Harmonic transients in time signals - Signal Processing in MRS with Biomedical Applications - page 159

are retained as physical/genuine, whereas those frequencies that are altered
when passing from s = 1 to s = 2 are rejected as unphysical/spurious. Sim
ilarly, in the nonstatespace variant of the FPT, one uses different powers s
of the evolution operator U that are implicit in the paradiagonal elements
[(n + s−1)/n] G (u) of the Pade table corresponding to the delayed counter
part G (s) (u) of the Green function G(u) from (4.12). In this case, one selects
several values of s to discriminate between physical and spurious eigenroots
of the denominator polynomials that are the said characteristic polynomi
als. The eigenroots that are stable/unstable for different s are considered as
physical/unphysical, respectively. These practical advantages lend support to
taking the initial times t s different from the conventional one, t 0 = 0.

4.6 The Rutishauser quotient-difference recursive algo-
rithm
The Pade approximant can alternatively be written in the form of continued
fractions, i.e., the CFs, as a staircase with descending quotients. There exist
several equivalent symbolic notations for a given CF and two of them read as
A 1

≡ A 1

A 2
B 2 +

A 3
B 3 +· · ·

(4.16)

B 1 +

A 2

B 1 +

A 3
B 3 + . . .

B 2 +

where the plus signs on the rhs of (4.16) are dropped, i.e., lowered to point at
a 'stepdown' process in forming the CF. The lhs of (4.16) represents quite a
natural way of writing the staircaseshaped continued fraction. However, for
frequent usage, the rhs of (4.16) is obviously more economical since it takes
less space. The rhs of (4.16) could be alternatively written by employing the
ordinary plus signs via A 1 /(B 1 + A 2 /(B 2 + A 3 /(B 3 +))).
The infinite and m th order delayed CF [5] related to the time series (4.11)
are introduced as
G CF(s) (u) = a (s)

a (s)
2r+1
u

a (s)
2
1

a (s)
3
u −· · ·−

a (s)
2r
1

1
u −

(4.17)

− − −· · ·

a (s)
2m+1
u

(u) = a (s)

a (s)
2
1

a (s)
3
u −· · ·−

a (s)
2m
1

G CF(s)
m

1
u −

(4.18)

− −

where a (s n are the expansion coe cients. Further, we can define the even (e)
and odd (o) contracted continued fractions (CCF) of order n that contain 2n
and 2n + 1 terms from the corresponding CFs
G CCF(s)
e,n

(u) = G CF(s)
2n

(u) = G CF(s) G CCF(s)

o,n

(u)

2n+1 (u)

(4.19)