Digital Signal Processing Reference
In-Depth Information
the impulse response of infinite length {
h
(
k
)} appearing in equation [4.5], is visible
when we expand into a series:
q
∑
−
n
bz
n
+∞
∑
()
n
=
0
()
−
n
[4.9]
hz
=
=
hnz
p
∑
−
n
n
=
0
az
n
n
=
0
which will not be reduced to a polynomial except for a trivial case when the
numerator is divisible by the denominator.
The particular case of FIR mentioned earlier is only the one where
a
n
= 0 for
n
≠
0. In this case, the transmittance can be written in the form of a polynomial:
q
q
∑∑
()
−
n
()
−
n
[4.10]
hz
=
bz
hnz
n
n
=
0
n
=
0
We note that if the input
u
(
k
)
is considered inaccessible (it is a virtual input), but
some of its probabilistic characteristics are known, we can perhaps calculate some
probabilistic characteristics of
x
(
k
),
considering that we know how the said
characteristics propagate through an invariant linear system. The simplest case is
when we are only interested in the 2
nd
order characteristics (Wiener-Lee relations;
see [KAY 88, MAR 87]), which may be written in discrete time as:
1
⎛⎞
() ()()
Sz Szhzh
=
*
for
z
∈
⎜
⎝⎠
x
u
z
*
[4.11]
( ) ( ) ( )
2
i
2
π
v
i
2
π
v
i
2
π
v
i
2
π
v
Se
=
Se
he
for
ze
=
x
u
It is no doubt compelling to take for
()
u
Sz
the simplest PSD possible:
()
te
2
u
Sz C
σ
==
that is to say for
u
(
k
)
= n
(
k
)
a
zero mean white noise.
In this case, the PSD can be
written as:
( )
2
( )
i
2
π
v
2
i
2
π
v
x
Se
=
σ
he
[4.12]
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