Digital Signal Processing Reference
In-Depth Information
As discussed, even if different signal processors mutually agree, this still does
not necessarily imply that they retrieved the true spectral parameters. The
FPT
(+)
and FPT
(−)
possess a joint conceptual design for a system response
function in the GreenPade functions G
K
(z) and G
−
K
(z
−1
) within the same
general mathematical Pade methodology. These versions from the FPT are
computationally complementary and, therefore, they could rightly be regarded
as two distinct strategies for the same problem. As stated, the main difference
between these two variants is in that the FPT
(−)
accelerates slowly conver
gent series [18], whereas the FPT
(+)
converts divergent series into convergent
ones by the principle of the Cauchy analytical continuation [19]. Specifically,
for the diagonal case [K/K]
G
N
(z
−1
), the optimal performances of the FPT
(−)
and FPT
(+)
are expected for N = 2K and N > 2K, respectively. The two
requirements, N = 2K and N > 2K yield, respectively, the algebraically de
termined and overdetermined system of linear equations for the expansion
coe
cients p
±
r
and q
±
s
of the polynomials P
±
K
(z
±1
) and Q
±
K
(z
±1
). Determina
tion of these expansion coe
cients, as the critical part of computations, must
be performed with high accuracy to provide maximal precision for the recon
structed fundamental frequencies and amplitudes. Machine accuracy can be
achieved by the outlined algorithm of the FPT, as done in our computations
with the results that are illustrated in chapter 6.
After highly accurate extraction of all the fundamental frequencies and
amplitudes{ω
±
k
,d
±
k
}from the input FID, the GreenPade spectra can be
generated in their complex modes defined by the Heaviside partial fractions
P
±
K−1
(z
±1
)
Q
±
K
(z
±1
)
K
d
±
k
z
±1
−z
±
k
G
±
K
(z
±1
) =
=
.
(2.188)
k=1
One can also establish an alternative representation of the same spectrum by
using only the roots of the numerator and denominator polynomials. To this
end, we additionally need to solve the equation
P
±
K−1
(z
±1
) = 0
(2.189)
which yields the roots denoted by{z
±
k
}(1≤k≤K−1)
≡e
±iω
±
k
z
±
k
τ
.
(2.190)
Thus, employing the sets of the roots{z
±
k
}and{z
±
k
}it becomes feasible to
G
±
K
(z
±1
) in the canonical forms that are fully equivalent to (2.188)
express
P
±
K−1
(z
±1
)
Q
±
K
(z
±1
)
=
p
±
K−1
q
±
K
K−1
k=1
(z
±1
−z
±
k
)
.
(2.191)
K
k=1
(z
±1
−z
±
k
)
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