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5.1 Small Scale Fading Characterization● Rayleigh Distribution

● Ricean Distribution

● Nakagami Distribution

5.1.1 Rayleigh Fading

Rayleigh Fading is a statistical model for the effect of a propagation environment

on a radio signal.[wiki] This model is effective in describing small scale fading

from a large number of paths with no dominant LOS term. The received signal

(R) can be described as the summation of all incoming replicas, in terms of their 

amplitude and phase.

r is the amplitude envelope, theta is the phase

,

, Gaussian distribution, mean 0

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5.1.1.1 Envelope PDF(r) Rayleigh Distribution

The Probability Density Function (PDF) of the amplitude envelope is represented

by the Rayleigh Distribution Function.

Formula Sheet:

Average envelope power : ,

5.1.1.2 Phase PDF(theta)

Formula Sheet: 

5.1.1.3 Fading Margin

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5.1.1.4 Example - Fading Margin & Outage Probability

Formula Sheet:Example: What is the required Fading Margin in dB, against Rayleigh Fading, to

obtain an outage probability of 1%?

●

●

●Fading Margin: =1/-ln(0.99)=100,

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5.1.2 Ricean Fading

Ricean Fading is a model that describes small scale fading from a large

number of paths with a dominant LOS term.[Wiki]

5.1.2.1 Envelope PDF(r) Ricean Distribution

The LOS term ...

●

The ratio between the power of the LOS component and the diffuse component

is called the Ricean k-factor. If k is 1 then this is equivalent to Rayleigh

distribution.

●

●

The amplitude envelope is described using Ricean Distribution:

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Formula Sheet:

Average Envelope Power:

5.1.2.2 Phase PDF(theta)

A is the power in the LOS component.

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5.1.3 Nakagami Distribution

In many cases the received signal can not be described as pure LOS + diffuse

components.

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6 FadingFading is the deviation in attenuation a signal experiences while propagating

through a channel. Fading can vary with time, location, and frequency.

6.1 Delay Spread FadingThe propagation of a transmitted signal Tx through a channel causes delay

spreading with maximum delay . The inverse of reveals the channel’s 

coherence bandwidth ( ).

●

● Symbol bandwidth ( ) is the inverse of symbol duration

○

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6.1.1 Flat Fading

● Flat fading occurs when the bandwidth of the Tx symbol ( ) is less than

the coherence bandwidth ( ) of the channel.

○

● No Inter Symbol Interference (ISI) since the delayed replicas arrives before

the start of the next symbol.

○

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6.1.2 Frequency Selective Fading

● Frequency selective fading occurs when the Tx symbol bandwidth ( ) isgreater than the coherence bandwidth ( ) of the channel.

○

● Inter Symbol Interference (ISI) occurs because the delayed replicas arrives

after the start of the next symbol.

○

○ Equalization can be used to compensate for this effect.

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6.2 Doppler Spread FadingDoppler spreading causes frequency modulation of the received signal (time

varying phase shift).

Coherence Time ( ) is the time over which the channel's response to a sinusoid

is correlated.

●

○ Max doppler spread

■ , = speed of Rx

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6.2.1 Slow Fading

Slow fading occurs when the symbol time ( ) is less than the coherence time for 

the channel.

● Ts < Tc

○ Over Ts the channel is highly correlated

● Bs =1/Ts > 1/Tc = Vmax

○ There is not much effect of doppler spread

● Equalization is Optional

● Slow Power Control

6.2.2 Fast Fading

● Ts > Tc● Bs =1/Ts < 1/Tc = Vmax

○ Large Doppler Shift

● Requires Equalization

● Fast Power Control

● if outside tc the channel varies fast

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6.3 Large Scale Fading●  Assume small scale fading is handled

Shadowing Parameter 

Log normal distribution