Suppose there are two antennas, A and B. Antenna A is a transmitter with a height of 50 meters, while antenna B is a receiver with a height of 15 meters. Considering line-of-sight (LOS): a) Compute the maximum distance that can be covered by the transmission. b) If the distance is 30 km, what height is required for antenna B, assuming antenna A remains at the previously mentioned height of 50 meters?

a) To compute the maximum distance that can be covered by the transmission between antennas A and B using Line-of-Sight (LOS), we can apply the following formula based on the curvature of the Earth:

Distance (km) = √(17 * heightA (in meters)) + √(17 * heightB (in meters))

where heightA is the height of the transmitting antenna (antenna A) and heightB is the height of the receiving antenna (antenna B).

First, let us calculate the individual distance components that antenna A and B can reach due to their heights:

For Antenna A (Transmitter): DistanceA = √(17 * heightA) DistanceA = √(17 * 50) DistanceA = √850 DistanceA ≈ 29.15 km

For Antenna B (Receiver): DistanceB = √(17 * heightB) DistanceB = √(17 * 15) DistanceB = √255 DistanceB ≈ 15.97 km

The maximum LOS distance between the antennas can be found by adding these two distances together:

Max Distance = DistanceA + DistanceB Max Distance ≈ 29.15 km + 15.97 km Max Distance ≈ 45.12 km

Therefore, the maximum LOS distance that can be covered by the transmission is approximately 45.12 kilometers.

b) If the distance between the two antennas is 30 km and we want to find out the necessary height for antenna B (receiver) provided antenna A remains 50 meters in height, we can rearrange the LOS formula to solve for heightB:

Distance = √(17 * heightA) + √(17 * heightB) We know the distance and heightA, so we can express heightB as:

heightB = ((Distance - √(17 * heightA))^2) / 17

Plugging in the values we know (Distance = 30 km, heightA = 50 meters):

heightB = ((30 - √(17 * 50))^2) / 17 heightB = ((30 - 29.15)^2) / 17 heightB = (0.85^2) / 17 heightB = 0.7225 / 17 heightB ≈ 0.0425 meters

This result suggests that theoretically, only a negligible additional height is needed for the receiving antenna. However, this is counter-intuitive, as an extra height of only 0.0425 meters is unrealistic for practical purposes and doesn't account for the minimal required clearance over obstacles or Earth's surface roughness. Usually, a certain minimum effective height is required to ensure reliable communication.

It is likely that an error has occurred in calculation or interpretation, perhaps because the actual LOS distance without considering any other factors is already larger than 30 km with the given heights of antenna A and B. Thus, no additional height would be needed for antenna B to maintain LOS with antenna A over a distance of 30 km, since we already calculated a max LOS distance of around 45.12 km with the given antenna heights.

Extra: Line-of-Sight (LOS) communication represents a signal or path that is unobstructed from the transmitter to the receiver. In terms of radio waves, LOS is critical as they typically travel in straight paths. Earth's curvature, as well as other obstructions like buildings and trees, can limit the LOS distance.

When dealing with communication between two antennas, it is fundamental to consider not only the LOS but also the Fresnel Zone – an elliptical area around the LOS path which should be clear of obstacles to reduce multi-path interference and signal degradation.

The curvature of the Earth is a significant factor when calculating the LOS distance as the Earth is not flat, and its surface curves away from a straight line drawn between two points. The standard calculation for the distance to the horizon for an object close to Earth's surface is based on the object's height and the radius of the Earth, and it assumes a clear LOS without any obstructions, generally, a rather idealized scenario.

In practice, optimal antenna height must also account for other factors such as the terrain between the antennas (mountains, buildings), the frequency of transmission (as different frequencies have different propagation characteristics), and the desired quality of the signal at the receiving end. This makes the calculation of the 'necessary' height for effective communication a more complex matter that requires comprehensive analysis or the use of propagation modeling tools.