参见论文： Dual-Functional Artificial Noise (DFAN) Aided
Robust Covert Communications in Integrated
Sensing and Communications

理论

\boxed{} ​用于加框

Lemma 2. (BTI): For any $\mathbf{A}\in {\mathbb{C}}^{N\times N}$,$\mathbf{b}\in {\mathbb{C}}^{N\times 1}$,$c\in \mathbb{R}$,$\mathbf{x}\sim \mathcal{C}\mathcal{N}(0,\mathbf{I})$and$\rho \in [0,1]$,if there exist $x$ and $y$, such that

$$\begin{array}{rl}\mathrm{Tr}(\mathbf{A})-\sqrt{2\ln (\frac{1}{\rho })}x+\ln (\rho )y+c& \ge 0,\\ \sqrt{{\Vert \mathbf{A}\Vert }_F^2+2{\Vert \mathbf{b}\Vert }^2}& \le x,\\ y\mathbf{I}+\mathbf{A}⪰0,y& \ge 0,\end{array}$$
the following inequality holds true

Pr ⁡ ( x H A x + 2 R e { x H b } + c ≥ 0 ) ≥ 1 − ρ . \Pr(\mathbf{x}^H\mathbf{A}\mathbf{x}+2\mathrm{Re}\{\mathbf{x}^H\mathbf{b}\}+c\geq0)\geq1-\rho. Pr(xHAx+2Re{xHb}+c≥0)≥1−ρ.

应用例子2：

已知
$${\mathbf{h}}_{\mathrm{w}}={\hat{\mathbf{h}}}_{\mathrm{w}}+{\mathbf{\gamma }}_{\mathrm{w}}^{\frac{1}{2}}{\mathbf{e}}_{\mathrm{w}}$$
问题
$$\begin{array}{r}\mathrm{Pr}({\mathbf{h}}_w^H\mathbf{W}{\mathbf{h}}_w\le p_{\mathrm{th}})\ge 1-{\rho }_c,\end{array}$$
即
$$\begin{array}{r}\mathrm{Pr}({\mathbf{h}}_w^H(-\mathbf{W}){\mathbf{h}}_w+p_{\mathrm{th}}\ge 0)\ge 1-{\rho }_c,\end{array}$$
即
$$\begin{array}{r}\mathrm{Pr}(({\hat{\mathbf{h}}}_{\mathrm{w}}+{\mathbf{\gamma }}_{\mathrm{w}}^{\frac{1}{2}}{\mathbf{e}}_{\mathrm{w}}{)}^H(-\mathbf{W})({\hat{\mathbf{h}}}_{\mathrm{w}}+{\mathbf{\gamma }}_{\mathrm{w}}^{\frac{1}{2}}{\mathbf{e}}_{\mathrm{w}})+p_{\mathrm{th}}\ge 0)\ge 1-{\rho }_c,\end{array}$$
即
Pr ⁡ ( e w H ( γ w 1 2 ( − W ) γ w 1 2 ) e w + 2 R e { e w H γ w 1 2 ( − W ) h ^ w } + h ^ w H ( − W ) h ^ w + p t h ≥ 0 ) ≥ 1 − ρ c , \begin{aligned} \Pr( \mathbf{e}_{\mathrm{w}} ^H ( \boldsymbol{\gamma}_{\mathrm{w}}^{\frac{1}{2}} ( - \mathbf{W} ) \boldsymbol{\gamma}_{\mathrm{w}}^{\frac{1}{2}} ) \mathbf{e}_{\mathrm{w}} + 2\mathrm{Re}\{\mathbf{e}_{\mathrm{w}}^H \boldsymbol{\gamma}_{\mathrm{w}}^{\frac{1}{2}} ( - \mathbf{W} ) \hat{\mathbf{h}}_{\mathrm{w}} \} +\hat{\mathbf{h}}_{\mathrm{w}}^H ( - \mathbf{W} ) \hat{\mathbf{h}}_{\mathrm{w}} + p_{\mathrm{th}} \geq 0 ) \geq 1-\rho_c, \end{aligned} Pr(ewH​(γw21​​(−W)γw21​​)ew​+2Re{ewH​γw21​​(−W)h^w​}+h^wH​(−W)h^w​+pth​≥0)≥1−ρc​,​

可以等价于下列问题：
$$\begin{array}{rl}\mathrm{Tr}(\mathbf{A})-\sqrt{2\ln (\frac{1}{\rho })}x+\ln (\rho )y+c& \ge 0,\\ \sqrt{{\Vert \mathbf{A}\Vert }_F^2+2{\Vert \mathbf{b}\Vert }^2}& \le x,\\ y\mathbf{I}+\mathbf{A}⪰0,y& \ge 0,\end{array}$$

应用例子：

已知：
$$\mathrm{Pr}({\mathbf{h}}_{\mathrm{w}}^H{\mathbf{S}}_1{\mathbf{h}}_{\mathrm{w}}\le 0)\ge 1-{\rho }_c,$$

$$\begin{array}{c}\text{Recall that }{\mathbf{h}}_{\mathrm{w}}={\hat{\mathbf{h}}}_{\mathrm{w}}+{\mathbf{\gamma }}_{\mathrm{w}}^{\frac{1}{2}}{\mathbf{e}}_{\mathrm{w}}.\text{ As per Lemma 2, (28) can}\\ \text{be equivalently transformed to the following inequalities}\end{array}$$

$$\begin{array}{rl}\mathrm{Tr}({\mathbf{A}}_{\mathrm{w}})-\sqrt{2\ln (\frac{1}{{\rho }_c})}x+\ln ({\rho }_c)y+c_{\mathrm{w}}& \ge 0,\\ \sqrt{{\Vert {\mathbf{A}}_{\mathrm{w}}\Vert }_F^2+2{\Vert {\mathbf{b}}_{\mathrm{w}}\Vert }^2}& \le x,\\ y\mathbf{I}+{\mathbf{A}}_{\mathrm{w}}& ⪰0,y\ge 0,\end{array}$$

$$\begin{array}{rl}& \mathrm{where}{\mathbf{A}}_{\mathrm{w}}={\gamma }_{\mathrm{w}}^{\frac{1}{2}}(-{\mathbf{S}}_1){\gamma }_{\mathrm{w}}^{\frac{1}{2}},c_{\mathrm{w}}={\hat{\mathbf{h}}}_{\mathrm{w}}^H(-{\mathbf{S}}_1){\hat{\mathbf{h}}}_{\mathrm{w}}\mathrm{and}{\mathbf{b}}_{\mathrm{w}}=\\ & {\gamma }_{\mathrm{w}}^{\frac{1}{2}}(-{\mathbf{S}}_1){\hat{\mathbf{h}}}_{\mathrm{w}}.\end{array}$$

注意其中只有$x$和$y$是辅助变量。