This document discusses various wireless propagation channels including free space propagation, reflection, scattering, and diffraction. It covers reflection propagation mechanisms such as reflection from dielectrics and conductors. Reflection coefficients and Snell's law are explained. Models for reflection, including the two-ray ground reflection model, are provided. Diffraction models like knife-edge diffraction and multiple knife-edge diffraction using methods like Bollington's method are summarized. Scattering models including Kirchoff's theory and perturbation theory are covered. Common fading models for mobile radio like Rayleigh, Rician, and Doppler shift models are described. Finally, different types of wireless channels including time-selective, frequency-selective, general, and WSSUS channels are classified

1. Unit 2
WIRELESS PROPAGATION CHANNELS

2. Modes of propagation
• Free space propagation
• Reflection – Smooth Surfaces
• Scattering – Rough Surfaces
• Diffraction – Edge Surfaces

3. REFLECTION PROPAGATION
MECHANISM
• When a radio wave propagating in one
medium impinges upon another medium
having different electrical property the wave is
subjected for reflection
– Reflection from dielectric
– Reflection form perfect conductor
– Ground reflection

4. Reflection form dielectric
• When the wave impinges on a dielectric
object,
– Part of the energy is transmitted and part of the
wave of reflected back to the first
medium.(assuming no loss of energy)

5. Snell’s Law
• Snell’s law is a formula used to describe the
relation ship between the angle of incidence
and the angle of refraction.
• Snell’s Law States that the ratio of the sines of
the angle of incidence and the angle of
refraction are equal to the ratio of phase
velocities in two media

6. Reflection Co Efficient
• The angle of incidence and the angle of reflection are
closely related with the REFLECTION Co EFFICIENT
• The electric field intensity of the reflected wave and
transmitted wave are related to the incident wave
through a factor called Fresnel reflection CoEfficient.
• Ref. Co Eff. Depends upon
– Wave polarization
– Angle of incidence
– Frequency of propagation

7. Factor of Polarization
• In general the electromagnetic waves are
polarized.
• Ie., two signals are placed orthogonally (90
deg apart)
– Vertical or horizontal polarization
– LH Circular Polarization and RH Circular
Polarization

8. Visualize polarization

9. • In general only two orthogonal polarizations
are considered to solve general reflection
problems.
• Here it is parallel and perpendicular
polarizations (As shown in figure)

10. EM Wave incident on a plane

11. Calculating Ref. Coefficient

12. Terms involved…
• Angle of incident wave
• Angle of reflected wave
• Angle of transmitted wave
• Permittivity - How E field is affected by dielectric
• Permeability - Degree of magnetization
• Conductance - ability for electricity to flow

13. Reflection coefficient.

14. ii. Reflection from Conductor
• Generally the EM Waves do not pass through
Conductors ie., all the energy is reflected back
to the first medium.
• If the signals are vertically polarized:
• If the signals are Horizontally polarized:
& Ei = Er
& Ei = - Er

15. Ref CoEff. For Conductor
• For a conductor, the reflection coefficient is
given by
1II  1  

16. Ground Reflection Model

17. 2 Ray Model
• Free space propagation model is in accurate in
many if the cases when used alone.
• This model is designed for both LOS and
Reflected rays.
• This model is accurate for predicting the large
scale signal strength over distance of several
Kilometers.
• In most of the cases the T-R Separation is only
few tens of kilometers hence the earth is
assumed to be FLAT.

18. 2 Ray Model Understanding

19. 2 Ray Model

20. General Expression for E field wrt “d”
and “t”
• E Field in Free space Prop is Given by:
• Eo - Free Space E Field
• do - Reference Distance
Envelope of E Field

21. • Two waves arrives at the receiver:
– Direct wave at distance d’
– Reflected wave at distance d”

22. • The Resulting E Field is given by:
1  

23. Path difference calculation
• The line of sight rays and reflected rays have
different paths.
• This difference is calculated by the method of
imaging .

24. Method of images
• The method of images (or method of mirror
images) is a mathematical tool for
solving differential equations, in which the
domain of the sought function is extended by
the addition of its mirror image.
• Generally they are used for analysing the
charges and the magnetic substances.

26. Method of images

27. • When T-R Separation is very large compared
to ht + hr the equation can be simplified by
using Taylor’s series approximation

28. Phase Difference and Time Delay
• Once the path difference is known, then the
Phase Difference between the two E Field
Components and Time Delay between the
arrival of the
two components
can be easily
computed by the
following relations:

29. • When “d” becomes larger and larger the
differences between the d’ and d” becomes
very small.
• In this case the amplitude levels of both LOS
and Reflected Rays are virtually identical.

30. Final Receive Power
• Finally, the Received power at the distance d
from the transmitter for the 2 ray model is
given by:

31. • The Path Loss is Defined as:

32. END
OF
REFLECTION PROPAGATION

33. DIFFRACTION
• Diffraction are due to when the signals hits
the edge surfaces (Preferably Sharp Edges)
• Diffraction allows the signal to propagate
around the curved surfaces of the earth and
to propagate behind obstructions.

34. Huygens's Principle
• The phenomenon of diffraction is explained by
Huygens's principle.
• “Which states that all points on a wave front
can be considered as a point sources for
production of secondary wavelets, and these
wavelets combine to produce a new wave
front in the direction of propagation”

35. Huygens Principle

36. Huygens's Secondary Source

37. Knife Edge Diffraction Model

39. • The difference between the direct path and
the diffracted path is called as “Excess Path
Length” which is given by:

40. • The angles of the diffracted waves are related
as:

41. • The final normalized equation is given by
“Fresnel – Kirchoff” Diffraction Parameter (v)
• Fresnel – Kirchoff Parameter is a dimension
less quantity.
• It characterizes the phase difference between
two propagating paths.
• It is used to characterize the diffraction losses
in general situation.

42. Multiple Knife Edge Diffraction
• In practical situations, especially in hilly terrains
the propagation path may consists of more than
one obstruction
• In this case the total diffraction loss due to all the
obstacles must be computed
• There are several methods available to analyse
the total diffraction loss in the system, they are
– Bollington's Method
– Epstein – Peterson Method
– Deygout’s Method

43. Bollington's method
• This is the simplest method
• The concept of this method is to replace the
multiple edges with single equivalent edge.
– Draw a tangential line from Tx to all the edges and
choose the steepest line so that all other remaining
lines falls under the chosen line.
– Repeat the same procedure from the Rx Side and
choose the steepest line so that all other lines falls
under the chosen line.
– Now extend the two selected lines and find the
intersecting point which gives the height of the single
equivalent edge.

45. 2. Epstein Peterson Method
• The previous method is inaccurate due to the
approximation made.
• In this method the diffraction loss is calculated
for each obstacles present between the Tx and
the Rx.
• Finally all the values are approximated and the
average value is taken as the overall diffraction
loss.
• This method is accurate than the former one but
also they have their own drawbacks due to the
approximations considered.

46. 3.Deygout’s Method
• The algorithm of deygout’s method is as followes:
– As a first step compute the attenuation between the Tx
and Rx by assuming that only ith screen is present (For all i)
– The screen that causes the largest attenuation is named as
the MAIN SCREEN
– Now compute the attenuation between the Tx and the
MAIN SCREEN. The largest value is named as the
SUBSIDARY MAIN SCREEN
– Similarly compute the attenuation between the MAIN
SCREEN and the RX. The largest value is named as the
SUBSIDARY MAIN SCREEN
– Add up all the losses for all considered screens.

47. END
OF
DIFFRACTION MODELS

48. SCATTERING
• Scattering occurs in such conditions where the
surface of the obstacle is rough, in addition to this
the surface dimensions should be small compared to
the wave length.
• Analysing the scattered have a vital role in RADAR
applications.

49. • Scattering method of propagation can be
studied by 2 different theories, they are as
follows,
– Kirchoff’s theory
– Perturbation theory

50. Kirchoff’s Theory
• This theory is very simple as it uses only
minimum amount of information such as
Probability density function of the surface.
• The main assumptions made in this theory is that
the heights of the rough surfaces are so small so
that the points will not interfere with each other.
• Here the power of the reflected signal is given by
the roughness of the surface as expressed below.

51. Perturbation Theory
• The perturbation theory generalizes the Kirchhoff
theory, using not only the probability density
function of the surface height but also its spatial
correlation function.
• In other words, it takes into account the question
“how fast does the height vary if we move a certain
distance along the surface?

52. END
OF
SCATTERING MODELS

53. PROPAGATION EFFECTS WITH MOBILE
RADIO

54. Introduction
• These multipath propagation leads to fading.
– Slow fading
– Fast fading

55. TYPES OF FADING

57. Fading Models
• Generally in most of the conditions it is too
complex to describe all the reflection, diffraction
and scattering signals which provides enormous
numbers of signals . Thus we consider some
STATISTICAL MODELS for the analysis purpose.
– Rayleigh fading model
– Rician fading model
– Doppler shift model
– Fast fading model

58. Rayleigh fading Model
• This is a small signal fading model.
• This model assumes that there is NO
DOMINANT PATH present in the system.
• Dominant path refers to the LINE OF SIGHT
SIGNAL.
• All the signals received by the receiver are the
reflected signals.
• The receiving terminal is considered as the
PORTABLE Terminal

60. • The transmitted signal reaches the receiver via
multiple paths.
• The complex form of those signals are
expressed as:
Electric field strength of nth path
Relative phase of the Nth path

61. • The small difference in the path length can
cause larger difference in the phase.
• Hence we assume that the phase is
UNIFORMLY distributed over [0,2pi]
• In the above expression if the value of the N
becomes very large then as per the central
limit theorem the random variables will follow
the approach of Gaussian Random Variables.
Gaussian Random Variables

62. • Now we consider only one single component to find the
Expectation of each component
• Since the Expected value of single component is ZERO then
the expected value of complex envelope is also ZERO

63. • The variance (POWER) in the envelope is given
by the mean square value.

64. • Since the mean value of the envelope is zero
the probability density function of the signal is
represented as Gaussian density function
• The amplitude of the signal is given by

65. • The probability density function of the
amplitude is given by
• This equation is referred to as RAYLEIGH
PROBABILITY DENSITY FUNCTION

66. • Integrating the probability density function
over the limit yields Cumulative probability
density function

67. • Now the mean and mean square values are
found easily by simple expectation operators

68. Amplitude distribution of Rayleigh
fading Model

69. END
OF
RAYLEIGH FADING MODEL

70. RICIAN FADING MODEL
• In this model it is assumed that the line of
sight path is also present along with the
reflected paths (Multipath)
• In such scenarios , Random multipath
components arriving at the receiver from
various angels are superimposed on a
stationary dominant signal
• Thus the complex form of envelop signal is
given by
Constant Term (LOS
Signal)
Collection of
Reflected paths

71. • The power levels of the different path signal
plays a vital role in analysis.
• The ratio between the power levels are
referred to as “Rician – K – Factor”

72. • The probability density function of the rician
fading model is given by

73. • The amplitude distribution of the rician fading
model is given by

74. END
OF
RICIAN FADING MODEL

75. DOPPLER SHIFT MODEL
• The term Doppler refers to the change or shift in
frequency due to the movement.
• The strength of the received signal is very high
and the frequency is also high when the receiver
is moving towards the source.
• The strength of the received signal is low and the
frequency is also low when the receiver is moving
away from the source.
• These movements results in the change in
frequency which is referred to as DOPPLER SHIFT

77. • In our case we consider the Doppler effect
with fixed transmitter and moving antenna as
depicted in the figure:

78. • From the fig. the receiver moves at the
constant velocity.
• If the complex envelope of the signal emitted
by the transmitter is
• Then the signal at any point along the x axis is
given by
Amplitude as a
function of
distance
Speed of
Light
Position of the
receiver

79. • Form the previous expression it is clear that
the phase of the signal varies according to the
distance of the receiver.
• Then the distance of the receiver is given by:
Amplitude
Part Phase Part
Initial position of
the receiver
Velocity of the
moving receiver

80. • Substituting the value of “x” in the above
expression we get
We have

81. • From the resulting expression the received
frequency is given by:
• The Doppler shift is given by:
• The relationship between the velocity and
Doppler shift is given by:

82. END
OF
DOPPLER SHIFT MODEL

83. FAST FADING
• In the previously discussed models the
terminals assumed is portable type.
• If we analyse the received signal strength for
the mobile terminal there will be a rapid
change in the received power, this
phenomenon is referred to as the fast fading.
• This model can be framed just by considering
the Rayleigh model and in addition to that
dopper effect is to be included.

84. • Hence the envelope is given by,
• Hereby we assume that all the rays are from
horizontal direction. The resulting model is
referred to as CLARKE’s Model

85. Illustration of local multipath arriving
at the receiver

86. • Time dependency of the fading model is
analysed by the auto correlation function
given by:
• The cross correlation between the pair of
signals is given by

87. END
OF
FAST FADING MODEL

88. CHANNEL CLASSIFICATIONS

89. INTRODUCTION
• Generally the wireless channels are classified
in to two types:
• Time invariant: the channel is said to be time
invariant when the BS,MS,IO’s are static.
• Time variant: the channel is said to be time
variant when either BS or the MS is moving.
• In real world scenarios the time invariant
channels are not possible for the wireless
communications.

90. • The channels are measure with an impulse
response h(t,ᵹ).
• Consider the system with following
paramenters:
• Input signal : S(t) – impulse signals with delay
• Channel response : h(t,ᵹ)
• Output signal : X(t)

91. • Then the output of the channel is given by
• Based on the characteristics of the
channels are classified in to
– Time selective channels
– Frequency selective channels
– General channels
– WSSUS channels

92. 1. Time selective channels
• They are also called as the frequency
dispersive channels.
• If the performance of the channel is varying
along with time then they are referred to as
the time selective channels.
• To analyse the response of the channel
consider the transmitted signal shown below

93. • Then the received signal is given by:
• In this model we assume that all different path
having the same length and delay. And the
channel impulse response is given by

94. Observations
• Each frequency component received by the
receiver approximately have the same
frequency. Thus they are called as the
frequency flat channel
• All FFC are not necessarily time selective
channel and similarly the vice – versa.

95. 2. Frequency selective channels
• This is also known as time dispersive channel
• If the impulse response of the channel
completely depends upon the frequency then
they are referred to as the frequency selective
channels
• In this model we assume that the reflected
signals arrive the receiver through different
path have different lengths (path lengths)
• The output of the system is given by:

96. • The impulse channel response is given by:

97. 3. General channels
• In some practical scenarios the channel can be
neither time selective nor frequency selective.
• These model is the combination of the time
selective model and the frequency selective
models
• The output of these channels are given by :

98. • The impulse response of the channel is given
by:
• In many cases the input to the system can be
given in continuous signals, it is given by:

99. 4. WSSUS Channels
• Wide Sense Stationary Uncorrelated Scattering
Channels.
• In most of the mobile communications the signals
spreads. They are of 2 types
• Delay spread:
– Because of these spread the received signals becomes
longer than the transmitted signals. This phenomenon
is referred to as the time dispersion
• Doppler spread:
– Because of these spread the bandwidth of the
received signal is very larger than the transmitted
signals. This phenomenon is referred to as the
frequency dispersion

100. • The above mentioned dispersion will create the
distortion in the received signal and the amount
of distortion depends on the design of the signal.
• These dispersions are measured by 2 parameters
they are:
– Coherence time.
• It is the time separation at which the amplitude of 2
received signals becomes completely uncorrelated
• It is a measure of length of time for which the channel can
be assumed as approximately constant.
– Coherence bandwidth.
• It is the frequency separation at which the attenuation of
two frequency domain samples of the channel becomes
completely uncorrelated
• It is the measure of approximate bandwidth within which
the channel is assumed to be constant.

101. • In order to understand the WSSUS Channel,
this can be understood as WSS separately and
US Seperately.
• The channel is assumed to be wss channel.
• The requirements of the channel to be a wss
channel is the autocorrelation function only
depends upon the time difference. They are
expressed as below

102. • In multipath propagation, the gain and the
phase of one signal will not be correlated with
the other signal . This phenomenon is referred
to as the “Uncorrelated Scattering”.
• They are expressed as follows.
• The combination of WSS channel and the
Uncorrelated Channel is called as WSSUS
Channel

103. Some effects
• Coherence time:
– If it is finite – channel is said to be time variant.
– If it is infinite – channel is said to be time invariant.
• Coherence bandwidth:
– If it is smaller, when compared to the bandwidth of
the transmitted signal then the channel is said to be
frequency selective.
– If it is larger, when compared to the bandwidth of
the transmitted signal then the channel is said to be
frequency non selective or frequency flat channels.

104. END
OF
CHANNEL CLASSIFICATION

105. CHANNEL MODELS
• For design, simulation and planning of
wireless systems we need models for the
propagation channels.
• There are 2 main approaches
– Design , testing and approval of wireless systems
– Designers are very much benefitted in optimizing
the system at computer level before
implementing.

106. Types of channel models
• Narrow band models
• Wideband models
• Ultra wideband models

107. Narrow band models
• These models are used for modeling of small
scale fading and large scale fading.
• These models focuses on fading and path loss.
• The various models available in this sector are
as followes:
– Path loss models
• The okumura model
• Walsh ikegami model
• Motley keenan model

108. Path loss model
• The very basic path loss model is break point
model.
• This model is described by the path loss
exponent (n)
• N=2 is valid for the distance below D Break
• N=4 is valid for the distance beyond D Break
• This is the most simplest model and many
more sophisticated models are also available
which are discussed in the upcoming slides

109. The Okumura - Hata Model
• The expression for this model is given by
• Where A,B,C are the factors that varies with the
frequency and antenna height.
• A increases with carrier frequency and
decreases with height of BS or MS
• B they are proportional to the path loss
exponent
• And so on..

110. Walfish – ikegami Model
• These models are more suitable for the
microcells and small macro cells regions.
• In this model the path loss consists of the
various parameters like
– Free space path loss Plo
– Multi screen Loss Lmsd
– Attenuation from roof tops Lrts

111. • The figure shows the modeling of walfish
model

112. The Motley – Keenan Model
• These models are especially for the indoor
systems.
• Suitable for within a home or office
environments.
• As similar to the wall fish model but in this
case the attenuation from the wall and floor
are also considered.
• The total path loss is expressed as

113. • This model is also called as site specific model.
• This model requires the knowledge of the
location of BS and MS and also the Building
plan.
• These models are not so accurate as they do
not include the “Go Around” Signals

114. Wide band Models
• These models are especially for the wide band
signals.
• There are various models available they are :
– Tapped delay line model
– Models for the power delay profile
– Models for the arrival time of the rays and clusters
– Standardized channel Model

115. Tapped delay line models
• They are also called as the n –Tap Rayleigh Fading
Model
• The response characteristics of this channel is given by
• For simplification purpose we analyze this model in 2
categories:
– In case 1: The number of Taps is limited to N=2. it is also
assumed that both the signals are reflected signals and
LOS is not available
– In case 2: The number of Taps is limited to N=1. it is
assumed that the received only signal is LOS signal.

116. Models for the power delay profile
• In this model the delay in power of each signal
is analyzed.
• The power delay profile is expressed as
• Based on the various environments the delay
gets varied. Some of the delay are mentioned
here.

118. Models for the arrival time of Rays and
Clusters
• In previous models the PDP are considered as
the continuous function of the delay.
• It is advantageous to describe the PDP by
arrival times.
• There are 2 models available:
– Delta K Model
– Saleh – Valenzuela Model (SV Model)

119. Delta – K Model
• This model consists of two states S1 and S2.
• When ever the power is received the
transition occurs from S1 to S2 for limited
period.
• On uniform reception of signal the system
stays at S2.
• If there is no signal received for a particular
period, then the transition occurs from S2 to
S1

120. SV Model
• This model have slighter different model than
the other models
• This model previously assumes that the
cluster of rays are available. They are
distributed over the Poisson Distribution.
• Now when ever the signal is received they are
distributed over the Poisson distribution with
the difference in time interval.

121. Standardized Model
• This model predicts the standard decay of the
signal over various regions.
– Typical urban
– Bad urban
– Rural area
– Hilly terrain

123. END
OF
CHANNEL MODELS
&
END OF UNIT - 2