## 5 Performance Analysis
### Probability of Error from Source to Relay Node  
According to Theorem 1, the received signal at the relay is the linear combination of original signals with additive Gaussian noise. Therefore, an error occurs if the noise vector is outside the fundamental Voronoi region [14] of the lattice, \(\Lambda^{\prime}\), in Equation (23), denoted by \(\mathcal{V}(\Lambda^{\prime})\). The probability of error in decoding a linear equation in \(n-\)dimensional lattice at the relay is given by:  
\[P_{1}(\mathbf{h},n)=\text{Pr}(\parallel\mathbf{z}_{R}\parallel\
otin\mathcal{ V}(\Lambda^{\prime})) \tag{25}\]  
The vector, \(\mathbf{z}_{R}\), has Gaussian distribution in each dimension with zero mean and variance, \(\sigma_{1}^{2}\). It is well known that the above probability expression depends on the shape of the Voronoi region. The bounds of such probability of error can be found in [16]. For simplicity, we now consider a subset of an integer lattice, namely, \(\mathcal{L}=\mathbb{Z}^{n}\), to draw the original signals, \(s_{l}\). This is because the Voronoi region for the integer lattice is a hypercube, and hence, it can be used to obtain an exact probability of the error expression. The following theorem gives this probability. **Theorem 2**: _The probability of error in decoding a linear equation of signals drawn from \(\mathcal{L}=\mathbb{Z}^{n}\) using IFP is given by:_  
\[P_{1}(\mathbf{h},n)=1-\left(\text{erf}\left(\frac{c}{2\sqrt{2}\sigma_{1}} \right)\right)^{n}\]  
_where \(c=\text{gcd}(h_{1}^{z},h_{2}^{z},\ldots,h_{L}^{z})\)_  
_Proof:_ We firstly use induction to prove the that the Voronoi region of \(\Lambda^{\prime}=\{\lambda^{\prime}\mid\lambda^{\prime}=\sum_{l=1}^{L}h_{l}^{z }\lambda_{l},h_{i}^{z}\in\mathbb{Z},\lambda_{i}\in\Lambda\}\) is a hypercube of side \(c\). Let \(L=2\) and \(n=1\). The Bezout's lemma [17] states that for any two (non-zero) integers, \(p\) and \(q\), there exist two integers, \(u\) and \(v\), such that  
\[pu+qv=d,\]  
where \(d\) is the common divisor of \(p\) and \(q\). In addition, if \(d>0\) is the greatest common divisor (gcd) of (\(p,q\)), it is the smallest positive integer satisfying this equation for any \((u,v)\) integer pair. Therefore, using [PERSON]'s lemma, any point in \(\Lambda^{\prime}\) can be written as:  
\[\lambda^{\prime}=b_{1}\lambda_{1}+b_{2}\lambda_{2}=d\mathbb{Z}\]  
where \(d=\text{gcd}(b_{1},b_{2})\). Similarly, this result can be extended to \(L=L+1\) by induction. Since this result is independent of the integers, \(\lambda_{i}\), therefore, it can be extended to \(n-\)dimensional lattices. Using this result, we can write:  
\[\Lambda^{\prime}=c\mathbb{Z}^{n}\]  
where \(c=\text{gcd}(h_{1}^{z},h_{2}^{z},\ldots,h_{L}^{z})\). Hence, the Voronoi region of \(\Lambda^{\prime}\) is a hypercube of side \(c\). To prove the second part of the theorem, we rewrite Equation (25) as:  
\[P_{1}(\mathbf{h},n)=1-\text{Pr}(\parallel\mathbf{z}\parallel \in\mathcal{V}(\Lambda^{\prime}))\] \[=1-\left(\frac{1}{\sqrt{2\pi\sigma^{2}}}\int\limits_{-c/2}^{c/2} e^{-\frac{x^{2}}{2\sigma_{1}^{2}}}du\right)^{n}\]\[P_{1}(\mathbf{h},n)=1-\left(\text{erf}\left(\frac{c}{2\sqrt{2}\sigma_{1}}\right) \right)^{n} \tag{26}\]  
where \(\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt\). This concludes our proof. For the special case of \(n=1\), Equation (26) reduces to:  
\[P_{1}(\mathbf{h},1)=\text{erfc}\left(\frac{c}{2\sqrt{2}\sigma_{1}}\right) \tag{27}\]  
where \(\text{erfc}(x)=1-\text{erf}(x)\). Note that in the proposed scheme, the effective behavior of the channel is reduced to a point-to-point AWGN channel with no channel fading. This makes its characterization possible. In the previous formulations of CF, the effective noise is Gaussian noise along with the self-noise, which results in the non-Gaussian distribution of total effective noise [5]. Hence, the characterization in existent formulations has been done by providing limits of error probability and not the accurate value.