Recapitulation of Pad´e-optimized processing of biomedical time signals - Signal Processing in MRS with Biomedical Applications - page 380

residual differences which remain after the values of the poles are subtracted
from the zeros, implying that d ± k are proportional to z ± k,Q −z ± k,P via

d ± k ∝(z ± k,Q −z ± k,P ).

(12.27)

Thus, the distances between poles z ± k,Q and zeros z ± k,P are proportional to the
amplitudes d ± k . Distances 2 have the meaning of a metric; they are related to
the socalled measure in a given vector space. Indeed, as shown in Ref. [5],
amplitudes d ± k

are present in the complexvalued Lebesgue measures dσ ± (z)

K

dσ ± (z) = ρ ± (z)dz σ ± (z) = d ± k ϑ(z ±1 −z ± k,Q ) (12.28)

k=1

K

ρ ± (z) = d ± k δ(z ±1 −z ± k,Q ) (12.29)

k=1

1, z ±1 ∈K ± G
0, z ±1 /∈K ± G

ϑ(z ±1 −z ± k,Q ) = (12.30)

with dϑ(z)/dz = δ(z), where δ(z) and ϑ(z) are the Dirac and Heaviside com
plex functions, respectively. Here,K ± G denotes the set of K G genuine sig

nal poles{z ± k,Q }(1≤k≤K G ) from the investigated FID. The differentials

dσ ± (z), i.e., the measures, are complexvalued and, therefore, the correspond
ing norms are also complex numbers.
Despite the appearance of complex measures and norms, the orthogonality
relationships of the Pade denominator polynomials in the FPT (±) are still pre
served. This is important, since distributions and locations of spurious poles
depend on features of the orthogonality relations satisfied by the Pade denom
inator polynomials. Clearly, it is vital to have full control over the locations
of all the zeros of Q ± K (z ±1 ) in the Pade quotients P ± K (z ±1 )/Q ± K (z ±1 ) from
the FPT (±) . Such a control is possible in the FPT (+) and FPT (−) , because
all the genuine zeros of Q K (z) and Q − K (z −1 ) are inside and outside the unit
circle, respectively. However, despite our prior knowledge about such precise
locations before reconstructing these zeros, as soon as the systematically in
creased degree K of Q ± K (z ±1 ) surpasses the unknown true order K G , spurious
roots{z ± k,Q }of the characteristic equations Q ± K (z ±1 ) = 0 would inevitably

appear. For the same reason, spurious zeros{z ± k,P

}will also emerge from the
accompanying secular equations of the numerator polynomials P ± K (z ±1 ) = 0.
This is where the Froissart concept comes into play to take advantage of

2 In complex vector spaces, distances should be taken in a generalized sense which, of course,
need not necessarily be reduced to a literal distance in units of physical length.