# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## 2 Model Formulation
### The MISO Fade Channel with Noisy Feedback  
The MISO FRC with noisy feedback studied in this subsection is also depicted in Figure 1b, which consists of a Source equipped with \(B\)\((B>1)\) antennas, a Relay, and a Destination with a single-antenna, respectively. The channel gains of the Source=Relay channel, the Relay=Destination channel, the Source=Destination channel and the Destination=Source channel are denoted by \(\mathbf{h}_{\text{sr}}\in\mathbb{C}^{1\times B},h_{\text{rd}}\in\mathbb{C}^{ 1\times 1},\mathbf{h}_{\text{sd}}\in\mathbb{C}^{1\times B},\mathbf{h}_{\text{ds}} \in\mathbb{C}^{8\times 1}\), respectively. In this work, the channels of these four links are independent and experience quasi-static fading; i.e., the channel state of each link is constant during one block and varies independently for the next. So the channel input-output relationships are given by  
\[\text{Relay:}\quad Y_{r,n}=\mathbf{h}_{\text{sr}}\mathbf{X}_{n}+ \eta_{r,n}\ \,\ \ 1\leq n\leq N, \tag{6}\] \[\text{Destination:}\quad Y_{n}=\mathbf{h}_{\text{sd}}\mathbf{X}_{n}+ \eta_{rd}\mathbf{X}_{r,n}+\eta_{n}\ \,\ \ 1\leq n\leq N,\] \[\text{Source:}\quad Y_{f,n}=\mathbf{h}_{\text{ds}}X_{f,n}+\mathbf{\eta }_{f,n}\ \,\ \ 1\leq n\leq N-1,\]  
where \(\mathbf{X}_{n}\in\mathbb{C}^{B\times 1}\) (resp. \(Y_{n}\in\mathbb{C}^{1\times 1}\)) and \(X_{r,n}\in\mathbb{C}^{1\times 1}\) (resp. \(Y_{r,n}\in\mathbb{C}^{1\times 1}\)) are the feedforward inputs (resp. outputs), and \(X_{f,n}\in\mathbb{C}^{1\times 1}\) and \(\mathbf{Y}_{f,n}\in\mathbb{C}^{8\times 1}\) are the feedback channel input and output, respectively. The channel noises of \(\eta_{n}\) and \(\eta_{r,n}\) are i.i.d. as \(\mathcal{N}(0,\sigma^{2})\), \(\mathcal{N}(0,\sigma^{2}_{r})\), respectively, and the elements of \(\mathbf{\eta}_{f,n}\in\mathbb{C}^{\mathbb{B}\times 1}\) are i.i.d. as \(\mathcal{N}\Big{(}0,\sigma^{2}_{f}\Big{)}\). Definition 4: For the MISO FRC with noisy feedback in Figure 1b, a \((N,|\mathcal{M}|,P,P_{r},P_{f})\)-code with the average power constraints consists of  
* A uniformly distributed message \(M\), which takes values over the set \(\mathcal{M}=\{1,2,\ldots,|\mathcal{M}|\}\). * Encoder \(1\) with output \(\mathbf{X}_{n}=\mathbf{f}_{1,n}(M,\mathbf{Y}_{f}^{n-1})\), where \(\mathbf{f}_{1,n}(\cdot)\) is an encoding function of Source at the time index \(n\)\((1\leq n\leq N)\), satisfying the average power constraint \(\sum_{n=1}^{N}E[\mathbf{X}_{n}^{M}\mathbf{X}_{n}]\leq NP\). * Relay with output \(X_{r,n}=\mathbf{f}_{r,n}(\gamma_{r}^{n-1})\), where \(\mathbf{f}_{r,n}(\cdot)\) is an encoding function of Relay at time index \(n\)\((1\leq n\leq N)\), satisfying the average power constraint \(\sum_{n=1}^{N}E[|X_{r,n}|^{2}]\leq NP_{r}\). * Encoder \(2\) with output \(X_{f,n}=\mathbf{f}_{2,n}(Y^{n})\), where \(\mathbf{f}_{2,n}(\cdot)\) is an encoding function of Destination at time index \(n\)\((1\leq n\leq N-1)\), satisfying the average feedback power constraint \(\sum_{n=1}^{N-1}E[|X_{f,n}|^{2}]\leq(N-1)P_{f}\). * Decoder has output \(\widehat{M}=\psi(Y^{N})\), where \(\psi\) is the decoding function of Destination. The average decoding error probability is also defined as (2). Definition 5: The rate \(R\) is said to be \((N,\epsilon)\)-achievable if for given coding blocklength \(N\) and decoding error probability \(\epsilon\), there exists a \((N,|\mathcal{M}|,P,P_{r},P_{f})\)-code described in Definition 4 such that (3). The \((N,\epsilon)\)-capacity [29] of the FRC with noisy feedback is defined as the supremum over all achievable rates in Definition 4, denoted as \(\mathcal{C}_{\mathrm{FRC}}^{\mathrm{NRp}}(N,\epsilon)\).