# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## Appendix C Proof of Theorem 3  
For the MISO FRC case, at time \(n\), according to (A22) and (62), the signals received by Eavesdropper can be rewritten as  
\[ \begin{split}&\mathbf{Z}_{n}=\mathbf{g}_{\text{se}}\frac{\mathbf{h}_{ \text{sd}}^{\mathcal{H}}}{\|\mathbf{h}_{\text{sd}}\|}X_{n}^{{}^{\prime}}+ \mathbf{g}_{\text{ee}}\frac{h_{\text{rd}}^{\mathcal{H}}}{|h_{\text{rd}}|}{X_{r,n}}^{{}^{\prime}}+\mathbf{\eta}_{e,n},1\leq n\leq N,\\ &\mathbf{Z}_{f,n}=\mathbf{g}_{\text{de}}X_{f,n}+\mathbf{\eta}_{fe,n},1 \leq n\leq N-1.\end{split} \] (A47)  
The proof of Theorem 3 consists of the following two parts (I)-(II). 1. To analyze the secrecy level \(\Delta_{I}=\frac{H(M|\mathbf{Z}_{1}^{N},\mathbf{Z}_{f,1}^{N-1},\mathbf{h},\mathbf{g})}{ H(M)}\), \(H(M|\mathbf{Z}_{1}^{N},\mathbf{Z}_{f,1}^{N-1},\mathbf{h},\mathbf{g})\) can be bounded by \[ \begin{split}& H(M|\mathbf{Z}_{1}^{N},\mathbf{Z}_{f,1}^{N-1},\mathbf{h}, \mathbf{g})\\ \overset{\text{(a)}}{\geq}& H(\Theta|\mathbf{Z}_{1}^{N}, \mathbf{Z}_{f,1}^{N-1},\mathbf{h},\mathbf{g},\mathbf{\eta}_{fe},\mathbb{M}_{1}^{N-1}) \\ \overset{\text{(b)}}{=}& H(\Theta|\underbrace{ \mathbf{g}_{\text{se}}\mathbf{h}_{\text{sd}}^{\mathcal{H}}}_{\text{2}_{1}}X_{ 1}^{{}^{\prime}}+\mathbf{\eta}_{e,1}^{,1}}_{Z_{1}}\underbrace{\frac{\mathbf{g}_{ \text{se}}\mathbf{h}_{\text{sd}}^{\mathcal{H}}}{\|\mathbf{h}_{\text{sd}}\|}X _{2}^{{}^{\prime}}+\mathbf{\eta}_{e,2}^{,2}}_{Z_{2}},\underbrace{\frac{\mathbf{g}_ {\text{se}}\mathbf{h}_{\text{sd}}^{\mathcal{H}}}{\|\mathbf{h}_{\text{sd}}\|}X _{3}^{{}^{\prime}}+\frac{\mathbf{g}_{\text{ee}}h_{\text{rd}}^{\mathcal{H}}}{ \|h_{\text{rd}}\|}}_{Z_{3}}X_{r,n}^{{}^{\prime}}+\mathbf{\eta}_{e,3}}_{Z_{3}}, \cdots,\underbrace{\frac{\mathbf{g}_{\text{se}}\mathbf{h}_{\text{sd}}^{ \mathcal{H}}}{\|\mathbf{h}_{\text{sd}}\|}X_{N}^{{}^{\prime}}+\frac{\mathbf{g} _{\text{ee}}h_{\text{rd}}^{\mathcal{H}}}{\|\mathbf{h}_{\text{rd}}\|}}_{Z_{N}} X_{r,N}^{{}^{\prime}}+\mathbf{\eta}_{e,N},\\ &\underbrace{\mathbf{g}_{\text{de}}X_{f,1}+\mathbf{\eta}_{fe,1}}_{Z_{ f,1}},\cdots,\underbrace{\mathbf{g}_{\text{de}}X_{f,N-1}+\mathbf{\eta}_{fe,N-1}}_{Z_{f,N-1}}, \mathbf{h},\mathbf{g},\mathbf{\eta}_{fe},\mathbb{M}_{1}^{N-1})\\ \overset{\text{(c)}}{=}& H(\[PERSON]|\underbrace{ \mathbf{g}_{\text{se}}\mathbf{h}_{\text{sd}}^{\mathcal{H}}}_{\text{2}_{1}}X_{ 1}^{{}^{\prime}}+\mathbf{\eta}_{e,1})\\ \overset{\text{(d)}}{=}& H(\[PERSON])+h(\mathbf{\eta}_{e,1} )-h\left(\frac{\mathbf{g}_{\text{se}}\mathbf{h}_{\text{sd}}^{\mathcal{H}}}{ \|\mathbf{h}_{\text{sd}}\|}X_{1}^{{}^{\prime}}+\mathbf{\eta}_{e,1}\right)\\ \overset{\text{(e)}}{\geq}& H(\Theta)- \underbrace{\log\left(1+\frac{\|\mathbf{h}_{\text{sd}}\cdot\mathbf{g}_{\text{ ss}}^{\mathcal{H}}\|^{2}p}{\sigma_{e}^{2}}\right)}_{Information leaked by Source at time 1}\end{split} ,\] (A48)  
where  
1. follows from the fact that \(\Theta\) is a deterministic function of \(M,\mathbf{\eta}_{\ell}=(\eta_{1}^{N},\eta_{r,1}^{N},\mathbf{\eta}_{f,1}^{N-1},\mathbf{ \eta}_{e,2}^{N},\mathbf{\eta}_{fe,1}^{N-1})\), \(\mathbb{M}_{1}^{N-1}=(\mathbb{M}_{\text{A}}[\gamma_{1}\widehat{\Theta}_{1}+V_{1} ],\cdots,\mathbb{M}_{\text{A}}[\gamma_{N-1}\widehat{\Theta}_{N-1}+V_{N-1}])\) and conditioning reduces entropy;
2. follows from (A47);
3. follows from \(X_{r,1}^{{}^{\prime}}=X_{r,2}^{{}^{\prime}}=0\), \(X_{n}^{{}^{\prime}}(2\leq n\leq N)\) and \(X_{r,n}^{{}^{\prime}}(3\leq n\leq N)\) are deterministic functions of \((\eta_{n},\eta_{r,n},\mathbf{\eta}_{f,n},\mathbf{h})\). Additionally, \(\mathbb{M}_{n}(1\leq n\leq N-1)\) is only dependent on the complex dither signal sequence \(V_{1}^{N-1}\) (see Property (2) of Proposition 2), and \((\mathbf{h},\mathbf{g},\mathbf{\eta}_{\ell},\mathbb{M}_{1}^{N-1})\) is independent of \((\Theta,\mathbf{\eta}_{e,1})\);
4. follows from the fact that \(\Theta\) is independent of \(\mathbf{\eta}_{e,1}\);* follows from \[