1. Wireless Communication Channels: Small-Scale Fading

2. Wireless Communication Channels (Reminder)
Distance Pathloss
Mobile Speed 3 Km/hr
PL=
35.225log10(DKM) d
Lognormal Shadowing
Mobile Speed 3 Km/hr
ARMA Correlated Shadow Model
Rapid Changes in Signal Strength over a small traveling distances d
Small-Scale Fading
Mobile Speed 3 Km/hr
Jakes’s Rayleigh Fading Model d

3. Multi-Path Propagation
From “Wireless Communications” Edfors, Molisch, Tufvesson
Multi-Path in the radio channel creates small-scale fading. The three most important effects are:
Rapid changes in signal strength over a small travel distance or time interval
Random frequency modulation due to varying Doppler shifts on different multi-path signals
Time dispersion (echoes) caused by multi-path propagation delays

4. Multi-Path Propagation Modeling
Power
Multi-Path Components τ0 τ1 τ2
Time
Multi-path results from reflection, diffraction, and scattering off environment surroundings
Note: The figure above demonstrates the roles of reflection and scattering only on multi-path

5. Multi-Path Propagation Modeling
Power
Multi-Path Components τ0 τ1 τ2
Time
As the mobile receiver (i.e. car) moves in the environment, the strength of each multi-path component varies

6. Multi-Path Propagation Modeling
Power
Multi-Path Components τ0 τ1 τ2
Time
As the mobile receiver (i.e. car) moves in the environment, the strength of each multi-path component varies

7. Multi-Path = Frequency-Selective!
0.5
0.5 1 1
0.5
1 μs
1 μs
f=1 MHz 1
0.5
0.5 1
0.5
-0.5 -1 -1
1 μs
1 μs
f=500 KHz 1 1
0.5
0.5
0.5
-0.5 -1 -1
1 μs
1 μs

8. Multi-Path = Frequency-Selective!
h(t)
|H(f)|
0.5
0.5 1
f (MHz)
0.5 1
1.5 2
1 μs
A multi-path channel treats signals with different frequencies differently
A signal composed of multiple frequencies would be distorted by passing through such channel

9. Multi-Path Propagation Modeling
From “Wireless Communications: Principles & Practice” T. Rappaport

10. Power Delay Profile
The power delay profile depicts the spatial average of received power within the multi-path channel over a radius that is comparable to the signal wavelength
From “Wireless Communications: Principles & Practice” T. Rappaport
Multi-Path Profile from a 900 MHz cellular system in San Francisco

11. Parameters of Mobile Multi-Path Channels
The power delay profile is used to derive some parameters that can help characterize the effect of the wireless channel on signal communication
We will discuss the following:
Time dispersion parameters
Mean excess delay
Rms delay spread
Excess delay spread (X dB)
Coherence bandwidth
Doppler spread and coherence time

12. Time Dispersion Parameters
Mean Excess Delay
P(τ)
RMS Delay Spread τ0 τ1 τ2 τ3 τN τ
Power Delay Profile
Note: These delays are measured relative to the first detectable signal (multi-path component) arriving at the receiver at τ0=0
Maximum Excess Delay (XdB) or Excess Delay Spread (XdB):
Time delay during which multi-path energy falls to X dB below the maximum (Note that the strongest component does not necessarily arrive at τ0)

13. From “Wireless Communications: Principles & Practice” T. Rappaport
Example
From “Wireless Communications: Principles & Practice” T. Rappaport
Example of an Indoor Multi-Path Profile; rms delay spread, mean excess delay, maximum excess delay (10 dB)

14. Coherence Bandwidth
A statistical measure of the range of frequencies over which the channel is can be considered to be “flat” (i.e., a channel which passes all spectral components with approximately equal gain and linear phase)
Coherence Bandwidth over which the frequency correlation function is 0.9
Coherence Bandwidth over which the frequency correlation function is 0.5

15. Doppler Shift is Given by
The difference in path lengths traveled by the wave from source S to the mobile at X and Y is Δl
Note: Assume SX, SY >>d such that angle of arrival is nearly equal at X and Y S Δl θ θ
Phase Difference due to variation in path lengths X d Y v
Doppler Shift is Given by

16. Doppler Spread and Coherence Time
Doppler spread and coherence time are parameters which describe the time varying nature of the channel
Doppler spread BD is a measure of spectral broadening due to the Doppler shift associated with mobile motion
Coherence time is a statistical measure of the time duration over which the channel impulse response is essentially invariant
Coherence Time is inversely proportional to Doppler spread
Coherence Time over which the time correlation function is 0.5
where fm is the maximum Doppler shift given by fm=v/λ
A Common Rule:

17. Flat Fading Vs Frequency Selective Fading
P(τ)
Power Delay Profile
A Common Rule of Thumb:
TS>10σt  Flat fading τ0 τ1 τN τ
Symbol Time (Digital Communication) TS 1 1 +
Minimal ISI
Wireless Channel + τ0 τN τa

18. Flat Fading Vs Frequency Selective Fading
P(τ)
Power Delay Profile
A Common Rule of Thumb:
TS<10σt  Frequency Selective Fading τ0 τ1 τ2 τ3 τN τ
Symbol Time (Digital Communication) TS 1 1 +
Significant ISI
Wireless Channel + τ0 τa τN

19. Frequency Selective Channel Simulation

20. Slow Fading Vs Fast Fading
P(τ0,t)
Power Delay Profile
P(τ)
P(τ0,TC)
P(τ0,2TC)
P(τ0,3TC)
P(τ0,KTC) τ0 τ t TC
2TC
3TC
KTC
Consider a wireless channel comprised of a single path component.
The power delay profile reflects average measurements
P(τ0) shall vary as the mobile moves
Fast Fading
Slow Fading
Frequency dispersion (time selective fading)

21. Stationary Receiver Tx signal = cos 2t
Assume v = 0, phase difference of two paths = 2.8 radians
Rx signal = cos 2t + cos (2t+2.8) S2 S1 θ Ф X d Y v
Note: S1 and S2 represent two paths that arrive at the receiver (example two ray model)

22. Slow Fading Tx signal = cos 2t
Assume slow fading, phase difference of two paths = 2.8
Rx signal = cos 2.01t + cos (2.02t+2.8) S1 θ Ф X d Y v
Note: S1 and S2 represent two paths that arrive at the receiver (example two ray model)

23. Fast Fading Tx signal = cos 2t
Assume slow fading, phase difference of two paths = 2.8
Rx signal = cos 2.1t + cos (2.2t+2.8) S1 θ Ф X d Y v
Note: S1 and S2 represent two paths that arrive at the receiver (example two ray model)

24. Stationary Receiver Vs Slow Vs Fast Fading

25. Types of Small-Scale Fading
From “Wireless Communications: Principles & Practice” T. Rappaport

26. Types of Small-Scale Fading
From “Wireless Communications: Principles & Practice” T. Rappaport