Digital Signal Processing Reference
In-Depth Information
Thus
s
(
n
)=
r
(
n
)mod
V
=
y
(
n
)+
q
(
n
)] mod
V
=
x
(
n
)+
h
(1)
x
(
n
−
1) +
...
+
h
(
L
)
x
(
n
−
L
)+
q
(
n
)] mod
V
[
x
(
n
)+
h
(1)
x
(
n −
1) +
...
+
h
(
L
)
x
(
n − L
)] mod
V
+
q
(
n
)mod
V
mod
V
=
=
s
(
n
)+
q
(
n
)] mod
V
where we have used Eq. (5.95). This proves Eq. (5.94), which reduces to
Eq. (5.93) when there is no noise.
The idea of introducing a pre-equalizer with modulo arithmetic was introduced
by Tomlinson [1971] and Harashima and Miyakawa [1972]. It is therefore referred
to as the Tomlinson-Harashima-Miyakawa precoding system. Apparently it
has its origin in the work of Gerrish and Howson [1967]. The MIMO version
of the Tomlinson-Harashima-Miyakawa precoder has important applications in
multiuser systems operating in the broadcast mode [Proakis and Salehi, 2008].
5.9 Controlled ISI and partial-response signals
Returning again to the digital communication system of Fig. 4.1 with prefilter
F
(
jω
)
,
equalizer
G
(
jω
)
,
and channel
H
(
jω
)
,
let us re-examine the product
H
c
(
jω
)=
F
(
jω
)
H
(
jω
)
G
(
jω
)
.
(5
.
96)
The channel
H
(
jω
) is given, and the filters
F
(
jω
)and
G
(
jω
) are designed based
on some requirements. The more flexibility we have in the shape of the product
H
c
(
jω
)
,
the easier it is to design the analog filters
F
(
jω
)and
G
(
jω
)
.
Since the
product
H
c
(
jω
) determines the equivalent digital communication channel with
impulse response
h
d
(
n
)=
h
c
(
nT
), its shape determines the properties of the
digital channel
H
d
(
z
)=
n
h
d
(
n
)
z
−n
.
(5
.
97)
If
H
c
(
jω
) is required to be such that ISI is eliminated (i.e., the zero-forcing
condition is satisfied), then
h
d
(
n
)=
δ
(
n
)
,
or equivalently
H
c
j
ω
+
2
πk
T
=1
.
∞
1
T
(5
.
98)
k
=
−∞
Figure 5.26(a) shows a typical example of
H
c
(
jω
) and its shifted versions taking
part in the summation above. If the bandwidth of
H
c
(
jω
)islessthan2
π/T
it
is clear that the ISI-free condition (5.98) cannot be satisfied. In a
minimum-
bandwidth
system,
H
c
(
jω
) has total bandwidth exactly equal to 2
π/T
.Inthis
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