Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 22

represent, respectively, the position and the width of the corresponding peak
in the cross section Q(E) as a function of the incident energy E. This res
onant maximum in Q(E) at E≈E k is manifested through a peak which
is superimposed on top of a smoothly varying background described by a
regular function f(E). Since they are located in the complex energy plane,
the resonance energies could be generally classified as sharp and broad. The
mentioned singularities in Q(E) associated with E k near the real energy axis
(small Γ k ) generate the welldelineated (isolated) resonances. In the energy
spectrum, they are seen as sharp peaks. The area underneath of such peaks
represents the resonant cross sections

Q k (E)≡Q(E k ). (1.6)

In such a case, the total cross section Q(E) near the resonant energy in the
single and multilevel BW formulae can be approximated by

Q(E)≈f(E) + Q 1 (E)

K

Q(E)≈f(E) + Q k (E) (1.7)

k=1

respectively. The nonsingular function f(E) is associated with a background
which could contain the contributions from broad resonances with large widths
Γ k . As stated, the physical interpretation of the BW formula is that it de
scribes absorption of an incident particle (or a wave) by a target. Due to
these circumstances, Q k (E) is proportional to the absorption spectrum A(E),
which represents the real part of the complexvalued spectrum F(E)

C 1
(E−E 1 ) + iΓ 1 /2

F(E)≡i ≡A 1 (E) + iD 1 (E) (1.8)

where C 1 is a constant. The imaginary part of F(E), denoted by D 1 (E), is
the dispersion spectrum

Γ 1 /2
(E−E 1 ) 2 + Γ 1 /4

A 1 (E) = C 1

E−E 1
(E−E 1 ) 2 + Γ 1 /4 .

D 1 (E) = C 1 (1.9)

Hence, for a singlelevel BW formula, the function F(E) is recognized as the
simplest PA to Q(E) with a numerator and denominator polynomial of the
degree 0 and 1, respectively, as symbolized by [0/1] Q (E)
F(E) = [0/1] Q (E).

(1.10)

Similarly, the manylevel BW formula for F(E) is obviously the PA of the
order [(K−1)/K] to Q(E)

K

C k
(E−E k ) + iΓ k /2

F(E) = i (1.11)

k=1