Biomedical Engineering Reference
In-Depth Information
a
RETRIEVED GENUINE and SPURIOUS PARAMETERS in FAST PADE TRANSFORMS FPT
(
±
)
: NOISE−CORRUPTED FID
FROISSART DOUBLETS (SPURIOUS RESONANCES) : CONFLUENCE of PADE POLES and PADE ZEROS GIVING NULL AMPLITUDES
+
k,Q
= [ −i /(2
)]
ln
(z
+
k,Q
) [ pFPT
(+)
], Pade zeros (
+
k,P
= [ −i /(2
)]
ln
(z
+
k,P
) [ zFPT
(+)
], Input poles (x) :
Pade poles (o) :
ν
πτ
•
) :
ν
πτ
ν
k
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Total Number of Frequencies
K
T
= K = 128
Number of Froissart Frequencies
K
F
= 103
Number of Genuine Resonances
K
G
= K
T
−K
F
= 25
B
0
= 1.5T
Partial
FID Length Used
N
P
= 2K
T
= N/4 = 256
25
Water
1
Lipid
PADE : FPT
(+)
CONVERGED
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
−3
+
k
) (ppm)
Re(
ν
b
−
k,Q
= [ i /(2
)]
ln
(z
−
k,Q
) [ pFPT
(−)
], Pade zeros (
−
k,P
= [ i /(2
)]
ln
(z
−
k,P
) [ zFPT
(−)
], Input poles (x) :
Pade poles (o) :
ν
πτ
•
) :
ν
πτ
ν
k
0
0.05
0.1
25
Toral Number of Frequencies
K
T
= K = 128
Number of Froissart Frequencies
K
F
= 103
Number of Genuine Frequencies
K
G
= K
T
−K
F
= 25
Water
B
0
= 1.5T
Partial
FID Length Used
N
P
= 2K
T
= N/4 = 256
0.15
1
Lipid
0.2
PADE : FPT
(−)
0.25
CONVERGED
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
−3
−
k
) (ppm)
Re(
ν
c
Pade (o) : |d
k
| = | P
K
(z
k,Q
) / [(d/dz
k,Q
)Q
K
(z
k,Q
)]| = |(p
K
/q
K
)
K
(z
k,Q
−z
±
m,P
) / [(z
k,Q
−z
±
m,Q
)]
m
≠
k
| , Input (x) : |d
k
|
Π
m=1
0.18
PADE : FPT
(
±
)
0.16
Total Number of Amplitudes
K
T
= 128
Number of Froissart Amplitudes
K
F
= 103
Number of Genuine Amplitudes
K
G
= K
T
−K
F
= 25
0.14
1
CONVERGED
Lipid
0.12
B
0
= 1.5T
Partial
FID Length Used
N
P
= 2K
T
= N/4 = 256
0.1
0.08
25
0.06
Water
0.04
0.02
0
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
−3
Chemical shift (ppm)
Fig. 25.2
Reconstruction of frequencies and amplitudes in the whole Nyquist interval by the
FPT
.˙/
at the signal length
N=4 D 256 .N D 1024/
for the noise-corrupted synthesized FID with
K D
25 complex damped harmonics. Panels (
a
)and(
b
) show genuine and spurious frequencies,
whereas the corresponding amplitudes are given on panel (
c
). In panel (
a
), the FPT
.C/
completely
separates the genuine from spurious frequencies into two disjoint regions Im
.
k
/>0
and
.
k
/<0;
Im
respectively. In the FPT
./
from panel (
b
), the imaginary parts of the genuine and
spurious frequencies have the same sign, Im
.
k
/>0:
The joint feature of panels (
a
)and(
b
)
is a clear and large set of pole-zero coincidences that define spurious resonances. In both panels
(
a
)and(
b
), the immediate neighborhood of the interval with the genuine resonances is the least
infiltrated with Froissart doublets. The reconstructed converged amplitudes associated with the
genuine resonances are identical in the FPT
.C/
and FPT
./
:
Panel (
c
) shows that all the Froissart
amplitudes are zero-valued
by
in exact agreement with the corresponding value of the
input data. Overall, Froissart doublets simultaneously achieve three important goals:
(i) noise reduction, (ii) dimensionality reduction as per (
25.8
) and (iii) stability
enhancement. Stability against perturbations of the physical time signal under study
K
G
D 128 103 D 25;
