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of redundant information. That is why PCA is often applied to extract, from the
original feature vectors, uncorrelated components that describe most of the vari-
ance. PCA has been employed to solve problems of overdetermined BSS (the case
where more sensors than source signals are available) [
23
] and to improve the
classification accuracy (see for instance [
24
,
25
]). PCA is equivalent to expressing
model in Eq. (
5.11
) in a subspace of properly selected dimension where most of
the original variance is explained
PV
i
¼
X
F
PV
ij
¼
X
F
¼
X
F
PF
0
,
V
ð
PCA
Þ
i
H
ð
PCA
Þ
ij
F
ð
PCA
Þ
0
PH
ij
P
T
;
ð
5
:
12
Þ
j
¼
0
j
¼
0
j
¼
0
where P is a projection unitary matrix (P
T
P
¼
1) provided by PCA that allows the
observation vectors to be ordered by their powers. PCA finds a rotated orthogonal
system such that the elements of the original vectors in the new coordinates
become uncorrelated, so the redundancy induced by correlation is removed [
3
].
Dimension of V
ð
PCA
Þ
i
depends on the variance to described on it with respect to the
variance on the original vector. For example, in the experiments, we reduce the
number of features from 512 (the first half of the DFT) to only 20, thus retaining
95 % of the total variance of the data.
In order to obtain one only feature vector corresponding to the complete
MIMO-LTI model of the multichannel impact-echo setup, let us form one vector
from all vectors V
ð
PCA
Þ
i
i
¼
1...N
2
4
3
5
P
F
H
ð
PCA
Þ
1j
F
ð
PCA
Þ
0
2
4
3
5
V
ð
PCA
Þ
1
j
¼
0
.
V
ð
PCA
Þ
N
.
,
V
ð
PCA
Þ
¼
P
F
H
ð
PCA
Þ
Nj
F
ð
PCA
Þ
0
j
¼
0
2
4
3
5
H
ð
PCA
Þ
1j
.
H
ð
PCA
Þ
Nj
¼
X
¼
X
F
F
F
ð
PCA
Þ
0
H
ð
PCA
Þ
j
F
ð
PCA
Þ
0
ð
5
:
13
Þ
j
¼
0
j
¼
0
Vector V
ð
PCA
Þ
implies a dimension increase by a factor N in comparison with
vectors V
ð
PCA
Þ
i
. Moreover, depending on the sensor spatial distribution, some
correlation may exist between components of V
ð
PCA
Þ
corresponding to different
sensors. Hence, a new PCA projection matrix P
1
should be applied in a similar
manner to Eq. (
5.12
), resulting in:
P
1
V
ð
PCA
Þ
¼
X
F
,
V
ðð
PCA
ÞÞ
¼
X
F
P
1
H
ð
PCA
Þ
P
1
F
ð
PCA
Þ
0
H
ðð
PCA
ÞÞ
j
F
ðð
PCA
ÞÞ
0
P
1
:
j
j
¼
0
j
¼
0
ð
5
:
14
Þ
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