# Performance Analysis of Troposphere Scattering Communication Channel with Chirp-BOK Modulation
## 2 System Model with Chirp-BOK Modulation
### Tropocatter Channel Model  
The main characteristics of tropospheric scattering propagation encompass transmission loss and fading. Transmission loss is influenced by various factors, such as the distance between the transmitter and receiver, the geographical location, and the scattering properties of scatterers. There are some classical methods for estimating scattering transmission losses, mainly the NBS-101 model and ITU-R model. NBS-101 is a widely used calculation method to estimate transmission losses in tropospheric scatter communication links. The NBS-101 prediction method considers the impact of nonlinear variations in the atmospheric refraction index when utilizing the equivalent earth radius. It appropriately adjusts this factor during the calculation process to align with actual link conditions, resulting in a prediction of scattering transmission loss that is more consistent with measured values. However, this method involves complex calculations and requires consultation of numerous charts, making its engineering application relatively challenging. The ITU-R P.617 model proposes the data and propagation prediction techniques required for the design of a radio relay system beyond the horizon, as well as a transmission loss prediction model for troposcatter signals. Many changes have been made since the P.617 recommendation was first proposed in 1992. From P.617-1 in 1992 to P.617-3 in 2013, this model was mainly used to predict the performance of a radio relay system from 50% to 99.999% of the time percentage. As it was updated to P.617-4 in 2017 and P.617-5 in 2019, the model was improved to predict performance for a time percentage of 0.001% to 99.999%. This pattern can be expressed as follows [24]:  
\[L=F+22\log f+35\log\theta+17\log d+L_{c}-Y_{p}, \tag{5}\]  
In Equation (5), \(L\) denotes the median of transmission loss (dB), \(F\) is weather factor (dB), \(f\) stands for frequency (MHz), \(\theta\) is the scatter angle (mrad), \(d\) is the circuitry length (km), \(L_{c}\) is the surface medium coupling loss (dB), and \(Y_{p}\) is the conversion factor that does not exceed the estimated \(p\) time percentage. The calculation process of the ITU-R617 prediction method is relatively simple, and most of the unknowns can be obtained through the corresponding calculation formula, accurately calculating the transmission loss of tropospheric scattering communication, which is more in line with the practical applications in China. Using the tranmission loss, as computed in Equation (5), the receiver signal-to-noise ratio (SNR) \(\gamma\) can be expressed as [25]  
\[\gamma=10\text{log}_{10}P_{t}-L-L_{F}-NF-10\text{log}_{10}(KT_{1}B), \tag{6}\]  
where \(P_{t}\) is the transmitter power, \(L_{F}\) is the transmitter and receiver feeder losses, \(NF\) is the receiver noise figure, \(K\) is Boltzman's constant, \(T_{1}\) is the noise temperature in Kelvin, and \(B\) is the transmission bandwidth. Fading is a typical characteristic of tropospheric scattering signals, which can be classified into fast fading and slow fading according to the duration of the fading period [26]. Slow fading reflects the attenuation caused by changes in the average value of a received level over a medium range of hundreds of wavelengths, primarily associated with climate conditions. Statistical findings indicate that the slow fading signal power obeys a log-normal distribution. Fast fading comprises frequency-selective fading and time-selective fading, which reflects the fast amplitude changes over a short period of time. The statistical experimental results show that fast fading always follows the Rayleigh distribution [27]. The received SNR \(\gamma\) will be used to analyze and simulate modulation performance over a toposcatter fading channel. The troposcatter channel is commonly referred to as Wide-Sense Stationary Uncorrelated Scattering (WSSUS), which follows Rayleigh distributions [28]. It is a linear time-varying (LTV) system that can be characterized as a continuous time-impulse response \(\bar{c}(\tau,t)\)[29].