# Cooperative Reception of Multiple Satellite Downlinks
## 2 System Description and Design
### SISO System  
Figure 1 simply visualizes the single radio link between the satellite as a transmitter and the ground station as a receiver. Such radio communication system utilizing a BFSK and considering an AWGN channel would be configured as in Figure 2.  
The simulation starts with streaming the input bits, i.e., the message signal (\(m\)), in a unipolar form. Then, since BFSK is a two-dimensional modulation scheme which maps input data into orthogonal representations, an input (1) is mapped into (1) whereas input (0) is mapped into (\(j\)). These complex symbols are sent from the transmitter's antenna to the receiver through the free space as a channel, which was a AWGN channel in our simulation. Through signal's propagation, noise is added to the signal and since the transmitted signal is complex, the noise addition in the simulation is complex as well. Therefore, adding the complex noise (\(n\)) to the complex transmitted signal (\(X\)) represents the received signal (\(r\)) as shown in Equation (1):  
\[r=X+n \tag{1}\]  
As mentioned earlier, the transmitted signal \(X\) has two representations and consequently, we have two states for the received complex signal as follows:  
For input 1, the received signal is: \(r=((1+n)+(0+n)\;j)\) and similarly, for input 0, \(r=((0+n)+(1+n)\;j)\). That would create the detection hypothesis at the receiver to  
Figure 1: Single downlink communication.  
Figure 2: Block diagram of SISO system utilizing BFSK and considering AWGN channel.  
compare the real part to the imaginary part of the received signal and decide whether it is (1) if the real part is greater than the imaginary part and (0) if the other way around. The detection hypothesis is therefore:  
\[\begin{array}{ll}&1\\ Real\ (r)&\begin{array}{l}>\\ <\\ 0\end{array}&\begin{array}{l}Imag(r)\\ 0\end{array}\end{array} \tag{2}\]  
The theoretic BER of the BFSK system was calculated using the error function as well as the Q function to be plotted and compared to the simulation BER [37; 38].  
\[\begin{array}{l}\text{BER}=\frac{1}{2}\ erfc\left(\sqrt{\frac{E_{b}}{2N_{o}}}\right) \end{array} \tag{3}\]  
\[\begin{array}{l}\text{BER}=Q\left(\sqrt{\frac{E_{b}}{N_{o}}}\right)\end{array} \tag{4}\]  
The simulation and theory BER curves matched, as illustrated in Figure 3, with the same expected behavior where the system performance was improved when increasing the energy per bit. Moreover, the performance was further observed over different data rates at \(E_{b}\)/\(N_{o}\) of 10 dB and the resulting curve is presented in Figure 4.  
Therefore, in order for SISO systems to function efficiently, we must either increase the transmitted power or to stream out at low signaling rates. However, that would be totally impractical and against the requirements of new generations' communications of high data rate signaling and hostile environment serving. This simply summarizes the fact that SISO systems are not promising schemes to effectively fulfill the demands of the upcoming technology trends and thus alternative techniques are needed.  
Figure 3: Simulation BER versus theory BER curves.