Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 23

F(E) = P K−1 (E)
Q K (E)

= [(K−1)/K] Q (E) (1.12)

where P K−1 (E) and Q K (E) are the numerator and denominator polynomial
of degree K−1 and K, respectively. This complexvalued spectrum can also
be written as a sum of its multilevel BW absorption and dispersion spectra
A(E) and D(E)

K

C k
(E−E k ) + iΓ k /2

F(E) = i ≡A(E) + iD(E) (1.13)

k=1

K

A k (E)≡ K

Γ k /2
(E−E k ) 2 + Γ k /4

A(E) = C k (1.14)

k=1 k=1

K

D k (E)≡ K

E−E k
(E−E k ) 2 + Γ k /4 .

D(E) = C k (1.15)

k=1 k=1

Naturally, the same BW formulae for one or many levels are also directly
applicable to spectroscopy.
The above outlines clearly show that the Pade approximant is an integral
part of the adequate description of the physics of resonance phenomena. Thus,
from the outset, the PA is optimally suited to yield the proper theoretical pre
dictions and physical interpretation of resonance data that are measured in
experiments. This is indeed the case not only in physics, but also across inter
disciplinary research fields [5]. This briefly expounded quantummechanical
concept of resonances is deeply rooted in one of the most fundamental quanti
ties in physics, termed the scattering or S−matrix [92, 93]. The S−matrix is
employed to map the incoming to the outgoing total scattering wave function
Ψ (−) and Ψ (+) , respectively, for the initial and final state of the whole system
under consideration [93]

Ψ (+) (E) = S(E)Ψ (−) (E). (1.16)

The incoming and outgoing particle fluxes Φ (−) and Φ (+) can be computed
from Ψ (+) and Ψ (−) , respectively. This implies that the cross section Q(E)
from (1.5) can be extracted from S(E). Hence, the elements of the S−matrix
contain all the physical properties of cross sections, including resonances. Im
portantly, S(E) is an analytic function in the complex energy plane with
singularities represented by poles and branch cuts 3 . In particular, the poles
of S(E) lead to the Lorentzianshaped resonances in Q(E). Clearly, a function
which possesses poles ought to be a rational function, and such is S(E). Hence,
it is not surprising that the matrix elements of the S−matrix are the Pade

3 It is well-known from the Cauchy analysis that any analytical function can be represented
by a sum of its poles [44, 47].