The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 83

description in detail.
In order to interconnect some of the subsections from this section, it is
important to ask here whether the phenomenon of Froissart doublets, as man
ifested through (2.226), is consistent with the concept of spuriousness viewed
from the standpoint of linear dependence? This sought consistency can indeed
be confirmed by the argument which runs as follows. In general, as mentioned,
the discussed linear dependence in mathematical modeling means, vaguely
speaking, that something was added which carries no new information. Such
an addition is a linear combination of the already included information. Then,
naturally, this extra information put into the considered model is redundant
and leads to a spurious part in the overall output data of the analysis. This
spuriousness need not be due to computational noise nor to noise in the input
data, as illuminated by pure analytical means from Ref. [35]. Rather, it is a
consequence of the lack of prior knowledge of the exact number K of the con
stituent components of the investigated time signal. In modeling via the FPT,
the fact that we do not know K in advance of the analysis prompts the addi
tion of more superfluous information through increasing the polynomial order.
This is what is done in (2.226) by passing from P K /Q K to P K+m /Q K+m ,
which is equivalent to writing P K+m /Q K+m = P K A m /[ Q K B m ], where we go
beyond K for m≥1, with A m and B m being the two extra polynomials.
Since the unknown exact spectrum is supposed to be P K /Q K , by having
the surpluses A m and B m , we effectively introduce spuriousness. In spite of
this latter obstacle, the FPT manages to properly detect the K genuine com
ponents by forcing A m and B m to coincide with each other, A m = B m ≡C m ,
so that cancellation takes place in the Pade quotient P K+m /Q K+m . By such
a forced confluence of A m and B m , the FPT successfully recovers the exact
spectrum P K /Q K via P K+m /Q K+m = P K A m /[ Q K B m ] = P K C m /[ Q K C m ] =
P K / Q K , as in (2.226) where, for simplicity, P K / Q K is relabeled as P K /Q K .
Moreover, the precise kind of spuriousness brought by A m and B m can now
be determined by reference to the algorithm which produces these two extra
polynomials. To this end, let us take as an example the simplest case m = 1,
where we enlarge the two systems of equations for the p's and q's (expansion
coe cients) of the P's and Q's (numerator and denominator polynomials)
from K to K + 1. By reference to subsection 2.8.3 , this automatically causes
the corresponding (K + 1)×(K + 1) Gram determinant of the augmented
system to be equal to zero, which implies that the system is singular and,
hence, linearly dependent.
Therefore, the origin of spuriousness brought into the FPT by the emergence
of A 1 and B 1 through augmentation P K /Q K →P K+1 /Q K+1 = P K A 1 /[ Q K B 1 ]
is in the linear dependence of the two systems of K + 1 equations for the
polynomial expansion coe cients. A similar argument is extended to the
case P K+m /Q K+m with m > 1. This is the essence of Froissart doublets
for noiseless time signals as mediated by polezero cancellation in (2.226).
Moreover, it is actually immaterial whether we explicitly computed poles and
zeros at all, since the extra polynomials A m and B m cancel out any way as a