Digital Signal Processing Reference
In-Depth Information
and
N
−
1
m
)
2
r
xy
[
=
m
2
=
1
)
(m
−
m
m
]
=−
(N
−
variance of abscissa
=
va
−
m
.
N
−
1
m
1
)
r
xy
[
m
]
=−
(N
−
(12.4)
In the present analysis, above three quantitative time-domain descriptors of the
cross-correlograms are used for the classification of gait signals.
12.3.2 Frequency Domain Features
As discussed earlier, one can analyze cross-correlation in frequency domain and
extract meaningful features from it by computing Fourier transform of each cross-
correlation sequence. The Fourier transform of cross-correlation sequence
r
xy
[
]
m
is
called the cross-spectral density
(S
xy
)
, which is defined as [
6
,
38
,
41
]:
∞
e
−
j
2
πfm
.
S
xy
(f)
=
r
xy
[
m
]
(12.5)
m
=−∞
The features extracted from the cross-spectral density
(S
xy
)
are called frequency-
domain features. These features should be ideally well-suited for characterizing a
bioelectric signal, but with a reduced dimension. From the cross-spectral density
information, one can create the corresponding magnitude and phase cross-spectral
density, i.e.,
|
S
xy
(f)
|
and
∠
S
xy
(f)
feature vectors. Then the features extracted from
|
S
xy
(f)
|
and
S
xy
(f)
can be given as:
fl
_
mag(n)
=
S
xy
(f)
f
=
nf
0
,n
=
∠
1
,
2
,
3
,...,
(12.6)
fl
_
phase(n)
= ∠
S
xy
(f)
|
f
=
nf
0
,n
=
1
,
2
,
3
,...,
(12.7)
=
fl
_
mag(
1
),fl
_
mag(
2
),...,fl
_
mag(n),...,
fl
_
phase(
1
),fl
_
phase(
2
),...,fl
_
phase(n),...
.
(12.8)
fl
_
composite
Here,
fl
_
mag(n)
denotes the magnitude of the cross-spectral density at the
n
th fre-
quency sample. Similarly,
fl
_
phase(n)
denotes the phase of the cross-spectral den-
sity at the
n
th frequency sample. Then the composite feature vector
fl
_
composite
can
be formed, considering all
fl
_
mag
and
fl
_
phase
coefficients. Figures
12.5
and
12.6
show the plots of the sample
S
xy
(f)
curves for the cross-correlation
sequences of the left stance interval up to the 30th frequency sample.
|
S
xy
(f)
|
and
∠
12.4 Elman's Recurrent Neural Network Based Classification
Recurrent neural networks (RNNs) are particularly useful for learning both temporal
and spatial patterns. As opposed to a multilayer perceptron (MLP), which employs
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