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)