Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 185

5.7 Signal-noise separation with exclusive reliance upon
resonant amplitudes
The derived proof of determining the exact number of genuine resonances
is based exclusively on the retrieved signal poles z ± k,Q , i.e., the quantities
that include only the complex frequencies, with no information about the
associated complex amplitudes d ± k . However, it is likewise important to know
whether the genuine and spurious resonances could also be distinguished by
their amplitudes. To investigate this subject, it would be advantageous to
have closed, analytical expressions for the amplitudes d ± k corresponding to the
signal poles z ± k,Q . Recall that the amplitudes d ± k are defined as the Cauchy
residues of the rational polynomial P ± K (z ±1 )/Q ± K (z ±1 ). Such residues for the
simple poles of Q ± K (z ±1 ) are defined by the expressions

(z ±1 −z ± k,Q ) P ± K (z ±1 ) d ± k

= lim

z ±1 →z ± k,Q

. (5.22)

Q ± K (z ±1 )

With the help of the canonical form (5.11), we can perform the limit in (5.22)
with the results

= p ± K
q ± K

d ± k

8
<

9
=

K

[z ±1 −z ± k,Q ] (z ±1 −z ± r,P )

r=1

×lim
z ±1 →z ± k,Q

:

A(z ±1 −z ± k−1,Q )[z ±1 −z ± k,Q ](z ±1 −z ± k+1,Q )B

;

where A = (z ±1 −z ± 1,Q ) and B = (z ±1 −z ± K T ,Q ). In these formulae for d ± k , we
can cancel out the common terms [z ±1 −z ± k,Q ] in the square brackets from the
numerator and denominator.

Therefore, in the remaining expressions, z ±1

can be replaced by z ± k,Q

so that

K

(z ± k,Q −z ± r,P )

= p ± K
q ± K

r=1

d ± k

k∈K

T .

(5.23)

K

(z ± k,Q −z ± s,Q )

s=1,s=k

More concisely, and similarly to (5.10), we can rewrite (5.23) as

K

(z ± k,Q −z ± k ′ ,P )

= p ± K
q ± K

d ± k . (5.24)

(z ± k,Q −z ± k ′ ,Q ) k ′ =k

k ′ =1