Harmonic transients in time signals - Signal Processing in MRS with Biomedical Applications - page 150

pronounced stability. This stability is the main reason for wide applications
of the Pade approximant via continued fractions for computation of virtu
ally all elementary and special functions. As mentioned earlier, the diagonal
[K/K] f (z) and paradiagonal [(K±1)/K] f (z) Pade approximant to a given
function f(z) are equivalent to the continued fractions [5].
In MRS, virtually all spectra have rolling backgrounds that can be naturally
described by the offdiagonal FPT. This is due to the fact that the quotient
P L /Q K for L > K can always be expressed as a sum of a polynomial B L−K
and the diagonal FPT which is the remaining quotient A K /Q K (see also
chapter 2, section 2.6)

G L,K (ω)∝ P L (ω)
Q K (ω)

= B L−K (ω) + A K (ω)

Q K (ω) .

(4.2)

The polynomial B L−K describes the background, and the new diagonal ratio
nal function A K /Q K is responsible for the polar structures of the spectrum,
i.e., its resonances. Crucially, the particular mathematical form (4.1) of the
quantummechanical rational response function to the given external pertur
bations or excitations is dictated by the intrinsically quantum origin of time
signals from MRS. The same form is also prescribed by the resolvent structure
of the Green operator, which generates the entire dynamics of the investigated
general system [5].
The specific rational form of a generic response function R(ω) in many
research fields is also implied by a standard connection between the input
I(ω) and the output O(ω) data in linear systems

Output = Response×Input O(ω) = R(ω)I(ω). (4.3)

Therefore, for a general system, R(ω) ought to be a rational algebraic function
as a ratio of the output and input data

Output
Input

R(ω) = O(ω) Response =

I(ω) .

(4.4)

The simplest form for the response function is given by a quotient of two
polynomials O L (ω) and I K (ω). In such a case, R(ω) as denoted by R L,K (ω)
becomes

R L,K (ω)∝ O L (ω)
I K (ω)

(4.5)

and this is precisely the FPT from (4.1). Nevertheless, simplicity is not the
sole reason which guides us to a rational polynomial for the response function.
More importantly, the rational functional form is dictated by the physics of the
response mechanism of any studied system upon exposure to external pertur
bations. This occurs because a system excited by an external field is automat
ically set into an oscillatory type of motion for which a differential equation of
a definite order represents the appropriate mathematical description. A like