Mobile radio propagation: Small-scale fading

1
Mobile radio propagation Small-scale fading

• Fading is the rapid fluctuation of the radio
  signal amplitude over a short period of time .
• Fading is caused by interference between two or
  more versions of transmitted signal, which arrive
  at the receiver at slightly different times.
• These multipath waves combine at the receiver to
  give a resultant signal, which can vary in delay,
  amplitude and phase.

2
Multipath effects

• Rapid changes in signal strength over a small
  distance or time interval.
• Random frequency modulation due to varying
  Doppler shift on different multipath signals.
• Time dispersion (echoes) caused by multipath
  propagation delay.

3
Causes of fading

• In urban areas, fading occurs because the height
  of mobile is lesser than the height of
  surrounding structures, such as buildings and
  trees.
• Existence of several propagation paths between
  transmitter and receiver.

4
Factors influencing small signal fading

• Multipath propagations
• Speed of mobile (Doppler shift)
• Speed of surrounding objects.
• Transmission bandwidth of signal and bandwidth of
  channel.

5
Analysis of multipath channel
Transmitter

• Receiver

Spatial position
d
6
Convolution model for multipath signal
A2 x(t- t 2)
LOS
T
R
x(t)
A1 x(t- t 1)

• Received signal y(t) A0 x(t) A1 x(t - t 1)
  A2 x(t - t 2) ...

7
Time varying system model for channel

• For a fixed position d, the channel between
  transmitter receiver can be modulated as a
  linear time varying system (LTV system).
• Impulse response of the LTI system can be given
  as h(d,t).
• If x(t) is the transmitted signal, the received
  signal can be represented as
• y(d,t) x(t) h(d,t)
• denotes convolution

8
System definition

• Linear Time Varying (LTV) System

h(t, t)
x(t)
y(t)
9
Signal definitions

• Transmitted signal

C(t)

• j 2 ? fct
• x(t) Re c(t) e

10
...Signal definitions

• Received signal

• j2p fct
• y(t) Re r(t) e
• j2p fct
• h(t, t) Re hb(t, t) e

• Impulse response

11
Base band equivalent channel impulse response
model
hb(t, t)
c(t)
r(t)
12
Modeling of the base band impulse response model

• Mathematical model

• r(t) c(t) hb(t, t)

hb(t,t)
t3
t2
t1
t0
t N-2 t N-1
t o t 1 t 2
13
Excess delay concept

• The delay axis t, t olt t lt t n-1 is divided
  into equal time delay segments called excess
  delay bins.
• t 0 0
• t 1 ? t
• t 2 2 ? t
• t N-1 (N-1)? t

14
Excess delay concept

• All multipath signals received within the bins
  are represented by a single resolvable multipath
  component having delay t i .
• This model can analyze transmitted signals having
  bandwidths less than 2/ ? t.

15
Simplified mathematical model for baseband
impulse response

• If the channel is assumed to be time invariant,
  over a small period of time or over small
  distance interval
• N-1 j ? i
• hb(t) ? ai e ?t ti
• i 0?????
• Output r(t) c(t) hb(t)

16
...Simplified mathematical model for base band
impulse response

• N-1 j? i
• r(t) ? ai e ct ti
• i 0?????
• For measuring or predicting the impulse response
  a probing pulse c(t) ?t is used.

17
Relationship between bandwidth and received power
- Wideband signals

• N-1 j ? i
• Received signal r(t) ? ai e ptti
• i 0
• Instantaneous received power amplitude
• N-1
• r(t)2 ? ak2
• k 0
• gtTotal received power sum of the power of
  individual multipath components.

18
Average small-scale received power

• N-1
• Ea,? PwB Ea,? ? ai exp j?i2
• i 0
• N-1 ---
• ? ai2
  
• i 0
• Ea,? average overall possible values of ai
  and ? I in a local area.
• ai 2 sample average area local measurement
  area, generally measured using multipath
  measurement equipment.

19
Relationship between bandwidth and received power
- Narrowband signals

• N-1 j?i
• Received signal r(t) ? ai e
  ptti
• i
  0
• Instantaneous received power

N-1 j?i r(t)2 ? ai e
2 i 0
20
Instantaneous Multipath received power amplitude

• N-1 j?i (t,t)
• r(t)2 ? ai e 2
• i 0

21
Average power over a local area

• N-1 j?i (t, t)
• Ea,? PwB Ea,? ? ai e
  2
• i 0

22
Conclusions

• When the transmitted signal has?a wide bandwidth
  gtgt bandwidth of the channel multipath structure
  is completely resolved by the receiver at any
  time and the received power varies very little.
• When the transmitted signal has a very narrow
  bandwidth (example the base band signal has a
  duration greater than the excess delay of the
  channel) then multipath is not resolved by the
  received signal and large signal fluctuations
  occur (fading).

23
Example

• Assume a discrete channel impulse response is
  used to model urban radio channels with excess
  delays as large as 100 ?s and microcellular
  channels with excess delays not larger than 4 ?s.
  If the number of multipath bins is fixed at 64
  find
• (a) ? t
• (b) Maximum bandwidth, which the two models can
  accurately represent.

24
Solution

• Delays in channel ? t, 2 ? t . N ? t
• Maximum excess delay of channel
• t N N ? t 100 ? s.
• 
  
• N 64
• ?t tN /N 100 ?s /64 1.5625 ?s
• Maximum bandwidth represented accurately by model
  2/ ?t
• 1.28 MHz

25
For microcellular channel

• Maximum excess delay of channel
• t N N ? t 4 ? s.
  
• N 64
• ? t t N /N 4 ? s /64 62.5 ns
• Maximum bandwidth represented accurately by model
  2/ ? t
  
  32 MHz

26
Example

• Assume a mobile traveling at a velocity of 10m/s
  receives two multipath components at a carrier
  frequency of 1000MHz.
• The first component is assumed to arrive at t 0
  with an initial phase of 0? and a power of
  70dBm.
• The second component is 3dB weaker than the first
  one and arrives at t 1? s, also with the
  initial phase of 0?.

27
...Example

• If the mobile moves directly in the direction of
  arrival of the first component and directly away
  from the direction of arrival of the second
  component, compute the following
• (a) The narrow band and wide band received power
  over the interval 0-0.5s
• (b) The average narrow band received power.

28
Narrow band instantaneous power

• N-1 j?i (t,t)
• r(t)2 ? ai e 2
• i
  0
• Now 70dBm gt 100 pw so a1 v 100 pw
• and 73dBm gt 50 pw so a2 v 50 pw
• ?i 2pd/? 2pvt/?
• ? (3108)/(100106) 0.3 m
• ?1 2p10t/0.3 209.4 t rad.

29

• ?2 -?1 -209.4 t rad.
• t 0
• r(t)2? v100 ? v50 2 291pw
• t 0.1
• r(t)2 v100. e j209.4 x 0.1 v50. e -j209.4
  x 0.1
• 78.2pw
• t 0.2
• r(t)2 v100. e j209.4 x 0.2 v50. e -j209.4
  x 0.2
• 81.5pw

30

• t 0.3
• r(t)2 291pw
• t 0.4
• r(t)2 78.2pw
• t 0.5
• r(t)2 81.5pw

31
Wideband instantaneous power

• N-1
• r(t)2 ? ak2 100 50 150 pW
• k 0

32
Average narrow band received power

• Ea,? PCW 2(291) 2(78.2) 2(81.5) /6
• 150.233pw
• The average narrow band power and wideband power
  are almost the same over 0.5s.
• While the narrow band signal fades over the
  observation interval, the wideband signal remains
  constant.

33
Small-scale multipath measurements

• Direct Pulse Measurements
• Spread Spectrum Sliding Correlator Measurement
• Swept Frequency Measurement

34
Types of Small Scale Fading
Doppler Spread
Multipath time delay
Fast Fading
Slow fading
Flat fading
Frequency Selective Fading
35
...Types of Small Scale Fading

• 2 main propagation mechanisms
• Multipath time delay spread
• Doppler spread
• Two types of fading are independent of each other.

36
Multipath terms associated with fading

• Ts Symbol period or reciprocal bandwidth
• Bs Bandwidth of transmitted signal
• Bc coherence bandwidth of channel
• Bc 1/(50??)??where ??? is rms delay spread

37
...Multipath terms associated with fading

• __ _
• ?? 2 ? 2 - ( ??)2
• _
• ? (? ak2 ??) / (? ak2) mean Excess delay
• __
• ? 2 (? ak2 ??2) / (? ak2)

38
Fading effects due to Doppler spread

• fc frequency of pure or transmitted sinusoid
• Received signal spectrum
• fc /- fd, fd
• Doppler shift

fc
S
?
V
39
Doppler spread and coherence time

• Doppler frequency shift fd (v / ?) cos ? ,
  
• Where Wavelength ? c / fc meters
• Doppler Spread BD fm Maximum Frequency
  deviation v / ?
• Coherence time Tc 0.423 / fm

40
Types of fading

• Flat fading
• Mobile channel has constant gain and linear phase
  response.
• Spectral characteristics of the transmitted
  signal are maintained at receiver.
• Bs ltlt Bc
• gt Ts gtgt ??

41
...Types of fading

• Frequency selective fading
• Mobile channel has a constant gain and linear
  phase response over a bandwidth.
• Bs gt Bc
• gt Ts lt ??
• Common rule of thumb
• If Ts gt 10 ?? gt Flat fading
• If Ts lt 10 ?? gt Frequency selective fading

42
How to decide flat or frequency selective fading?

• Common rule of thumb
• If Ts 10 ?? gt Flat fading
• If Ts lt 10 ?? gt Frequency selective fading

43
Fast fading channel

• The channel impulse response changes rapidly
  within the symbol duration.
• This causes frequency dispersion due to Doppler
  spreading, which leads to signal distortion.
• Ts gt Tc
• Bs lt BD

44
Slow fading channel

• The channel impulse response changes at a rate
  much slower than the transmitted signal s(t).
• Ts ltlt Tc
• Bs gtgt BD
• Velocity of mobile (or velocity of objects in
  channel) and base band signaling determines slow
  fading or fast fading.

45
Rayleigh and Ricean distributions...

• Rayleigh fading distribution
• In mobile radio channels, the Rayleigh
  distribution is commonly used to describe the
  statistical time varying nature of the received
  envelope of flat fading signal.

46

• Pdf (Probability density function)
• p(r) (r/?2) e (r2/2?2) (0 r ?)
• 0 r lt 0
• ? ? rms value of received voltage before envelope
  detection.

47
Cumulative distribution function (cdf)

• R
• P (R) P( r ? R) ? p(r) dr
• 0
• 1 e (R2/2?2)
• 8
• Mean Value ER ? r p(r) dr
• 0
• ???/2
• 1.25336 ?

48

• ?R2 ER2 - ?E(R)?2
• 8
• ? r2 p(r) dr - ?2p/2
• 0
• 0.4292 ?2
• Median value for r gt ½
• ? pr dr gt r (median) 1.77?
• rmedian
• Median value for r ? p(r ) dr gt rmedian
  1.77?
  
  0

49
Ricean fading distribution

• When there is a dominant (non fading) signal
  component present such as LOS propagation path,
  the small scale fading envelope distribution is
  Ricean.
• This can be modeled as random, multipath
  components arriving at different angles
  superimposed on a stationary dominant signal.

50
...Ricean fading distribution

• p(r) (r/?2)e (r2 A2) / (2?2) Io(Ar/?2 )
• for A ? 0, r ? 0
• p(r) 0 for r lt 0
• Io ? Modified Bessel function of first kind and
  zero order

51
Ricean factor

• K(dB) 10 log(A2/2?2) dB
• 10 log (Deterministic signal power/ variance
  of multipath)

52
Level crossing and fading statistics

• Level crossing rate (LCR) is defined as the
  expected rate at which the normalized Rayleigh
  fading envelope, crosses a specified level in a
  positive going direction.

53
Simplified equation for LCR

• NR ?(2?) fm ?e-?2
• fm Maximum Doppler frequency
• ? R/Rms is the value of the specified level R,
  normalized to the local rms amplitude of the
  fading signal

54
Example
For a Rayleigh fading signal, compute the
positive going level crossing rate for ? 1,
when the maximum Doppler frequency (fm) is 20
Hz. What is the maximum velocity of the mobile
for this Doppler frequency if the carrier is 900
MHz?
55
Solution
? 1 fm 20 Hz The number of zero level
crossings is NR ?2? (20) e-1
18.44 Crossings/Sec Maximum velocity of mobile
fd ? 20 (3 X 108)/(900X106) 6.66 m/s
56
Average fade duration

• Average period of time for which the received
  signal is below a specified level R.

57
Formula for Average fade duration

• ? e?2 1
• ________
• ?fm?2?

58
Example

• Find the average fade duration for threshold
  level ? 0.01, ? 0.1 and ? 1, when the
  Doppler frequency is 20Hz.

59
Solution

• ? ? e?2 1
• ?fm?2?
• 0.01 19.9?s
• 0.1 200?s
• 1.0 3.43ms

60
Statistical methods for multipath fading channels

• Clarks model for Flat Fading
• Two-Ray Rayleigh Fading Model
• Saleh and Valenzuela Indoor statistical Model
• SIRCIM (Simulation of Indoor Radio Channels
  Impulse Response Models)
• SMRCIM (Simulation of Mobile Radio Channel
  Impulse-Response Models)