Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 86

where u k = e −iω k τ

|> 1, which is implied by the maintained in
equality Re(ω k ) > 0. In other words, when the harmonic variable z −1

with |u k

is used

from the outset, the absolute value|c n |of every element c n of the entire set

{c n |<∞, but only if we redefine

z k , i.e., the old definition z k = exp (iω k τ) from (3.2) cannot remain intact.
The necessary redefinition is z k

}(0≤n≤N−1) will still remain finite,|c n

≡u k , as directly prescribed by (3.3). Si
multaneously and advantageously, by this redefinition we return to the old
z−variable instead of introducing the new u−variable. Thus, if we want the
fundamental harmonic variables z k

≡exp (−iω k τ) to be the new signal poles,
rather than the old ones z k = exp (iω k τ), such that in both cases Re(ω k ) > 0,
then the following “new” representation of the FID must consistently be em
ployed

K

d k (z k ) −n ≡e −iω k τ

c n = z k Re(ω k ) > 0 |z k |> 1 (3.4)

k=1

where|c n |<∞. It is seen here that all the new signal poles z k = e −iω k τ lie
outside the unit circle because|z k |=|e −iτ Re(ω k )+τ Im(ω k ) |> 1 for Re(ω k ) > 0.
Working backwards by substituting the redefined signal poles z k = e −iω k τ
into the “new” representation c n =

K
k=1 d k (z k ) −n

from (3.4) yields c n =

K
k=1 d k exp (inω k τ), which is the old representation from (3.2), as expected.
Likewise, when the old signal poles z k = e iω k τ are inserted into the old repre

K
k=1 d k (z k ) n from (3.2), the expression (2.193) is obtained.
Hence, the two representations (3.2) and (3.4) are, in fact, the same. They are
written as two formally different expressions depending whether z k = e iω k τ
or z k = e −iω k τ is used for the signal poles.
When displayed as the corresponding Argand plots in the complex z−plane,
the FID points{c n

sentation c n =

}in the representation (3.2) are built from the signal

poles z k = e iω k τ

|< 1. Alternatively, if
depicted as the associated Argand plots in the complex z −1 −plane, the FID
points{c n

that all lie inside the unit circle,|z k

}in the representation (3.4) are generated using the signal poles
z k = e −iω k τ

that are all located outside the unit circle,|z k |> 1. Suppose that

the input signal poles z k = e iω k τ

|< 1 inside the unit circle from the
representation (3.2) are all located, e.g., in the first quadrant of the z−plane,
as in our illustrations. Then, the locations of the alternative input signal poles
z k = e −iω k τ from the representation (3.4) will be automatically found in the
fourth quadrant of the z −1 −plane outside the unit circle. This follows from
the relationship z − k = 1/z k .
When dealing with the input data, the two representations (3.2) and (3.4)
are a pure formality, of course. Nevertheless, the above outlines are made in
order to clearly trace the origin of the two variants of the fast Pade transform,
the FPT (+) and FPT (−) , that construct the response functions using the
harmonic variable z and z −1 , respectively. Specifically, using the same FID,
the FPT (+)

with|z k

are set to reconstruct the poles z k and z − k and FPT (−) as the

Pade approximations to the corresponding input data z k and z −1 via z k

≈z k

k