Large Scale fading is predominantly due to path loss phenomenon.

Large Scale fading manifests itself as scintillations in the voltage envelope of the received signal over relatively large distances between the transmitter and the receiver.

It is common to model large scale fading effects using a path loss model that obeys some sort of power law.

For a line of sight link, the Friis free space model can be used.  (See Antenna Engineering Handbook, 2^nd ed., by Johnson and Jasik, pp. 1-12).

P_r = ( lambda / 4*Pi*R)² * G_t * G_r * P_t   where:

P_r is the received power

R is the distance (range) between transmitter and receiver

G_t is the gain of the transmitting antenna (in the direction of the receiving antenna)

G_r is the gain of the receiving antenna (in the direction of the transmitting antenna)

and P_t is the transmitted power

The receiver must be in the radiating far field of the transmitting antenna for the Friis model to be accurate.

An accepted value for the radiating far field boundary is (2 D^2 / lambda) where:

D =is the largest dimension of the antenna and lambda is the operating wavelength.

Moreover, the Friis model does not apply when R=0 (or very close to the transmitting antenna) so that an additional factor is introduced to accommodate the case when the receiver is very near the transmitter (but still in the far field).

In general, the average Path Loss is usually expressed as a function of distance (range) as PL(R) ~  (R/Ro) ^n where:

R  is the distance (range) between transmitter and receiver.

Ro is the compensating close distance factor.  This can be determined experimentally.

and the exponent n is a function of the operating channel and the environment.

in dB: PL(R) ~ PL(Ro) + 10*n log(R/Ro)

The basic model above does not account for several important factors, including the terrain profile.

Various distributions have been extensively studied for their ability to model large scale path loss as a function of terrain, obstructions, Earth curvature and other variables.  See for example "A Unified Approach to the Performance Analysis of Digital Communication over Generalized Fading Channels", by Simon and Alouini, Proceedings of the IEEE, September 1998, pp. 1860.

These include (see above paper): Log normal (urban shadowing), Nakagami-n (Rice) (one LOS plus diffuse scatterers), Nakagami-m, etc...

Small Scale fading is predominantly due to two constituent mechanisms, and manifests itself as scintillations in the voltage envelope of the received signal over short distances (on the order of operating wavelengths).

The first Small Scale constituent factor is called multipath.

When the signal arrives at the receiver from two or more distinct paths in the channel, the individual components tend to interfere with each other. The interference can be constructive or destructive, resulting in fluctuations in the voltage envelope of the received signal. This is because the signals arrive with different phase shifts (relative to the line of sight path) due to path length differences.

If the operating channel exhibits a constant gain response and a linear phase response across a bandwidth greater than the bandwidth of the propagating signal, then the voltage envelope of the received signal will exhibit flat fading. This is a type of narrow band channel. Across time, only the amplitude of the received signal can vary (due to the constructive/destructive interference). The spectral content of the transmitted signal does not change.

If the operating channel does not exhibit a constant gain response and a linear phase response across a bandwidth greater than the bandwidth of the propagating signal, then the voltage envelope of the received signal will exhibit frequency selective fading. This is a type of wide band channel. Across time, the received signal will express both muted copies of the transmitted signal as well as delayed copies of the transmitted signal. This is due to the time dispersion of source symbols within the channel (some delayed multipath components have delays greater than the symbol period) and is the cause of ISI at the receiver.

Thus, the relationship between the gain/phase response of the channel and the transmitted signal bandwidth governs whether the signal undergoes flat or frequency selective fading.

The second Small Scale constituent factor is called Doppler Spread.

If certain channel characteristics that are dependent upon receiver velocity (Doppler spectrum) change within a symbol period, then the channel exhibits fast fading. This causes fluctuations in the voltage envelope of the received signal due to Doppler spreading.

If certain channel characteristics that are dependent upon receiver velocity (Doppler spectrum) don't change within a symbol period (static), then the channel exhibits slow fading. In other words, the Doppler spread is less than the bandwidth of the transmitted signal.

Specifically, the rate of change of the Doppler Spectrum of the operating channel is a function of receiver relative velocity. In other words, if a signal consisting of one frequency is injected into the channel, a mobile receiver will capture a signal composed of a spectrum ( Doppler Spectrum) ranging +/- fd around the original frequency, where fd is equal to the Doppler shift. The Doppler shift is itself a function of the relative receiver velocity and the angle of arrival of the incident rays. Thus, the relationship between the receiver velocity and the transmitted signal bandwidth governs whether the signal undergoes fast or slow fading.

For example, it is common to model the flat fading channel with a Rayleigh distribution. The probability density function of the Rayleigh distribution is expressed as (r/ð²)*e^{(-r^2/(2ð^2))} for r>=0 and 0 elsewhere.  (See Digital Communications by Simon Haykin, pp.565).

You can use our simple graphing calculator to explore distributions such as these as follows:

Hit Add Function.

Plug in x*e^ (-(x^2/2)). This normalizes the average power to 1.

Hit Rescale.

Set xMin = 0

Set yMin = -1.0

Set yMax = 1.0

Set yscale = 0.1

Note that the maximum occurs when the abscissa is equal to 1 and that the maximum magnitude of the distribution is roughly equal to .6

These are correct values for the Rayleigh Distribution when the average power of the received signal is normalized to 1.