Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 208

EXACT RECONSTRUCTION of the TRUE NUMBER of GENUINE FREQUENCIES and AMPLITUDES by FPT (+) : NOISELESS FID
FREQUENCIES in pFPT (+) : ν

+
k,Q = [ −i /(2 πτ )] ln (z + k,Q ) , zFPT (+) : ν

+
k,P = [ −i /(2 πτ )] ln (z + k,P ) , INPUT POLES (x) : ν k

AMPLITUDES in FPT (+) : d +

k = P + K (z + k,Q ) / [(d/dz + k,Q )Q + K (z + k,Q )] = (p + K /q +

K

(z +

k,Q −z + m,P ) / [(z + k,Q −z + m,Q )] m ≠ k , INPUT (x) : d k

GENUINE FREQUENCIES: POSITIVE IMAGINARY PARTS & CHEMICAL SHIFTS witin 0.985 ppm (Lipid : # 1) − 4.68 ppm (Water : # 25)
FROISSART DOUBLETS ( ν

K ) Π m=1

+
k,Q ; POLES = ZEROS ): NEGATIVE IMAGINARY PARTS & ZERO−VALUED AMPLITUDES
GENUINE & SPURIOUS ( FROISSART ) RESONANCES SEPARATED in 2 DISJOINT REGIONS, Im( ν

+
k,P = ν

+
k ) > 0 and Im( ν

+
k ) < 0

+
k,Q [ pFPT (+) ] , Pade zeros (

+
k,P [ zFPT (+) ] , Input poles (x) :

Argand plot for frequencies ; Pade poles (o) : ν • ) : ν ν k

−0.1

−0.05

0

PADE : FPT (+)
TOTAL NUMBER of
FREQUENCIES
K T = K = 128
NUMBER of
FROISSART DOUBLETS
K F = 103
NUMBER of
GENUINE FREQUENCIES
K G = K T − K F = 25

0.05

0.1

25

Water

B 0 = 1.5T

0.15

1

PARTIAL
FID LENGTH USED
N P = 2K T = 256

Lipid

0.2

0.25 CONVERGED

12 11 10 9 8 7 6 5 4 3 2 1 0 −1 −2 −3

+
k ) (ppm)

(i) Re( ν

Absolute values of amplitudes ; Fast Pade Transform, FPT + (o) : |d +
k | , Input (x) : |d k |

0.18

PADE : FPT (+)
TOTAL NUMBER of

0.16

0.14

AMPLITUDES
K T = K = 128

1

Lipid

0.12

NUMBER of

0.1

ZERO−VALUED

B 0 = 1.5T

0.08

FROISSART AMPLITUDES
K F = 103

25

0.06 Water PARTIAL

FID LENGTH USED
N P = 2K T = 256

NUMBER of

0.04

GENUINE AMPLITUDES
K G = K T − K F = 25 0.02

CONVERGED

0

12 11 10 9 8 7 6 5 4 3 2 1 0 −1 −2 −3

(ii) Chemical shift (ppm)

FIGURE 6.2
Froissart doublets for unequivocal determination of the exact number K G of
the genuine frequencies and amplitudes from the total number K T ≡K of the

spectral parameters reconstructed by the FPT (+) for the noiseless FID with
input data from Table 3.1. In panel (i), the FPT (+) separates the genuine
from spurious frequencies in the two nonoverlapping regions, Im(ν + ) > 0 and
Im(ν + ) < 0, respectively. In panel (ii), all the spurious (Froissart) amplitudes
are unambiguously identified by their zero values.