The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 40

of being a veritable alternative to the Schrodinger equation, especially for
spectral analysis and signal processing. Furthermore, it is advantageous to
solve the Green eigenvalue problem (2.29) instead of the original Schrodinger
equation (2.1) because G(ω) from the former equation is a bound operator,
whereas ˆ from the latter one is an unbound operator 1 . This means that the
eigenvalues{ω k }obtained from the Green eigenproblem can simultaneously

possess the lower and upper bounds or limits, whereas only the lower bounds
could be provided by the Schrodinger equation. In the Green eigenproblem
(2.29), frequency ω is considered as a scaling parameter, which can be taken to
be ω = ω 0 , where ω 0 is any selected fixed number. Upon finding the solution
of (2.29) for ω = ω 0 , the sought eigenfrequencies{ω k }can be deduced from

the eigenvalues λ k (ω 0 ) through the following relationship

− 1 ω k = ω 0

λ k (ω 0 ) .

(2.31)

Advantageously, the corresponding eigenfunctions{|Υ k )}are the same in
the Green and the Schrodinger eigenvalue problem. This feature has already
been encountered previously in the case of the general equation (2.13).

2.2 Expansion methods for signal processing
2.2.1 Non-classical polynomials
Herein, we will employ real orthogonal polynomials{Q r (x)}as basis functions
to establish an expansion method for computation of the most important phys
ical quantities, such as Green functions, density of states, integrated density
of states, etc. Such polynomials are introduced on the real axis in the interval
[a,b] which can be finite or infinite. The expansion coe cients of these poly
nomials are also realvalued. However, since Q r (x) is an analytic function,
the quantity Q r (z) exists for a complex variable z in the limit Im(z)−→0.
For a real x and a complex z we have

Q ∗ n (x) = Q n (x) Q ∗ n (z) = Q n (z ∗ )

(2.32)

where the star superscript denotes the usual complex conjugation. By dσ(x)
we denote a nonnegative Riemann measure on [a,b]

dσ(x) = W(x)dx (2.33)

1 An operator is said to be bound if and only if its norm is limited both from below and
above. However, an operator bound only from below (as is actually the case with most
Hamiltonian operators) is an unbound operator.