Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 222

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

0.05

PADE : FPT (−)

TOTAL NUMBER of

0.1

25

Water

FREQUENCIES

K T = K = 90

0.15

NUMBER of

B 0 = 1.5T 1

FROISSART FREQUENCIES

Lipid

K F = 66

0.2 PARTIAL

NUMBER of

FID LENGTH USED

GENUINE FREQUENCIES

0.25

K G = K T − K F = 24 N P = 2K T = 180

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

)] ln (z k,Q ) [ pFPT (−) ] , Pade zeros (

−
k,P = [ i /(2

)] ln (z k,P ) [ zFPT (−) ] , Input poles (x) :

Pade poles (o) : ν πτ • ) : ν πτ ν k

0

0.05

PADE : FPT (−)

TOTAL NUMBER of

0.1

25

Water

FREQUENCIES

K T = 110

0.15

NUMBER of

B 0 = 1.5T

1

FROISSART FREQUENCIES

Lipid

K F = 85

0.2 PARTIAL

NUMBER of

FID LENGTH USED

GENUINE FREQUENCIES

0.25

K G = K T − K F = 25 N P = 2K T = 220

CONVERGED

12 11 10 9 8 7 6 5 4 3 2 1 0 −1 −2 −3

−
k ) (ppm)

(ii) Re( ν

FIGURE 6.13
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, 220.