The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 38

When (2.21) is inserted into (2.28), the result (2.18) for the autocorrelation
function C(t) is obtained as the sum of K damped complex exponentials.
This is enabled by passing to the complex frequency plane and employing
the Cauchy residue theorem to perform the integral exactly through the sum
of poles at ω = ω k . Functions G(ω) and C(t) are associated with the same
operator ˆ as the only source of the whole information about the system.
Such a property and the possibility of switching from C(t) to G(ω) or the
other way around is a guarantee that the information is fully preserved when
passing from the time to the frequency domain or vice versa. The spectral
resonance parameters from (2.24) are the position, width, height and phase of
the k th peak that are equal or proportional to Re(ω k ), Im(ω k ),|d k |, Arg(ω k ),

respectively. By definition (2.19), the residue d k is a measure of the extent
of the squared projection of the state |Υ k ) onto (Φ 0 |. In other words, the

amplitudes{d k }are the weights that contain information about the extent

of the contributions from individual fundamental frequencies{ω k }to C(t) in
(2.18). Furthermore, the magnitudes{|d k |}are the intensities of the harmon
ics{exp (−iω k t)}that are the principal components in the autocorrelation
function C(t). Moreover, φ k = Arg(d k ) is the k th phase of complexvalued
C(t). The realvalued quantities of a particular interest, such as the magni
tude |G(ω)|, power |G(ω)| 2 , absorption Re(G(ω)) and dispersion Im(G(ω))
spectra are all available as soon as the Green function G(ω) is constructed.
This illuminates the key role of the Green function.
The summation from (2.21) represents the Heaviside partial fraction rep
resentation of the Green operator G(ω). This sum is also the spectrum in
terms of a mixed operatorscalar valued function of frequency ω. The outlined
derivation makes no approximation and, therefore, the obtained Heaviside
representation is the exact spectrum. Furthermore, the sum over k in (2.21)
can be performed exactly via

K K K

π k
ω−ω k

|Υ k )(Υ k |
ω−ω k

|Υ k )(Υ k |(ω1− ˆ ) −1 = 1(ω1− ˆ

) −1

= =

k=1 k=1 k=1

where the ǫ−factor is omitted. Here, we employed the following Green eigen
value problem where ω appears as a parameter

1
ω−ω k

G(ω)|Υ k ) = λ k (ω)|Υ k )

λ k (ω) = (2.29)

in accordance with the relationship (2.13) for the particular choice f( ˆ ) =

−i(ω1− ˆ ) −1 . Therefore, we finally arrive at the result

G(ω) =−i1(ω1− ˆ

=−i(ω1− ˆ ) −1 1≡−i(ω1− ˆ ) −1 ) −1

(2.30)

where the unity operator 1 can be placed either before or after the inverse of
the Schrodinger operator ω1− ˆ depending upon whether the Green eigen
value problem is used for|Υ k ) or (Υ k |. Of course, both states|Υ k ) and (Υ k |are