# Performance Analysis of Troposphere Scattering Communication Channel with Chirp-BOK Modulation
## 4 Simulation and Discussion  
In the subsequent simulation, the setting parameters of the multipath Rayleigh fading channel are shown in Table 1.  
Equation (14) states the average BER output from the troposcatter channel with chirp-BOK modulation in the absence of ISI (i.e., Rayleigh flat fading), which can be expressed as \(P_{e}=\frac{1}{2}\Big{(}1-\sqrt{\frac{\bar{\gamma}}{2+\bar{\gamma}}}\Big{)}\), where \(\bar{\gamma}\) denotes the average SNR. A comparison between the simulation result and theoretical result is shown in Figure 3; the simulation results were verified through numerous simulations, revealing that when the parameter \(D\) is set to 6, there is a high degree of consistency with the theoretical results. When tap delay \(T\) is equal to 10 ns and 50 ns, respectively, with Path-Gain-Vector1, the influence of different Doppler shifts \(f_{d}\) on the system BER performance is shown in Figure 4. As shown in Figure 4, the BER curves are basically consistent in the range of \(f_{d}\in[1\) Hz, 100 Hz], and the influence of \(f_{d}\) on the BER is relatively small. The subsequent simulation and analysis are based on the assumption that \(f_{d}=10\) Hz. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Maximum Doppler Shift & \(f_{d}\in[1\) Hz, 100 Hz\(]\) \\ Delay-Vector & \([0\ 1\ 2\ 3\ 4\ 5\ 6]\times T(\text{ns})\) \\ Path-Gain-Vector1 & \([-10\ -3\ 0\ -2\ -5\ -8\ -10]\ (\text{dB})\) \\ Path-Gain-Vector2 & \([10\ -3\ 0\ -2\ -5\ -8\ -10]\ (\text{dB})\) \\ Time-Bandwidth Product & \(D=BT_{b}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters for simulation. Figure 3: Comparison of theoretical and simulated values. The effect of varying tap delay vector \(T\) on BER performance of the system with \(D\) equal to 6 and 60 is illustrated in Figure 5a and Figure 5b, respectively. As illustrated in Figure 5a, under the premise of a small \(D\), the BER curve exhibits a flattening trend as the value of \(T\) increases. When \(T=1\) ns and \(T=\) ns, the two BER curves are essentially identical, indicating that the ISI in the system is small in the presence of a relatively short time interval. This can be regarded as frequency flat fading. As the value of \(T\) increases to 10 ns, 30 ns, and 50 ns, the BER performance of the system declines continuously. This is due to the fact that as \(T\) increases, the ISI phenomenon in the channel becomes more pronounced, resulting in a transition from a frequency flat fading channel to a frequency-selective fading channel. Figure 5b shows that the BER performance improves with an increase in \(T\). This is due to the fact that a larger time-bandwidth product results in a more rectangular spectral characteristic of the chirp signal, rendering it more resistant to frequency-selective fading. Consequently, the system is more effectively capable of suppressing frequency-selective fading caused by different paths when \(T\) continuously increases, thus enhancing the reception quality and BER performance. The system BER performance for tap delays \(T\) of 5 ns, 10 ns, 30 ns, and 50 ns is analyzed in Figures 6-9 when \(D\) is 8, 20, 40, 60, and 120, respectively  
Assume that target bit error rate is \(10^{-4}\) and the reference \(D=8\) is used as a basis for the gain calculation. In Path-Gain-Vector1, the optimal selection to achieve the target BER is 1.7 dB less than the SNR required for \(D=8\) when the tap delay \(T\) is 5 ns. Similarly, the optimal selections to achieve the target BER are 5 dB, 9.5 dB and 10.0 dB less than \(D=8\) when the tap delay \(T\) is 10 ns, 30 ns and 50 ns, respectively. In Path-Gain-Vector2, the optimal configuration to reach the target BER is 1.7 dB less than the SNR required for \(D=8\) when the tap delay \(T\) is 5 ns. Similarly, the optimal configurations to reach the target BER are 3.9 dB, 7.7 dB, and 9.8 dB less than \(D=8\) when the tap delay \(T\) is 10 ns, 30 ns and 50 ns, respectively. Figure 10 is a BER performance comparison between chirp-BOK, BPSK, QPSK and 16-QAM modulation schemes.