Digital Signal Processing Reference
In-Depth Information
could be small. This limitation calls for savings on the number of multiplications (by
far, the more demanding operations). In this way and as discussed above, the SEA
and SDA become natural choices.
We will consider a channel given by the following transfer function:
2
z
−
1
5
z
−
2
z
−
3
5
z
−
4
H
(
z
)
=
0
.
5
+
1
.
+
1
.
−
+
0
.
.
(4.47)
The training sequence is known at the transmitter and receiver and it is a binary signal
(BPSK) which could take the values 1 and -1 with equal probability. The variance
of additive noise
v
2
v
001. With these choices, the SNR at the
channel input is set at 30dB. The length of the adaptive filters is fixed at 2
L
(
n
)
is fixed at
σ
=
0
.
15.
In Fig.
4.4
we see the learning curves for the SEA and SDA. Although in the
+
1
=
next sections we will give some insights on the problem of obtaining
E
|
2
analytically, here we will obtain that curve using averaging. The curve is the result
of the averaging of the instantaneous values
e
(
n
)
|
2
over 200 realizations of the
equalization process (using different realizations of the noise and training sequence
in each of them). With this, we can see the transient behavior and the steady state
performance of both algorithms for this application. The values of the step sizes for
the SEA and SDA are chosen as 0.0015 and 0.018 respectively. They were chosen to
obtain the same level of steady state performance in both algorithms. As explained
in Sects.
4.4.1
and
4.4.2
, we see that the SDA presents a faster speed of convergence
than the SEA. We also see the transient behavior of the SEA mentioned above (slow
convergence speed at the beginning and a faster behavior at the end of the adaptation
process). In Figs.
4.5
and
4.6
we see the symbol scatter plots corresponding to the
transmission of a quadrature phase shift keying
8
(QPSK) [
15
] sequence of symbols,
before and after equalization. Before equalization we see that for a detector it could
be very hard to determine the exact symbol transmitted. On the other hand, we
clearly see four clusters after equalization, each centered around the corresponding
true symbol. It is clear that in this case the performance of the detector will be
significantly improved.
|
e
(
n
)
|
4.5 Convergence Analysis of Adaptive Filters
Adaptive filters are stochastic systems in nature. For this reason, they are clearly
erties of SD can be analyzed using standard mathematical tools, convergence
properties of adaptive filters must be analyzed using probability theory tools. This
is also because adaptive filters can be thought as estimators that use input-output
samples in order to obtain a vector that will be, in an appropriate sense, close to the
optimal Wiener solution. Hence, the adaptive filter solution is a random variable,
8
QPSK is a digital constellation composed by four symbols, which can be represented as complex
quantities:
e
j
π/
4
,
e
j
3
π/
4
,
e
j
5
π/
4
and
e
j
7
π/
4
.
