Digital Signal Processing Reference
In-Depth Information
EXACT RECONSTRUCTION of the TRUE NUMBER of GENUINE FREQUENCIES and AMPLITUDES by FPT
(−)
: NOISELESS FID
FREQUENCIES in pFPT
(−)
:
ν
−
k,P
= [ i /(2
πτ
)]
ln
(z
−
k,Q
) , zFPT
(−)
:
ν
−
k,P
= [ i /(2
πτ
)]
ln
(z
−
k,P
) , INPUT POLES (x) :
ν
k
AMPLITUDES in FPT
(−)
: d
−
k
= P
K
(z
k,Q
) / [(d/dz
k,Q
)Q
K
(z
k,Q
)] = (p
K
/q
K
)
Π
m=
K
(z
k,Q
−z
m,P
) / [(z
k,Q
−z
m,Q
)]
m
≠
k
, INPUT (x) : d
k
GENUINE FREQUENCIES: POSITIVE IMAGINARY PARTS & CHEMICAL SHIFTS within 0.985 ppm (Lipid : # 1) − 4.68 ppm (Water : # 25)
FROISSART DOUBLETS (
−
k,Q
; POLES = ZEROS ): POSITIVE IMAGINARY PARTS & ZERO−VALUED AMPLITUDES
GENUINE & SPURIOUS ( FROISSART ) RESONANCES MIXED in the SAME REGION, Im(
ν
−
k,P
=
ν
ν
−
k
) > 0 and Re(
ν
−
k
)
∈
[0.985, 4.68] ppm
−
k,Q
[ pFPT
(−)
] , Pade zeros (
•
) :
ν
−
k,P
[ zFPT
(−)
], Input poles (x) :
ν
k
Argand plot for frequencies ; Pade poles (o) :
ν
0
0.05
PADE : FPT
(−)
TOTAL NUMBER of
FREQUENCIES
K
T
= K = 128
NUMBER of
FROISSART DOUBLETS
K
F
= 103
NUMBER of
GENUINE FREQUENCIES
K
G
= K
T
− K
F
= 25
0.1
25
Water
0.15
B
0
= 1.5T
1
Lipid
0.2
PARTIAL
FID LENGTH USED
0.25
CONVERGED
N
P
= 2K
T
= 256
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
−3
−
k
) (ppm)
(i) Re(
ν
Absolute values of amplitudes ; Fast Pade Transform, FPT
−
(o) : |d
+
k
| , Input (x) : |d
k
|
0.18
PADE : FPT
(−)
0.16
TOTAL NUMBER of
0.14
AMPLITUDES
K
T
= K = 128
1
Lipid
0.12
NUMBER of
0.1
ZERO−VALUED
B
0
= 1.5T
0.08
FROISSART AMPLITUDES
K
F
= 103
NUMBER of
25
0.06
PARTIAL
Water
FID LENGTH USED
N
P
= 2K
T
= 256
0.04
GENUINE AMPLITUDES
K
G
= K
T
− K
F
= 25
0.02
CONVERGED
0
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
−3
(ii) Chemical shift (ppm)
FIGURE 6.6
Froissart doublets for unequivocal determination of the exact number K
G
of
the genuine frequencies and amplitudes from the total number K
T
≡K of
the spectral parameters reconstructed by the FPT
(−)
for the noiseless FID
with input data from Table 3.1. In panel (i), the FPT
(−)
mixes the genuine
and spurious frequencies in the same range, Im(ν
−
) > 0. In panel (ii), all the
spurious (Froissart) amplitudes are unambiguously identified by their zero
values.
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