The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 78

2.8.3 The role of linear dependence as spuriousness in de-
termining K within the state space-based perspective
of signal processing
The investigated FID from its geometric progression c n =

K

k=1 d k z k can also
be viewed from a state spacebased perspective in signal processing. This can
be done by introducing a K−dimensional linear vector spaceKwith the sym
metric inner product (2.3) and a basis{Υ k }(1≤k≤K) which is a complete

set of the solutions of the stationary Schrodinger equation (2.6). The cor
responding nonstationary solution|Φ(t)) of the timedependent Schrodinger
equation (2.1) is given by (2.4). Its discretized version is|Φ n ) = U n |Φ 0 ) where
U is the time evolution operator U≡U(τ) = exp (−i ˆ τ). Here, the continu
ous time variable t is discretized via t = nτ (0≤n≤N−1), where τ is the
sampling rate. The two equivalent representations Φ and Υ can be related to
each other. This is done by using the spectral decomposition (2.16) of the time
evolution operator in Φ n to arrive at the expansion Φ n =

K

k=1 d 1/2

u k

|Υ k ),
where u k = exp (−iω k τ) and d k is the amplitude/residue from (2.19), i.e.,
d k = (Φ 0 |Υ k ) 2 . To cohere with the notation from the other subsections in
this section, we shall switch from the u−to the z−variable, in which case Φ n
will be replaced by Φ ′ n where|Φ ′ n ) = exp (in ˆ τ)|Φ 0 ) and

k

K

d 1/2
k

Φ ′ n = z k d k = (Φ ′ 0 |Υ k ) 2 .

|Υ k ) (2.215)

k=1

This K−dimensional state vector Φ ′ n is defined through its K components or
“coordinates”

Φ ′ 0 ={d 1/2 ,d 1/2

2

,d 1/2
3

,,d 1/2
K

}

1

Φ ′ 1 ={d 1/2 z 1 ,d 1/2 z 2 ,d 1/2 z 3 ,,d 1/ K z K }

1 2 3

Φ ′ 2 ={d 1/ 1 z 1 ,d 1/ 2 z 2 ,d 1/ 3 z 3 ,d 1/2

K z 2 K

}

.

Φ ′ n ={d 1/2 z 1 ,d 1/2 z 2 ,d 1/2 z 3 ,,d 1/ K z K }. (2.216)

1 2 3

} k=1 In other words, since{Υ k is a basis in the spaceK, any function from

this space including Φ ′ n

∈Kcan be expanded in terms of the elements Υ k
such that the k th component/coordinate of Φ ′ n is d 1/2

k

u k

as in (2.215), or

equivalently
Φ ′ n = d 1/ 1 z 1 Υ 1 + d 1/ 2 z 2 Υ 2 + d 1/ 3 z 3 Υ 3 ++ d 1/2

K z K Υ K . (2.217)

Taking the scalar product between|Φ ′ n ) and (Φ ′ m |Φ ′ n ), setting the

normalization of Υ k to unity through||Υ k ||≡1 and using the orthogonaliza
tion (Υ k ′ |Υ k ) = δ k,k ′ , it follows from (2.216) and (2.217)

|via (Φ ′ m

K

S n,m ≡(Φ ′ m |Φ ′ n ) = d k z n+m

. (2.218)

k

k=1