28-11-2012, 05:11 PM
ADJOINT LMS: AN EFFICIENT ALTERNATIVE TO THE
FILTERED-X LMS AND MULTIPLE ERROR LMS ALGORITHMS
AN EFFICIENT ALTERNATIVE.pdf (Size: 324.57 KB / Downloads: 18)
INTRODUCTION
The Filtered-x LMS algorithm is currently the most
popular method for adapting a filter when there exists
a transfer function in the error path. Such instances
arise, for example, in active control of sound and vibration.
For multiple-input-multiple-output systems
the Multiple Error LMS Algorithm is a generalization
of Filtered-x LMS. The derivation of both algorithms
rely on several assumptions, including linearity of the
adaptive filter and error channel. Furthermore, in the
Multiple Error LMS Algorithm the desirable order N
computational complexity of LMS is lost, resulting in
a prohibitive cost in certain DSP implementations.
In this paper, we describe a new algorithm termed adjoint
LMS which provides a simple alternative to the
previously mentioned algorithms. In adjoint LMS, the
error (rather than the input) is filtered through an adjoint
filter of the error channel. Performance regarding
convergence and misadjustment are equivalent. However,
linearity is not assumed in the derivation of the
algorithm. Furthermore, equations for single-inputsingle-
output (SISO) and multiple-input-multiple-output
(MIMO) are identical and both remain order N.
SIMULATIONS AND CONCLUSIONS
A SISO simulation illustrating the similar performance
of adjoint LMS to filtered x-LMS is shown in Figure 3.
Learning curves are remarkably alike even though the
individual stochastic weight updates are not identical.
Our conclusions are that adjoint-LMS has an equivalent
rate of convergence and misadjustment to filtered
x-LMS with a substantial computational savings over
Multiple Error LMS. The only trade-off, in certain cases,
is a slightly tighter restriction for stability on the learning
parameter as might be expected due to the delayed
weight update [3]. This causes a slight increase in misadjustment
for large learning parameters as illustrated
in Figure 4. Fortunately, this occurs beyond the desirable
operating range for the algorithms.