The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 36

time t, as per (2.2). At any two times t ′ and t with t < t ′ , the state|Φ(t ′ ))
can be viewed as a delayed copy of |Φ(t)). Hence, the overlap (Φ(t ′ )|Φ(t))
taken at t < t ′ indeed gives a measure of correlation between the state|Φ(t))
and its delayed copy|Φ(t ′ )). The special case t ′ = 0 gives the autocorrelation
function, C(t) = (Φ(0)|Φ(t)).
Once (2.16) is inserted into (2.17), the following expression is obtained

K

e −iω k t d k C(t) = (2.18)

k=1

where Im(ω k ) < 0, as in (2.16). During the derivation of (2.18), the quantity
d k is identified as the matrix element of the projection operator π k taken over
the initial state

d k ≡(Φ 0 |π k |Φ 0 ) = (Φ 0 |Υ k ) 2 (2.19)
where symmetry (2.3) of the inner product is used. The result d k = (Φ 0 |Υ k ) 2

is the residue linked to the eigenfrequency ω k .
The usefulness of the spectral representation (2.16) can be seen by introduc
ing the Green operator as the operatorvalued Fourier integral of the evolution
operator

G(ω)≡ ∞

dt e iωt U(t). (2.20)

0

The operator integral (2.20) can be reduced to the ordinary integration by
substituting the spectral representation of

U(t) into (2.20).

The resulting

scalar integral is carried out via
∞ ∞

K K

dt e iωt U(t) = dt e i(ω−ω k +iǫ)t = π k (ω−ω k + iǫ) −1 . π k

k=1 k=1

0 0

Here, ǫ−→0 + is an infinitesimally small positive number known as the Dyson
exponential damping factor, which is introduced to secure convergence at the
upper limit t =∞. As such, the final result is understood with a limit being
taken for ǫ to approach zero through positive numbers, as indicated by the
plus superscript on zero, 0 + . Thus, the Green operator (2.20) becomes

K

π k
ω−ω k + iǫ

G(ω) =−i

ǫ−→0 + . (2.21)

Im(ω k ) < 0

k=1

Now, in principle, the ǫ−damping might safely be set to zero (as will be done
in the sequel), since ω k is already a complex number with Im(ω k ) < 0, as
per (2.16). The sum over k in (2.21) can explicitly be performed to yield the
equivalent exact expression

P K−1 (ω)
Q K (ω) .

G(ω) =−i

(2.22)