The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 71

resonance are given by φ ± k ≡Arg(d ± k ). The absolute values of the amplitudes,

|d ± k |, are not the only spectral parameters that determine the height of the

corresponding resonance peak. This becomes clear from the ersatz spectrum

K

|d k |
ω−ω k

E

K (ω) =−i

. (2.184)

k=1

As before, this Heaviside partial fraction representation can explicitly be
summed up to produce a quotient of a numerator and denominator polynomial
of degree K−1 and K, respectively. As such,E K (ω) is seen to be a para
diagonal FPT in the variable ω [5]. For the sake of brevity, the superscripts
±in (2.184) are dropped. The absorption ersatz spectrum corresponding to
(2.184) is a sum of K pure Lorentzians

K K

|
ω−ω k

|d k |Im(ω k )

[(ω−Re(ω k )] 2 + [Im(ω k )] 2 .

|d k

Re −i = (2.185)

k=1 k=1

From here, the height h k of the k th Lorentzian peak is given by the expression
|d k

|Im(ω k )/{[(ω−Re(ω k )] 2 + [Im(ω k )] 2 }taken at ω = Re(ω k ) via

|d k |Im(ω k )
[(ω−Re(ω k )] 2 + [Im(ω k )] 2

|d k |
Im(ω k ) .

h k ≡ lim
ω−→Re(ω k )

= (2.186)

Thus, the height h k of the k th Lorentzian peak is determined by the two spec
tral parameters,|d k |and Im(ω k ), and not by|d k |alone, as further illustrated
through the figures in chapter 3. The briefly outlined procedure completes the
determination of all the fundamental complex frequencies and the associated
complex amplitudes. Such reconstructed spectral data yield the sought four
peak parameters (position, width, height, phase) for each of the K Lorentzian
resonances in the spectra computed by both variants of the FPT.
Given the input Maclaurin sum (2.162), the uniqueness of the FPT guaran
tees that the FPT (+) and FPT (−) must reconstruct the identical frequencies

and amplitudes{ω k ,d k }and{ω − k ,d − k }. Furthermore, these retrieved spectral

parameters ought to be equal to the true complex frequencies and amplitudes
{ω k ,d k }from the input FID
ω k = ω − k d k = d − k

= ω k = d k . (2.187)

Such equalities will be fulfilled if the values for all the reconstructed spectral
parameters{ω ± k ,d ± k }have converged as a function of the increased number

of signal points, as shown in chapter 3. Moreover, during this selfcontained
crosschecking, the FPT can also discriminate with certainty between the
genuine and spurious resonances by relying upon the concept of Froissart
doublets [44] or polezero cancellations [45].
The overall meaning of the explained intrinsic crossvalidation in the FPT (±)
is in avoiding altogether the customary need to compare different processors.