Digital Signal Processing Reference
In-Depth Information
the point z
∞
located at infinity, z = z
∞
≡∞. Then, the ensuing first L + K
expansion coe
cients of the generated series for R
(PA)
(z
−1
) would exactly
coincide with F
M
(z
−1
) from (12.4)
∞
P
−
L
(z
−1
)
Q
−
K
(z
−1
)
b
−
n
z
−n
=
(12.13)
n=0
L+K
L+K
b
−
n
z
−n
=
c
n
z
−n
∴
(12.14)
n=0
n=0
b
−
n
∴
= c
n
0≤n≤L + K.
(12.15)
Hence M = L + K (QED). Relationship (12.15) follows from the uniqueness
theorem of power series expansions. Thus, the PA from (12.2) and the original
infinite sum F(z
−1
) from (12.4) are in the best contact since they exhibit exact
agreement to within the first L+K terms of the Maclaurin expansion (12.13) of
P
−
L
(z
−1
)/Q
−
K
(z
−1
). Moreover, the result M = L+K simultaneously provides
the Pade estimate of the difference between (12.4) and (12.2), as expressed
symbolically by
F(z
−1
)−
P
−
L
(z
−1
)
Q
−
K
(z
−1
)
=O
−
(z
−L−K−1
)
z−→∞ (12.16)
whereO
−
(z
−L−K−1
) is the remainder of power series expansions around
z = z
∞
=∞. The functionO
−
(z
−L−K−1
), as an explicit error of the ap
proximation F(z
−1
)≈P
−
L
(z
−1
)/Q
−
K
(z
−1
), itself represents a power series
with expansion coe
cients{a
−
n
}multiplied by z
−L−K−m
(m = 1, 2, 3,...,∞)
∞
O
−
(z
−L−K−1
) =
a
−
n
z
−n
a
−
n
= c
n
−b
−
n
(12.17)
n=L+K+1
where, in general, a
−
n
= c
n
for L + K + 1≤n≤∞. In other words, the
mentioned “figure of merit” is explicitly given by the easily obtainable error
termO
−
(z
−L−K−1
), which is an infinite sum with higherorder expansion
terms than those retained in the Maclaurin series (12.13) for the polynomial
quotient P
−
L
(z
−1
)/Q
−
K
(z
−1
) from the PA.
The definition (12.16) of the PA is reminiscent of a variancetype estimate
for the difference between the input data (observed, measured), F(z
−1
), and
output (modeled) function, P
−
L
(z
−1
)/Q
−
K
(z
−1
)
Pade Variance =O
−
(z
−L−K−1
)
= F(z
−1
){input data (observed, measured)} (12.18)
−
P
−
L
(z
−1
)
Q
−
K
(z
−1
)
{output data (modeled, objective function)}.
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