# Cooperative Reception of Multiple Satellite Downlinks
## 3 Results and Discussion
### The Best Stream Is Repeatedly Injected  
The third optional technique that could be implemented in the proposed cooperative system is when the best stream is repeatedly injected. Out of the received versions, the one with the lowest BER could be repeatedly tailed to the combining matrix, as visualized in Figure 8c, at each combining trial. This could gradually impose more detection weight to the best stream and hence achieve less errors at the combined stream. The improvement techniques can be modeled the same as the raw combining in Equations (16) and (17) except the required pre-combining procedures of each technique. For instance, in the first technique, the worst stream should be excluded out of the combining matrix and then the same modeling applied for combining the new \((N-1)\)-by-\(L\) combining matrix \(C\), as demonstrated earlier, to get the following ML criteria:  
\[r_{c_{ML}}=\underset{r_{c}\
otin r_{N-1}}{argmin}\left|\left|r_{1:N-1}-Hr_{c} \right|\right|^{2} \tag{18}\]  
where the initial assumption of \(r_{pre\_c}=r_{1}\) can be also valid then modified to get the \(r_{c}\) based on:  
\[r_{c}(k)=\left\{ \begin{array}{ll}\dfrac{r_{pre_{c}}(k)}{r_{pre_{c}}(k)},&if \text{ mode }\{\ r_{1}(k),\ r_{2}(k),\ \ldots,\ r_{N-1}(k)\}=r_{pre_{c}}(k)\\ \dfrac{r_{pre_{c}}(k)}{r_{pre_{c}}(k)},&Otherwise\end{array} \right. \tag{19}\]  
Table 1 lists the achieved BER of each suggested SP technique versus the raw combining BER values. Excluding the worst stream out of the combined streams reflected slight improvement compared to the original combining BER values. Replacing that worst stream with the best stream improved the performance even more. The best achieved BER values were when always injecting the best stream into the combining matrix which reached zero error at six receiving sites combining. Figure 9 presents the simulation BER results of the original combining algorithm as well as the three proposed algorithms in linear and logarithmic scales. The three SP algorithms improved the system performance in achieving less detected errors compared to the raw combining. Rejecting the worst quality stream slightly enhanced the system performance with more than 50% errors reduction in the combining trials of more than seven sites. Replacing the worst stream with the best stream led to even better performance. In the last proposed technique, the best stream was repeatedly tailed to the combining matrix, imposing more detection weight for the least corrupted received signal version. Doing so, the system achieved zero errors detection as early as when only six sites were engaged. Figure 9b does not show that achieved zero BER due to the infinitive logarithmic quantities which explains why linear scale curves are also provided. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No. of Sites** & **Raw** & **Worst Excluded** & **Worst Replaced** & **Best Tailed** \\ \hline
1 & 0.26862 & 0.26862 & 0.26862 & 0.26862 \\
2 & 0.25414 & 0.21686 & 0.11775 & 0.05261 \\
3 & 0.13884 & 0.10295 & 0.0468 & 0.00645 \\
4 & 0.12037 & 0.08797 & 0.05483 & 0.00044 \\
5 & 0.05619 & 0.03321 & 0.01966 & \(2\times 10^{-5}\) \\
6 & 0.04439 & 0.025 & 0.01785 & 0 \\
7 & 0.01624 & 0.00694 & 0.00501 & 0 \\
8 & 0.01131 & 0.00445 & 0.00362 & 0 \\
9 & 0.00301 & 0.00087 & 0.00069 & 0 \\
10 & 0.00183 & 0.00041 & 0.0004 & 0 \\
11 & 0.00029 & \(5\times 10^{-5}\) & \(6\times 10^{-5}\) & 0 \\
12 & 0.00013 & \(2\times 10^{-5}\) & \(4\times 10^{-5}\) & 0 \\
13 & \(3\times 10^{-5}\) & 0 & 0 & 0 \\
14 & \(1\times 10^{-5}\) & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: BER results from raw receive diversity and proposed SP algorithms. The Euclidean distance (ED) can be measured to indicate how close the received versions as well as the combined one are from the original message stream through:  
\[ED_{a,b}=sqrt\left((a-b)*(a-b)^{\prime}\right)\]  
where \(a\) and \(b\) can be any signal vectors of the same length.