NMR Water Artifact Removal - Biomedical Signal Analysis: Contemporary Methods and Applications - page 387

damped complex harmonic functions
S ( t 1 ,t 2 )=
i

M i exp( − i Ω 1i t 1 )exp( − λ 1i t 1 )exp( − i Ω 2i t 2 )exp( − λ 2i t 2 )

on to which Gaussian noise is superimposed. Signal processing is per-
formed by Fourier analysis, resulting in spectra consisting of sums of
Lorentzian-shaped resonance lines [76] given by
f ( ω 1 ,ω 2 )=
i

M i 1
i ΔΩ 1i + λ 1i

1
i ΔΩ 2i + λ 2i

.

Statistical independence of two signals requires their scalar product to
be zero both in the time domain and in the frequency domain. Therefore
nonoverlapping resonance lines should be reasonably independent. But
because the limited range of chemical shifts (i.e. the spread of the
proton resonances on the frequency scale) is rather limited compared
to individual resonance line widths, statistical independence is hard
to assure in general. second-order techniques like the GEVD using
matrix pencils discussed above, as well as many others, exploit some
weaker conditions for the separation of sources, assuming that they
have a temporal structure with different autocorrelation functions or,
equivalently, different power spectra.

14.5 Results and Discussion

EDTA spectra
First, 2-D NOESY spectra of simple solute molecules such as EDTA were
analyzed. Presaturation of the water resonance was applied in all cases.
FIDs S ( t 1,j ,t 2 ) recorded at fixed evolution times t 1,j , j =1 , ...., m
were sampled over time spans t 2 and have been Fourier transformed
with respect to both time domains to obtain corresponding spectra
S ( ω 1 ,ω 2 ) which could be corrected for any phase distortions. Data
matrices X = x 1 , ..., x N were then formed with one row representing a
single spectrum S ( ω 2 ,t 1,j ) corresponding to a fixed evolution time t 1,j .
The final m N matrix X then contained as many rows as there were

different evolution times t 1,j according to the experimental protocol.
Typically m = 128 evolution periods were considered and N = 2048
data points were sampled from each spectrum in the t 2 domain. Due to

×