## 3 Performance Analysis
### Bob's Average Bit Error Probability  
Based on assumption of (15), the upper bound of ABER at Bob can be formulated as  
\[P_{B_{k}}\leqslant\frac{1}{R_{p}2^{R_{p}}}\sum_{\mathbf{x}_{\mathcal{I}_{g_{k }}}^{s}\in\mathbb{X}}\mathbf{x}_{\mathcal{I}_{2g_{k}}}^{s}\sum_{\hat{\mathcal{ X}}\
eq\mathbf{x}_{\mathcal{I}_{g_{k}}}^{s}}d\Big{(}\mathbf{x}_{\mathcal{I}_{g_{k }}}^{s}\rightarrow\mathbf{x}_{\mathcal{I}_{2g_{k}}}^{n}\Big{)}P\Big{(}\mathbf{ x}_{\mathcal{I}_{g_{k}}}^{s}\rightarrow\mathbf{x}_{\mathcal{I}_{2g_{k}}}^{n}\Big{)}, \tag{21}\]  
where \(R_{p}\) denotes the transmission rate based on (13), \(d\Big{(}\mathbf{x}_{\mathcal{I}_{g_{k}}}^{s}\rightarrow\mathbf{x}_{\mathcal{I }_{2g_{k}}}^{n}\Big{)}\) represents the Hamming distance between vectors \(\mathbf{x}_{\mathcal{I}_{g_{k}}}^{s}\) and \(\mathbf{x}_{\mathcal{I}_{2g_{k}}}^{n}\), \(P\Big{(}\mathbf{x}_{\mathcal{I}_{g_{k}}}^{s}\rightarrow\mathbf{x}_{\mathcal{I }_{2g_{k}}}^{n}\Big{)}\) is the pairwise Error probability (PEP). In particular, the PEP in (21) can be given by  
\[\begin{split}& P\Big{(}\mathbf{x}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}\left({\mathbf{x}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}}}}}}}-{\mathbf{A}}\mathbf{x}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}^{n}}}}}}}}}}}}}}}}}}}}}}}}}}}}} \|\|\|\|\|\|\|\right.\\ &=\mathcal{\mathcal{R\left[\left[{ \mathbf{R}\left[\mathbf{n}_{B}^{H}\left(\mathbf{H}_{\Lambda}\mathbf{x}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}} \left({\mathbf{x}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}}}}}}-{\mathbf{A}}\mathbf{x}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}}}}}}}}}}}}}}}}}\right)\right. \right] \right\right\}}}}\right\right\right\}}\right\right\right\right\}\\ \right\end{\] (\)where \(\mathbf{H}_{\Lambda}=\mathbf{H}(\Theta_{B})\mathbf{W}\Lambda}}\) is defined and \(\mathbf{n}_{B}\sim\mathcal{CN}\big{(}\mathbf{0},\sigma_{B}^{2}\mathbf{I}_{N _{r}\times N_{r}}\big{)}}\). Noting that \(\mathcal{R}\Big{[}\mathbf{n}_{B}^{H}\Big{(}\mathbf{H}_{\Lambda}\mathbf{x}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}}}}}-\mathbf{H}_{ \Lambda}\mathbf{x}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}}}}\Big{)}\Big{]}\) is a Gaussian random variable obeying \(\mathcal{CN}\sim\left(0,\frac{1}{2}\sigma_{n}^{2}\right\|\mathbf{H}_{\Lambda} \Big{(}\mathbf{x}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}}}}- \mathbf{x}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}}}}\Big{)}}\Big{\|}^{2}\)), PEP can be obtained as  
\[P\Big{(}\mathbf{x}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}}^{s}\to\mathbf{x}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}}}}}\Big{)}=Q\left(\sqrt{\frac{\left\| \mathbf{H}_{\Lambda}\Big{(}\mathbf{x}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}-\mathbf{x}_{\mathcal{I}_{\mathcal{I }_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}} }}}}}}}^{s}\Big{)}\right\|^{2}}{2\sigma_{n}^{2}}}\right). \tag{23}\]  
After that, the upper bound of ABEP of Bob at the \(k\)-th slot can be obtained by substituting (23) to (21). However, the signal conveyed at Bob is associated with the mapped covert information and it will affect the practical ABEP.