# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## 2 Model Formulation
### The Gaussian Relay Channel with Noisy Feedback  
In this subsection, the communication scenario is depicted in Figure 1a, which consists of a Source, a Relay, and a Destination. Source wishes to transmit the message to Destination with the help of Relay, and the message received by Destination can be sent to Source through an AWGN feedback channel. The channel gains of the Source-Relay channel, the Relay-Destination channel, the Source-Destination channel and the Destination-Source channel are denoted by \(h_{sr},h_{rd},h_{sd},h_{ds}\), respectively. So the channel inputs-outputs relationships are given by  
\[ \begin{array}{ll}\text{Relay:}&Y_{r,n}=h_{sr}X_{n}+\eta_{r,n},\ \ 1\leq n\leq N,\\ \text{Destination:}&Y_{n}=h_{sd}X_{n}+h_{rd}X_{r,n}+\eta_{n},\ \ 1\leq n\leq N,\\ \text{Source:}&Y_{r,n}=h_{ds}X_{f,n}+\eta_{f,n},\ \ 1\leq n\leq N-1.\end{array}  \tag{1}\]  
where \(X_{n}\) (resp. \(Y_{n}\)) and \(X_{r,n}\) (resp. \(Y_{r,n}\)) are the feedforward inputs (resp. outputs), and \(X_{f,n}\) and \(Y_{f,n}\) are the feedback channel input and output, respectively. The channel noises of \(\eta_{n},\eta_{r,n},\eta_{f,n}\) are independent and identically distributed (i.i.d.) as \(\mathcal{N}(0,\sigma^{2})\), \(\mathcal{N}(0,\sigma^{2}_{r})\), \(\mathcal{N}\Big{(}0,\sigma^{2}_{f}\Big{)}\), respectively. **Definition 1**. : _For the GRC with noisy feedback in Figure 1a, a (\(N\), \(|\mathcal{M}|\), \(P\), \(P_{r}\), \(P_{f}\))-code with the average power constraints consist of:_  
* _A uniformly distributed message_ \(M\)_, which takes values over the set_ \(\mathcal{M}=\{1,2,\ldots,|\mathcal{M}|\}\)_._
* _Encoder_ \(\mathcal{I}\) _with output_ \(X_{n}=f_{1,n}\left(M,Y_{f}^{n-1}\right)\)_, where_ \(f_{1,n}(\cdot)\) _is an encoding function of Source at the time index_ \(n\)__\((1\leq n\leq N)\)_, satisfying the average power constraint_ \(\sum_{n=1}^{N}E\left[X_{n}^{2}\right]\leq NP\)_._
* _Relay with output_ \(X_{r,n}=f_{r,n}\left(Y_{r}^{n-1}\right)\)_, where_ \(f_{r,n}(\cdot)\) _is an encoding function of Relay at time index_ \(n\)__\((1\leq n\leq N)\)_, satisfying the average power constraint_ \(\sum_{n=1}^{N}E\left[X_{r,n}^{2}\right]\leq NP\)_._  
Figure 1: The information-theoretic schematic diagram. * _Encoder 2 with output_ \(X_{f,n}=f_{2,n}(Y^{n})\)_, where_ \(f_{2,n}(\cdot)\) _is an encoding function of Destination at time index_ \(n\)__\((1\leq n\leq N-1)\)_, satisfying the average feedback power constraint_ \(\sum_{n=1}^{N-1}E\left[X_{f,n}^{2}\right]\leq(N-1)P_{f}\)_._
* _Decoder has output_ \(\bar{M}=\psi(Y^{N})\)_, where_ \(\psi\) _is the decoding function of Destination. The average decoding error probability is defined as_ \[P_{e}=\frac{1}{|\mathcal{M}|}\sum_{m\in\mathcal{M}}\Pr\Bigl{\{}\psi\Bigl{(}Y^{ N}\Bigr{)}\
eq m|m\text{ was sent}\Bigr{\}}.\] (2)  
**Definition 2**. : _The rate \(R\) is said to be \((N,e)\)-achievable if for given coding block length \(N\) and decoding error probability \(e\), there exists a \((N,|\mathcal{M}|,P,P_{r},P_{f})\)-code described in Definition 1 such that_  
\[\frac{\text{log}|\mathcal{M}|}{N}\geq R-\epsilon,\ P_{e}\leq\epsilon. \tag{3}\]  
_The \((N,e)\)-capacity [29] of the GRC with noisy feedback is defined as the supremum over all achievable rates in Definition 2, denoted as \(\mathcal{C}_{\text{GRC}}^{\text{NFD}}(N,e)\)._  
**Definition 3**. : _The modulo-d function [30; 31] (also named the one-dimensional modulo-lattice function) is defined as_  
\[M_{d}[x]=x-d\cdot\left\lfloor\frac{y}{d}+\frac{1}{2}\right\rfloor, \tag{4}\]  
_where \(d>0\) is the step size of the one-dimensional modulo-lattice and \(\lfloor x\rfloor\) represents the largest integer that is smaller than \(x\). Proposition 1 provides the properties of the modulo-d function [30; 31]._  
**Proposition 1** ([30; 31]). : \(\quad\)__  
1. \(\quad M_{d}[x]\in[-\frac{d}{2},\frac{d}{2}).\)__
2. \(\quad M_{d}[M_{d}[x]+y]=M_{d}[x+y].\)__
3.