Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 179

through the following representation

K

zFPT (±) ∝g ± K,P (z ±1 −z ± r,P )

Spectra in (5.3)

r=1

or the poles{z ± k,Q

}via

K

in pFPT (±) ∝g ± K,Q (z ±1 −z ± s,Q )

Spectra (5.4)

s=1

where g ± K,P and g ± K,Q represent the socalled gain factors. Clearly, if zeros

{z ± k,P }and poles{z ± k,Q }are simultaneously employed within shape spectra

and/or in quantification, the usual composite representations FPT (±) are ob
tained through the union of the two new constituent representations, the
zFPT (±) and pFPT (±) .
Alternatively, the zFPT (±) and pFPT (±) can be investigated via the canon
ical forms of the polynomials P ± K (z ±1 )

K

P ± K (z ±1 ) = p ± K (z ±1 −z ± r,P )

(5.5)

r=1

and Q ± K (z ±1 )

K

Q ± K (z ±1 ) = q ± K (z ±1 −z ± s,Q ). (5.6)

s=1

With these formulae it becomes possible to write the corresponding expres
sions for the general derivatives of the polynomials P ± K (z ±1 ) and Q ± K (z ±1 ).
For instance, the first derivatives of Q ± K (z ±1 ), which will be needed in this
chapter at z ±1 = z ± k,Q , can be obtained by the following explicit expressions

d
dz ±1 Q ± K (z ±1 )

′ (z ± k,Q )≡

Q ± K

z ±1 =z ± k,Q

K

= q ± K (z ± k,Q −z ± s,Q ). (5.7)

s=1,s=k

Here, it is seen that for simple poles, defined as the nonconfluent zeros z ± k,Q =

z ± k ′ ,Q (k ′ = k) of the denominator polynomial Q ± K (z ±1 ), the first derivative
Q ± K ′ (z ± k,Q ) is always different from zero

Q ± K ′ (z ± k,Q ) = 0.

(5.8)