Digital Signal Processing Reference
In-Depth Information
ABSOLUTE VALUES of AMPLITUDES RECONSTRUCTED by FPT
(+)
: NOISELESS FID
Pade (o) : |d
+
k
| = | P
+
K
(z
+
k,Q
) / [(d/dz
+
k,Q
)Q
+
K
(z
+
k,Q
)]| = |(p
+
K
/q
+
K
(z
+
k,Q
−z
+
m,P
) / [(z
+
k,Q
−z
+
K
)
Π
m=1
m,Q
)]
m
≠
k
| , Input (x) : |d
k
|
0.18
PADE : FPT
(+)
0.16
TOTAL NUMBER of
0.14
AMPLITUDES
1
K
T
= K = 90
Lipid
0.12
NUMBER of
0.1
ZERO−VALUED
0.08
FROISSART AMPLITUDES
K
F
= 66
B
0
= 1.5T
25
0.06
Water
NUMBER of
PARTIAL
0.04
GENUINE AMPLITUDES
FID LENGTH USED
0.02
K
G
= K
T
− K
F
= 24
N
P
= 2K
T
= 180
0
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
−3
(i) Chemical shift (ppm)
Pade (o) : |d
+
k
| = | P
+
K
(z
+
k,Q
) / [(d/dz
+
k,Q
)Q
+
K
(z
+
k,Q
)]| = |(p
+
K
/q
+
K
(z
+
k,Q
−z
+
m,P
) / [(z
+
k,Q
−z
+
K
)
Π
m=1
m,Q
)]
m
≠
k
| , Input (x) : |d
k
|
0.18
PADE : FPT
(+)
0.16
TOTAL NUMBER of
0.14
AMPLITUDES
1
K
T
= K = 130
Lipid
0.12
NUMBER of
0.1
ZERO−VALUED
0.08
FROISSART AMPLITUDES
K
F
= 105
B
0
= 1.5T
25
0.06
Water
NUMBER of
PARTIAL
0.04
GENUINE AMPLITUDES
FID LENGTH USED
0.02
K
G
= K
T
− K
F
= 25
N
P
= 2K
T
= 260
CONVERGED
0
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
−3
(ii) Chemical shift (ppm)
FIGURE 6.11
Absolute values of amplitudes as a function of chemical shifts, Re(ν
+
), for
the noiseless FID with input data from Table 3.1.
|
are denoted by×. Symbols◦represent the corresponding absolute values of
amplitudes{|d
k
Exact input data |d
k
|}reconstructed by the FPT
(+)
at the partial signal lengths
N
P
= 180, 260.
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