Biomedical Engineering Reference
In-Depth Information
3
Multiple channels (multivariate) signals
The technological progress in the biomedical field has led to the construction of
recording equipment which allows recording of activity from multiple sites. Today
a typical dataset contains not only two or four but dozens or hundreds of channels.
This is especially the case for EEG, MEG, and sometimes also for ECG and EMG
signals. Analysis of multichannel data can give a better insight into the relations be-
tween the investigated sites, but it is a challenging task. Besides many experimental
and computational difficulties, the problem quite often lies in the proper application
of existing mathematical tools. The techniques capitalizing on the covariance struc-
ture of the multichannel (multivariate ) data are especially useful in this respect. In
this chapter an introduction to basic aspects of multichannel data processing will be
presented.
3.1 Cross-estimators: cross-correlation, cross-spectra, coherence
(ordinary, partial, multiple)
Joint moments of the order two: cross-correlation
R
xy
and cross-covariance
C
xy
,
were defined in Sect. 1.1 (equations 1.11 and 1.12) by means of ensemble averaging
formalism. Under the assumption of ergodicity they can be expressed by the formu-
las:
Z
∞
R
xy
(
τ
)=
x
(
t
)
y
(
t
+
τ
)
dt
(3.1)
−
∞
Z
∞
C
xy
(
τ
)=
∞
(
x
(
t
)
−
μ
x
)(
y
(
t
+
τ
)
−
μ
y
)
dt
(3.2)
−
Cross-correlation and cross-spectrum are bound by means of Fourier transform
and inverse Fourier transform (cf. Sect. 2.3.2.1.3):
Z
∞
e
−
i
2π
f
τ
d
τ
S
xy
(
f
)=
R
xy
(
τ
)
(3.3)
−
∞
Z
∞
e
i
2π
f
τ
df
R
xy
(
τ
)=
S
xy
(
f
)
(3.4)
−
∞
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