Variable Step-size and Regularization Parameters for NLMS
In echo cancellation, proper control of step-size and regularization
parameters of the normalized least mean squares (NLMS) can improve the
overall performance of the system. The filter update equation for
NLMS is
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h̄(n
+ 1) =
h̄(n)
+ μ(n)
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e(n)x̄(n)
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‖x̄(n)‖²
+ ε(n)
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(1)
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where μ(n) is the variable step-size and
ε(n) is the variable regularization. The main
goals of these control parameters and of any adaptive algorithm is to
increase the convergence speed, minimizing the system distance and
decrease the divergence rate in the presence of noise. Analysis of
how these parameters affect the performance of the adaptive algorithm
reveals how these parameters can be made time variable to achieve
these goals, and thus greatly improve the overall performance of an
echo canceller.
Adaptive filter theory shows us the relationship the step-size and
regularization parameters have on the mis-adjustment, and the
convergence speed of the echo cancellation system. For example, the
convergence speed is
where Τmax, is the maximum time constant and
λmin is the minimum eigenvalue of the
autocorrelation matrix R. As we can observe from (2) is that
convergence speed is inversely proportional to the step-size. The
mis-adjustment is described as
where M is the mis-adjustment and Tr(R) is the
trace of the autocorrelation matrix. This shows that the
mis-adjustment is proportional to the step-size. The translation of
these results for the regularization parameter results in reversing
the proportionality. Combining the results of (2) and (3), results in
an engineering trade-off between convergence speed and the system
distance. Therefore, for initial convergence or re-adaptation, the
step-size should be close to one and the regularization should be
close to zero, and as the mis-adjustment approaches zero, the
step-size should approach zero and the regularization should approach
infinity. In acoustic echo cancellation, the ability to vary the
step-size (or lower the regularization value) is an important
characteristic for environments in which the echo path is
time-varying.
In addition to controlling the convergence speed and mis-adjustment,
the step-size and regularization parameters can be used to control the
influence of noise into the system. This noise includes the presence
of the near-end talker and/or local background noise. As discussed in
the Double-Talk
Detection in Echo Cancellation, when significant levels of
near-end and far-end signals are present simultaneously, adaptation
needs to be frozen in order to prevent divergence. Therefore, an
optimal step-size parameter should be decreased by the presence of
noise. To put this result together from the results of the previous
paragraph, the optimal step-size parameter should be
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μopt(n) =
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E{e
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2
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(n)}
+
E{n
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2
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(n)}
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res
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(4)
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where
is the expectation value of the residual error power, and
is expectation value of the noise power. As one can observe, when the
residual error signal is large, the step-size will be close to one,
and when the presence of local noise is large, the step-size is close
to zero. This scaling is important in hands-free communication
systems set in environments with large levels of background noise,
such as drive-thru's and factory settings.
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