Digital Signal Processing Reference
In-Depth Information
coe
cients{c
n
}from the input Maclaurin series (12.4) are time signal points
or FIDs, or equivalently, autocorrelation functions. These are given by linear
combinations of decaying trigonometric functions that are complexvalued
damped exponentials called fundamental harmonics (transients)
K
d
k
z
k
z
k
= e
iω
k
τ
c
n
=
Im(ω
k
) > 0.
(12.5)
k=0
Here, τ is the sampling or dwell time, whereas{ω
k
,d
k
}are the nodal angular
frequencies and the associated amplitudes, respectively. By inserting (12.5)
into F(z
−1
) from (12.4), the infinite sum over n can be carried out using
the exact result for the geometric series
∞
n=0
(z
k
/z)
−n
= 1/(1−z
k
/z) =
z/(z−z
k
). The obtained fraction z/(z−z
k
) is the simplest 1st order di
agonal (L = K = 1) rational polynomial in the variable z
+1
≡z. To co
here with (12.5), general variable z can also be written in the harmonic
form z = exp (iωτ), where ω is a running complex angular frequency. Obvi
ously, a sum of K elementary fractions z/(z−z
k
), as implied by F(z
−1
) =
K
k=1
zd
k
/(z−z
k
) is the K th order
diagonal (L = K) rational polynomial P
K
(z)/Q
K
(z)
K
k=1
d
k
∞
n=0
(z
k
/z)
n
=
∞
n=0
c
n
z
−n
=
K
≡
P
K
(z)
Q
K
(z)
zd
k
z−z
k
F(z
−1
) =
(12.6)
k=1
L
K
P
L
(z) =
p
r
z
r
Q
K
(z) =
q
s
z
s
.
(12.7)
r=1
s=0
Therefore, for the expansion coe
cients{c
n
}in the form of geometric pro
gression (12.5), the exact result for the infinite sum in (12.4) is given precisely
by the rhs of (12.6), which can alternatively be rewritten as
F(z
−1
) = R
(PzT)
(z)
(12.8)
R
(PzT)
(z)≡
P
K
(z)
Q
K
(z)
.
(12.9)
Here, the acronym PzT stands for the socalled Pade ztransform. Distin
guishing PA from PzT is essential due to the subtle, but critical differences
(i) and (ii) between these two methods:
•(i) The standard Pade approximant is invariably introduced in the litera
ture on this method as the rational polynomial P
−
L
(z
−1
)/Q
−
K
(z
−1
) from (12.2)
in the same variable z
−1
as the original function F(z
−1
) from (12.4). On the
other hand, we can alternatively interpret (12.4) as the usual ztransform in
variable z
−1
. As such, subsequently using geometric progression (12.5) for the
c
n
's, the resulting rational function R
(PzT)
(z) = P
K
(z)/Q
K
(z) from (12.9)
becomes the exact Pade polynomial quotient, but in the new variable z rel
ative to the initial z−transform F(z
−1
). Thus, given F(z
−1
), the first key
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