A
(
t
)=
H
A
s
A
(
t
)
.
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