## 2 Theoretical Model of a System in the Presence of Gaussian Noise
### Single-Correlator Receiver  
In this case the spreader is represented by a multiplier, as shown on the left hand upper side of Figure 1, and a single correlator with output \(w_{1}\) at the receiver side. The chip sequence \(m(k)\) is split into in-phase and quadrature sequences using the demultiplexer block (DEMUX) in such a way that the even-indexed chip sequence \(m_{t}(k)\) modulates the in-phase carrier (\(\sqrt{E_{c}}\sqrt{2/M}\ \mathrm{cos}\ \Omega_{c}k\)) and odd-indexed chip sequence \(m_{Q}(k)\) modulates the quadrature carrier (\(\sqrt{E_{c}}\sqrt{2/M}\ \mathrm{sin}\ \Omega_{c}k\)), where \(E_{c}\) is the energy per chip and \(M\) is the number of interpolated samples contained in one chip interval. Therefore, the transmitted signal can be defined as  
\[s(k)=m_{t}(k)\sqrt{E_{c}}\sqrt[2]{2\over M}\ \mathrm{cos}(\Omega_{c}k)+m_{Q}(k) \sqrt{E_{c}}\sqrt[2]{2\over M}\ \mathrm{sin}(\Omega_{c}k), \tag{1}\]where \(m_{I}(k)\) and \(m_{Q}(k)\) are in-phase and quadrature chip sequences expressed in discrete time domain respectively and \(\Omega_{c}\) is the frequency of the carrier. It is important to note that \(k\) is a discrete time variable. The noise is to be generated at discrete time instants defined by \(k\). Thus, the band-limited pass-band noise is expressed as  
\[n(k)=n_{I}(k)\sqrt{E_{N}}\sqrt{\frac{2}{M}}\cos(\Omega_{c}k)-n_{Q}(k)\sqrt{E_{N }}\sqrt{\frac{2}{M}}\sin(\Omega_{c}k), \tag{2}\]  
where the energy of the noise samples inside a chip interval is \(E_{N}=N_{0}/2\) and \(n_{I}\) and \(n_{Q}\) are in-phase and quadrature noise samples of zero mean and unit variance. The block schematic of this noise generator is presented inside Figure 1. The noise is expressed in this form to comply with the applied signal processing demodulation procedure of both the signal and noise at the receiver side. Namely, in simulation, it is important to achieve that the power of the noise in respect to the power of the signals are controlled at all times at the transmitter side. The received noisy signal \(s_{R}(k)=s(k)+n(k)\) is demodulated using a correlator that consists of a multiplier and an adder. It is sufficient to present the processing of the signal in one branch of the demodulator because the processing in both branches is equivalent. The signal at output of the receiver multiplier is  
\[y_{I}(k)=[s(k)+n(k)]\sqrt{\frac{2}{M}}\cos\Omega_{c}k). \tag{3}\]  
The samples of this signal are added in the chip interval (corresponds to integration in continuous time systems). Because, in the case of a single-correlator receiver, the source generates binary signal the output of the transmitting spreader was the first spreading sequence \(c_{i1}(k)\), _i.e._, \(m(k)=c_{i1}(k)\), and a random sample \(z_{i}\) of a random process \(Z_{i}\) in \(I\) branch is obtained, _i.e._,  
Figure 1: Block schematic of communication system. \[z_{il}=\sum_{1}^{M}y_{l}(k)=\sqrt{E_{c}}c_{i1}+\sqrt{E_{N}}n_{il},\text{for}\ i=2,4,6,...,\beta, \tag{4}\]  
and in \(Q\) branch as  
\[z_{iQ}=\sqrt{E_{c}}c_{i1}+\sqrt{E_{N}}n_{iQ},\text{for}\ i=1,3,5,...\beta. \tag{5}\]  
A multiplexer (MUX) is used to combine in-phase and quadrature sequences back into a \(2\beta\)-chip sequence \(z_{i}\)  
\[z_{i}=\sqrt{E_{c}}c_{i1}+\sqrt{E_{N}}n_{i},\text{for}\ i=1,2,3,...\,2\beta\ \ \, \tag{6}\]  
where \(n_{  
\[ \begin{split}\sigma_{w1}^{2}=2\beta E_{c}E&\{c_{i1}^{4} \}-2\beta E_{c}E^{2}\{c_{i1}^{2}\}+2\beta E_{N}\sigma_{ni}^{2}E\{c_{i1}^{2}\} \\ &=2\beta E_{c}[E\{c_{i1}^{4}\}-E^{2}\{c_{i1}^{2}\}]+2\beta E_{N} \sigma_{ni}^{2}E\{c_{i1}^{2}\}\end{split}