# Effects of Interference and Mitigation Using Notch Filter for the DVB-S2 Standard
## 3 The Effects of Interference
### Signal Constellation  
A constellation diagram is a representation of a digitally modulated signal in a two-dimensional plane [29]. The general effect of CWI on the power spectral density, received SNR and signal constellation will be investigated in this section. Considering that transmitted signal \(S\) is the output of a QPSK signal constellation, then, with CWI, the received signal is:  
\[\textit{Received Signal}=\textit{Signal}+\textit{CWI}=S+\textit{a-exp}\Big{(}2 \pi j\cdot f_{j}\cdot t\Big{)} \tag{1}\]  
where \(a\) and \(f_{j}\) are amplitude and center frequency of CWI, respectively. For simplicity, consider that the transmitted signal \(S\) has normalized power. Then the power of the two different parts are:  
\[\begin{array}{l}P_{\textit{Signal}}=E^{2}(S)=1\\ P_{\textit{Interference}}~{}=~{}E^{2}(\textit{CWI})=~{}a^{2}\end{array} \tag{2}\]  
Then, Signal to Jamming Ratio (SJR) and Jamming to Signal Ratio parameters (JSR) are equal to:  
\[\textit{JSR}~{}=~{}\frac{P_{\textit{Interference}}}{P_{\textit{signal}}}~{}=~ {}a^{2},~{}\textit{JSR}(dB)~{}=~{}-20 log_{10}^{a} \tag{3}\]  
Mathematically, it can be proven that adding a CWI would make the symbols circle around their constellation point. Since the transmitted signal is in the form \(\mathrm{S}=I(t)+\mathrm{jQ(t)}\), it can be transformed into complex cartesian form:  
\[S=\Big{(}I(t)+\textit{a}\textit{c}\textit{c}\textit{s}\textit{s}\textit{s} \textit{s}\textit  
This part tries to calculate the Signal to Jamming Noise Ratio (SJNR) based on the noise and interference power. Figure 4 shows how AWGN and CWI affect the desired signal. In the channel, first the AWGN with variance \(\sigma_{N}^{2}\) is added to the signal with normalized power:  
\[SNR=\ \frac{P_{signal}}{P_{Noise}}=\ \frac{1}{\sigma_{N}^{2}} \tag{6}\]  
Calculating the signal to jamming noise ratio (SJNR), a new term of the total variance \(\sigma_{T}^{2}\) is defined as:  
\[\begin{array}{c}\sigma_{T}^{2}=\sigma_{N}^{2}+a^{2}=\frac{1}{SNR}+\frac{1}{5 \beta R}\\ SJNR_{dB}\ =\ log_{10}\big{[}\text{{dBtoPower}}\big{(}\frac{1}{SNR}\big{)}+a^{2} \big{]}\end{array} \tag{7}\]  
The equations above (6) and (7) explains received SNR reduction based on the CWI power. Figure 5 displays received SJNR based on SJR in different SNR. The simulation results completely match the obtained equations since both theoretical and simulation curves are perfectly overlapping. When the SJR is large enough, the SJNR curve converges to the SNR value. For larger SNRs, the system experiences more distortion than with a smaller SNR value. For example, in SJR = 15 dB, the curve of SNR = 20 dB decreases by 6 dB, however, the curve of SNR = 10 dB decrease just by 1 dB. Thus, the systems with higher QoS experience more distortion.  
Overall, this section covered the effect of the interference on the constellation diagram, QoS (in terms of received SNR) and PSD relative to its power. These results show that the CWI plays an important role in the overall SJNR value, especially for high values of SJR.  
Figure 4: Utilized scenario to measure the CWI effect on received SNR.  
Figure 3: Effect of CWI with different power on QPSK signal constellation.