Algorithm 1 Reweighted ${\ell }_1$ Norm Based Sparse Precoding Design.

Initilization: Set iteration counter $t=1$, Lagrangian dual variable ${\mathit{\lambda }}^{\left(0\right)}>0$, $l=1,\cdots ,L$. Repeat: Calculate ${\mathbf{\Psi }}^{\left(t\right)}$ and ${\mathbf{S}}_k^{\left(t\right)}$, $k=1,\cdots ,K$, according to (15) and (27), respectively. Update $\left\{{\lambda }_l\right\}$ according to (28). Update the iteration counter $t\leftarrow t+1$ and $$r=\sum _{l\in \mathcal{L}}{|{\lambda }_l\sum _{k\in \mathcal{K}}\mathrm{Tr}\left\{{\mathbf{B}}_l{\tilde{\mathbf{V}}}_k{\mathbf{Q}}_k{\tilde{\mathbf{V}}}_k^{†}\right\}|}^2.$$ Stop if $r<\epsilon$, where $\epsilon$ is a pre-defined tolerance threshold.