WIRELESS NETWORKS AND MOBILE COMMUNICATIONS

SMALL SCALE FADING

The statistical properties of small-scale fading are often described by probability distributions. The two most common models are Rayleigh and Rician fading.

1. Rayleigh Fading 

Rayleigh fading is a statistical model that applies when there is no direct line-of-sight (LOS) path between the transmitter and receiver. This is common in congested urban environments or indoor settings where the signal must bounce off numerous things before reaching the receiver.

The received signal is the sum of several reflected, dispersed, and diffracted waves, each of which has nearly equal power.
The amplitude of the received signal's envelope has a Rayleigh distribution.
This model is frequently regarded as a worst-case situation since the lack of a strong LOS path makes the signal extremely vulnerable to deep and frequent fades.

2. Rician Fading

Rician fading is a broader model that takes into account both multipath components and a strong line-of-sight (LOS) path. This is frequent in suburban or open rural locations where a straight path exists. The received signal consists of a strong, consistent LOS component and several lesser, dispersed components.
The amplitude of the received signal's envelope has a Rician distribution. The Rician K-factor, which is the power in the LOS component divided by the power in the scattered components, determines the severity of the fading. A high K-factor indicates that the LOS path is dominating, and fading is less severe. As the K-factor approaches 0, the LOS path's power decreases and the model approaches a Rayleigh distribution.

The Doppler Effect and Coherence Time
The mobility of the transmitter, receiver, or objects in the environment adds another layer of complexity to small-scale fading. This movement causes the Doppler effect.

1. The Doppler Effect
The Doppler effect is the change in frequency of sound waves as they reflect from moving objects, increasing in frequency when approaching and decreasing when moving away, allowing for the determination of flow direction and velocity.

A signal path from a moving object approaching the receiver will experience a positive Doppler shift or higher frequency, whereas a signal path from a moving object moving away will experience a negative Doppler shift or lower frequency. And the collective effect of these individual shifts is a spread of the received signal's spectrum, known as the Doppler spread (BD​). A larger Doppler spread indicates that the channel is changing more rapidly over time.

2. Coherence Time
Coherence time (Tc​) is a measure of how long the wireless channel's characteristics remain statistically constant. It is a crucial parameter for understanding the nature of fading.

The coherence time is inversely proportional to the Doppler spread.
Tc​≈fm​1​, where fm​ is the maximum Doppler shift.
The relationship between coherence time and the duration of a transmitted symbol (Ts​) determines whether the channel experiences slow fading or fast fading:

a) Slow Fading: If the symbol duration is much shorter than the coherence time (Ts​<<Tc​), the channel's state is essentially constant for the duration of one or several symbols. The signal experiences a single fade level during the transmission.
b) Fast Fading: If the symbol duration is longer than the coherence time (Ts​>Tc​), the channel's characteristics change significantly within the time it takes to transmit a single symbol. Different parts of the same symbol experience different fading levels, leading to signal distortion