Resolving multipath interference when modelling radio wave propagation

Question

I am creating a raytracer to model the propagation of radio waves from a simple router.

I am assuming that the rays have a frequency of 2.4GHz and a velocity of the speed of light. The router has an initial power of 1mW.

For simplicity I use FSPL (free-space path loss) to model the degradation of the signal over distance.

When one ray intersects a point, it is easy to calculate the RSS (received signal strength) in dBm. You take the distance the ray has traveled, plug it into FSPL, and convert the result from watts to dBm.

However, it is unclear how to resolve the effects of multipath interference when 2 or more rays intersect at a given point. I am not sure, for example what the amplitude of the radio waves are, or if there any dampening effects. Can someone explain how to model this phenomenon accurately?

Answer 1

When two waves arrive at the same point the resulting signal is the addition of two electromagnetic waves. The resulting signal depends of the phase difference between signals. This phase difference is equal to the path length difference in unit of the wave length of the signal λ.

For example, lets consider a one dimensional case of 2 electromagnetic signals interfering together. If λ = 12,5 cm (this is the wave length of a 2,4 GHz signal), and if the lengths of the paths from the emitting antenna are:

l₁ = 120 m
l₂ = 123 m

then Δl = 300 cm

then Δφ = 2π(Δl/λ) mod 2π = 2π(300/12,5) mod 2π = 0

the 2 signals will fully add and the resulting signal amplitude is twice the original signal amplitude.

If on the other hand:

l₁ = 120 m
l₂ = 120,0625 m

then Δl = 6,25 cm

then Δφ = 2π(Δl/λ) mod 2π = 2π(6,25/12,5) mod 2π = π

the 2 signals are in phase opposition and will cancel each others. The resulting signal amplitude will be 0.

To compute a real case which is 4 dimensionnal (interference with the previous waves after reflections should be computed),

consider this basic documentation: Interference (wave propagation),

and look toward ray tracing computation: Ray tracing (physics).