Stage inputs and initialization $$\bar {b}^{1}_{m}(n)=b^{m}_{\ell -1}(n), \bar {f}^{1}_{m}(n)=f^{m}_{\ell -1}(n), \bar {e}^{1}_{m}(n)=e^{m}_{\ell -1}(n)$$ (T.1.1) $$\gamma ^{f}_{\ell -1,1}(n)=\gamma _{\ell -1}(n-1), \gamma ^{b}_{\ell -1,1}(n)=\gamma _{\ell -1}(n)$$ (T.1.2) $$r^{b}_{\ell -1,k}(-1)=r^{f}_{\ell -1,k}(-1)=\delta, (k=1,\ldots,M)$$ (T.1.3) $$\bar {\kappa }^{b}_{kj}(-1)=\bar {\kappa }^{f}_{kj}(-1)=\Delta ^{e}_{k\upsilon }(-1)=\Delta ^{f}_{k\upsilon }(-1)=\Delta ^{b}_{k\upsilon }(-1)=0.0$$ (T.1.4) (k=1,…,M),(j=k+1,…,M),(υ=1,…,M) For k=1,…,M Computations at SOPs $$\hat {b}_{\ell -1}^{k}(n)=\bar {b}^{k}_{k}(n), \hat {f}_{\ell -1}^{k}(n)=\bar {f}^{k}_{k}(n)$$ (T.1.5) $$r^{b}_{\ell -1,k}(n) = \lambda \ r^{b}_{\ell -1,k}(n-1) + \gamma ^{b}_{\ell -1,k}(n) \ \mid \hat {b}_{\ell -1}^{k}(n) \mid ^{2}$$ (T.1.6) $$\gamma ^{b}_{\ell -1,k+1}(n)=\gamma ^{b}_{\ell -1,k}(n) - \mid \gamma ^{b}_{\ell -1,k}(n) \mid ^{2} \mid \hat {b}_{\ell -1}^{k}(n)\mid ^{2}/r^{b}_{\ell -1,k}(n)$$ (T.1.7) $$r^{f}_{\ell -1,k}(n) = \lambda \ r^{f}_{\ell -1,k}(n-1) + \gamma ^{f}_{\ell -1,k}(n) \ \mid \hat {f}_{\ell -1}^{k}(n) \mid ^{2}$$ (T.1.8) $$\gamma ^{f}_{\ell -1,k+1}(n)=\gamma ^{f}_{\ell -1,k}(n) - \mid \gamma ^{f}_{\ell -1,k}(n)\mid ^{2} \mid \hat {f}_{\ell -1}^{k}(n) \mid ^{2}/r^{f}_{\ell -1,k}(n)$$ (T.1.9) For j=k+1,…,M $$\bar {b}^{k+1}_{j}(n)=\bar {b}^{k}_{j}(n) - \bar {\kappa }^{b^{*}}_{kj}(n-1) \ \hat {b}_{\ell -1}^{k}(n)$$ (T.1.10) $$\bar {\kappa }^{b}_{kj}(n)= \bar {\kappa }^{b}_{kj}(n-1) + \gamma ^{b}_{\ell -1,k}(n) \ \bar {b}^{k+1^{\ast }}_{j}(n) \hat {b}^{k}_{\ell -1}(n)/r^{b}_{\ell -1,k}(n)$$ (T.1.11) $$\bar {f}^{k+1}_{j}(n)=\bar {f}^{k}_{j}(n) - \bar {\kappa }^{f^{*}}_{kj}(n-1) \ f_{\ell -1}^{k}(n)$$ (T.1.12) $$\bar {\kappa }^{f}_{kj}(n)= \bar {\kappa }^{f}_{kj}(n-1) + \gamma ^{f}_{\ell -1,k}(n) \ \bar {f}^{k+1^{\ast }}_{j}(n) \hat {f}^{k}_{\ell -1}(n)/r^{f}_{\ell -1,k}(n)$$ (T.1.13) End For υ=1,…,M Joint process estimation (ROP) $$e_{\upsilon }^{k+1}(n)=e_{\upsilon }^{k}(n) - \Delta ^{{e}^{*}}_{k\upsilon }(n-1) \ \hat {b}_{\ell -1}^{k}(n)$$ (T.1.14) $$\Delta ^{e}_{k\upsilon }(n)= \Delta ^{e}_{k\upsilon }(n-1) + \gamma ^{b}_{\ell -1,k}(n) \ e^{k+1^{\ast }}_{\nu }(n) b^{k}_{\ell -1}(n)/r^{b}_{\ell -1,k}(n)$$ (T.1.15) Forward error prediction (ROP) $$f^{k+1}_{\upsilon }(n)=f^{k}_{\upsilon }(n) - \Delta ^{f^{*}}_{k\upsilon }(n-1) \ \hat {b}_{\ell -1}^{k}(n-1)$$ (T.1.16) $$\Delta ^{f}_{k\upsilon }(n)= \Delta ^{f}_{k\upsilon }(n-1) + \gamma ^{b}_{\ell -1,k}(n-1) \ f^{k+1^{\ast }}_{\nu }(n) b^{k}_{\ell -1}(n-1)/r^{b}_{\ell -1,k}(n-1)$$ (T.1.17) Backward error prediction (ROP) $$b^{k+1}_{\upsilon }(n)=b^{k}_{\upsilon }(n-1) - \Delta ^{b^{*}}_{k\upsilon }(n-1) \ \hat {f}_{\ell -1}^{k}(n)$$ (T.1.18) $$\Delta ^{b}_{k\upsilon }(n)= \Delta ^{b}_{k\upsilon }(n-1) + \gamma ^{f}_{\ell -1,k}(n) \ b^{k+1^{\ast }}_{\nu }(n) f^{k}_{\ell -1}(n)/r^{f}_{\ell -1,k}(n)$$ (T.1.19) End End Stage outputs $$b^{m}_{\ell }(n)=b^{M+1}_{m}(n), \ f^{m}_{\ell }(n)=f^{M+1}_{m}(n),$$ (T.1.20) $$e^{m}_{\ell }(n)=e^{M+1}_{m}(n), \ \gamma _{\ell }(n)=\gamma ^{b}_{\ell -1,M+1}(n)$$ (T.1.21)