The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 64

dent from the following two reasons. For any given time signal c n , represented
by a sum of the K damped complex exponentials c n =

K

k=1 d k u k with, e.g.,

K
k=1 d k /(u−u k )
at any u = exp (−iωτ). Here, the sum can be carried out analytically to yield
the polynomial quotient P K−1 (u)/Q K (u), which is recognized as the para
diagonal FPT. A like rationale also holds true for a more general time signal
in the form of a linear combination of attenuated complex exponentials with
nonstationary polynomialtype amplitudes. Hence, the FPT is optimal for
processing time signals modeled as described. This is expected, since the FPT
is the exact theory for a function which is itself a rational polynomial.
From an algorithmic viewpoint, the FPT gives the unique quantification in
MRS by solving the characteristic equation Q K (u) = 0. The results of such
rooting are the retrieved K values of the complex harmonics{u k

stationary amplitudes{d k }, the exact spectrum reads as

}from the

input time signal. }are re

constructed with no extra labor in computations by using the Cauchy residue
formula d k

Furthermore, the corresponding amplitudes{d k

∝P K−1 (u k )[(d/du)Q K (u)] u=u k for all the distinct roots (u p =

u q , p = q). Once the spectral parameters{u k ,d k }are obtained in this way,

the associated nondegenerate Lorentzian spectrum in the complex mode is
constructed by means of the Heaviside partial fractions

K
k=1 d k /(u−u k ).
Such an analysis extends easily to the case of multiple roots of Q K (u) to
describe fully overlapping (degenerate) resonances.
On the other hand, if no model is presupposed for c n , the resulting ex
act spectrum is the Green function given by the Maclaurin expansion G(u) =

∞

n=0 c n u −n−1 . Here again the FPT is optimal, since its nondiagonal polyno
mial quotient P L (u)/Q K (u) has the best contact with G(u) through the first
L+K expansion terms of the Maclaurin series. This follows from the exact ac
cord between the partial sum

∞ L+K

n=0 c n u −n−1 which
is truncated at n = L + K and the original Maclaurin series of P L (u)/Q K (u)
via the first L+K terms, as per the definition of the FPT, i.e., P L (u)/Q K (u)−
G(u) =O(u −L−K−1 ). As usual, the symbolO(u −L−K−1 ) represents the re
mainder which is itself a series in terms of u −L−K−2−n (n≥0) with calculable
expansion coe cients. For consistency, the outlined two circumstances (with
and without an explicit modeling of the examined FID) need to be inter
related.
With this goal, we substitute the modeled time signal c n =

n=0 c n u −n−1 in G(u) =

K

k=1 d k u k

into

∞

n=0 c n u −n−1 . The obtained infinite
Maclaurin sum from the Green function can be performed exactly by means
of the geometric series with the result G(u) =

the exact modelfree spectrum G(u) =

K
k=1 d k /(u−u k ), which rep
resents the paradiagonal FPT, i.e., P K−1 (u)/Q K (u). Thus, the general spec
trum G(u) without an explicit modeling of the time signal is reduced, when
the signal is afterwards assumed to be modeldependent, to the result which
coincides with the corresponding finding from a different and independent
analysis. Hence consistency of the two representations.
Of course, according to section 2.3 , the Pade frequency spectrum in the form