Can you hear me now?

In 2003, Martin Cooper wrote a nice article on 'Smart Antennas' for the magazine Scientifc American. In the article he illustrates how signals from multiple radio transmitters add up and are received by antennas.

Waves transmitted by the transmitters 'add up.' This implies that if they arrive 'in-sync' (peaks line up), the signal is very strong. Conversely, when the waves are completely 'out of sync,' (peaks coincide with valleys), they cancel each other out.



If we keep things very simple and we assume that:

• The surface or field across which transmitting and receiving occurs is flat and featureless so that nothing disturbs the transmitter signals.
• There is no signal attrition; i.e., signals do not weaken or change wavelength as they travel.
• We have (only) two transmitters.
• Both transmitters transmit at the same wavelength.

Then the formula for the (relative) signal strength at any point in the field is:
1 + sin((π / 2) + ((dist_1 - dist_2) / wl * 2 * π), where
• dist_1: distance from the point of reception to transmitter 1.
• dist_2: distance from the point of reception to transmitter 2.
• wl: wavelength of the transmissions.

As illustrated below, the pattern of signal strength across the field can become interesting and complex. The signal strength varies from very high (white) to zero (black).

Below you can play with this.

Wavelength: Transmitter x and y coordinates: Transmitter 1: (1 ≤ x ≤ 1000)   (1 ≤ y ≤ 1000) Transmitter 2: (1 ≤ x ≤ 1000)   (1 ≤ y ≤ 1000)    Compute!