Table 1 Summary of the DT-ACMO RLS algorithm

1:

Initialize: \({\boldsymbol {P}}[0]=\delta ^{-1}{\boldsymbol {I}}_{N(\lambda _{max}+T) \times N(\lambda _{max}+T)}\),

\({\boldsymbol {Z}}\left [0\right ] = {\boldsymbol {I}}_{N(\lambda _{max}+T) \times N(\lambda _{max}+T)}\),

2:

Generate Φ[0] randomly with the power constraint

\({\text {Tr}}\left ({\boldsymbol {\Phi }}_{{eq_{k}}_{MAS}}[i]\boldsymbol {\Phi }^{{H}}_{{eq_{k}}_{MAS}}[i]\right) \leq {\mathrm {P}_{\mathrm {R}}}\).

3:

For each instant of time, i=1, 2, …, compute

4:

\({\boldsymbol {k}}[i]=\frac {\lambda ^{-1}{\boldsymbol {P}}[i-1] {\boldsymbol {r}}_{k_{MAS}}[i]}{1+\lambda ^{-1}{\boldsymbol {r}}_{k_{MAS}} ^{H}[i]{\boldsymbol {\Psi }}^{-1}[i-1] \boldsymbol {r}_{k_{MAS}}[i]}\),

5:

\(\boldsymbol {\Phi }^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i] = {\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i-1]+\lambda ^{-1}({\boldsymbol {r}}_{e_{MAS}}[i]-{\boldsymbol {Z}}[i-1]{\boldsymbol {k}} [i])\)

\({\boldsymbol {r}}_{k_{MAS}}^{{H}}[i]{\boldsymbol {P}}[i-1]\),

6:

Update: \({\boldsymbol {P}}[i]=\lambda ^{-1}{\boldsymbol {P}}[i-1]-\lambda ^{-1}{\boldsymbol {k}}[i]{\boldsymbol {r}}_{k_{MAS}} ^{{H}}[i]{\boldsymbol {P}}[i-1]\),

7:

Update: \({\boldsymbol {Z}}[i] = \lambda {\boldsymbol {Z}}[i-1]+{\boldsymbol {r}}_{e_{MAS}}[i] {\boldsymbol {r}}_{k_{MAS}}^{{H}}[i]\).

8:

Normalization: \({\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i] = \frac {{\sqrt {\mathrm {P}_{\mathrm {R}}}}{\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i]} {\sqrt {\sum _{k=1}^{n_{r}}{\text {Tr}}\left ({\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i] \left ({\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i]\right)^{{H}}\right)}}\).