Biomedical Engineering Reference
In-Depth Information
fd
k
;
z
k;Q
g:
d
k
via
By definition, the corresponding amplitudes
are the Cauchy
z
k
0
;Q
/ ¤
z
k;Q
k
0
¤ k;
residues of the spectra from (
25.5
). For non-degenerate roots
.
d
k
D
lim
z
˙1
!
z
k;Q
f.
z
˙1
z
k;Q
/ŒP
K
.
z
˙1
/=Q
K
.
z
˙1
/g;
we have,
so that:
Y
P
K
.
z
k;Q
/
z
k;Q
z
k
0
;P
p
K
q
K
d
k
D
D
(25.6)
d
z
k;Q
/Q
K
.
z
k;Q
/
z
k;Q
z
k
0
;Q
/
k
0
¤k
.
d
=
.
k
0
D1
d
k
z
k;Q
z
k;P
/;
H)
/ .
(25.7)
d
z
k;Q
/Q
K
.
z
k;Q
/
where
is the first derivative of the denominator polynomials.
Thus, each amplitude has the meaning of a metric in the sense of the distance
given by the separation between the pole and zero,
.
d
=
d
k
z
k;Q
z
k;P
:
This is
consistent with the mathematical complex analysis according to which the Cauchy
residue describes the behavior of line integrals of a meromorphic function around
the given pole. This completes the reconstruction of the
/
2K
complex fundamental
G
.C/
K
G
.
K
.
f!
k
;d
k
g
in the FPT
.˙/
:
z
1
/
parameters
Both
.
z
/
and
from (
25.3
)
z
1
/:
and (
25.4
) approximate the same Green function
G.
The Maclaurin series
z
1
/
G.
is convergent for
j
z
j >1
and divergent for
j
z
j <1;
i.e., outside and
G
.
K
.
inside the unit circle, respectively. The FPT
./
z
1
/
via
is defined in terms
G
.
K
.
G
.
K
.
of the same variable
z
1
z
1
/
z
1
/
as
. Therefore,
converges outside the
unit circle,
but does so faster than the original Maclaurin series. Hence,
convergence acceleration of
j
z
j >1;
z
1
/
by the FPT
./
:
On the other hand, the FPT
.C/
G.
G
.C/
K
.
/
through
z
employs the variable
z
and, and as such, converges inside the unit
j
j <1
G.
z
1
/
circle (
z
), where the exact Green function
diverges. Hence, analytical
Overall, the FPT
.C/
and FPT
./
are optimally
suited to work inside and outside the unit circle, respectively. Nevertheless, by the
Cauchy concept of analytical continuation, they are both well defined everywhere
in the complex plane with the exception of the pole positions that are located at
z
˙1
G.
z
1
/
by the FPT
./
:
continuation
G
.˙/
K
z
k;Q
:
z
˙1
/ D P
K
.
z
1
/=Q
K
.
z
1
/
D
However, physical spectra
.
in the
FPT
.˙/
are perfectly well defined even at the poles, since
z
˙1
z
k;Q
¤ 0
for
real frequencies that are of interest in practice. The internal cross-validation in
the fast Pade transform is achieved upon full convergence in the FPT
.˙/
leading
!
k
!
k
d
k
d
k
to the agreements
are
the genuine amplitude and frequencies from the time signal (
25.1
). It is in this
straightforward way that the FPT
.˙/
is able to solve exactly the harmonic inversion
problem (quantification) by using only the sampled time signal
!
k
and
d
k
where
f!
k
;d
k
g
fc
n
g
to reconstruct
all its constituent fundamental frequencies and amplitudes
f!
k
;d
k
g
according
to (
25.1
).
The spectra
P
K
.
z
˙1
/=Q
K
.
z
˙1
/
are meromorphic functions, since poles are their
z
k;Q
g
z
k;P
g
only singularities. Poles
f
and zeros
f
of these spectra are the roots of
Q
K
.
z
˙1
/ D 0
P
K
.
z
˙1
/ D 0
and
, respectively. Here, as elsewhere, the harmonic
