## 3 Proposed Analog RF Beamforming
### Destination Analog RF Combiner  
Similar to the common analog precoder at the source for transmit beamforming gain over all the channels \(\textbf{H}_{k},\ k\in\big{\{}1,\ldots,K\big{\}}\). There is a definite need to derive the common analog combiner \(\textbf{W}_{RF}\in\mathbb{C}^{N_{r}\times N_{t}^{RF}}\) at the destination for the RF receive processing over all the channels from the \(K\) relay nodes to the destination, i.e., \(\textbf{G}_{1},\ \textbf{G}_{2},\ldots,\textbf{G}_{K}\). This target can be achieved by minimizing the reconstruction loss of \(\textbf{W}_{RF}\) over the set of optimal combiners  
\[\Big{\{}\textbf{W}_{opt1}\in\mathbb{C}^{N_{r}\times N_{t}^{RF}},\ \textbf{W}_{opt2}\in\mathbb{C}^{N_{r}\times N_{t}^{RF}},\ldots,\textbf{W}_{optk }\in\mathbb{C}^{N_{r}\times N_{t}^{RF}}\Big{\}}\]  
corresponding to the aforementioned channels \(\textbf{G}_{k},\ k\in\big{\{}1,\ldots,K\big{\}}\). Therefore, a problem is formulated that attempts to minimize the error of a single matrix over several matrices simultaneously. The solution of this problem leads to the design of the required RF combining matrix. The problem formulation is given as  
\[\begin{array}{c}\min_{\textbf{W}_{RF}}\Big{\{}\|\textbf{W}_{RF}-\textbf{W}_ {opt1}\|_{F}^{2}+\cdots+\|\textbf{W}_{RF}-\textbf{W}_{optk}\|_{F}^{2}\Big{\}},\\ s.t.\ |\textbf{W}_{RF}(x,y)|=\frac{1}{\sqrt{N_{r}}}\ \forall\ x,y.\end{array} \tag{26}\]  
It is known that \(\|A\|_{F}^{2}+\|B\|_{F}^{2}\geq\|A+B\|_{F}^{2}\), and therefore, the cost function in (26) can be written as  
\[\|\textbf{W}_{RF}-\textbf{W}_{opt1}\|_{F}^{2}+\cdots+\|\textbf{W}_{RF}- \textbf{W}_{optK}\|_{F}^{2}\geq\|\textbf{W}_{RF}-\bigg{(}\frac{\textbf{W}_{ opt1}+\cdots+\|\textbf{W}_{optK}}{K}\bigg{)}\|_{F}^{2}. \tag{27}\]  
Let  
\[\textbf{W}_{opt}=\bigg{(}\frac{\textbf{W}_{opt1}+\cdots+\textbf{W}_{optK}}{K} \bigg{)}. \tag{28}\]Using (27) and (28), the problem (26) can be re-translated into the following form  
\[\min_{\mathbf{W}_{RF}}\lVert\mathbf{W}_{RF}-\mathbf{W}_{opt}\rVert_{F}^{2}\]  
\[s.t.\ |\mathbf{W}_{RF}(x,y)|=\frac{1}{\sqrt{N_{r}}}\,\forall\ x,y. \tag{29}\]  
The global optimal solution to the problem (29) i.e., \(\lVert\mathbf{W}_{RF}-\mathbf{W}_{opt}\rVert_{F}^{2}=0\) is possible only when the element-wise constant amplitude constraint associated with \(\mathbf{W}_{RF}\) is relaxed, but it is not allowed to relax the constant modulus constant while designing \(\mathbf{W}_{RF}\), and therefore, \(\lVert\mathbf{W}_{RF}-\mathbf{W}_{opt}\rVert_{F}^{2}\approx 0\) is expected. This condition leads to the sub-optimal solution of (29) as  
\[\lVert\mathbf{W}_{RF}-\mathbf{W}_{opt}\rVert_{F}^{2}=\lVert\mathbf{W}_{RF}\rVert_{F}^{2}+ \lVert\mathbf{W}_{opt}\rVert_{F}^{2}-2 tr\Big{\{}\Re\Big{[}\mathbf{W}_{RF}\mathbf{W}_{opt}^ {H}\Big{]}\Big{\}}, \tag{30}\]  
where \(\Re[.]\) represents the real part of the input entity, whose minimum is achieved when \(\mathbf{W}_{RF}\) has the same element-wise phase as that of \(\mathbf{W}_{opt}\), thus  
\[\mathbf{W}_{RF}=\bigg{(}\frac{1}{\sqrt{N_{r}}}\bigg{)}exp\big{\{}j\ arg\big{(} \mathbf{W}_{opt}\big{)}\big{\}}, \tag{31}\]  
where \(arg(.)\) denotes the argument operator.