Calculation of angular parameters¶
This example demonstrates the calculation of angular parameters related to the positioning of transmitters and receivers, as well as the estimation of their accuracy with respect to antenna pattern characteristics and signal-to-noise ratio (SNR).
This example performs:
- Computation of the angles of place (βM and βN) and angle of sight (ψN) based on trigonometric functions that take into account geocentric and oblique distances.
- Calculation of antenna pattern width at different frequencies (it is used to calculate angular errors).
- Calculation of RMS error is defined as the ratio of the radiation pattern width to the square root of the specific signal-to-noise ratio.
Let's move on to the implementation of the algorithm.
The first step is to determine the parameters of the system.
DH = 39609 # наклонная дальность, км
theta0_deg = 56.24 # геоцентрический угол, градусы
RZ = 6371 # средний радиус Земли, км
theta_a1_deg = 16 # ширина диаграммы направленности на 30 МГц, градусы
theta_a2_deg = 0.9 # ширина диаграммы направленности на 3000 МГц, градусы
q1 = 4.91 # отношение сигнал/шум на частоте 49 МГц
q2 = 10132 # отношение сигнал/шум на частоте 50 МГц
deg_to_rad = pi / 180 # перевод градусов в радианы
Calculate the angle to the location of the means from the transmitter location.
betaM_deg = atan(DH/RZ)/deg_to_rad # Преобразуем результат в градусы
println("Угол места средства βM = ", betaM_deg, " градусов")
Calculate the angle to the transmitter location from the transmitter location.
betaN_deg = -(betaM_deg + theta0_deg)
println("Угол места передатчика βN = ", betaN_deg, " градусов")
Calculation of the angle of sight of the transmitter from the location point.
psiN_deg = betaN_deg + 90
println("Угол визирования передатчика ψN = ", psiN_deg, " градусов")
Calculation of the RMS bearing error.
theta_a1_rad = theta_a1_deg * deg_to_rad
theta_a2_rad = theta_a2_deg * deg_to_rad
sigma_psi1 = theta_a1_rad / (sqrt(2 * q1))
println("Среднеквадратическая ошибка для 49 МГц: ", sigma_psi1, " рад")
sigma_psi2 = theta_a2_rad / (sqrt(2 * q2))
println("Среднеквадратическая ошибка для 50 МГц: ", sigma_psi2, " рад")
Conclusion¶
Based on the results of calculations, we see: increasing the frequency and signal-to-noise ratio leads to a decrease in the angular error, which confirms the theoretical expectations.
This is critical for satellite communication and navigation systems, where even small angular errors can significantly affect the accuracy of data transmission or positioning.
Thus, the analysis indicates the need to use high frequencies and provide SNR to maximise accuracy in systems with high angular measurement requirements.