Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 218

ARGAND PLOTS for LINEAR FREQUENCIES RECONSTRUCTED by FPT (+) : NOISELESS FID

+
k,Q = [ −i /(2 πτ )] ln (z + k,Q ) [ pFPT (+) ] , Pade zeros ( • ) : ν

+
k,P = [ −i /(2 πτ )] ln (z + k,P ) [ zFPT (+) ] , Input poles (x) : ν k

Pade poles (o) : ν

−0.1

−0.05

0

PADE : FPT (+)
TOTAL NUMBER of 0.05

FREQUENCIES
K T = K = 90

0.1

25

Water

B 0 = 1.5T

NUMBER of

0.15 FROISSART FREQUENCIES

K F = 66

1

PARTIAL

Lipid

0.2 NUMBER of FID LENGTH USED

N P = 2K T = 180

GENUINE FREQUENCIES
K G = K T − K F = 24

0.25

12 11 10 9 8 7 6 5 4 3 2 1 0 −1 −2 −3

+
k ) (ppm)

(i) Re( ν

+
k,Q = [ −i /(2 πτ )] ln (z + k,Q ) [ pFPT (+) ] , Pade zeros ( • ) : ν

+
k,P = [ −i /(2 πτ )] ln (z + k,P ) [ zFPT (+) ] , Input poles (x) : ν k

Pade poles (o) : ν

−0.1

−0.05

0

PADE : FPT (+)

0.05 TOTAL NUMBER of

FREQUENCIES
K T = 130

25

0.1

Water

B 0 = 1.5T

NUMBER of

0.15 FROISSART FREQUENCIES

K F = 105

1

PARTIAL

Lipid

0.2 NUMBER of FID LENGTH USED

N P = 2K T = 260

GENUINE FREQUENCIES
K G = K T − K F = 25

0.25

CONVERGED

12 11 10 9 8 7 6 5 4 3 2 1 0 −1 −2 −3

+
k ) (ppm)

(ii) Re( ν

FIGURE 6.10
Distribution of poles and zeros in the complex ν + −plane via the Argand plot
for complex frequencies ν in rectangular coordinates for the noiseless FID
with input data from Table 3.1. Exact input frequencies are denoted by×.
Symbols◦and•show the poles and zeros, respectively, reconstructed by the
FPT (+) at the partial signal lengths N P = 180, 260.