Calculation of angular parameters
This example demonstrates the calculation of angular parameters related to the positioning of transmitters and receivers, as well as an assessment of the accuracy of their operation, taking into account the characteristics of the antenna pattern and the signal/noise level (SNR).
In this example, the following is performed:
- Calculation of elevation angles (βM and βN) and angle of sight (ψn) based on trigonometric functions that take into account geocentric and oblique distances.
- Calculation of the width of the antenna pattern at various frequencies (it is used to calculate angular errors).
- The calculation of the RMS error is defined as the ratio of the width of the radiation pattern 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 system parameters.
DH = 39609 # inclined range, km
theta0_deg = 56.24 # geocentric angle, degrees
RZ = 6371 # average radius of the Earth, km
theta_a1_deg = 16 # the width of the radiation pattern at 30 MHz, degrees
theta_a2_deg = 0.9 # the width of the radiation pattern at 3000 MHz, degrees
q1 = 4.91 # signal-to-noise ratio at 49 MHz
q2 = 10132 # signal-to-noise ratio at 50 MHz
deg_to_rad = pi / 180 # converting degrees to radians
Calculation of the angle to the vehicle location from the location of the transmitter.
betaM_deg = atan(DH/RZ)/deg_to_rad # Convert the result to degrees
println("The angle of the vehicle seat is βM = ", betaM_deg, " degrees")
Calculation of the angle to the location of the transmitter from the location of the facility.
betaN_deg = -(betaM_deg + theta0_deg)
println("The angle of the transmitter site is βN = ", betaN_deg, " degrees")
Calculation of the viewing angle of the transmitter from the location.
psiN_deg = betaN_deg + 90
println("The viewing angle of the transmitter ψN = ", psiN_deg, " degrees")
Calculation of the standard deviation of the bearing.
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("RMS error for 49 MHz: ", sigma_psi1, " glad")
sigma_psi2 = theta_a2_rad / (sqrt(2 * q2))
println("RMS error for 50 MHz: ", sigma_psi2, " glad")
Conclusion
Based on the calculation results, we see that an increase in frequency and signal-to-noise ratio leads to a decrease in angular error, which confirms theoretical expectations.
This is critically important for satellite communications 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 achieve maximum accuracy in systems with high requirements for angular measurements.