Table 1 Algorithm 1

Input: \({N_t}\), \({N_r}\), K, N, M, \({{\mathbf {H}}}\).

Analog stage:

According \({C_{n,{\mathrm{max}} }} \in {\mathrm{max}} \left\{ {\left. {\begin{array}{*{20}{l}}{{C_1}}&{{C_2}}&{\begin{array}{*{20}{l}} \cdots&{{C_N}}\end{array}}\end{array}} \right\} } \right.\), determine the optimization order.

for \(1 \le n \le N\) do

1) Update matrix \({{\mathbf {T}}}_{n - 1}\), obtain the submatrix \({{{\mathbf {G}}}_{n - 1}}.\)

2) Obtain the value of \({{{{\bar{{\mathbf {a}}}}}}_{n,opt}}.\)

3) Compute \({{{{\bar{{\mathbf {a}}}}}}_n} = \frac{1}{{\sqrt{M} }}{e^{j\angle {{{{{\bar{{\mathbf {a}}}}}}}_{n,opt}}}}\) and get \({{{\mathbf {F}}}_{\mathrm{RF}}}\) from (9).

End for

End stage

Digital stage:

4) Define the transit matrix \({{{\mathbf {H}}}_{{\mathop {\mathrm{int}}} ,k}} = {{{\mathbf {H}}}_k}{{{\mathbf {F}}}_{\mathrm{RF}}},k \in \{1,2, \ldots ,K\} .\)

5) Define the matrix \({{{{\tilde{{\mathbf {H}}}}}}_k} = {\left[ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {{{\mathbf {H}}}_{{\mathop {\mathrm{int}}} ,1}^T,}&\cdots \end{array},}&{{{\mathbf {H}}}_{{\mathop {\mathrm{int}}} ,k - 1}^T,}&{{{\mathbf {H}}}_{{\mathop {\mathrm{int}}} ,k + 1}^T,}&\cdots \end{array},}&{{{\mathbf {H}}}_{{\mathop {\mathrm{int}}} ,K}^T} \end{array}} \right] ^T}.\)

6) Compute digital precoding matrix \({{{\mathbf {F}}}_{\mathrm{BB}}}{{\mathrm{=}}}\left[ {\begin{array}{*{20}{l}}{{{\mathbf {F}}}_{\mathrm{BB}}^1}&{{{\mathbf {F}}}_{\mathrm{BB}}^2}&\cdots&{{{\mathbf {F}}}_{\mathrm{BB}}^K}\end{array}} \right] .\)

End stage

Obtain the total equivalent baseband channel \({{{\mathbf {H}}}_{total}} = {{\mathbf {H}}}{{{\mathbf {F}}}_{\mathrm{RF}}}{{{\mathbf {F}}}_{\mathrm{BB}}}.\)

Compute \({{\mathbf {P}}}\) by using water-filling power allocation of the total equivalent channel \({{{\mathbf {H}}}_{total}}.\)

End stage

Output: \({{{\mathbf {F}}}_{\mathrm{RF}}},\;{{{\mathbf {F}}}_{\mathrm{BB}}},\;{{\mathbf {P}}}.\)