Fading factors and Doppler shift derivation

Fading is the rapid fluctuation of the amplitude, phase or delay of a received radio signal over a short time or travel distance, caused by interference between multiple versions of the transmitted signal (multipath) arriving at slightly different times.

Factors influencing small-scale fading

1. Multipath propagation: reflecting and scattering objects create several waves with different amplitudes, phases and delays. Their vector sum varies rapidly, and the spread of delays causes time dispersion and ISI.
2. Speed of the mobile: motion gives each multipath component a different Doppler shift, causing random frequency modulation. Faster movement means faster fading.
3. Speed of surrounding objects: moving vehicles, people and doors make the channel time-varying. If objects move faster than the mobile, they dominate; otherwise they can be ignored.
4. Transmission bandwidth of the signal: if signal bandwidth is larger than the coherence bandwidth, the signal is distorted (frequency selective); if smaller, only its amplitude fades (flat fading).

Derivation of Doppler shift

A mobile moves with constant speed v from X to Y (distance l) in time Δt, receiving from a distant source S at angle θ to the direction of motion.

             S (distant)
            /
           /   rays nearly parallel
          / θ
   X ------------- Y  →  v
   |<---- l ----->|
   Δl = l cosθ

Distance moved:       l  = v·Δt
Path difference:      Δl = l·cosθ = v·Δt·cosθ
Phase change:         Δφ = 2πΔl/λ = 2π·v·Δt·cosθ / λ
Doppler shift:        fd = (1/2π)(Δφ/Δt)
                      fd = (v/λ)·cosθ

• Towards the source (θ = 0): fd = +v/λ (frequency increases).
• Away from the source (θ = 180°): fd = −v/λ.
• Maximum Doppler shift: fm = v/λ.

Example: fc = 900 MHz (λ = 0.333 m), v = 72 km/h = 20 m/s → fm = 20/0.333 = 60 Hz.