A DCA-Based Approach for Outage Constrained Robust Secure Power-Splitting SWIPT MISO System

Abstract

This paper studies the worst-case secrecy rate maximization problem under the total transmit power, the energy harvesting and the outage probability requirements. The problem is nonconvex, thus, hard to solve. Exploiting the special structure of the problem, we first reformulate as a DC (Difference of Convex functions) program. Then, we develop an efficient approach based on DCA (DC Algorithm) and alternating method for solving the problem. The computational results confirm the efficiency of the proposed approach.

Appendix A

First, we transform the EH constraint. According to [4], we rewrite $\vert \mathbf h _{jk}^H\mathbf w _j\vert ^2 = \mathbf w _j^H (\overline{\mathbf{H }}_{jk} + \mathbf d _{jk})\mathbf w _j,$ where $\overline{\mathbf{H }}_{jk} = \overline{\mathbf{h }}_{jk}\overline{\mathbf{h }}_{jk}^H, \ \mathbf d _{jk} = \overline{\mathbf{h }}_{jk}\varDelta \mathbf h _{jk}^H + \varDelta \mathbf h _{jk} \overline{\mathbf{h }}_{jk}^H + \varDelta \mathbf h _{jk}\varDelta \mathbf h _{jk}^H$. By applying the triangle inequality and the Cauchy-Schwarz inequality, we have

$$\begin{aligned} \Vert \mathbf d _{jk} \Vert \le \Vert \overline{\mathbf{h }}_{jk}\Vert \Vert \varDelta \mathbf h _{jk}^H\Vert +\Vert \varDelta \mathbf h _{jk}\Vert \Vert \overline{\mathbf{h }}_{jk}^H\Vert +\Vert \varDelta \mathbf h _{jk}\Vert \Vert \varDelta \mathbf h _{jk}^H\Vert \\ \Vert \mathbf d _{jk} \Vert \le \Vert \mathbf Q _{jk} \Vert ^{-1}+ 2 \Vert \overline{\mathbf{h }}_{jk} \Vert \sqrt{\Vert \mathbf Q _{jk} \Vert ^{-1}} = \epsilon _{jk} \Rightarrow - \epsilon _{jk} {{\,\mathrm{{\mathbf I}}\,}}_N \le \mathbf D _{jk} \le \epsilon _{jk} {{\,\mathrm{{\mathbf I}}\,}}_N, \\ {{\,\mathrm{thus}\,}}, \ {{\,\mathrm{Tr}\,}}\left[ (\overline{\mathbf{H }}_{jk} - \epsilon _{jk} {{\,\mathrm{{\mathbf I}}\,}}_N) \mathbf W _j\right] \le \vert \mathbf h _{jk}^H\mathbf w _j\vert ^2 = {{\,\mathrm{Tr}\,}}\left[ (\overline{\mathbf{H }}_{jk} + \mathbf d _{jk}) \mathbf W _j\right] . \end{aligned}$$

Therefore, the EH constraint is recast as $\sum _{j=1}^K {{\,\mathrm{Tr}\,}}(\overline{\mathbf{H }}_{jk} \mathbf W _j - \epsilon _{jk} \mathbf W _j) \ge \frac{E_k}{(1-\rho _k)}.$

$$\begin{aligned} {{\,\mathrm{Next,}\,}} \ R_{I,k} \le \log \Big (1+\frac{\rho _k \mathbf h _{kk}^H\mathbf W _k\mathbf h _{kk}}{\rho _k\sum _{j\ne k}{} \mathbf h _{jk}^H\mathbf W _j\mathbf h _{jk}+\rho _k\sigma _{1k}^2+\sigma _{2k}^2}\Big ) \end{aligned}$$

(22)

is reformulated similarly to [1] as

$$\begin{aligned} \left\{ \begin{matrix} \beta _k \ge \dfrac{1}{\rho _k}, \ \sum _{j=1}^K f_{jk} +\sigma _{1k}^2+ \beta _k\sigma _{2k}^2 \ge a_k, \ \sum _{j\ne k}f_{jk} +\sigma _{1k}^2+ \beta _k\sigma _{2k}^2 \le b_k, \\ 0 < u \le a_k, \ v \ge b_k > 0, \ \lambda _{jk} \ge 0, \ x \ge R_{I,k}, \ 2^x v - u \le 0,\\ \mathbf h _{kk}^H\mathbf W _k\mathbf h _{kk} \ge f_{kk} \Leftrightarrow \begin{bmatrix}\lambda _{kk} \mathbf Q +\mathbf W _k &{} \mathbf W _k\overline{\mathbf{h }}_{kk} \\ \overline{\mathbf{h }}_{kk}^H\mathbf W _k &{}\overline{\mathbf{h }}_{kk}^H\mathbf W _k\overline{\mathbf{h }}_{kk} - f_{kk} - \lambda _{kk} \end{bmatrix} \succeq \mathbf 0 , \\ \mathbf h _{jk}^H\mathbf W _k\mathbf h _{jk} \le f_{jk} \Leftrightarrow \begin{bmatrix}\lambda _{jk} \mathbf Q -\mathbf W _j &{} -\mathbf W _j\overline{\mathbf{h }}_{jk} \\ -\overline{\mathbf{h }}_{jk}^H\mathbf W _j &{}-\overline{\mathbf{h }}_{jk}^H\mathbf W _j\overline{\mathbf{h }}_{jk} + f_{jk} - \lambda _{jk} \end{bmatrix} \succeq \mathbf 0 , \ j\ne k. \end{matrix}\right. \end{aligned}$$

The relaxed constraints hold with equalities at the optimal solution [1].

By using slack variables $\xi _k = \frac{2^{R_{I,k} - R_k} - 1}{{{\,\mathrm{Tr}\,}}(\mathbf G _{kk}{} \mathbf W _k)}$, the outage constraint is transformed into

$$\begin{aligned} \left\{ \begin{matrix} -\ln p_k - \xi _k \sigma _{3k}^2 - \sum _{j\ne k} \ln ( 1 + \xi _k{{\,\mathrm{Tr}\,}}(\mathbf G _{jk}{} \mathbf W _j) \le 0 , \\ \ {{\,\mathrm{Tr}\,}}(\mathbf G _{kk}{} \mathbf W _k) \le s_k, \ 0< s_k, \ 0 <\xi _k, \\ \xi _k \le \frac{2^{R_{I,k} - R_k} - 1}{s_k} \Leftrightarrow R_k - R_{I,k}+\log _{2}(1+\xi _k s_k) \le 0. \end{matrix}\right. \end{aligned}$$

At the optimum, $\xi _k = \frac{2^{R_{I,k} - R_k} - 1}{{{\,\mathrm{Tr}\,}}(\mathbf G _{kk}{} \mathbf W _k)}$ at optimum, if not, we can increase $R_k$. In addition, ${{\,\mathrm{Tr}\,}}(\mathbf G _{kk}{} \mathbf W _k) = s_k$ at the optimum, otherwise, we can decrease s leading to increase $R_k$ due to $\xi _k = \frac{2^{R_{I,k} - R_k} - 1}{{{\,\mathrm{Tr}\,}}(\mathbf G _{kk}{} \mathbf W _k)}$.

Such that, if the relaxed constraints do not hold with equalities at the optimum, the objective function can be further increased.