Digital Signal Processing Reference
In-Depth Information
s
(0)
(a)
input symbol stream
s
(1)
s
(2)
t
−
T
0
T
(b)
d
(
n
)
with real-time
dimension indicated
t
−
T
0
T
Figure 4.30
. (a) The input symbol stream with real-time dimensions, and (b) the
impulse response of
D
(
z
)=
C
L
(
z
)
G
L
(
z
) with real-time dimensions, which results in
perfect symbol recovery in absence of noise.
4.8.4 Non-uniqueness of FSE
The fact that the biorthogonal partner is not unique means that the FSE is
not unique for a given channel. Even though all FSEs are equivalent in the
absence of channel noise, this is not the case when there is noise. In practice it
is therefore wise to choose the FSE such that the effect of noise is minimized.
To see how this can be accomplished, consider the example where
L
=2. Inthis
case the channel
C
2
(
z
) and the FSE function
G
2
(
z
) have the polyphase form
C
2
(
z
)=
R
0
(
z
2
)+
z
−
1
R
1
(
z
2
)
,
G
2
(
z
)=
E
0
(
z
2
)+
zE
1
(
z
2
)
,
(4
.
55)
and the condition for perfect equalization is
R
0
(
z
)
E
0
(
z
)+
R
1
(
z
)
E
1
(
z
)=1
.
(4
.
56)
Note now that, given any transfer function
A
(
z
), the following identity holds
trivially:
R
0
(
z
)
R
1
(
z
)
A
(
z
)
−
R
1
(
z
)
R
0
(
z
)
A
(
z
)=0
.
(4
.
57)
This means that, if we define a new transfer function
G
2
(
z
) with polyphase
components
E
0
(
z
)=
E
0
(
z
)+
R
1
(
z
)
A
(
z
)
,
E
1
(
z
)=
E
1
(
z
)
−
R
0
(
z
)
A
(
z
)
,
(4
.
58)
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