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In this thesis, two new types of LMS adaptive filtering
algorithmsare proposed. The first one is based on algebraic
reduction of adaptive filtering equation, while the second one
is based on extrapolation of the filter coefficients. By
rearranging the filtering equation, the algebraic-reduction LMS
(ARLMS) adaptive filtering algorithm costs 50% fewer
multiplications at the expense of 50% more additions than the
direct-form LMS (DLMS) algorithm. In addition, the new
algorithm introduces an extra adaptation parameter other than
the filter coefficients. On the other hand, the extrapolated LMS
(ELMS) algorithm extrapolates the adaptation coefficients to
values close to the converged ones. As a result,
considerable adaptation iterations are saved. The thesis
first derives the optimal coefficient solution and minimal mean
square error (MMSE) of the algebraic-reduction adaptive
filtering (ARAF) structure. From the derivations, the
optimal coefficient solutions and MMSEs for ARAF structure and
direct-form Wiener filter are the same when system satisfies
a specific condition. Otherwise, their optimal coefficient
solutionsand MMSEs are different and the ARAF structure has a
smaller MMSE than that ofthe direct-form Wiener filter. The
specific condition is that the mean of input signal
multiplied by the conjugated summation of the adaptive
coefficient equals the mean of desired signal.
Secondly, we introduce the ARLMS algorithm and explore its
properties. FromARAF structure, an ARLMS algorithm is proposed,
in contrast to the DLMS algorithm which is a realization
of Wiener adaptive filter. The mean square errors of ARLMS and
DLMS algorithms are different, because the ARLMS
algorithmintroduces an extra adaptation parameter. Simulations
show that the ARLMS algorithm has a smaller steady-state MSE
than that of the DLMS algorithm when the step sizes are properly
chosen and system is not under the mentioned specific
condition. The conditions of convergence in mean and mean square
for for the ARLMS algorithm are derived under the conditions of
zero-mean input and desired signals. Accordingly, closed-form
steady-state mean square error (MSE) as a function of the LMS
algorithm step size μ and an extra compensation step
size α are derived, which is slightly larger than that of the
DLMS algorithm. Meanwhile, convergence bounds for μ and α are
also derived. It is shown that, when μ is small enough and
α is properly chosen, the ARLMS algorithm has comparable
performance to that of the DLMS algorithm. Correspondingly,
simple working rules and ranges for α and μ to make such
comparability are provided. For verification, the ARLMS
algorithm has been applied to HDTV and NTSC multi-path
equalizations, Echo cancellation and HDSL equalization.
Thirdly, the combined ARLMS algorithms with the signed LMS
algorithm (SA), signed regressor algorithm (SRA), signed-signed
(or signed product) algorithm (SSA) and the Duhamel''s fast exact
least mean square (FELMS) adaptive algorithm are
simulated to converge as fast as their counter DLMS variants.
They maintain comparable performances to those of the DLMS
variants when α isproperly chosen. Further, the ARLMS adaptive
algorithm is extended to two- dimensional adaptive filtering.
Simulations on image noise reduction also showthat the 2-D
algorithm has comparable performance to that of the conventional
2-D LMS algorithm when a is properly chosen.
Finally, the ELMS algorithms in various forms are detailed.
Simulations show that the ELMS algorithm is very sensitive to
step size, and the time instances that the extrapolation is
made. To avoid wide and/or incorrect extrapolation, pre-
cautious scheme is included, which improves the
extrapolation accuracy noticeably.
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