Harmonic transients in time signals - Signal Processing in MRS with Biomedical Applications - page 176

and, thus

G FPT(s)+
n

(u) = G LCF(s)
e,n

(u)

(u) = G CF(s)
2n

= G CCF(s)
e,n

(u) (n = 1, 2, 3 ...). (4.110)

We see from this result that the delayed diagonal fast Pade transform, FPT (s)+ ,
with the convergence region inside the unit circle (|u|< 1) coincides with the
even part of the delayed Lanczos continued fraction for any order n. Fur

thermore, in this method, the ensuing PadeGreen function G FPT(s)+

n (u) is
convergent for |u| < 1, as opposed to the original truncated Green func
tion (4.79) which is divergent in the same region, i.e., inside the unit circle.
Therefore, inside the unit circle, the delayed diagonal fast Pade transform
G FPT(s)+

n (u) employs the Cauchy concept of analytical continuation to in
duce or force convergence into the originally divergent series, which is the
Green function (4.77).
Overall, we can conclude that the introduction of G FPT(s)± (u ±1 ) is ex

n

tremely helpful in proving that the same LCF, i.e., G LCF(s) (u) contains implic

n

itly both G FPT(s)− (u −1 ) (as an accelerator of monotonically converging series)

n

and G FPT(s)+

n (u) (as an analytical continuator of divergent series/sequences).
Specifically, G FPT(s)−

(u −1 ) and G FPT(s)+

(u) are equal to the odd and even

n n

part of G LCF(s) (u), i.e., G FPT(s)− (u −1 ) = G LCF(s) (u) and G FPT(s)+ (u) =

n n o,n n

G LCF(s)

(u), respectively. This means that the FPT (s)− and FPT (s)+ contain e,n

twice as many terms as the original CF, i.e, G FPT(s)− (u −1 ) = G CF(s)

2n+1 (u) and

n

G FPT(s)+ (u) = G CF(s)

2n

(u). Hence, the FPT (s)− and FPT (s)+

are equal to the

n

odd and even contracted continued fractions, G FPT(s)− (u −1 ) = G CCF(s) (u)

n o,n

and G FPT(s)+ (u) = G CCF(s)

e,n (u), as per (4.96) and (4.110), respectively. When
such equivalences are established, it appears as optimal to employ only the
quantities G LCF(s)

n

(u) and G LCF(s)

(u), as the two equivalent expressions for

e,n o,n

G FPT(s)+ (u) = G CF(s)

e,2n (u) and G FPT(s)−

(u −1 ) = G CF(s)

o,2n+1 (u) for any fixed
s (s = 0, 1, 2,...), in order to generate the two sets of observables in the
FPT (s)+ and FPT (s)− that converge inside|u|< 1 and outside|u|> 1 the
unit circle, respectively. Crucially, the PadeGreen functions G FPT(s)+

n n

n (u) and
G FPT(s) n (u −1 ) are, respectively, the lower and upper bounds of the computed
observables (spectra, eigenfrequencies, density of states, etc.). For instance,
Re(G FPT(s)+ (u)) = Re(G LCF(s) (u)) and Re(G FPT(s)− (u −1 )) = Re(G LCF(s)

o,n (u))
represent the lower and the upper limits of the envelope of the absorption
total shape spectrum for a given time signal{c n+s

n e,n n

}(0≤n≤N−1).

Likewise, the eigenfrequencies and residues{ω (s)+
k

,d (s)+
k

}and{ω (s)−
k

,d (s)−
k

}
(k = 1, 2,...,K), that are obtained using the two variants of the FPT, i.e.,
G FPT(s)+
K

(u) = G LCF(s)
e,K

(u) and G FPT(s)−
K

(u −1 ) = G LCF(s)

o,K (u) are, respectively,
the lower and upper limits of the true, i.e., exact values{ω k ,d k }.