1. Wireless Communication Channels: Small-Scale Fading

2. Clarke’s Model for Flat Fading
Assumptions:
Mobile traveling in x direction
Vertically polarized wave
Multiple waves in the x-y plane arrive at the mobile antenna at the same time
Waves arrive at different angles α z y
in x-y plane α x
For N waves incident at the mobile antenna
Each wave arriving at an angle αn will experience a different Doppler shift fn
E0 amplitude of the local average E-field
Cn random variable representing the amplitude of individual waves
fc carrier frequency
φn random phase shift due to distance traveled by the nth wave

3. Clarke’s Model for Flat Fading
Given that:
Φn uniformly distributed over 2π
N is sufficiently large (i.e., the central limit theorem is applicable)
Therefore:
Both Tc(t) and Ts(t) may be modeled as:
Gaussian Random Processes

4. Clarke’s Model for Flat Fading
Power received at mobile antenna
Rayleigh Distribution

5. Rayleigh Fading Distribution
Main Assumption:
No LOS
All waves at the mobile
receiver experience
approximately the same
attenuation z y dα
in x-y plane α x
0.6065/σ
constant
p(r)
σ2: Time average received power
σ : rms value of received voltage r σ

6. Rayleigh Fading Statistics
Probability the received signal does not exceed a value R
Mean value of the Rayleigh distribution
Variance of the Rayleigh distribution
Median of the Rayleigh distribution

7. Ricean Fading Distribution
Main Assumption:
LOS
There is a dominant
wave component at the
mobile receiver in addition
to experience multiple
waves that experience
approximately the same
attenuation z y dα
in x-y plane α x
A : Peak amplitude of the dominant signal
I(.): Modified Bessel function of the first kind and zero-order
σ2: Time average received power of the non-dominant components

8. Riciean & Rayleigh Fading
Define K called the Ricean Factor:
The ratio between the deterministic signal power and the power of the non-dominant waves
p(r)
K=-∞ dB
Rayleigh Distribution
K=6 dB r

9. Level Crossing Rate and Mean fade Duration for Rayleigh Fading Signals
Level Crossing Rate Statistic:
The expected rate at which Rayleigh fading envelope normalized to local rms level crosses a specified level in a positive–going direction
ρ:= R/Rrms
fm: Maximum Doppler shift
Mean Fade Duration Statistic:
The average period of time for which the received signal is below a specified level R
Mean Fade duration is a very important statistic that helps define the time correlation behavior of BER performance at the receiver

10. How Wireless Channels Components Fit Together
Distance Pathloss
Mobile Speed 3 Km/hr
PL=
35.225log10(DKM) d
Lognormal Shadowing
Mobile Speed 3 Km/hr
ARMA Correlated Shadow Model d
Small-Scale Fading
Mobile Speed 3 Km/hr
Jakes’s Rayleigh Fading Model d

11. How Wireless Channels Components Fit Together
PTGT GR
Wireless Channel
PR=PTGTGR x Distance Pathloss x Shadowing Parameters x Small-Scale Fading Power

12. System Modeling of Wireless Networks: Example

13. Diversity Techniques

14. What is Diversity?
Diversity techniques offer two or more inputs at the receiver such that the fading phenomena among these inputs are uncorrelated
If one radio path undergoes deep fade at a particular point in time, another independent (or at least highly uncorrelated) path may have a strong signal at that input
By having more than one path to select from, both the instantaneous and average SNR at the receiver may be improved

15. Diversity Techniques: Space Diversity
Receiver Space Diversity
M different antennas appropriately separated deployed at the receiver to combine uncorrelated fading signals 1
Transmitter
Receiver 2 M

16. Diversity Techniques: Space Diversity
Transmitter Space Diversity
M different antennas appropriately separated deployed at the transmitter to obtain uncorrelated fading signals at the receiver
The total transmitted power is split among the antennas 1
Receiver
Transmitter 2 M

17. Diversity Techniques: Frequency Diversity
Modulate the signal through M different carriers
The separation between the carriers should be at least the coherent bandwidth Bc
Different copies undergo independent fading
Only one antenna is needed
The total transmitted power is split among the carriers f
Δf>Bc t

18. Diversity Techniques: Time Diversity
Transmit the desired signal in M different periods of time i.e., each symbol is transmitted M times
The interval between transmission of same symbol should be at least the coherence time Tc
Different copies undergo independent fading f
Δt>Tc t

19. Diversity Combining Techniques

20. Selection Combining Select the strongest signal Transmitter Receiver
Channel 1
Channel 2
Channel M
SNR
Monitor
Select MAX
SNR=γmax

21. Selection Combining
Consider M independent Rayleigh fading channels available at the receiver
Average SNR at all Diversity Branches
SNR = Γ
Instantaneous SNR at Diversity Branch i
SNR = γi
Rayleigh Fading Voltage means
Exponentially Distributed Power
Outage Probability of
a Single Branch
Outage Probability of
of Selection Diversity Combining

22. Maximal Ratio Combining
Selection Combining does not benefit from power received across all diversity branches
Maximal Ratio Combining conducts a weighted sum across all branches with the objective of maximizing SNR G1 r1
Channel 1 ∑ G2 r2
Channel 2
Transmitter
Receiver GM rM
Channel M

23. Maximal Ratio Combining
Consider M independent Rayleigh fading channels available at the receiver
Envelope applied to
receiver detector
Total Noise Power
applied to detector
SNR at the
receiver detector
Cauchy’s Inequality
γMRC is maximized when Gi=ri (MRC requires channel measurements)

24. Maximal Ratio Combining
γMRC is maximized when Gi=ri
(MRC requires channel measurements)
Rayleigh Fading Voltage means
Exponentially Distributed Power
SNR γMRC is Gamma distributed (sum of M exponential random variables)
Outage Probability of
of Maximal Ratio Diversity Combining

25. Equal Ratio Combining
Maximal Ratio Combining requires estimation of the channel across all diversity branches
Equal Gain Combining conducts a sum across all branches (i.e. Gi=1 for all i) r1
Channel 1 ∑ r2
Channel 2
Transmitter
Receiver rM
Channel M

26. Equal Gain Combining
Consider M independent Rayleigh fading channels available at the receiver
Envelope applied to
receiver detector
Total Noise Power
applied to detector
SNR at the
receiver detector
EGC is a special case of MRC with Gi=1
SNR and outage probability performance in
EGC is inferior to that of MRC