Digital Signal Processing Reference
In-Depth Information
ARGAND PLOTS for HARMONIC VARIABLES RECONSTRUCTED by FPT
(−)
: NOISELESS FID
PADE POLES (o) : z
−
k,Q
[ pFPT
(−)
] , PADE ZEROS (
) : z
−
k,P
[ zFPT
(−)
]
•
B
0
= 1.5T
TOTAL NUMBER of
HARMONICS
K
T
= 90
PARTIAL FID LENGTH
N
P
= 2K
T
= 180
1
0.8
0.6
ZEROS of SPECTRUM
P
−
K
(z
−1
)/Q
−
K
(z
−1
) :
0.4
P
−
K
(z
−1
) = 0
P
−
K
(z
−
≡
k,P
)
0.2
z
−
k,P
= exp(−2i
πτ
ν
−
k,P
)
Water
C
0
25
POLES of SPECTRUM
P
−
−0.2
K
(z
−1
)/Q
−
K
(z
−1
) :
−0.4
Q
−
K
(z
−1
) = 0
Q
−
K
(z
−
≡
k,Q
)
z
−
k,Q
= exp(−2i
πτ
ν
−
k,Q
)
−0.6
FPT
(−)
*
−0.8
NUMBER of
GENUINE HARMONICS
K
G
= K
T
− K
F
= 24
NUMBER of
FROISSART HARMONICS
K
F
= 66
−1
1
Lipid
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(i) Re(z
k
)
PADE POLES (o) : z
−
k,Q
[ pFPT
(−)
] , PADE ZEROS (
) : z
−
k,P
[ zFPT
(−)
]
•
B
0
= 1.5T
TOTAL NUMBER of
HARMONICS
K
T
= 110
PARTIAL FID LENGTH
N
P
= 2K
T
= 220
1
0.8
0.6
ZEROS of SPECTRUM
P
−
K
(z
−1
)/Q
−
K
(z
−1
) :
0.4
P
−
K
(z
−1
) = 0
P
−
K
(z
−
≡
k,P
)
0.2
z
−
k,P
= exp(−2i
−
k,P
)
πτ
ν
Water
0
C
25
POLES of SPECTRUM
P
−
−0.2
K
(z
−1
)/Q
−
K
(z
−1
) :
−0.4
Q
−
K
(z
−1
) = 0
Q
−
K
(z
−
≡
k,Q
)
z
−
k,Q
= exp(−2i
−
k,Q
)
πτ
ν
−0.6
FPT
(−)
*
−0.8
NUMBER of
GENUINE HARMONICS
K
G
= K
T
− K
F
= 25
CONVERGED
NUMBER of
FROISSART HARMONICS
K
F
= 85
−1
1
Lipid
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(ii) Re(z
−
k
)
FIGURE 6.12
Distribution of poles and zeros in the complex z
−
−plane via the Argand
plot for complex harmonics z
−1
= exp (−2iπν
−
) in polar coordinates for
the noiseless FID with input data from Table 3.1. Symbols◦and•show
the poles and zeros, respectively, reconstructed by the FPT
(−)
at the partial
signal lengths N
P
= 180, 220.
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