# How can you calculate the minimal cone angle between two satellites to prevent interference?

I have a model of 2 ground stations (G1 and G2) and 2 LEO satellites (S1 and S2). G1 and G2 are located close to each other. S1 transmits data to G1 and S2 transmits to G2 with almost the same frequency. Thus, if those 2 satellites are too close to each other during transmission, one may interfere with the other.

The question is, how close can the satellites be to the other during the transmission without interfering? How can the minimum allowed cone angle be calculated?

Would appreciate, if you give some links.

• This depends on the exact radiation pattern of the antenna and the sophistication of the receiver to discriminate between signals. en.wikipedia.org/wiki/Radiation_pattern It's more of an EE/DSP question than a space exploration question. – Russell Borogove Mar 7 '18 at 17:36
• @RussellBorogove Thanks, I would be happy to get more detailed information about all dependencies, if possible – Tarlan Mammadzada Mar 7 '18 at 17:37

Based on the information in the question, which is nearly nothing, let's assume the ground stations and satellites are all using circular dish antennas. Then we can use physics to approximate the radiation patterns with an Airy function.

$$I(\theta) = \left(\frac{2J_1(x)}{x} \right)^2$$

$$x = ka\sin(\theta)$$

where the wavenumber is given by $k = 2\pi/\lambda = 2 \pi f / c$ ($\lambda$ is of course the wavelength, c the speed of light and f is the frequency) and $a$ is the radius of the aperture or in this case the dish.

So say for example at $f =$ 10 GHz and $a =$ 1.5 meters (3 meter diameter) you can plot a simple radiation pattern for a circular aperture or dish. Of course in reality its more complex and ugly, but then you'd have to use a measured radiation pattern.

I'm pretty sure you can do the trigonometry based on the distances to the satellites based on your previous questions.

I have heard numbers like 10 dB for the threshold of interference using satellite TV receivers. I have no idea how close that is, so perhaps 20 dB would be better. These days the signals are digital and there is so much error correction, it's likely to be somewhere in that range.

def I_airy(ka, theta):
x = ka * np.sin(theta)
I = (2.*spj1(x)/x)**2
return I

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import j1 as spj1

halfpi, pi, twopi = [f*np.pi for f in 0.5, 1, 2]

f = 10E+09  # Hz
c = 3E+08   # m/s
D = 3.0     # meters
a = 0.5*D

ka = twopi * f * a / c

thetadegs = np.linspace(-2, 2, 600)  # Skip Zero!

I = I_airy(ka, theta)
I_db = 10 * np.log10(I)

plt.figure()
plt.subplot(2, 1, 1)