The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 35

whole information about the considered system. As stated, this is known as
completeness of the quantummechanical description of general phenomena in
nature, and transcribed mathematically by the closure relation

K

π k = 1

π k =|Υ k )(Υ k | 1≤k≤K (2.14)

k=1

where π k is the conventional projection operator.
Clearly, equation (2.14) is also the spectral decomposition of the unity oper
ator, 1. This allows the derivation of the corresponding spectral representation
of f( ˆ ) via multiplication of both sides of equation (2.13) by (Υ k

|. The sub

sequent usage of the closure relation (2.14) gives

K

f( ˆ ) = f(ω k )π k . (2.15)

k=1

For f( ˆ ) = e −i Ωt , the general formula (2.15) leads to the spectral decompo
sition of the evolution operator

U(t) via

K

U(t) =

e −iω k t π k Im(ω k ) < 0. (2.16)

k=1

Hence, the exact spectral representation of the time evolution operator is given
precisely by the sum of K complex damped exponentials with the amplitudes
in the form of the projectors π k . Since the operators ˆ and U(t) are linear,
the same conclusion also extends to a scalar variant of (2.16). This follows by
taking the matrix element of U(t) between the states (Φ 0

|and|Φ 0 )

|U(t)|Φ 0 ). C(t)≡(Φ 0 (2.17)

The obtained quantummechanical quantity C(t) represents the continuous
autocorrelation function. Its physical meaning is understood by the argument
which runs as follows. As seen in (2.5), the state|Φ(t)) at time t is generated
by propagation of the initial wave packet|Φ 0 ) from t = 0 to t through U(t).
Given a fixed |Φ 0 ) at t = 0, there will be a nonzero chance of detecting
the system in the state|Φ(t)) at the subsequent time t > 0, if the overlap
between these two wave packets is not zero. In quantum mechanics, this type
of overlap is defined via the scalar product of|Φ(t)) and (Φ 0 |. The result is the
matrix element (Φ 0 |Φ(t)) which is C(t) from (2.17) by reference to (2.5). The
autocorrelation function C(t) indicates the degree of correlations between
the states |Φ(t = 0)) and |Φ(t = 0)) in the course of the exposure of the
system to the action of the dynamical operator ˆ . The operator ˆ itself is
responsible for the difference between|Φ(t)) and|Φ 0 ), as seen in (2.4). When
the dynamics are turned off by setting ˆ = 0, the system would indefinitely
remain in the initial state|Φ 0 ). This yields C(t) = C 0 = (Φ 0 |Φ 0 ) = 0 at any