Digital Signal Processing Reference
In-Depth Information
GENUINE and SPURIOUS PARAMETERS in FAST PADE TRANSFORM ; FID LENGTH: N
P
= 256
FROISSART DOUBLETS (SPURIOUS RESONANCES): CONFLUENCE of PADE POLES & ZEROS GIVING NULL AMPLITUDES
Argand Plot: Input Exact Poles (x), Pade Poles (o), Pade Zeros (
•
)
−0.05
Froissart Doublets
0
0.05
25
0.1
Water
0.15
1
Lipid
0.2
B
0
= 1.5T
N
P
= 256
PADE: FPT
(+)
Converged
0.25
6
5
4
3
2
1
0
+
k
) (ppm)
(i) Re(
ν
Argand Plot: Input Exact Poles (x), Pade Poles (o), Pade Zeros (
•
)
−0.05
Froissart Doublets
0
0.05
25
0.1
Water
0.15
1
Lipid
0.2
B
0
= 1.5T
N
P
= 256
PADE: FPT
(−)
Converged
0.25
6
5
4
3
2
1
0
−
k
) (ppm)
(ii) Re(
ν
Absolute Values of Amplitudes: Input Exact Data (x), Pade (o)
B
0
= 1.5T
N
P
= 256
PADE: FPT
(
±
)
Converged
0.15
1
Lipid
0.1
Water
25
0.05
0
Zero Amplitudes in
Froissart Doublets
6
5
4
3
2
1
0
ν
k
) (ppm)
(iii) Chemical shift, Re(
FIGURE 3.19
A subset of the whole set of Froissart doublets in the FPT
(±)
at a quarter of
the FID length, N/4 = 256 (N = 1024). On panel (i), the FPT
(+)
achieves a
total separation of genuine from spurious resonances that are mixed together
in the FPT
(−)
on panel (ii). Panel (iii) shows genuine and spurious amplitudes
in the FPT
(±)
. The reconstructed converged amplitudes are identical in the
FPT
(+)
and FPT
(−)
. All the spurious amplitudes are zerovalued.
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