Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 189

residues d ± k are indeed found to be zerovalued, this could only be due to the
zero distance between the corresponding poles and zeros z ± k,Q

−z ± k,P = 0, as
per (5.30). With such a graphic picture, it becomes crystal clear why Frois
sart doublets must have zero amplitudes. Such an occurrence was already
encountered in chapter 2 via expression (2.227) and illustrated in chapter 3
through a preview via Fig. 3.19.

5.8 Pade partial fraction spectra
When the spectral parameters{z ± k,Q ,d ± k

}are reconstructed by the FPT (±)
following the explained procedure, yet another form for the Pade complex
mode spectra P ± K (z ±1 )/Q ± K (z ±1 ) can be obtained. These spectra are the
Heaviside or Pade partial fractions that have the following representations for
the diagonal versions of the FPT (±)

where the numerator and denominator

polynomials have the same degree K

K

P ± K (z ±1 )
Q ± K (z ±1 )

d ± k z ±1
z ±1 −z ± k,Q

= b ± 0 + . (5.35)

k=1

The factored terms b ± 0 in (5.35) are called baseline constants that describe
the corresponding flat backgrounds

≡ p ± 0
q ± 0

b ± 0 . (5.36)

It is possible to invert the frequency spectra from (5.35) by the inverse fast
Pade transforms (IFPT). These inversions yield the corresponding time signals

K K

d ± k e ±2iπnτ ν ± k,Q c ± n = b ± 0 δ(n) + d ± k z ±n = b ± 0 δ(n) +

(5.37)

k,Q

k=1 k=1

where the n th power of z ± k,Q ≡(z ± k,Q ) n . The symbol δ(n) is

the usual discrete unit impulse (or discrete unit sample, or discrete unitstep
time signal). The meaning of this quantity is the same as in the Kronecker
δ−symbol δ n,0 , i.e., δ(n) = 1 for n = 0 and δ(n) = 0 for n = 0. For this
reason, δ(n) is alternatively called the Kronecker discrete time sequence. In
practice, care must be exercised not to interpret δ(n) as a sampled version of
the associated continuous Dirac delta function δ(t). The latter function δ(t)
cannot be sampled, because it is infinite at time t = 0 [5, 84].
Within the FPT (+) , the formulae (5.35) and (5.37) for the spectra and the
corresponding reconstructed time signal, respectively, can be simplified by

is denoted by z ±n
k,Q