Table 3 The R-CMLF algorithm

Stage inputs

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

(T.3.1)

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

(T.3.2)

\( \Delta ^{e}_{\ell,k}(-1)=\Delta ^{f}_{\ell,k}(-1)=\Delta ^{b}_{\ell,k}(-1)= \Delta ^{eq}_{\ell }(0)=0.0, (k=1,\ldots,M) \)

(T.3.3)

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.3.4)

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

(T.3.5)

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

(T.3.6)

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

(T.3.7)

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.3.8)

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

(T.3.9)

Combination processing (type-1 processor)

\( e_{\ell }^{eq}(n) = e_{\ell }^{eq}(n) + v^{k}(n) \ e_{\ell }^{k}(n) \)

(T.3.10)

\( \Delta ^{eq}_{\ell }(n) = \Delta ^{eq}_{\ell }(n) + v^{k}(n) \ \Delta ^{e}_{\ell,k}(n) \)

(T.3.11)

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.3.12)

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

(T.3.13)

Backward error prediction (ROP)

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

(T.3.14)

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

(T.3.15)

End

Combination processing (type-2 processor)

For k=1,…,M

\( a^{k}(n+1)=a^{k}(n)-\mu _{a} \ e^{eq}_{N}(n) \ \left (e_{N}^{k}(n) - e^{eq}_{N}(n)\right) \ v^{k}(n) + \rho \ \left (a^{k}(n)-a^{k}(n-1)\right)\)

(T.3.16)

\( \beta (n+1)= \beta (n+1) + e^{a^{k}(n+1)} \)

(T.3.17)

\( v^{k}(n+1)= \frac {e^{a^{k}(n+1)}}{\beta (n+1)} \)

(T.3.18)

End

Stage outputs

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

(T.3.19)