Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 186

We see that the denominator in (5.23) and (5.24) is never zero in the case of
simple poles, similarly to (5.7) and (5.8). Moreover, the numerator in (5.23)
represents the canonical form of P ± K (z ± k,Q ) by reference to (5.5). Similarly,
the denominator in (5.23) is the canonical form of the first derivative with
respect to z ±1 of Q ± K (z) taken at z ±1 = z ± k,Q

according to (5.7). This yields

the following equivalent formulae for d ± k

= P ± K (z ± k,Q )
Q ± K

d ± k (5.25)

′ (z ± k,Q )

where we always have Q ± K ′ (z ± k,Q ) = 0, as in (5.8). The formulae from (5.25)
could also be established by using definition (5.22) without referring to any
special representation of the pertinent polynomials. With this goal, we employ
the characteristic equation Q ± K (z ± k,Q ) = 0 from (5.2) and the definition of the
first derivative via Q ± K

′ (z ± k,Q ) = lim z ±1 →z ± k,Q [Q ± K (z ±1 )−Q ± K (z ± k,Q )](z ±1 −z ± k,Q ).
This precisely reproduces (5.25) as follows

(z ±1 −z ± k,Q ) P ± K (z ±1 ) d ± k = lim

z ±1 →z ± k,Q

Q ± K (z ±1 )

−1

Q ± K (z ±1 )−Q ± K (z ± k,Q )
z ±1 −z ± k,Q

= P ± K (z ± k,Q )

lim
z ±1 −z ± k,Q

= P ± K (z ± k,Q )
Q ± K

(QED).

′ (z ± k,Q )

To find out whether the amplitudes d ± k can be used to distinguish between
the genuine and spurious resonances, we employ the same prescription as in
(5.17) to rewrite (5.23) as

K T

(z ± k,Q −z ± r,P )

= p ± K
q ± K

d ± k

r=1

(5.26)

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=1

k∈K T .

: ;

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

Since the two setsK G andK F are disjoint, we know that if k∈K G (respec
tively, k∈K F ), then the amplitudes d ± k

on the lhs of (5.26) will correspond