Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 187

to the genuine (respectively, spurious) resonances. Hence, for genuine reso
nances, i.e., k∈K

F , i.e., k = k ′ G (which automatically implies that k /∈K

for k ′ ∈K

F ), the genuine amplitudes can be obtained from (5.26) via

K T

(z ± k,Q −z ± r,P )

= p ± K
q ± K

r=1

d ± k

(5.27)

K T

(z ± k,Q −z ± s,Q )

s=1,s=k

8
<

9
=

K G +K F

K G

(z ± k,Q −z ± r ′ ,P )

(z ± k,Q −z ± r,P )

= p ± K
q ± K

r ′ =K G +1,r ′ =k

r=1

k∈K G .

: ;

K G K G +K F

(z ± k,Q −z ± s,Q ) (z ± k,Q −z ± s ′ ,Q )

s=1,s=k s ′ =K G +1,s ′ =k

r ′ ,s ′ ∈K F

In (5.27), the rational polynomial in the curly brackets is equal to unity,
because of the coincidence of the constituent polynomials in the numerators,

K G +K F K G +K F

r ′ =K G +1,r ′ =k (z ± k,Q

−z ± r ′ ,P ) and denominators,

s ′ =K G +1,s ′ =k (z ± k,Q

−z ± s ′ ,Q ).

This maps (5.27) into the form

K G

(z ± k,Q −z ± r,P )

= p ± K
q ± K

d ± k

r=1

k∈K G (genuine). (5.28)

K G

(z ± k,Q −z ± s,Q )

s=1,s=k

Similarly, for k∈K

F , the Froissart or spurious amplitudes are identified from

(5.26) as

K T

(z ± k,Q −z ± r,P )

= p ± K
q ± K

r=1

d ± k

(5.29)

K T

(z ± k,Q −z ± s,Q )

s=1,s=k

8
<

9
=

K G +K F

K G

(z ± k,Q −z ± r ′ ,P )

(z ± k,Q −z ± r,P )

= p ± K
q ± K

r ′ =K G +1,r ′ =k

r=1

k∈K F .

: ;

K G K G +K F

(z ± k,Q −z ± s,Q ) (z ± k,Q −z ± s ′ ,Q )

s ′ =K G +1,s ′ =k

s=1,s=k

r ′ ,s ′ ∈K F

The polynomial quotients in the curly brackets from (5.29) are equal to
zero. This is due to the fact that the corresponding numerator polynomials