Digital Signal Processing Reference
In-Depth Information
On the other hand, the SDA does not suffer from the slow convergence of the SEA.
Similarly to what was previously done for the SEA, the SDA can also be interpreted
in terms of the LMS in the following way:
w
(
n
)
=
w
(
n
−
1
)
+
M
(
n
)
x
(
n
)
e
(
n
),
where
M
.
This means that each coefficient of the filter has its own step size. Although this is
also a time varying step size as in the SEA, its dynamics are independent on the filter
convergence, in contrast with the SEA. Actually, the convergence of SDA and LMS
can be very similar [
14
].
(
n
)
is a diagonal matrix with its
i
-th entry being
μ
i
(
n
)
=
μ/
|
x
(
n
−
i
)
|
4.4.3 Example: Adaptive Equalization
Channel equalization is a very important topic in digital communications [
15
,
16
].
In band limited communications systems, a particular important impairment is the
distortion introduced by the channel (besides the additive noise in it). This is reflected
in the phenomenon known as
intersymbol interference
(ISI). As more input symbols
are introduced in the channel per unit of time, the output pulses (each corresponding
to an input symbol) begin to overlap considerably. In this way, the input symbols
begin to interfere with each other and the receiver is not able to distinguish the
individual pulses corresponding to each of the input symbols. The reason for this
phenomenon is the limited bandwidth of real channels, and their departure from an
ideal flat frequency response [
15
].
In precise terms, we can write the action of a communication channel over a
sequence of input symbols (after a proper sampling procedure) as
M
y
(
n
)
=
x
(
n
−
i
)
h
i
+
v
(
n
)
i
=
0
M
=
h
0
x
(
n
)
+
x
(
n
−
i
)
h
i
+
v
(
n
),
(4.43)
i
=
1
ISI
where
h
=[
h
0
,
h
1
,...,
h
M
]
represents the finite length impulse response of the
channel, and
v
is the additive noise introduced to the channel. Assuming that the
desired symbol by the receiver at time
n
is
x
(
n
)
(
n
)
, it is clear that the effect of the ISI
term given by
i
=
1
x
h
i
will decrease the quality of service required. The
detector at the receiver will use
y
(
n
−
i
)
. Because of the ISI
term, it will see an increased noise floor which will degrade its performance. It is
important to note, that in contrast with the impairment caused by the noise
v
(
n
)
to estimate the symbol
x
(
n
)
,the
problem of the ISI cannot be solved increasing the energy of the transmitted symbols.
(
n
)
