Table 1 Date level recursive MSNWF

Forward recursion for i = 1, 2, ⋯, D

\( {\mathbf{t}}_i={\displaystyle {\sum}_{n=1}^{L-1}{d}_{i-1}^{*}(n){\mathbf{x}}_{i-1}(n)} \)

t _i  = t _i /‖t _i ‖ ₂

\( {d}_i(n)={\mathbf{t}}_i^{\mathrm{H}}{\mathbf{x}}_{i-1}(n),\kern0.24em n=1,\cdots, L-1 \)

x _i (n) = x _i− 1(n) − d _i (n)t _i , n = 1, ⋯, L − 1

ε _D (n) = d _D (n), n = 1, ⋯, L − 1

Backward recursion for i = D − 1, ⋯ , 2, 1

\( {\omega}_{i+1}={\displaystyle {\sum}_{n=1}^{L-1}{d}_i(n){\varepsilon}_{i+11}^{*}\left[n\right]/}{\displaystyle {\sum}_{n=1}^{L-1}{\left|{\varepsilon}_{i+1}(n)\right|}^2} \)

ε _i (n) = d _i (n) − ω _i  + ₁ ε _i+ 1(n), n = 1, ⋯, L − 1

Calculate the Wiener filter coefficient

\( {\mathbf{w}}^{(D)}={\displaystyle {\sum}_{i=1}^D{\left(-1\right)}^{i+1}\left\{{\displaystyle {\prod}_{l=1}^i}\left(-{\omega}_l\right)\right\}\times {\mathbf{t}}_i} \)