Harmonic transients in time signals - Signal Processing in MRS with Biomedical Applications - page 155

∞

n=0 x n

where the geometric series = 1/(1−x) is used, so that

∞ K

d (s)
k

c n+s u −n−1 =

u−u k .

(4.14)

n=0

k=1

This expression represents the most important property of the mathematical
model (4.7). Here, the lhs of (4.14) is recognized as the exact delayed spec
trum which corresponds to the delayed signal (4.7). In theory, the exactness
stems from the presence of an infinite sum, i.e., a series over signal points. We
see that the model (4.7) is indeed powerful, since it does not use any approx
imation to reduce the infinite sum over the signal points

∞

n=0 c n+s u −n−1

to

K

k=1 d (s)

the finite Heaviside partial fractions,

k /(u−u k ), with the retrieved

spectral parameters{u k ,d (s)

}. Naturally, in practice, only a finite number of
signal points is conventionally available and, therefore, the exact spectrum

k

∞
n=0 c n+s u −n−1

cannot be obtained. However, even in this case, the finite

N−1
n=0 x n = (1−x N )/(1−x) can be used to derive the following
exact result for the truncated spectrum

geometric sum

N−1 K

1−(u k /u) N
u−u k

d (s)
k

c n+s u −n−1 =

. (4.15)

n=0

k=1

This formula tends to (4.14) as N−→∞if|u k /u|< 1 (1≤k≤K). In the
response functions (4.14) and (4.15) associated with the infinite (N =∞) and
finite (N <∞) signal length, respectively, the exact delayed spectrum for the
model (4.7) is represented by rational functions. Specifically, the two rational
functions from the rhs of (4.14) and (4.15) are polynomial quotients that
are seen as the Pade approximants for the input sums

∞
n=0 c n+s u −n−1

and

N−1

n=0 c n+s u −n−1 , respectively. Hence, for any processing method employed
to recover the spectral parameters{u k ,d (s) }from the input delayed time

signal of the form (4.7), the ensuing spectrum invariably coincides with the
Pade approximant through the unique quotient of two polynomials for the
given power series expansion such as the Green function (4.12). This shows
that among all the signal processors, the FPT indeed appears as optimal for
parametric estimations of spectra based upon the time signal (4.7).

k

4.5 The key prior knowledge: Internal structure of time
signals
As stated, by construction, the Pade approximant becomes the exact theory
if the input function is a rational function given as a quotient of two poly
nomials, i.e., a rational polynomial.

Of course, the exactness overrides the