# Effects of Interference and Mitigation Using Notch Filter for the DVB-S2 Standard
## 3 The Effects of Interference
### Symbol Timing Recovery  
The STR block is an important part of the digital receivers. Gardner (GA) algorithms [24] is widely employed and well known for its great efficiency and high performance in acquisition procedures. It could be considered as the heart of the different types of receivers like DVB and time-division multiple access (TDMA) systems. Figure 6 displays the structure of a Gardner symbol timing recovery [24]. The timing error is calculated in two parts: the real (\(\mathrm{e_{R}}\)) and the imaginary (\(\mathrm{e_{I}}\)). The sum of those two is the total timing error (\(\mathrm{e_{total}}\)), and they are estimated as:  
\[ \begin{array}{l}e_{R}=\{x[nT]-x[(n-1)T]\}\times x\Big{[}(n-\frac{1}{2})T]\\ e_{I}=\{y[nT]-y[(n-1)T]\}\times y\Big{[}(n-\frac{1}{2})T\Big{]}\\ e_{total}=e_{R}+e_{I}\end{array}  \tag{8}\]  
where \(x[nT]\) and \(y[nT]\) are the real and imaginary parts of the received samples and \(T\) is the sample duration. Although, when the system is affected by a CWI, its real and imaginary components are added to the samples. \[ \begin{array}{l}x^{\prime}[nT]=\ x[nT]+\alpha cos\Big{(}2\pi f_{j}.T.n\Big{)} \\ y^{\prime}[nT]=\ y[nT]+\alpha sin\Big{(}2\pi f_{j}.T.n\Big{)}\end{array}  \tag{9}\]  
By substitution of (9) within (8), timing error under the jamming situation can be calculated as the following:  
\[ \begin{array}{l}e^{\prime}_{R}=\Big{\{}x[nT]-x[(n-1)T]+a\Big{[}cos\Big{(}2 \pi f_{j}Tn\Big{)}-cos\Big{(}2\pi f_{j}T(n-1)\Big{)}\Big{]}\Big{]}\times\\ \Big{[}x[nT-T/2]-acos\Big{(}2\pi f_{j}T(n-1/2)\Big{)}\Big{]}\\ e^{\prime}_{I}=\Big{\{}y[nT]-y[(n-1)T]+a\Big{[}sin\Big{(}2\pi f_{j}Tn\Big{)}- sin\Big{(}2\pi f_{j}T(n-1)\Big{)}\Big{]}\Big{]}\times\\ \Big{[}y[nT-T/2]-asin\Big{(}2\pi f_{j}T(n-1/2)\Big{)}\Big{]}\\ e^{\prime}_{Total}=\ e^{\prime}_{R}+e^{\prime}_{I}\end{array}  \tag{10}\]  
Figure 5: Received SJNR based on the SJR value in different SNR. Figure 6: Structure of the symbol time recovery. One can factorize and estimate the Final Error (FE) of the signal with a CWI:  
\[ \begin{array}{ll}FE=e_{total}+2\pi a&f_{j}T\\ &\times\left[\left[x[nT]-x[(n-1)T]\right]\times cos\left(2\pi f_{j}Tn\right)+ \left\{y[nT]-y[(n-1)T]\right\}\right]\\ &\times sin\left(2\pi f_{j}Tn\right)\right]\end{array}  \tag{11}\]  
Based on this estimation, it is clarified that the deviation from the correct timing error is proportional to the center frequency of a CW signal and its power. From these results, CWI with higher frequencies has a larger effect on the STR stage than the ones with lower frequencies. DVB-S2 signal is a narrowband signal, which means that out of band interferences cannot affect its performance. For in-band CWI, multiplication of \(f_{j}T\) has maximum a value of 0.175. DVB-S2 is a narrowband signal, for in-band CWI, \(f_{j}\) must be smaller than the maximum frequency of the transmitted signal. DVB-S2 signal benefits from the pulse shaping based on the raised cosine filtering with a maximum roll factor of \(\beta\) = 0.35. Thus, the maximum bandwidth of the signal is:  
\[BW=\ \frac{1+\beta}{T_{s}}=\ \frac{1+\beta}{N_{up}.T} \tag{12}\]  
where T\({}_{s}\) is the symbol duration and N\({}_{up}\) is an upsampling factor. For in-band CWI, the maximum value of multiplication of \(f_{j}\times T\) can be calculated as the following equation:  
\[f_{j}<\frac{BW}{2}\ \Rightarrow\ f_{j}\times T<\frac{BW.T}{2}\ <\frac{1+\beta}{2N_{up}} \tag{13}\]  
Considering the maximum roll of factors and \(N_{up}=4\), multiplication of \(f_{j}T\) has a maximum value of 0.175. This criterion helps the Gardner algorithm to be naturally robust against the CWI. Figures 7 and 8 show the estimated timing error with the Gardner algorithm while CWI is added to the input samples with different power and frequencies. In these figures, the input signal has a delay equal to half of the symbol period. Simulation results completely match the theoretical analysis. In Figures 7 and 8, for the same power, CWI with higher frequency has more effect on timing error detection.