Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 209

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

K (z)/Q + K (z) = (p + K /q +

K

(z−z +

k,P )/(z−z + k,Q )

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

K )

Π k=1

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

k,P in FPT (+) via P +

•

K (z) = 0

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

k,Q in FPT (+) via Q + K (z) = 0

FROISSART DOUBLETS (POLE − ZERO CANCELLATIONS): z +

k,P = z +

k,Q (confluence of

′•′ & ′ o ′ )

GENUINE HARMONICS in the FIRST QUADRANT, INSIDE the UNIT CIRCLE C ( |z| < 1 ), RANGING from # 1 (Lipid) to # 25 (Water)

FROISSART DOUBLETS OUTSIDE C : GENUINE & SPURIOUS HARMONICS SEPARATED in 2 DISJOINT REGIONS, |z| < 1 & |z| > 1

B 0 = 1.5T

TOTAL NUMBER of PARTIAL FID LENGTH USED

N P = 2K T = 256

1

HARMONICS
K T = K = 128

1
Lipid

0.8

PADE : FPT (+)

0.6

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

0.4

CONVERGED

0.2

C

0

25
Water

−0.2

−0.4

−0.6

−0.8

NUMBER of

FROISSART DOUBLETS
K F = 103

−1

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

Re(z k )

FIGURE 6.3
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 FID with input data from Table 3.1 cor
rupted with random noise. The FPT (+) separates the genuine from spurious
harmonics in the two nonoverlapping regions, inside and outside the unit
circle C, respectively.