The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 79

The rhs of (2.218) is recognized as the geometric series representation of the
time signal point c n+m so that

c n+m = (Φ ′ m |Φ ′ n ) = S n,m . (2.219)

Thus, in the basis{Υ k }(1≤k≤K), the time signal c n+m appears as the

matrix element of vectors |Φ ′ n ) and (Φ ′ m |. Since, in general, c n+m = 0 for

arbitrary nonnegative integers n and m, functions |Φ ′ n ) and (Φ ′ m

| are not
orthogonal to each other and, as they stand, they do not form a basis set.
The matrix element between|Φ ′ n ) and (Φ ′ m |from (2.219) represents, in fact,

the socalled overlap S n,m ≡(Φ ′ m |Φ ′ n ). This overlap S n,m is nothing but the

signal point c n+m by virtue of (2.219). Note that the set of K vectors from
(2.216) could be orthogonalized by using, e.g., the GramSchmidt or Lanczos
orthogonalization. However, orthogonalization is a convenience, but not a
necessity in the expansion methods.
Although they do not represent a basis, the K−dimensional vectors
{Φ ′ 1 , Φ ′ 2 , Φ ′ 3 ,..., Φ ′ K

}from (2.216) can be easily shown to be linearly indepen
dent 3 . By contrast, the augmented set of vectors{Φ ′ 1 , Φ ′ 2 , Φ ′ 3 ,..., Φ ′ K , Φ ′ K+1 }

is linearly dependent and so is any other sequence with more added Φ ′ −terms,
{Φ ′ 1 , Φ ′ 2 , Φ ′ 3 ,..., Φ ′ K+m

}(m = 1, 2,...). This follows from the wellknown fact
that the necessary and su cient condition for a set of vectors to be linearly
dependent is that the Gram determinant with the elements S n,m = (Φ ′ m |Φ ′ n )

is equal to zero

det{S n,m }= 0 (n,m = 1, 2,...,K + 1). (2.220)

Due to (2.218), the lhs of (2.220) is equal to the Hankel determinant H K+1 (c 0 ).
Therefore (2.220) can be rewritten as H K+1 (c 0 ) = 0

det{S n,m }= H K+1 (c 0 ) = 0 (n,m = 1, 2,...,K + 1). (2.221)

This coincides with the condition (2.204). Therefore, we see that the previous
checkings for the fulfillment of the condition (2.204), by which the exact num
ber K of the harmonic components in the time signal c n can be determined, is
entirely equivalent to verifying whether the vectors{Φ ′ 1 , Φ ′ 2 , Φ ′ 3 ,..., Φ ′ K , Φ ′ K+1 }

are linearly dependent. Hence, this is yet another way of arriving to the same
criterion (2.204) for determining the exact number K of true resonances.
Recall that, by reference to (2.198), zerovalued amplitude d K+m for m > 1
represents a signature of spuriousness in the time signal c n with precisely K

3 Let S ={ξ 1 , ξ 2 , ξ 3 , ..., ξ M }be a finite set of M distinct elements in a linear spaceM. The
set S is said to be linearly dependent if there exists a final set of scalars{a 1 , a 2 , a 3 , ..., a M }
(not all zero) such that P m=1 a m ξ m = 0. Conversely, the set S will be linearly independent,
if it is not linearly dependent. In other words, when a set S is linearly independent, then
for any choices of distinct elements{ξ 1 , ξ 2 , ξ 3 , ..., ξ M }in S and scalars{a 1 , a 2 , a 3 , ..., a M },
the relation P m=1 a m ξ m = 0 could be fulfilled if and only if all the M values of the scalars
{a m } m=1 are simultaneously equal to zero, a m = 0 (m = 1, 2, 3, ..., M ).