Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 212

EXACT RECONSTRUCTION of the TRUE NUMBER of GENUINE HARMONICS by FPT (−) : NOISELESS FID
POLE − ZERO CANONICAL FORM of FPT (−) : P −

K (z −1 )/Q − K (z −1 ) = (p − K /q −

K

(z −1 −z −

k,P )/(z −1 −z − k,Q )

zFPT (−) and pFPT (−) ; TWO COMPLEMENTARY SPECTRAL REPRESENTATIONS of FPT (−) :
zFPT (−) (

K )

Π k=1

) : ZEROS of FPT (−) ; ARGAND PLOT for ALL ZEROS z −

k,P in FPT (−) via P − K (z −1 ) = 0

•

pFPT (−) (o) : POLES of FPT (−) ; ARGAND PLOT for ALL POLES z −

k,Q in FPT (−) via Q − K (z −1 ) = 0

FROISSART DOUBLETS (POLE − ZERO CANCELLATIONS): z −

k,P = z −

k,Q (confluence of

& o )

′•′ ′ ′

GENUINE HARMONICS in the FOURTH QUADRANT, OUTSIDE the UNIT CIRCLE C ( |z| > 1 ), RANGING from # 1 (Lipid) to # 25 (Water)
FROISSART DOUBLETS OUTSIDE the UNIT CIRCLE C : GENUINE and SPURIOUS HARMONICS MIXED in the SAME REGION, |z| > 1

B 0 = 1.5T

TOTAL NUMBER of PARTIAL FID LENGTH USED

N P = 2K T = 256

1

HARMONICS
K T = K = 128

0.8

0.6

0.4

0.2

Water

C

0

25

−0.2

−0.4

−0.6

PADE : FPT (−)

NUMBER of

−0.8

NUMBER of
FROISSART DOUBLETS
K F = 103

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

−1

1
Lipid

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

Re(z −

k )

FIGURE 6.5
Froissart doublets for unequivocal determination of the exact number K G of
the genuine harmonics from the total number K T ≡K of all the harmonics
reconstructed by the FPT (−) for the noiseless FID with input data from Table
3.1. The FPT (−) mixes together the genuine and spurious harmonics in the
same region outside the unit circle C. The ghost zero z = 0 from the FPT (+)
does not appear any longer at the center of C in the FPT (−) . Rather, it is
moved to infinity (in theory, or to very large distance, in practice), since the
FPT (−)

employs the inverse variable z −1 .