The document discusses small-scale fading or multipath fading in wireless channels. It defines key terms like delay spread and Doppler spread caused by multipath propagation. It describes the time-varying channel impulse response and how multipath components combine constructively or destructively. It also introduces parameters used to characterize mobile multipath channels, such as coherence bandwidth, Doppler spread, and coherence time.

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The Wireless Channel (2)
Wireless Communication

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
phaseshiftt of
D   2 f D  dt
N N
t
 1 2 1  2  Bu1
If reflector cluster exists, two multipath components
with delay and are resolvable if
 If t criteriat isnot 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 sometimescalled
excessdelay 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

20. Time Dispersion Parameters

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 seenthat
 The mean excess delay:

1 5   0.11   0.1 2    0.01 0   4.38 s
 0.01  0.1  0.1  1
 The second moment
1 5   0.11   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

25. Types of Small-Scale Fading

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

33. Summary

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

38. Rayleigh Fading
 The corresponding Rayleigh pdf is

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
atrR=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  201 e1  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

46. Rician Fading Distribution
 The Rician pdf is

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