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Table 1 NPVSS-NLMS algorithm

From: An overview on optimized NLMS algorithms for acoustic echo cancellation

 

Initialization:

 

\(\widehat {\mathbf {h}}(0)=\mathbf {0}_{L \times 1} \)

 

\(\widehat {\sigma }_{e}^{2}(0) = 0 \)

 

Parameters:

 

\(\lambda = 1 - \frac {1}{KL},\) weighting factor with K>1

 

\({\sigma _{v}^{2}},\) noise power known or estimated

 

δ>0, regularization

 

ζ>0, very small number to avoid division by zero

 

For time index n=1,2,…:

 

\(e(n) = d(n) - \mathbf {x}^{T}(n) \widehat {\mathbf {h}}(n-1) \)

 

\(\widehat {\sigma }_{e}^{2}(n) = \lambda \widehat {\sigma }_{e}^{2}(n-1) + (1-\lambda) e^{2}(n) \)

 

\(\alpha _{\text {NPVSS}}(n) =1- \frac {\sigma _{v} }{ \zeta + \widehat {\sigma }_{e}(n)} \)

 

\(\mu _{\text {NPVSS}}(n) = \left \{ \begin {array}{lll} \alpha _{\text {NPVSS}}(n) \left [ \delta + \mathbf {x}^{T}(n) \mathbf {x}(n) \right ]^{-1}, & \text {if} \ \alpha _{\text {NPVSS}}(n) > 0 \\ \\ ~~~~~~0, & \text {otherwise} \end {array}\right.\)

 

\(\widehat {\mathbf {h}}(n) = \widehat {\mathbf {h}}(n-1) + \mu _{\text {NPVSS}}(n) \mathbf {x}(n) e(n)\)