Mixed l_2-l_p Norm Adaptive Algorithms for System Identification: Application to Echo Cancellation Azzedine Zerguine

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Mixed l_2-l_p Norm Adaptive Algorithms for
System Identification Application to Echo
CancellationAzzedine Zerguine
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Abstract

• In this work, the performance evaluation of lp
  based algorithms (l1, l2, l4) is investigated.
• This study is carried out for noise with uniform
  distribution and Gaussian distribution.
• Detailed simulations are performed to assess the
  behavior of these algorithms under these noise
  environments.

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Outline

• Most widely used algorithms.
• Problems of the LMS algorithm.
• The LMF algorithm.
• The mixed L2-L4 adaptive algorithm.
• The proposed algorithm.
• Conclusion

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Most widely used algorithms

• LMS and its children.
• RLS and its variants.

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The most widely used algorithms

• Sign error
• Sign data
• Sign error and data

• Fast Kalman
• FTF

• Advantages
• Fast Convergence

• Advantages
• Stable
• Simple

• Problems
• Stability
• Complexity

• Problems
• Slow convergence
• Sensitive to noise

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Problems of the LMS algorithm
Solution
Solution
Solution
Mixed-Norm algorithm
Variable step size algorithm
Normalized LMS
Problem
Problem
Higher steady state value than the LMS
Fixing the min. max.values of the step size
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k1
k2


LMS algorithm
LMF algorithm
Optimum only when noise statistics are Gaussian
Optimum if noise statistics are different from
Gaussian
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The mixed l2-l4 adaptive algorithm
?
? 0
? 1
LMS algorithm
LMF algorithm
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The update scheme for the l2-l4 algorithm
If e(n) gt 1 ? The algorithm goes unstable.
Solution
Increase ?
? very small
Convergence
LMS algorithm
A better is the l2-lp algorithm
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The proposed algorithm
0 ? ? lt 1
?
? 0
? 1
LMS algorithm
LMP algorithm
p 2
p 1
p 4
Sign error LMS
LMS algorithm
LMF algorithm
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Simulation Results

• Echo canceller with both Near End (NE) and Far
  End (FE) sections.
• A bulk delay between NE and FE sections.
• Performance measure Normalized Weight Error.
• Two scenarios are considered, one when the noise
  is Gaussian and the other when it is uniformly
  distributed.

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Conclusion

• This work has treated in details the performance
  evaluation of lp based algorithms l1, l2 and l4
  which correspond to the sign-error LMS algorithm,
  LMS algorithm and LMF algorithm, respectively.
• This study is carried out for noise with uniform
  distribution and Gaussian distribution.
• Detailed simulations are performed to assess the
  behavior of these algorithms under these
  circumstances.

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• It was shown that the LMS algorithm outperforms
  the other two algorithms and this due to its
  robustness and well behaved structure.
• The LMF algorithm was much performing well than
  the sign-error LMS algorithm and this was
  expected by the virtue of the fact that it has a
  simplified version of the LMS algorithm
• Other important observations were noticed as well
  in this work, such as the performance of the LMF
  algorithm under uniform noise statistics over its
  Gauussian counterpart.