Digital Signal Processing Reference
In-Depth Information
s
1
s
2
x
1
x
2
s
1
s
2
s
s
(
s
)
(
x
)
s
N
x
M
s
N
x
FIGURE 1.1
General scheme.
transmission channel is band-limited, i.e., it necessarily imposes distortion
over the transmitted signal if such signal exceeds the allowed passband.
Moreover, the channel presents additional impairments since its frequency-
response in the passband is often not flat, and is also subject to noise. In the
most treatable case, the channel is assumed linear and time-invariant, i.e.,
the output is obtained by a temporal convolution, and the noise is assumed
Gaussian and additive.
In analog communications systems, channel impairments lead to a
continuous-time distortion over the transmitted waveform. In digital com-
munication, information is carried by a sequence of symbols, instead of a
continuous waveform. Such symbols constitute a given transmission signal
in accordance with a given modulation scheme. Hence, the noxious effect
of the channel impairments in digital communications is a wrong symbol
decision at the receiver.
Since information is conveyed by a sequence of symbols, it is suitable to
employ a discrete-time model for the system, so that both the channel and
the equalizer may be viewed as discrete-time filters, and the involved signals
are numerical sequences. So, the problem may be represented by the scheme
in Figure 1.2, where
s
(
n
)
is the transmitted signal; ν
(
n
)
is the additive noise;
x
istheestimateof
the transmitted signal, provided by the equalizer through the mapping
(
n
)
is the received signal, i.e., the equalizer input; and
ˆ
s
(
n
)
ˆ
s
(
n
)
=
W
[
x
(
n
)
]
(1.3)
Since the channel is linear, we can characterize it by an impulse response
h
(
n
)
so that the mapping provided by the channel may be expressed by
H
[
s
(
n
)
=
s
(
n
)
∗
h
(
n
)
(1.4)
]
x
(
n
)
s
(
n
)
s
(
n
)
ˆ
Σ
ν(
n
)
FIGURE 1.2
Equalization scheme.
