The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 80

components. The mechanism of this signature is in the condition (2.204).
The same condition (2.204) determines that the vectors of K +m components
{Φ ′ 1 , Φ ′ 2 , Φ ′ 3 ,..., Φ ′ K+m }for m≥1 are linearly dependent. On the other hand,

the overlap (Φ ′ m |Φ ′ n ) of these vectors is equal to the signal point c n+m accord

ing to (2.218). Thus, in the representation (2.218), the signal c n+m with K
harmonics is viewed as being built from the overlap of the two state vectors
|Φ ′ n ) and (Φ ′ m |, where according to (2.216) each Φ ′ r (r = n,m) has K compo

={d 1/2
1

z 1 ,d 1/2 z 2 ,d 1/2 z 3 ,,d 1/ K z r K nents Φ ′ r

}. Therefore, linear dependence

2 3

of vectors{Φ ′ 1 , Φ ′ 2 , Φ ′ 3 ,..., Φ ′ K , Φ ′ K+m

}for m > 1 represents the same kind of
spuriousness as the one due to d K+m = 0 for m > 1.
In practice, while using the FPT (−) , we search for stabilization of all the
genuine spectral parameters and for identification of every spurious resonance.
Such a stabilization occurs at those Pade orders K ′ that are larger than the
sought true number K of resonances, K ′ ≡K + m (m > 1). For K ′ > K,
spurious resonances stem from a linear dependence of their state vectors{Φ s }
(spurious states for K ′ < K will be discussed in chapter 6). Due to zero am
plitudes of all the components of a spurious vector|Φ ′ s ) =

K+m
k=1

d 1/2
k

z k

|Υ k )
for m > 1, its overlap with any genuine (or spurious) state is equal to zero
(Φ ′ s

|Φ ′ r ) = c r+s = 0∀r,s (for Φ ′ r genuine or spurious & Φ ′ s spurious)
where r = s is permitted if Φ ′ r and Φ ′ s are both spurious states. IfAandB
are the two disjoint vector spaces of genuine and spurious states, respectively,
then it follows from the preceding expression that

A⊥B (Agenuine &Bspurious vector spaces). (2.222)

However, c r+s = 0 from (Φ ′ s |Φ ′ r ) = 0 should not be confused with zero filling

from the FFT, in which N zeros c n = 0 (n > N−1) are added to the input
time signal c n (0≤n≤N−1) to achieve a sinctype interpolation. When Φ ′ r
and Φ ′ s

coincide, the overlap (Φ ′ s |Φ ′ s ) = c 2s = 0 becomes the squared norm

||Φ ′ s |Φ ′ s ) 1/2 = 0. A

normalized state is given by N m Φ ′ m where N m is the normalization constant.
An unnormalized physical state Φ ′ m

|| 2 . Therefore, spurious states have zero norm||Φ ′ s ||≡(Φ ′ s

is normalized to the unit particle flux

via (N m Φ ′ m |N m Φ ′ m ) = 1, which gives N m = 1/||Φ ′ m ||. Spurious states{Φ ′ s

}
cannot be normalized, since their normalization constants are equal to infinity
{Φ ′ s

} normalized = N s {Φ ′ s Φ ′ s

} un−normalized (N s =∞for spurious) (2.223)

N s ≡ 1
||Φ ′ s || (||Φ ′ s

Φ ′ s ||= 0 for spurious). (2.224)

Zero norms, or equivalently, infinite normalization constants of spurious states
correspond to the zero flux of particles. Thus, spurious states do not describe
any physical particle. Overall, the lesson from this subsection is that linearly
dependent vectors or equations bring no new information whatsoever to the
analysis. Rather, when they are added to the remaining genuine, linearly
independent vectors or equations, spurious Froissart doublets are obtained.