Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 217

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 USED
N P = 2K T = 180

1

1
Lipid

0.8

FPT (+)

*

0.6

ZEROS of SPECTRUM
P +

NUMBER of
GENUINE HARMONICS
K G = K T − K F = 24

K (z)/Q +

K (z) :

0.4

P +
K (z) = 0

P +

K (z +

≡

k,P )

0.2

z +
k,P = exp(2i πτ ν

+
k,P )

C

0

25
Water

POLES of SPECTRUM
P +

−0.2

K (z)/Q +

K (z) :

−0.4

Q +
K (z) = 0

Q +

K (z +

≡

k,Q )

z +
k,Q = exp(2i πτ ν

+
k,Q ) −0.6

−0.8

NUMBER of
FROISSART HARMONICS
K F = 66

−1

−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 = 130

PARTIAL FID LENGTH
N P = 2K T = 260

1

1
Lipid

0.8

FPT (+)

*

0.6

ZEROS of SPECTRUM
P +

NUMBER of
GENUINE HARMONICS
K G = K T − K F = 25
CONVERGED

K (z)/Q +

K (z) :

0.4

P +
K (z) = 0

P +

K (z +

≡

k,P )

0.2

z +
k,P = exp(2i

+
k,P )

πτ ν

0 C

25
Water

POLES of SPECTRUM
P +

−0.2

K (z)/Q + K (z) :

−0.4

Q +
K (z) = 0

Q +

K (z +

≡

k,Q )

z +
k,Q = exp(2i

+
k,Q )

πτ ν

−0.6

−0.8

NUMBER of
FROISSART HARMONICS
K F = 105

−1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

(ii) Re(z +

k )

FIGURE 6.9
Distribution of poles and zeros in the complex z + −plane via the Argand plot
for complex harmonics z + = 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, 260.