Algorithm 2 Robust Joint Beamforming and Sensing Design under Imperfect CSI

Require: ${\{{\tilde{\mathbf{h}}}_m,{\tilde{\mathbf{g}}}_m\}}_{m=1}^M$, ${\tilde{\mathbf{h}}}_s$, ${\tilde{\mathbf{g}}}_s$; $\left\{{\sigma }_m^2\right\}$, ${\sigma }_s^2$; $\{{\epsilon }_m,{\epsilon }_s,{\delta }_m,{\delta }_s\}$; $r_0$; $\left\{({\theta }_{\ell },{\mathsf{\Gamma }}_{\ell })\right\}$; $P_s$, $P_T$; K Ensure: $\mathbf{v}$, $\left\{{\mathbf{w}}_m\right\}$, ${\mathbf{V}}_0$ 1.Find a feasible initial point ${\mathbf{V}}^{\left(0\right)},{\mathbf{V}}_0^{\left(0\right)},\left\{{\mathbf{W}}_m^{\left(0\right)}\right\}$ and set $k=0$ 2.while $k<K$ and $\mathsf{\Delta }R\ge \nu$ do 3.   Update the approximation parameters $y^{\left(k\right)}$ and $\left\{b_m^{\left(k\right)}\right\}$ according to (58) and (59) 4.   Construct convex approximations of the exponential constraints using (69) and (70) 5.   Apply the S-procedure to transform the robust constraints into LMIs in (66)–(68) 6.   Solve the convex semidefinite programming problem (73) to obtain $({\mathbf{V}}^{*},{\mathbf{V}}_0^{*},\left\{{\mathbf{W}}_m^{*}\right\})$ 7.    Update $${\mathbf{V}}^{(k+1)}={\mathbf{V}}^{*},{\mathbf{V}}_0^{(k+1)}={\mathbf{V}}_0^{*},{\mathbf{W}}_m^{(k+1)}={\mathbf{W}}_m^{*}$$ 8.   Compute the sum rate improvement $\mathsf{\Delta }R$ 9.   $k=k+1$ 10.Extract $\mathbf{v}$ and $\left\{{\mathbf{w}}_m\right\}$ as the principal eigenvectors of ${\mathbf{V}}^{\left(k\right)}$ and $\left\{{\mathbf{W}}_m^{\left(k\right)}\right\}$