Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 178

used [5]. In this way, spurious content (noise and noiselike pollution) which is
usually viewed as a burden in every data can, in fact, be advantageously used
to tease out the genuine information from the output results of the analysis.

5.1 Critical importance of poles and zeros in generic
spectra
As stated, a Froissart doublet [44] is a spectral pair consisting of a pole
and a zero that coincide with each other. Hence, for an investigation of
Froissart doublets, one needs zeros and poles of the complex Pade spectra
P ± K (z ±1 )/Q ± K (z ±1 ) in the FPT (±) . Such zeros and poles are defined as the
solutions of the characteristic equations for the polynomials in the numerator

P ± K (z ±1 ) = 0

(5.1)

and denominator

Q ± K (z ±1 ) = 0. (5.2)

Thus far, the solutions of the numerator and denominator characteristic equa
tions from (5.1) and (5.2) were denoted by z ±1
k

in (2.190) and z ±1
k

in (2.181).

However, from now on we shall use the alternative notations z ±1
k,P

≡z ± k,P and

z ±1
k,P

≡z ± k,Q for the zeros of polynomials P ± K (z ±1 ) and Q ± K (z ±1 ), respectively.
Here, the second subscript P in z ± k,P and likewise Q in z ± k,Q is introduced to in
dicate that these eigenroots z ± k,P = exp (±iω ± k,P τ) and z ± k,Q = exp (±iω ± k,Q τ)
satisfy their respective characteristic equations P ± K (z ± k,P ) = 0 and Q ± K (z ± k,Q ) =
0, according to (5.1) and (5.2).

5.2 Spectral representations via Pade poles and zeros as
pFPT (±) and zFPT (±)
By studying polezero cancellations, we are naturally led to the introduction
of the two pairs of complementary representations of the FPT (±) . One of these
pairs is zFPT (±) named the 'zeros of the FPT (±) ', whereas the other pair is
pFPT (±) called the 'poles of the FPT (±) '. The zFPT (±) and pFPT (±) can au
tonomously either the zeros{z ± k,P give spectra by employing exclusively }