Biomedical Engineering Reference
In-Depth Information
convergence of the envelope spectrum at
N
P
D 180
is the fact that no difference
exists between any two envelope spectra computed at
N
P
D 180; 220; 260:
This is
so in particular for
because the area of the 12th peak is overestimated
precisely by the amount of the corresponding area of the missing 11th peak on
panels (d). The two identical total shape spectra at
N
P
D 180
N
P
D 220
on panels (f) contain 24 and 25 resonances, respectively. Such a discrepancy in
the number of reconstructed resonances is not detected by the residual or error
spectra (not shown). This implies that it is not reliable to use the converged total
shape spectra and the related residual or error spectra as the only criterion for
determining the number of reconstructed resonances. Precisely this latter criterion
is used in all fittings, within MRS and beyond, that rely heavily upon the residual
spectrum defined as the difference between the spectrum from the FFT and a
modeled spectrum. The right column of Fig.
25.1
clearly shows how precarious
it is to surmise which components are hidden underneath a spectral structure. Thus,
rather than reconstructing the mobile lipids under the two broad structures in the
range 1-2 ppm, as done unambiguously by the FPT, equally acceptable (in the least-
square sense) results of fitting by the usual methods from MRS could “reconstruct”
two, three, four or more peaks that would all give the same absorption total shape
spectrum from 1 ppm to 2 ppm. This is reminiscent of the Lanczos paradox [
1
]of
fitting the same experimental data with 3 identical curves with widely different set of
parameters. Even more serious problems with clinically unacceptable uncertainties
stemming from fittings are found in other parts of the spectrum from panel (c) in
Fig.
25.1
. In particular, any attempts to use fitting to ascertain that the peaks close
to 2.7 ppm are, in fact, almost degenerate would be practically impossible.
We also consider a noise-corrupted time signal
N
P
D 180
and
fc
n
C r
n
g;
where
fc
n
g
is the
same noiseless FID employed in Fig.
25.1
. Here,
is zero-mean complex-valued
random Gaussian white noise (orthogonal in its real and imaginary parts). The
standard deviation
fr
n
g
is set to be 0.00289 RMS, where RMS is the root-mean-
square of the noise-free FID,
of
fr
n
g
Reconstructions by the FPT
.C/
are illustrated in
Fig.
25.2
for genuine and spurious resonances that appear in the entire Nyquist
range. The only difference between Froissart doublets for the noise-free
fc
n
g:
fc
n
g
and
noise-corrupted
time signals from Figs.
25.1
and
25.2
is that the latter
are more irregularly distributed. This is expected due to the presence of the random
perturbation
fc
n
C r
n
g
in the noisy FID. However, this difference is irrelevant, since the
only concern to SNS is that noise-like or noisy information is readily identifiable
by pole-zero coincidences. Once the Froissart doublets are identified and discarded
from the whole set of results, only the reconstructed parameters of the genuine
resonances will remain in the output data. Crucially, the latter set of Pade-retrieved
spectral parameters also contains the exact number
fr
n
g
K
G
of genuine resonances as the
difference between the total number
K K
T
of all the found resonances and the
number
K
F
of Froissart doublets,
K
G
D K
T
K
F
:
In Fig.
25.2
, we used a quarter
of the full signal length
N
P
D N=4 D 1024=256;
which corresponds to the Pade
P
K
.
z
˙1
/=Q
K
.
z
˙1
/:
polynomial degree
K D 128
in spectra
In the whole Nyquist
range, the FPT
.C/
and FPT
./
find 103 Froissart doublets,
K
F
D 103:
Therefore, the
of genuine resonances reconstructed by the FPT
.C/
and FPT
./
is given
number
K
G
