Table 2 The modified SPMLS algorithm

Stage inputs and initialization

\( \gamma ^{f}_{\ell -1,1}(n)=\gamma _{\ell -1}(n-1), \gamma ^{b}_{\ell -1,1}(n)=\gamma _{\ell -1}(n)\)

(T.2.1)

\( r^{b}_{\ell -1,k}(-1)=r^{f}_{\ell -1,k}(-1)=\delta, (k=1,\ldots,M) \)

(T.2.2)

\( \Delta ^{e}_{\ell,k}(-1)=\Delta ^{f}_{\ell,k}(-1)=\Delta ^{b}_{\ell,k}(-1)=0.0, \ (k=1,\ldots,M) \)

(T.2.3)

For k=1,…, M

Computations at SOPs

\( r^{b}_{\ell -1,k}(n) = \lambda _{k} \ r^{b}_{\ell -1,k}(n-1) + \gamma ^{b}_{\ell -1,k}(n) \ \mid b_{\ell -1}^{k}(n) \mid ^{2} \)

(T.2.4)

\( \gamma ^{b}_{\ell -1,k+1}(n)=\gamma ^{b}_{\ell -1,k}(n) - \mid \gamma ^{b}_{\ell -1,k}(n) \mid ^{2} \mid b_{\ell -1}^{k}(n)\mid ^{2}/r^{b}_{\ell -1,k}(n) \)

(T.2.5)

\( r^{f}_{\ell -1,k}(n) = \lambda _{k} \ r^{f}_{\ell -1,k}(n-1) + \gamma ^{f}_{\ell -1,k}(n) \ \mid f_{\ell -1}^{k}(n) \mid ^{2} \)

(T.2.6)

\( \gamma ^{f}_{\ell -1,k+1}(n)=\gamma ^{f}_{\ell -1,k}(n) - \mid \gamma ^{f}_{\ell -1,k}(n)\mid ^{2} \mid f_{\ell -1}^{k}(n) \mid ^{2}/r^{f}_{\ell -1,k}(n) \)

(T.2.7)

Joint process estimation (ROP)

\( e_{\ell }^{k}(n)=e_{\ell -1}^{k}(n) - \Delta ^{e^{*}}_{\ell,k}(n-1) \ b_{\ell -1}^{k}(n) \)

(T.2.8)

\( \Delta ^{e}_{\ell,k}(n)= \Delta ^{e}_{\ell,k}(n-1) + \gamma ^{b}_{\ell -1,k}(n) \ e^{k^{\ast }}_{\ell }(n) b^{k}_{\ell -1}(n)/r^{b}_{\ell -1,k}(n) \)

(T.2.9)

Forward error prediction (ROP)

\( f_{\ell }^{k}(n)=f_{\ell -1}^{k}(n) - \Delta ^{f^{*}}_{\ell,k}(n-1) \ b_{\ell -1}^{k}(n-1) \)

(T.2.10)

\( \Delta ^{f}_{\ell,k}(n)= \Delta ^{f}_{\ell,k}(n-1) + \gamma ^{b}_{\ell -1,k}(n-1) \ f^{k^{\ast }}_{\ell }(n) b^{k}_{\ell -1}(n-1)/r^{b}_{\ell -1,k}(i-1) \)

(T.2.11)

Backward error prediction (ROP)

\( b^{k}_{\ell }(n)=b_{\ell -1}^{k}(n-1) - \Delta ^{b^{*}}_{\ell,k}(n-1) \ f_{\ell -1}^{k}(n) \)

(T.2.12)

\( \Delta ^{b}_{\ell,k}(n)= \Delta ^{b}_{\ell,k}(n-1) + \gamma ^{f}_{\ell -1,k}(n) \ b^{k^{\ast }}_{\ell }(n) f^{k}_{\ell -1}(n)/r^{f}_{\ell -1,k}(n) \)

(T.2.13)

End

Stage outputs

\( \gamma _{\ell }(n)=\gamma ^{b}_{\ell -1,M+1}(n)\)

(T.2.14)