Digital Signal Processing Reference
In-Depth Information
phase cycling every fourth spectrum has been considered yielding only
data matrices of size (
m
2048).
A matrix pencil (
C
1
,
C
2
) comprised two covariance matrices
C
of
the data where the second covariance matrix
C
2
represented a delayed
or filtered version of
R
1
. With zero mean data the covariance matrices
C
of the data equaled their correlation matrices
C
=
R
. The latter were
of dimension 32
×
N
=32
×
×
32, and the expectations were estimated according to
N
1
N
<x
i
x
j
>
=
x
i
(
n
)
x
j
(
n
)
(14.11)
n=1
with
N
= 2048 representing the number of samples in the
ω
1
domain
in the case of
R
1
. The second correlation matrix
R
2
of the pencil was
obtained in two different ways:
•
First, by collecting spectral data at frequencies below the water reso-
nance (i.e., only data points between 1285 and 2048) were used to cal-
culate the expectations in the covariance matrix
R
2
of the pencil. That
amounts to low-pass filtering the whole spectrum. Any smaller frequency
shifts did not yield reasonable results (i.e., a successful separation of the
water and the EDTA resonances could not be obtained).
•
A second procedure consisted in bandpass filtering the water resonance
in the frequency domain with a narrow-band filter which removed only
the water resonance. The spectra were then converted to the time
domain with an inverse Fourier transform, and corresponding correlation
matrices were calculated with time domain data for both correlation
matrices of the pencil. Even in the case of
R
1
the data had to be Fourier-
transformed first to be able to effect a phase correction to the spectra,
which then were subjected to an inverse Fourier transform to obtain
suitably corrected time domain data.
The matrix pencil thus obtained was treated in the manner given
above to estimate the independent components of the EDTA spectra and
the corresponding demixing matrix. Independent components showing
spectral energy only in the frequency range of the water resonance were
related to the water artifact. To effect a separation of the water artifact
and the EDTA spectra, these water-related independent components
were deliberately set to zero. Then the whole EDTA spectrum could be
reconstructed with the estimated inverse of the demixing matrix and the
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