## 3 Phase I : Proposed Precoding-Based Source to Relay Transmission
### Proposed Integer Forcing Precoder  
We aim to precode the signals, such that a linear combination of original signals can be obtained at the relay at the maximum rate. It is well known that zero forcing (ZF) precoding is a standard suboptimal approach for precoding, which is known to provide a promising trade-off between complexity and performance [11]. The traditional zero forcing precoder is designed to eliminate the effect of the channel fading, while satisfying constraints of power optimally as per certain performance measures. The precoder required in implementing CF has similar design requirements as ZF precoding, but with an important difference. In CF, the precoder is required to eliminate only the non-integral part of the channel, while retaining the integral part in the precoded signal. In this subsection, we show the design of a naive precoder to implement CF. A sub-optimal integer forcing precoder (IFP) that maximizes the rate of computation, while eliminating the non-integral part of the channel, is given by the solution to the following optimization problem:  
\[\max_{\mathbf{h}^{z}\in\mathbb{Z}^{L},\mathbf{h}^{z}\
eq 0}\ R( \mathbf{h},\mathbf{w})\] (14) subject to \[\ E[\mid w_{l}\mid^{2}]\leq\gamma_{w} \tag{15}\]In order to solve this problem, we relaxed condition (15), making it an unconstrained optimization problem. A direct differentiation of \(R(\mathbf{h},\mathbf{w})\) with respect to \(\mathbf{W}\) gives:  
\[\mathbf{W}_{opt}=diag(\mathbf{h}^{-1}\mathbf{h}_{opt}^{\mathbf{z}})\]  
where \(\mathbf{h}_{opt}^{\mathbf{z}}\) is the solution to:  
\[\max_{\mathbf{h}^{\mathbf{z}}\in\mathbb{Z}^{L},\mathbf{h}^{\mathbf{z}}\
eq 0} \text{ log}^{+}\left(\frac{\parallel\mathbf{h}^{z}\parallel^{2}}{\sigma_{1}^{2}}\right) \tag{16}\]  
\[\text{ subject to }\ \ \ E[\parallel h_{l}^{z}h_{l}^{-1}\parallel^{2}<\gamma_{w} \ \ \ \ \ \ \text{ for }l=1\ldots L \tag{17}\]  
Hence, the integer part of the channel, \(\mathbf{h}^{\mathbf{z}}\), should be chosen, such that, within the constraint of maximum precoding power, \(\gamma_{w}\), the computational rate is maximized. A possible solution is given by (rounding off using the Babai estimate [12]):  
\[\mathbf{h}^{\mathbf{z}}_{opt}=[\gamma_{w}\mathbf{h}] \tag{18}\]  
where \([. ]\) represents the closest integer. Without loss of generality, one of the solutions can be taken as \(\mathbf{h}^{\mathbf{z}}_{opt}=[\mathbf{h}]\). The resultant precoder, which also imposes the constraint, \(\mathbf{h}^{\mathbf{z}}\
eq 0\  
where \(r=h_{l}^{\overline{z}}/h_{l}^{z}\). Since \(h_{l}^{z}\in\mathbb{Z}\) and \(h_{l}^{\overline{z}}\in(-0.5,0.5)\) for \(\mid h_{l}\mid>0.5\), therefore, the range of \(r\) is given by \(r\in(-0.5,0.5)\). Putting \(r\) in the expression of \(w_{l}\) above, we get the range of \(w_{l}\) as \(w_{l}\in(\frac{2}{3},2)\). Consequently, the range of \(w_{l}^{2}\) is given by \(w_{l}^{2}\in\left(\frac{4}{9},4\right)\). In the limiting condition, the upper bound of the average precoder power, \(E[\mid w_{l}\mid^{2}]=\gamma_{w}\) (using the power limitation from (3)), can be given as:  
\[\frac{4}{9}<\gamma_{w}<4 \tag{21}\]  
Hence, the precoded signal power using the proposed IFP is bounded for channel realization with reasonably good gains. This concludes our proof. _Remark_: From theorem 1, it can be seen that the use of IFP to implement CF effectively reduces the channel to a reliable AWGN (additive white Gaussian noise) channel. Therefore, the achievable rate using the above IFP is given by:  
\[R_{IFP}(\mathbf{h})\leq\text{log}^{+}\left(\frac{\parallel\mathbf{h}^{z}\parallel ^{2}}{\sigma_{1}^{2}}\right) \tag{22}\]  
From Equations (2) and (21), it is clear that when \(\gamma_{w}=1\), the power consumed by the precoded signal is equal to the power consumed by the original signal under limiting conditions. However, when \(\gamma_{w}<1\) (or equivalently, \(h_{l}^{z}<h_{l}\)), the precoded signal consumes less power than the original signal. On the other hand, when \(\gamma_{w}>1\), the precoded signal consumes additional power.