Harmonic transients in time signals - Signal Processing in MRS with Biomedical Applications - page 151

differential equation also describes the internal state of the system prior to any
external perturbation. Algebraically, these differential or difference equations
are equivalent to the corresponding characteristic or secular equations. This
follows from the fact that the degree of a given characteristic polynomial is
equal to the order of the corresponding differential equation. Thus, solving a
differential equation is equivalent to obtaining all the roots of the associated
characteristic polynomial. Physically, the roots of the characteristic polyno
mial Q K (ω) represent the fundamental frequencies{ω k }(1≤k≤K) of the

intrinsic oscillations of the investigated system. Integer K is the total number
of fundamental damped harmonics which are given in terms of the complex
frequencies{ω k

}.
Thus, under the most general circumstances in virtually all research fields,
including MRS, the adequate physical considerations invariably lead to a poly
nomial quotient for the response function, or a frequency spectrum, as pre
scribed precisely by the FPT. This is anticipated, since the formalism of the
Green function is fully equivalent to the Schrodinger equation through which
quantum physics becomes applicable to any system, including living organ
isms. One of the justifications for this statement is the proven equivalence
of generic time signals with quantummechanical autocorrelation functions
[5]. Due to this fact, every data analysis based upon time signals is auto
matically within the realm of quantum mechanics. Since such signals are
abundantly present in living organisms, there is no need for any further proof
that quantum mechanics is applicable to life sciences. While direct solutions
of the Schrodinger equations are possible to find only for the simplest systems
of a very limited practical utility, the Green functions can be computed for
any physical system regardless of its complexity as long as, e.g., time signal
points are available. We have noted that such time signals that are precisely
of the same structure as autocorrelation functions from quantum physics are
encountered in MRS and in many other fields [5]. Therefore, the quantum
mechanical formalism of the Green functions implemented via, e.g., the FPT
can be used to spectrally analyze these time signals that are either generated
theoretically as autocorrelation functions or measured experimentally as free
induction decay curves.

}and the corresponding complex amplitudes{d k

4.2 The exact solution for the general harmonic inver-

sion problem

The adequacy of modeling any investigated phenomenon chiefly depends upon
the correct description of the underlying physical process. The succinct out
lines from the preceding section indicate that the FPT is the method of choice
for solving the quantification problem in MRS and in other fields that use time