# Cooperative Reception of Multiple Satellite Downlinks
## 2 System Description and Design
### SIMO System
Since the two terms \(P\left(r_{0}=r_{c}\right)\) and \(f_{r_{1:N}|H}(r_{1:N}|H)\) are independents of the aimed \(r_{c}\), the maximization of the probability to generate good estimated combined stream is therefore reduced to maximizing only the dependent PDF term:  
\[f_{r_{1:N/r_{0},H}}(r_{1:N}|r_{0}=r_{c},\ H) \tag{13}\]  
Thus, we can set our maximum likelihood (ML) goal in generating the combined stream as the combined stream as:  
\[r_{c_{ML}}=\underset{r_{c}\
otin r_{1:N}}{argmax}\ f_{r_{1:N/r_{0},H}}(r_{1:N}| r_{0}=r_{c},\ H) \tag{14}\]  
If we reform the conditional PDF term in Equation (14) using the general communication model of estimating the received signal in a MIMO system \(r_{1:N}=H\ r_{0}+n_{1:N}\)[4], we obtain:  
\[\underset{c\in(1:N)}{\text{\emph{min}}}\;\;\text{\emph{BER}}\;(r_{c}) \tag{6}\]0  
where \(f_{n}\) is the PDF of the white Gaussian noise (\(n\)), i.e., \(f_{n}(n)=\frac{1}{(\pi\sigma^{2})^{n}}e^{-\frac{1}{\sigma^{2}}|n|^{2}}\). Accordingly, the left side of Equation (15) can be maximized by reducing \(\|n\|^{2}\) for the aimed combined stream and hence the maximum likelihood can be achieved if the \((r_{1:N}-Hr_{c})\) is minimized based on:  
\[\underset{c\in(1:N)}{\text{\emph{min}}}\;\;\text{\emph{BER}}\;(r_{c}) \tag{6}\]1  
To meet the criterion of distance reduction, i.e., the term \(||r_{1:N}-Hr_{c}||^{2}\) in Equation (16), we applied the likelihood modal hypothesis on each column in the matrix \(\mathcal{C}\) to generate the combined stream \(r_{c}\). This combining can generate a stream which follows the tendency of the data towards the original data distribution. Hence, the combined stream can achieve lesser distance to the original stream than the distance from each received version to the original data. The proposed combining can be implemented by initially selecting the first received version out of the \(N\) received versions to be the pre-combined stream, \(r_{pre\_c}\), i.e., \(r_{pre\_c}=r_{1}\). Then, the combined stream \(r_{c}\) is derived from \(r_{pre\_c}\) or its complement according to:  
\[r_{c}(k)=\left\{ \begin{array}{ll}r_{pre_{c}}(k),&\mbox{if }mode\left\{\ r_{1}(k), \ r_{2}(k),\ \ldots,\ r_{N}(k)\right\}=r_{pre_{c}}(k)\\ \overline{r_{pre_{c}}(k)},&\mbox{Otherwise}\end{array} \right. \tag{17}\]  
Therefore, the combined stream can be derived from the received versions on a bit-by-bit inspection of the data tendency in all the received streams. This is promising to bring data into the combined stream that can better fit the original data distribution. In the results section, the Euclidean distance (ED) results of the \(r_{0}\)-to-\(r_{c}\) and \(r_{0}\)-to-\(r_{1:N}\) can verify how much reduction in that distance this algorithm can provide. The MATLAB simulation of the proposed SIMO system can be configured from multiple SISO links whose simulation steps are detailed in Section 2.1. These SISO links were degraded with different impairment levels to create independent and uncorrelated received versions of the same transmitted data. Therefore, the simulation started with generating random unipolar data and then mapping these input bits into complex BPSK symbols. In the next step, \(N\) complex AWGN streams were generated and added to form \(N\) multiple uncorrelated versions of the received signal to be fed to the detector. The detection process passed the detected bit stream from each received version to the VGS to establish the combining square matrix \(C\) where the \(n^{\text{th}}\) row corresponds to detected bits from the \(n^{\text{th}}\) receiving site. The VGS hence started applying the maximum likelihood hypothesis expressed in Equation (17) on that square matrix based on vertical bit disagreement rule. This maximum likelihood detection was performed at the combining matrix in an ascending tempo starting from the 1 st row to the \(N\)th one representing the possibilities of combining multiple streams gradually. Therefore, the first case represents conventional single stream detection as it performed detection on the first row only.