2. Review
What we discussed last lecture:
The large-scale fading because of path loss
The empirical path loss formulas
Today, we will discuss about the small-scale fading
and the statistical models represent it
3. Introduction
The small-scale fading is usually called “fading”
It is caused by multipath signal, so it is also called
“multipath fading”
Multipath signal causes constructive and destructive
addition of the received signal
4. Introduction
If a single pulse is transmitted in the multipath
channel, it will yield a train of pulses with delay time
LOS
Reflected components
Delay spreadTd ): the time delay between the
(
arrival of the first received signal component and the
last received signal component associated with a
single transmitted pulse
5. Introduction
If the delay spread is small compared to the (1/BW),
then there is little time spreading in the received
signal
If the delay spread is relatively large, there is little
time spreading of the received signal, i.e. signal
distortion
Multipath channel is also time-varying that means
either the transmitter or the receiver is moving
It also causes the location of the reflectors will
change over time
We will limit the model to be narrowband fading, i.e.
the BW is small compared to (1/delay spread)
6. Introduction
Physical factors influencing fading:
Multipath propagation
Speed of the mobile
Speed of surrounding objects
The transmision bandwidth of the signal
7. Review of Doppler Shift
The received signal may experience Doppler shift
v eff
If the receiver is moving towards the transmitter, the
Doppler freq is positive, otherwise it is negative
8. Example
Consider a transmitter which radiates a carrier of
1850 MHz. For a vehicle moving 26.82 mps,
compute the received carrier frequency if the mobile
is moving:
Directly towards the transmitter
Directly away from the transmitter
In a direction which is perpendicular to the direction of
arrival of the transmitted signal
9. Solution
Carrier freq = 1850 MHz
3 8
Wavelength = fc 18501010 0.162m
6
c
Vehicle speed = 26.82 m/s
Vehicle moving towards the transmitter means
positive Doppler frequency
26.82
f f c f d 1850 cos 0 1850.00016MHz
0.162
Vehicle moving towards the transmitter means
negative Doppler frequency
26.82
f f c f d 1850 cos 0 1849.999834MHz
0.162
Vehicle is moving perpendicular means90
26.82
f f c f d 1850 cos90 1850MHz
0.162
10. Doppler Spread
Doppler spread is given by
Ds : D2 D1
fv fv
Where, D1 and D2
c c
E.g. If the mobile is moving at 60 kph and f = 900
MHz, the the Doppler spread is
fv fv fv
Ds 2
c c c
900 106 16.67
2 100 Hz
3 108
11. Time-Varying Channel Impulse Response
We have already known that the transmitted signal is
s t u t e j 2 f t u t cos 2 fct u t sin 2 f ct
c
Then, the received signal in multipath channel is
n = 0 corresponds to the LOS path
N(t) is the number of resolvable multipath
components
is corresponding delay
is Doppler phase shift
is amplitude
12. Time-Varying Channel Impulse Response
The n-th resolvable multipath component may
correspond to the multipath associated with a single
reflector or multiple reflectors clustered together
13. Time-Varying Channel Impulse Response
If single reflector exists, the amplitude is based on
the path loss and shadowing, its phase change
n t with f t
associated e j 2delay
c n
and Doppler
phaseshiftt of
D 2 f D dt
N N
t
1 2 1 2 Bu1
If reflector cluster exists, two multipath components
with delay and are resolvable if
If t criteriat isnot satisfied, then it is nonresolvable
u the 1 u 2
since
1 2
The nonresolvable components are combined into a
single multipath component with delay and
an amplitude and phase corresponding to the sum of
different components
14. Time-Varying Channel Impulse Response
The amplitude of the summed signal will undergo
fast variations due to the constructive and
destructive combining of the nonresolvable multipath
components
Wideband channels have resolvable multipath
components the parameters change slowly
Narrowband channels tend to have nonresolvable
multipath components the parameters change
quickly
15. Time-Varying Channel Impulse Response
We can simplifyr t by letting
n t 2 fc n t D
n
The received signal is then
The received signal is obtained by convolving the
baseband input signal with equivalent lowpass time-
varying channel impulse response of the channel,
and then upconverting the carrier frequency
16. Time-Varying Channel Impulse Response
The c , t represents the equivalent lowpass
t
response of the channel at time t to an impulse at
time
17. Parameters of Mobile Multipath Channels
Time dispersion parameters
Coherence bandwidth
Doppler spread and coherence time
18. Time Dispersion Parameters
The time dispersive properties of wideband multipath
channels are most commonly quantified by their
mean excess delay and rms delay spread
The mean excess delay:
a 2
k k P k k
k
k
a k
2
k P
k
k
The rms delay spread is the square root of the
second central moment of the power delay profile
2
2
a 2 2
k k P k
2
k
2 k
k
a k
2
k P k
k
19. Time Dispersion Parameters
The delays are measured relative to the first
0
detectable signal arriving at the receiver at
The maximum excess delay (X dB) of the power
delay profile is defined to be the time delay during
which multipath energy falls to X dB below the
maximum.
X 0
The maximum excess delay sometimescalled
excessdelay spread, which can be expressed as
X
Where is the maximum delay at which a multipath
component is within0 X dB of the strongest arriving
multipath signal and is the first arriving signal
21. Coherence Bandwidth
Coherence bandwidth is a statistical measure of the
range of frequencies over which the channel can be
considered “flat”
Flat fading is a channel which passes all spectral
components with approximately equal gain and
linear phase
The coherence bandwidth can be expressed as
1 (above 90% correlation)
Bc
50
1
Bc (above 50% correlation)
5
22. Example
Compute the mean excess delay, rms delay spread,
and the maximum excess delay for the following
power delay profile
Estimate the 50% coherence bandwidth of the
channel
23. Solution
Using the definition of maximum excess delay (10
10 dB 4 s
dB), it can be seenthat
The mean excess delay:
1 5 0.11 0.1 2 0.01 0 4.38 s
0.01 0.1 0.1 1
The second moment
1 5 0.11 0.1 2 0.01 0
2 2 2 2
2 21.07 s 2
0.01 0.1 0.1 1
The rms delay spread:
21.07 4.38 1.37 s
2
The coherence bandwidth:
1 1
Bc 146kHz
5 5 1.37 s
24. Doppler Spread and Coherence Time
Doppler spread has been discussed before
The coherence time is related with Doppler spread
(Doppler shift)
0.423
Tc
v
26. Flat Fading
If the mobile radio channel has a constant gain and
linear phase response over a bandwidth which is
greater than the bandwidth of the transmitted
signal, then the received signal will undergo flat
fading
27. Flat Fading
Flat fading channels are also known as amplitude
varying channels
It is also sometimes referred to as narrowband
channels
The most common amplitude distributions are:
Rayleigh, Rician, and Nakagami
Summarize: a signal undergoes flat fading if
B B
s c
Ts
28. Frequency Selective Fading
If the channel has a constant-gain and linear phase
response over a bandwidth that is smaller than the
bandwidth of transmitted signal, then the channel
creates frequency selective fading on the received
signal
29. Frequency Selective Fading
The received signal includes multiple versions of the
transmited waveform which are attenuated and
delayed in time, and hence the received signal is
distorted
Frequency selective fading is due to time dispersion
of the transmitted symbols within the channel
Thus, the channel induces intersymbol interference
(ISI)
The modeling for this kind of channel is more difficult
since each multipath signal must be modeled and
channel must be considered to be a linear filter
The common model: 2-ray Rayleigh fading
30. Frequency Selective Fading
It is sometimes called wideband channels since the
bandwidth of the signal is wider than the bandwidth
of the channel impulse response
Summarize: a signal undergoes frequency selective
fading if
Bs Bc
Ts
31. Fast Fading
In a fast fading channel, the channel impulse
response changes rapidly within the symbol duration
In other words, the coherence time of the channel is
smaller than the symbol period of the transmitted
signal
This causes frequency dispersion (time selective
fading) due to Doppler spread, which lead to signal
distortion
Signal distortion due to fast fading increases with
increasing Doppler spread relative to the bandwidth
of the transmitted T T
signal
s c
Summarize: a signal undergoes fast fading if
Bs BD
32. Slow Fading
In a slow fading channel, the channel impulse
response changes at a rate much slower than the
transmitted signal
The channel may be assumed to be static over one
or several reciprocal bandwidth interval
The Doppler spread of the channel is much less than
the bandwidth of the baseband signal
Summarize: a signal undergoes slow fading if
Ts Tc
Bs BD
34. Remarks
When a channel is specified as a fast or slow fading
channel, it does not specify whether the channel is flat
fading or frequency selective
Fast fading only deals with the rate of change of the
channel due to motion
In flat fading channel, we can approximate the impulse
response to be simply delta function
A flat fading, fast fading channel is a channel in which the
amplitude of the delta function varies faster that the rate
of the transmitted baseband signal
A frequency selective, fast fading channel, the
amplitudes, phases, and time delays of any one of the
multipath components vary faster than the rate of change
of the transmitted signal
35. Rayleigh Fading
The Rayleigh distribution is commonly used to
describe the statistical time varying nature of the
received envelope of a flat fading signal
Rayleigh distributed signal:
36. Rayleigh Fading
The Rayleigh distribution has pdf
the rms value of the received voltage signal before envelope detection
2 the time-average power of the received signal before envelope detection
The probability that the envelope of the received
signal does not exceed a specified value R is
37. Rayleigh Fading
The mean value of Rayleigh distribution is
The variance of the Rayleigh distribution (represent
the ac power)
The median value is
The median is often used in practice
39. Level Crossing and Fading Statistics
The level crossing rate (LCR) is defined as the
expected rate at which the Rayleigh fading
envelope, normalized to the local rms signal
level, crosses a specified level in a positive-going
direction
The number of level crossing per second is given by
N R rp R, r dr 2 f D e
2
0
Where
r
p is r
time derivative of r(t) (the slope)
R,
r
is the joint density function of r and
atrR=RR
rms
40. Example
For a Rayleigh fading signal, compute the positive-
going level crossing 1 for
rate when the maximum
Doppler frequency is 20 Hz
What is the maximum velocity of the mobile for this
Doppler frequency if the carrier frequency is 900
MHz?
41. Solution
Use the equation for LCR
NR 2 201 e1 18.44
Use equation of Doppler frequency
v f D 20 1 3 6.66m / s
42. Level Crossing and Fading Statistics
The average fade duration is defined as the average
period of time for which the received signal is below
a specified level R.
For a Rayleigh fading signal, it is given by
1
Pr r R
NR
1
Pr r R i
T i
R
p r dr 1 exp 2
0
So, the average fade duration can be expressed as
2
e 1
f D 2
43. Example
Find the average fade duration for threshold levels0.01
when the Doppler frequency is 200 Hz
Solution
Average fade duration is
0.012
e 1
19.9 s
0.01 200 2
44. Rician Fading Distribution
When there is a dominant stationary (nonfading)
signal component present, such as line-of-sight
propagation path, the small-scale fading envelope
distribution is Rician
Random multipath componnets arriving at different
angles are superimposed on a stationary dominant
signal 2
r A
2
r Ar by
The Rician distribution isI 0given for A 0, r 0
p r 2 e 2
2
2
0 for r 0
45. Rician Fading Distribution
The Rician distribution is described in terms of a
parameter K
A2
K
2 2
A2
K dB 10 log 2
2
As K 0 we have Rayleigh fading
As K we have no fading, channel has no
multipath, only LOS component
47. Conclusions
Small-scale fading is variation of signal strength over
distances of the order of the carrier wavelength
It is due to constructive and destructive interference
of multipath
Key parameters:
Doppler spread coherence time
Delay spread coherence bandwidth
Statistical small-scale fading: Rayleigh fading and
Rician fading flat fading