Digital Signal Processing Reference
In-Depth Information
difference between the PA and PzT is that the former and the latter are de
fined in variables z
−1
and z, respectively. Of course, since the PzT is also
a rational polynomial, the PzT and PA both belong to the same family of
Pade approximants, albeit with two different tasks. To specify these tasks,
given (12.4), we can consider two regions|z|> 1 and|z|< 1 in the complex
z−plane. For|z|> 1 and|z|< 1, the series F(z
−1
) from (12.4) will converge
(say, slowly) and diverge, respectively. Therefore, the rational polynomial
P
−
K
(z
−1
)/Q
−
K
(z
−1
) from the usual PA in the same variable z
−1
with respect
to F(z
−1
) accelerates the already existing convergence of (12.4) for|z|> 1.
For the opposite case|z|< 1, the input series (12.4) diverges. However, for the
same case|z|< 1, the rational polynomial P
K
(z)/Q
K
(z) from the standard
PzT converges, as it is defined in terms of the variable z as opposed to z
−1
from F(z
−1
). In this way, by means of the Cauchy analytical continuation,
the PzT effectively induces convergence into the originally divergent series
F(z
−1
) for|z|< 1. This is how the same Pade methodology can achieve two
opposite mappings via transforming divergent series into convergent ones, and
converting slowly into faster converging series (hence acceleration).
•(ii) The numerator polynomial P
−
K
(z
−1
) in, e.g., the diagonal PA generally
possesses the free, constant expansion coe
cient (p
−
0
= 0), such that the sum
over r in (12.3) can start from r = 0 yielding
P
−
K
(z
−1
) = p
−
0
+ p
−
1
z
−1
+ p
−
2
z
−2
++ p
−
K
z
−K
.
(12.10)
However, by definition, the corresponding expansion coe
cient p
0
of the nu
merator polynomial P
K
(z) in the diagonal PzT is zero. Hence, this time, the
sum over r in (12.7) for P
K
(z) begins with r = 1 with no free, z−independent
term, thus producing
P
K
(z) = p
1
z + p
2
z
2
++ p
K
z
K
.
(12.11)
The mentioned uniqueness of the Pade approximant for the given input Maclau
rin series (12.4) presents a critical feature of this method. In other words, the
ambiguities encountered in other mathematical modelings are eliminated from
the outset already at the level of the definition of the PA. Moreover, this def
inition contains its “figure of merit” by revealing how well the PA can really
describe the function F(z
−1
) to be approximated. More precisely, given the
infinite sum F(z
−1
) via (12.4), the key question to ask is whether could it
be possible to determine the value of a positive integer M for which the PA
would be able to exactly reproduce the Maclaurin polynomial F
M
(z
−1
) term
by term? Here, F
M
(z
−1
) is a partial, finite sum from (12.4)
∞
M
F(z
−1
) = F
M
(z
−1
) +
c
n
z
−n
F
M
(z
−1
) =
c
n
z
−n
.
(12.12)
n=0
n=M +1
The answer to the posed question is in the a
rmative and can be found by
expanding R
(PA)
(z
−1
) from (12.2) as an infinite sum in powers of z
−1
around
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