Digital Signal Processing Reference
In-Depth Information
()
+
[
()
]
=
[
(
)
]
wn
b
b
if
otherwise
sgn
en
sgn
xn k
-
Ó
(
)
=
k
wn
+
1
(7.8)
k
()
-
wn
k
which is more concise from a mathematical viewpoint because no multiplica-
tion operation is required for this algorithm.
The implementation of these variants does not exploit the pipeline features of
the TMS320C6x processor. The execution speed on the TMS320C6x for these vari-
ants can be slower than for the basic LMS algorithm due to additional decision-type
instructions required for testing conditions involving the sign of the error signal or
the data sample.
The LMS algorithm has been quite useful in adaptive equalizers, telephone can-
celers, and so forth. Other methods, such as the recursive least squares (RLS) algo-
rithm [4], can offer faster convergence than the basic LMS but at the expense of
more computations. The RLS is based on starting with the optimal solution and then
using each input sample to update the impulse response in order to maintain that
optimality. The right step size and direction are defined over each time sample.
Adaptive algorithms for restoring signal properties become useful when an
appropriate reference signal is not available. The filter is adapted in such a way as
to restore some property of the signal lost before reaching the adaptive filter.
Instead of the desired waveform as a template, as in the LMS or RLS algorithms,
this property is used for the adaptation of the filter. When the desired signal is avail-
able, a conventional approach such as the LMS can be used; otherwise, a priori
knowledge about the signal is used.
7.3 ADAPTIVE LINEAR COMBINER
We will consider one of the most useful adaptive filter structures—the linear adap-
tive combiner. Two cases occur when using the linear combiner: (1) multiple inputs
and (2) a single input.
Multiple Inputs
The case of multiple inputs is described in Figure 7.6. The configuration consists of
K
independent input signals, each of which is weighted by
w
(
k
) and combined to
form the output,
K
Â
()
=
(
) (
)
yn
wknxkn
,
,
(7.9)
k
0
The input can be represented as a (
K
+
1)-dimensional vector,
[
]
T
()
=
(
)
( )
◊◊◊
(
)
X
nxnxn
0
,
1
,
xKn
,
(7.10)
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