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Normalized LMS

Module by: Douglas L. Jones. E-mail the author

In "normalized" LMS, the gradient step factor μμ is normalized by the energy of the data vector: μ NLMS =α X k H ⁢ X k +σ μ NLMS α X k H X k σ where αα is usually 12 1 2 and σσ is a very small number introduced to prevent division by zero if X k H ⁢ X k X k H X k is very small. W k + 1 = W k +1 X H ⁢X⁢ e k ⁢ X k W k + 1 W k 1 X H X e k X k The normalization has several interpretations

  1. corresponds to the 2nd-order convergence bound
  2. makes the algorithm independent of signal scalings
  3. adjusts W k + 1 W k + 1 to give zero error with current input: W k + 1 ⁢ X k = d k W k + 1 X k d k
  4. minimizes mean effort at time k+1 k 1
NLMS usually converges much more quickly than LMS at very little extra cost; NLMS is very commonly used. In some applications, normalization is so universal that "we use the LMS algorithm" implies normalization as well.

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