The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 31

Hence, the stated quantummechanical postulate on completeness implies that
everything one could possibly learn about any considered system is also con
tained in the autocorrelation function C(t). Despite the fact that the same full
information is available from Υ, Φ(t), ˆ , U(t) and C(t), the autocorrelation
functions are more manageable in practice, since they are observables. As a
scalar, the quantity C(t) has a functional form which is defined by its depen
dence upon the independent variable t. Hence, since C(t) ingrains the whole
information about the studied generic system, it is critically important to
have the quantummechanical prediction for the shape of the autocorrelation
function. To obtain the explicit form of the autocorrelation function C(t)
for varying time t, one can use the spectral representation of U(t). This rep
resentation is easily obtained via multiplication of the defining exponential
exp (−i ˆ t) for U(t) by the unity operator 1, which is taken from the com
pleteness relation. The end result is U(t) =

K
k=1 exp (−iω k t)π k . Therefore,
preservation of the entire information from the system imposes the form of
the spectral representation of the evolution operator given by the sum of K
damped complex exponentials with the operator amplitudes π k . Substituting
such a representation for U(t) into C(t) yields C(t) =

K
k=1 d k exp (−iω k t).
Thus, the obtained shape of the quantummechanical autocorrelation func
tions represents a linear combination of K complex damped exponentials with
the scalar amplitudes d k . Crucially, this is not a fitting model for C(t) in
troduced by hand. Rather, it is the shape of the autocorrelation function
demanded by the form exp (−i ˆ t) of the quantummechanical evolution op
erator U(t). The derived form of C(t) corresponds precisely to time signals
c(t) in many research branches. Crucially, such time signals are measured in
experiments across interdisciplinary fields.
The outlined procedure establishes the equivalence between time signals
and autocorrelation functions from quantum mechanics. This leads to a di
rect link between quantum mechanics and signal processing as two otherwise
separated branches. Such a connection is of fundamental importance, be
cause a correct mathematical modeling in signal processing can only be done
with a proper physical description of the investigated phenomenon. As seen,
quantum mechanics amply fulfills this strict requirement.

2.1 Direct link of quantum-mechanical spectral analysis
with rational response functions
In quantum physics, spectral analysis in the time domain employs the non
stationary Schrodinger equation [5]

i ∂
∂t

ˆ |Φ(t)).

|Φ(t)) = (2.1)