Digital Signal Processing Reference
In-Depth Information
2.7
Diagonal and para-diagonal fast Pade transform
In the rest of this topic, we will illustrate only the diagonal and paradiagonal
forms of the FPT
(±)
. The computational algorithm of the FPT
(±)
begins by
extracting the Pade polynomials P
±
K
(z
±1
) and Q
±
K
(z
±1
) from the given set of
the input time signal points{c
n
}. This is prescribed already in the definition
of the FPT
(±)
from (2.169) by which the expansion coe
cients{p
±
r
}and{q
±
s
}
of P
±
K
(z
±1
) and Q
±
K
(z
±1
) can be extracted. To this end within the FPT
(+)
or FPT
(−)
, we first multiply (2.165) for L = K by Q
K
(z) or Q
−
K
(z
−1
), and
then compare the coe
cients of the same powers of the expansion variables.
For the FPT
(+)
, the results of this procedure are the two systems of linear
equations for the polynomial coe
cients{q
s
}and{p
r
}
K
q
s
c
m+s
=−c
m
(0≤m≤I)
(2.176)
s=1
K−k
p
k
c
r
q
k+r
=
(1≤k≤K)
(2.177)
r=0
where I = N−K−1. Similarly, in the case of the FPT
(−)
, we obtain the two
systems of linear equations for the expansion coe
cients{q
−
s
}and{p
−
r
}
K
q
−
s
c
K+m−s
=−c
K+m
(1≤m≤I)
(2.178)
s=1
k
p
−
k
c
r
q
−
k−r
=
(0≤k≤K).
(2.179)
r=0
In (2.176) and (2.178), the expansion coe
cients q
±
0
can be set to unity. The
equivalent matrix forms of the systems of equations (2.176)-(2.179) will be
given in chapter 4. Notice that the expansion coe
cients{q
−
s
}and{q
s
}of the
polynomial Q
−
K
(z
−1
) and Q
K
(z) have the meaning of the forward and back
ward prediction coe
cients in the FPT
(−)
and FPT
(+)
, respectively. Thus,
when all the K coe
cients{q
−
s
}are determined, (2.178) can be employed
to predict the new, unavailable time signal points c
n
for n > N from a lin
ear combination of the known input data{c
n
}(0≤n≤N−1). This is the
essence of the powerful extrapolation feature of the FPT
(−)
directly in the
time domain. Likewise, if all the K coe
cients{q
s
}are computed, (2.176)
can be used to recur backwards and retrieve the old time signals points, as
in (2.159) and (2.161). Such retrieved signal points fully agree with the given
input data{c
n
}(0≤n≤N−1) and this is the proof that the computed
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