Digital Signal Processing Reference
In-Depth Information
that the original time signal c
n
simply does not possess such components,
{z
K
′
,d
K
′
}(K
′
> K). We see that, in the end, all such spurious components
{z
K+1
,d
K+1
; z
K+2
,d
K+2
; z
K+3
,d
K+3
; ...}of the estimate c
n
for the input
time signal c
n
with K harmonics{z
1
,d
1
; z
2
,d
2
; z
3
,d
3
; ...; z
K
,d
K
}are washed
out from the output data by their zerovalued intensities. This leads straight
to the exact result c
n
= c
n
, as per (2.211).
2.8.2
Exact number K and the existence of the solution of
ordinary difference equations
At this point, we make an apparent digression, and write the K th order
ordinary difference equation (OE) similarly to (2.137) and (2.138), except
for the z−variable, z = e
iωτ
K
K
d
k
e
inω
k
τ
q
k
c
n+k
= 0
∴
c(t) =
(2.212)
k=1
k=1
where{q
k
}
k=1
are the given constant coe
cients (not all of which are zero).
This is the discretized version of the associated ordinary differential equation
(ODE) defined by
K
k
K
−i
d
dt
d
k
e
iω
k
t
.
q
k
c(t) = 0
∴
c(t) =
(2.213)
k=1
k=1
All the constants{d
k
}(1≤k≤K) are obtained from the K boundary
conditions to (2.212) or (2.213) [5].
The ω
k
's are deduced from the roots
z
k
= e
iω
k
τ
of the characteristic or secular equation
K
q
k
z
k
z
k
= e
iω
k
τ
= 0
∴
(2.214)
k=1
which is automatically generated by (2.212). Crucially, the necessary and suf
ficient condition for the existence of the solution c
n
of (2.212) in the unique
form c
n
=
K
k=1
d
k
e
inω
k
τ
is given precisely by (2.204). Hence, our tempo
rary excursion to the OE was not a digression after all. Nowhere have we
mentioned the Pade approximant in this passage to the differential calculus,
and yet the nature of the invoked dynamics has naturally driven us to the PA
through the explicit appearance of the Pade denominator polynomial Q
K
(z)
disguised in the secular equation (2.214), which can equivalently be written
as Q
K
(z)≡
K
k=1
q
k
z
k
= 0, as in (2.180). It is remarkable that different
mathematical strategies in the time domain cohere while jointly determining
the exact number K of harmonics by the same condition (2.204).
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