## 4 Analysis of Anti-Jamming Performance  
In this section, we will focus on the resistance to interference in spread-spectrum communication systems. Coherent systems, which have knowledge or the ability to replicate the chaotic carrier, are generally better at resisting interference compared to non-coherent systems. However, chaos-based communication faces challenges with chaotic synchronization, which affects the performance of coherent systems. On the other hand, non-coherent systems, while inherently having poorer noise performance than coherent systems, offer practical advantages. Specifically, we present results and evaluations for the scenario where the interference signal is a narrowband sinusoidal interference signal. Our main objective in this chapter is to investigate the performance of chaos-based digital communication systems in the presence of strong interference signals and additive white Gaussian noise. Figure 8 depicts the block diagram of a chaotic communication system with added interference and noise. According to the impact of narrowband signal interference on communication systems presented in [2], and assuming the interference signal has a power of \(P_{jam}\) and is a sinusoidal wave with a carrier frequency of \(f\), i.e.,  
\[s^{\prime}(t)=\sqrt{2P_{jam}}sin(2\pi k\frac{F}{(PR+R)}+\theta) \tag{22}\]  
where \(\theta\) is any constant phase angle, \(F\) is the normalized interference frequency defined as \(F=fT_{b}\), and \(T_{b}\) is the duration of a symbol. Sinusoidal interference signals are one of the most commonly used interference signals and are easy to generate. Assuming the output of the transmitter is \(s(t)\), and considering the presence of both additive interference and noise, the signal at the input of the receiver \(r(t)\) is given using the following equation. \[r(t)=s(t)+s^{\prime}(t)+\xi(t) \tag{23}\]  
where \(\xi(t)\) is the Gaussian white-noise function. This signal will be used by the receiver to recover the digital signal that the transmitter has sent. Before deriving the bit error rate, let us first consider  
Figure 8: Block diagram of chaotic communication system with added interference and noise. the bit error rate for the \(l\)th bit. Assuming a receiver of the correlator type, the output of the correlator for the \(l\)th bit \(y_{l}\) is given using the following equation. \[ \begin{split} y_{l}=\sum_{p=1}^{P}\sum_{k=0}^{R-1}\Bigl{[}& x_{k,p,l}+\sqrt{2P_{jam}}sin\Bigl{(}\frac{\pi kT}{PR+R}+\theta\Bigr{)}+\xi_{k,p} \Bigr{]}\\ &\times\Bigl{[}b_{l}x_{k+R,p,l}+\sqrt{2P_{jam}}sin\Bigl{(}\frac{( k+PR+R)\pi F}{PR+K}+\theta\Bigr{)}+\xi_{k+R,p}\Bigr{]}\end{split}  \tag{24}\]  
Expanding the above equation and redefining  
\[ \begin{split} y_{l}&=\underbrace{b_{l}P\sum_{k=0}^{R -1}x_{k,l}^{2}}_{A}+\underbrace{\sqrt{2P_{jam}}P\sum_{k=0}^{R-1}x_{k,l}sin \Bigl{(}\frac{(k+PR)\pi F}{PR}+\theta\Bigr{)}}_{B}\\ &\qquad+\underbrace{P\sum_{k=0}^{R-1}+\sqrt{2P_{jam}}b_{l}\sum_ {k=0}^{R-1}x_{k+R,l}sin\Bigl{(}\frac{k\pi F}{PR}+\theta\Bigr{)}}_{C}\\ &\qquad+\underbrace{2P_{jam}P\sum_{k=0}^{R-1}sin\Bigl{(}\frac{k \pi F}{PR}+\theta\Bigr{)}sin\Bigl{(}\frac{(k+PR)\pi F}{PR}+\theta\Bigr{)}}_{D }\\ &\qquad+\underbrace{P\sum_{k=0}^{R-1}x_{k,l}\xi_{k+R}}_{G}+ \underbrace{b_{l}P\sum_{k=0}^{R-1}x_{k+R,l}\xi_{k}}_{H}+\underbrace{\sqrt{2P_{ jam}}P\sum_{k=0}^{R-1}\Bigl{[}\xi_{k+R}sin\Bigl{(}\frac{k\pi F}{PR}+\theta \Bigr{)}\Bigr{]}}_{C}\\ &\qquad+\underbrace{\sqrt{2P_{jam}}P\sum_{k=0}^{R-1}\xi_{k}sin \Bigl{(}\frac{(k+PR)\pi F}{PR}+\theta\Bigr{)}}_{K}+\underbrace{+P\sum_{k=0}^{R -1}\xi_{k}\xi_{k+R}}_{L}\end{split}  \tag{25}\]  
The following expression represents the decision criteria when the information signal outputs are, respectively, \"+1\" and \"\(-1\)\":  
\[ \begin{split} y_{l}|(b_{l}=+1)=A+B+C+D+G+H+J+K+L\\ y_{l}|(b_{l}=-1)=-A+B+C+D+G-H+J+K+L\end{split}  \tag{26}\]  
where \(A\) is the useful information signal, \(B\), \(C\), \(D\) is the narrowband interference signal, and \(G\), \(H\), \(J\), \(K\), \(L\) is the noise interference signal.