The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 56

2.3 Recurrent time signals and their generating fractions
as spectra with no recourse to Fourier integrals
We recall that one of the equivalent ways to introduce classical polynomials is
to use their socalled generating functions. For example, the Chebyshev poly
nomials of the second kind U n (x) can be introduced as the general expansion
coe cient in the following development

∞

f(x,y)≡ 1
1−2xy + y 2

U n (x)x n = (2.125)

n=0

sin(n cos −1 x)
sin(cos −1 x) .

U n = (2.126)

Here, 1/(1−2xy + y 2 ) is the generating function of U n (x). The generating
function f(x,y) is a rational function, which is a polynomial quotient and, as
such, it represents the PA of the order [0/1] in the variable x for a fixed y or of
order [0/2] in y for a fixed x. The simplest polynomial in x is obtained when
all its expansion coe cients are set to zero and simultaneously the coe cient
of the highest power is taken as one. This is the monomial or power function,
M n (x) = x n , and its generating function is 1/(x−y) according to the binomial
expansion

∞

g(x,y)≡ 1 M n (x)y −n−1 M n (x) = x n .

x−y =

(2.127)

n=0

This generating function g(x,y) is also a rational function. Moreover, 1/(x−y)
is the PA of order [0/1] in the variable x for a fixed y or likewise in y for a
fixed x. Such reasoning can be extended to the operatorvalued functions. For
example, the Green operator

G(u, U)≡G(u) = (u1−U) −1

is the generating

function of the operator monomial M n ( U) = U n

via

∞

1
u1−U

G(u) =

M n ( U)u −n−1 M n ( U) =

U n . (2.128)

=

n=0

|G(u)|Φ 0 ) is a generating function
for the autocorrelation functions or signal points (Φ 0

Likewise, the Green functionR≡(Φ 0

|{u1−U} −1 |Φ 0 ) =

|U n |Φ 0 ). To get an insight into the functional
dependence of the generating functionR(u) for the c n 's, one can use the
Schrodinger eigenvalue problem for the evolution operator U viz U|Υ k ) =
u k

∞

n=0 c n u −n−1

where c n = (Φ 0

|Υ k ). The eigenvectors{|Υ k )}form a basis and the completeness can be ex
pressed through the spectral decomposition of the unity operator 1 =

K
k=0 π k