# Cooperative Reception of Multiple Satellite Downlinks
## 2 System Description and Design
### SIMO System
In parallel, the system throughput, a QoS metric in satellite communications to indicate the number of received packets to the required processing time [40], can be maximized in the proposed scheme as the cooperative system receives \(N\) streams that can be simultaneously processed at the receiving sites. After these motivational probabilistic concepts on how this receive diversity scheme can significantly reduce system outages and enhance the overall performance, the next step was to analyze the proposed system model. Once the VGS receives the \(N\) downlink replicas with equal data length \(L\), the combining matrix \(C\) is generated to eventually have the dimensions of \(N\)-by-\(L\). Subsequently, the VGS triggers the combining algorithm to combine these multiple streams in a way so that the resulting combined stream, \(r_{c}\), would have lesser BER than all BERs from the received versions. If we structure the combining matrix as:  
\[C=\left[ \begin{array}{ccc}r_{1}(1)&\cdots&r_{1}(L)\\ \vdots&\ddots&\vdots\\ r_{N}(1)&\cdots&r_{N}(L)\\ \end{array} \right]\]  
Then, instead of just selecting the stream \(r_{n^{th}}\) out of the received versions based on the selection combiner (SC) method with respect to the criterion:  
\[\underset{n\in(1:N)}{\text{\emph{min}}}\;\;\text{\emph{BER}}\;(r_{n^{th}}) \tag{5}\]  
The aimed combined stream \(r_{c}\) should achieve even lesser BER and it should be none of the received versions, i.e.,  
\[\underset{c\in(1:N)}{\text{\emph{min}}}\;\;\text{\emph{BER}}\;(r_{c}) \tag{6}\]  
By interpreting the criterion in Equation (6) from the likelihood perspective, the combined stream's likelihood, \(L_{c}\), can be represented as:  
\[L_{c}\;(\;r_{c}|r_{1},\;r_{2},\;\ldots,\;r_{N}) \tag{7}\]  
Since the received versions, \(r_{1},\;r_{2},\;\ldots,\;r_{N}\), are independent, the likelihood is therefore equivalent to the joint probability mass function (PMF) of these \(N\) versions and can be alternatively expressed as:  
\[\prod_{i=1}^{N}L_{c}\;(r_{c}|r_{i}) \tag{8}\]  
Equation (8) represents the likelihood quantity the VGS should maximize in the combining process to provide as good of an estimate of the transmitted stream as possible. To this context, one concept to reach this goal is to combine these streams with respect to the maximization of the likelihood function through the selection of the more likely occurring bit of each column in \(C\) to generate \(r_{c}\) according to:  
\[r_{c}(k)=mode\;\{\;r_{1}(k),\;r_{2}(k),\;\ldots,\;r_{N}(k)\} \tag{9}\]  
where \(mode\) is the function that picks the most frequently occurring entry at each column in \(C\), and \(k\) = 1, 2, \(\ldots,\;L\). If we assume that the original transmitted stream is \(r_{0}\), the probability of detection errors can be expressed as [4]:  
\[P_{e}(k)=P\;(\;r_{c}(k)\
eq\;r_{0}(k)) \tag{10}\]In general, the minimization of the error probability \(P_{e}\) corresponds to the maximization of the probability to provide a good estimate \(P\) to the originally transmitted data. Therefore, in our combining, the virtual ground station should generate a combined stream \(r_{c}\) whose probability is  
\[P\left(\ r_{0}=r_{c}|r_{1:N},\ H\right) \tag{11}\]  
where \(H\) denotes the channels impulse response matrix. The probability in Equation (11) can be further expressed using the conditional probabilities [4] as follows:  
\[P\left(\ r_{0}=r_{c}|r_{1:N},\ H\right)=\frac{P\left(r_{0}=r_{c}\right)f_{r_{1: N/r_{0},H}}(r_{1:N}|r_{0}=r_{c},\ H)}{f_{r_{1:N}|H}(r_{1:N}|H)} \tag{12}\]  
where \(f_{r_{1:N}|H}\) is the conditional probability density function (PDF) of \(r_{1:N}\) given \(H\), while \(f_{r_{1:N/r_{0},H}}\) is the conditional PDF of \(r_{1:N}\) given \(r_{0}\) and \(H\).