Short notes about "Multi-user Radio Communications" part 2

Short Notes about
“Multiuser Radio Communications”
Part Two
Presented by:
Eng. Mohamed Mohy-El Din Shaheen
E-Mail; mohamedmohy24@gmail.com
Teaching Assistant, Dept. of Electrical and
Computer Engineering,
Higher Technological Institute,
Egypt

CONTENTS
8.5- Wireless Communication.
8.5.1 Propagation Effects.
8.6- Statistical Characterization of Multipath
Channels.
8.6.1- Delay Spread and Doppler Spread.
8.6.2- Classification of Multipath Channels.

8.5- WIRELESS COMMUNICATION
 In this section we study “Wireless
Communications”,
 Which is synchronous with mobile radio,
 Where a radio transmitter or receiver is
capable of being moved as shown in Fig
8.24.
 We have two important factors:
1. Median Signal Strength:
 Which predicts the minimum power,
 Needed to radiate from the transmitter,
 To provide an acceptable of coverage
service area.
2. Signal Variability:
 Characterizes the fading nature of the
channel.
Fig 8.24 Mobile Radio
Communication [14].

 A model for the Cellular
Radio System is shown in
Fig 8.25.
 It consists of an array of
hexagonal cells ,
 With a Base Station located
at the center of each cell
 Typical cell has a radius 1 to
12 [miles].
 The Base Station:
 Acts as an interface between
mobile subscriber and
cellular radio system.
 Is connected to a switching
center by wire lines.
Fig 8.25 Model of Cellular
Radio [15].

 Mobile Switching Center has two
important roles:
A. Acts as the interface between the
“Cellular Radio System” and the
“Public Switched Telephone
Network”.
B. It performs overall control of the
mobile communication.
 “MSC” performs “Hand-Off “operation
as shown in Fig 8.26,
 By monitoring (SNR) of a call in
progress as measured in the “Base
Station”.
 When the (SNR) falls below a
prescribed threshold,
 “MSC” switches the call to another
“Base Station”.
Fig 8.26 Hand-Off Operation [16].

 The Cellular Concept relies on Two
Essential Features;
1. Frequency Reuse
 Is the use of Radio channels on the
same carrier frequency ,
 To cover different areas,
 Which are physically separated from
each other,
 To prevent “Co-Channel Interference”
as shown in Fig 8.27.
 Frequency Reuse Objectives:
 Keep the transmitted power from
each Base Station to a minimum.
 Position the antennas of the Base
stations just high enough to provide
for the area coverage of the
respective cells.
Fig 8.27 Frequency Reuse
Concept [17].

2. Cell Splitting:
 Is used to handle the additional
growth in traffic within the particular
cell.
 A single cell can contain a number
of microcells as shown in Fig 8.28,
 Which have a smaller radius than
the original cells,
 The new Base stations have lower
transmitted power and antenna
height.
 And the same set of frequencies
are used with a new plan.
Fig 8.28 Cell Splitting Concept [18].

 Locating CO-Channel
Cells
 To find the nearest CO-Channel
neighbors,
 Do the following;
a) Move 𝑖 = 2 cells in the 𝑈
direction.
b) Turn 600
counterclockwise
and move 𝑗 = 1 cells in the
𝑉 direction as shown in Fig
8.29.
 In North America the band of
cellular system is (800-900)
MHz.
 This high frequency band
provides,
 A good portable coverage by
penetrating buildings.
Fig 8.29 Illustrating the
Determination of CO-Channel
Cells [19].

Fig 8.30: A signal from a base station can
travel either by a line-of- sight path (LOS) or
by non-line-of-sight (NLOS) paths [20].
 One of Propagation problems
in built-up areas is;
 There is no line of sight path
to the Base Station.
 Radio propagation takes
place by way of scattering,
 From the surfaces of the
surrounding buildings as
shown in Fig 8.30.

 Multipath Phenomenon:
 Means that the various incoming radio
waves reach their destination,
 From different directions and with
different time delays.
 Consider a static multipath
environment;
 Involving a stationary receiver and a
transmitted sinusoidal carrier signal.
 The effect of the different time delay,
 Is to introduce a relative phase shift
between the two components of the
received signal.
 If the phase shift is zero, then the two
components add constructively as in
Fig 8.31.
 If the phase shift is 180 degrees, then
the two components add destructively
Fig 8.31 Constructive and
Destructive forms of Multipath
Phenomenon for Sinusoidal
Signals[21].

 We may use “Phasors” to
demonstrate,
 The Constructive effects of
Multipath as shown in Fig
8. 32 a.
 And the Destructive effects
of Multipath as shown in
Fig 8.32 b.
(a) (b
)Fig 8.32 Phasor Representations of (a)
Constructive and (b) Destructive Forms of

 Consider a dynamic
multipath environment,
 In which the receiver is in
motion,
 The direct transmission
signal and the reflected
signal ,
 Reach the receiver via paths
of different lengths.
 Due to the motion of the
receiver,
 There is a continuous
change in the length of,
 Each propagation path.
 Hence, the Phase Shift
between the direct
transmission signal and the
reflected signal,
 Is a function of spatial
 Signal Fading:
 Fig 8.33 shows that, there is,
 Constructive Addition at some locations.
 And almost complete cancellation at some
other locations
Fig 8.33 Illustrating how the Envelope Fades as Two
Incoming Signals Combine with Different Phases.

 Fig 8.34 displays the
Fading nature of the
Received Signal.
 The received signal
Envelope is measured in
𝑑𝐵𝑚 .
 The unit 𝑑𝐵𝑚 is
defined as;
 10 𝑙𝑜𝑔10
𝑃
𝑃𝑜
 Where;
 𝑃 denotes the power
being measured.
 𝑃𝑜 = 1 𝑚𝑖𝑙𝑖𝑤𝑎𝑡𝑡 . Fig 8.34 Experimental Record of Received
Signal Envelope in an Urban Area.

Fig 8.35 Illustration the Calculation
of Doppler Shift [22].
 Consider the situation in Fig 8.35;
 The receiver is moving along the line
𝑿𝒀 .
 With a velocity 𝒗 .
 The received signal is due to a radio
wave from a scatter 𝑺 .
 We may write the following equation;
 Where;
 ∆𝑙 is the incremental change in the
path length of the radio wave.
 𝑑 is movement of the receiver from
point 𝑋 to point 𝑌 .
 𝛼 is the spatial angle between the
incoming radio wave and the direction
of motion of the receiver.
∆𝒍 = 𝒅 𝐜𝐨𝐬 𝜶
= −𝒗 ∆𝒕 cos 𝜶
(8.34)
 𝑣 is constant velocity.
 ∆𝑡 is time taken for the receiver to
move from point 𝑋 to point 𝑌 .

8.5.1. Propagation Effects
 We may write the following equation;
 Where;
 ∆∅ is the change in the phase angle
of the received signal at point 𝑌 with
respect to that at point 𝑋 .
 𝜆 is the radio wavelength.
 ∆𝑙 is the incremental change in the
path length of the radio wave.
move from point (𝑋) to point (𝑌).
∆∅ =
𝟐𝝅
𝝀
∆𝒍
= −
𝟐𝝅𝒗 ∆𝒕
𝝀
cos 𝜶
(8.35)
 𝛼 is the spatial angle
between the incoming radio
wave and the direction of
motion of the receiver.

8.5.1. Propagation Effects
 Then we can write the following
equation:
 Where;
 𝑉 is the Doppler Shift or the
Apparent Change in Frequency as
shown in Fig 8.36.
 ∆𝜙 is the change in the phase
angle of the received signal at point (
𝑌) with respect to that at point (𝑋).
move from point (𝑋) to point (𝑌).
 𝜆 is the radio wavelength.
𝑽 = −
𝟏
𝟐𝝅
∆∅
∆𝒕
=
𝒗
𝝀
𝐜𝐨𝐬 𝜶
(8.36)
 𝛼 is the spatial angle between
the incoming radio wave and
the direction of motion of the
receiver.
 Note:
 The Doppler Shift is positive,
 When the radio waves arrives
from a head of the mobile unit.
 The Doppler Shift is negative,
 When the radio waves arrives
from behind the mobile unit.
Fig 8.36 Doppler Shift Concept [23].

8.6- Statistical Characterization of
Multipath Channels

8.6- STATISTICAL CHARACTERIZATION OF
MULTIPATH CHANNELS
 We present a statistical characterization of a mobile
radio channel.
 Consider a mobile radio channel with multiple
propagation paths, then we can write;
 Where;
 𝑠 𝑡 is the Transmitted Band-Pass Signal.
 𝑠∼
𝑡 is The Complex Envelope of 𝑠 𝑡 .
 𝑓𝑐 is a Nominal Carrier Frequency.
𝒔 𝒕 = 𝑹𝒆 𝒔∼
𝒕 𝒆𝒙𝒑 𝒋𝟐𝝅𝒇 𝒄 𝒕 (8.37)

MULTIPATH CHANNELS
 The channel is time varying,
 Due to Multipath effects as shown in
Fig 8.37.
 The Impulse response of the channel is
delay dependent, then we can write;
 Where;
 ℎ 𝜏; 𝑡 is The Impulse Response of
the Channel.
 ℎ∼
𝜏, 𝑡 is The Complex Impulse
Response of the Channel and is called
“The Input Delay Spread Function of
the Channel”.
 𝜏 is Delay Variable.
 𝑓𝑐 is a Nominal Carrier Frequency.
𝒉 𝝉; 𝒕 = 𝑹𝒆 𝒉∼ 𝝉; 𝒕 𝒆𝒙𝒑 𝒋𝟐𝝅𝒇 𝒄 𝒕 (8.38)
Fig 8.37 Channel Impulse Response
[24]
 The Filtering Nature of the Channel is
caused by;
 The Summation of Amplitudes and Delays
of the Multiple Arriving Waves at any
Instant of Time.

MULTIPATH CHANNELS
 We may use the Convolution Integral as follows;
 Where;
 𝑡 is The Complex Envelope of the Channel Output.
𝒔∼
𝒕 =
𝟏
𝟐 −∞
∞
𝒔~
𝒕 − 𝝉 𝒉∼
𝝉; 𝒕 𝒅𝝉 (8.39)

1
2
is the result of using Complex Notation.
 𝑠∼ 𝑡 is The Complex Envelope of (𝑠(𝑡) Transmitted Band-Pass Signal.).
 𝑠~
𝑡 − 𝜏 is The Complex Envelope of (𝑠(𝑡) Transmitted Band-Pass Signal.)
but at Convolution Integral.
 ℎ~ 𝜏, 𝑡 is The Complex Impulse Response of the Channel and is called
“The Input Delay Spread Function of the Channel”.

MULTIPATH CHANNELS
 We can write the following equation;
 Where;
 𝐻∼
𝑓, 𝑡 is Time Varying Transfer
Function of the Channel.
 ℎ~
𝜏; 𝑡 is The Complex Impulse
Response of the Channel and is
called “The Input Delay Spread
Function of the Channel”.
 𝑓 is Frequency Variable.
 Note:
 Transfer Function as shown in Fig
8.38:
 Phase Vs. Frequency and,
 Magnitude Vs. Frequency.
𝑯∼
𝒇; 𝒕 =
−∞
∞
𝒉~
𝝉; 𝒕 𝒆𝒙𝒑 −𝒋𝟐𝝅𝒇𝝉 𝒅𝝉 (8.40)
Fig 8.38 Transfer Function and
Impulse Response [25].

MULTIPATH CHANNELS
 We can write the following equation;
 Where;
 𝜏1, 𝑡1; 𝜏2, 𝑡2 is the Definition for the Autocorrelation Function.
𝑹 𝒉~ 𝝉 𝟏, 𝒕 𝟏; 𝝉 𝟐, 𝒕 𝟐 = 𝑬 𝒉~∗ 𝝉 𝟏; 𝒕 𝟏 𝒉~ 𝝉 𝟐, 𝒕 𝟐 (8.41)
 𝜏1 is Propagation Delay of the 1st Path involved in the
Calculation.
 𝜏2 is Propagation Delay of the 2nd Path involved in the
Calculation.
 𝑡1 time at which the output of the 1st Path is observed.
 𝑡2 time at which the output of the 2nd Path is observed.
 𝐸 is Statistical Expectation Operator.
 ℎ~
𝜏1, 𝑡1 is The Complex Impulse Response of the Channel
and is called “The Input Delay Spread Function of the Channel”.
 ℎ∼∗ 𝜏2, 𝑡2 is the Complex Conjugation of “The Input Delay
Spread Function of the Channel”.

MULTIPATH CHANNELS
 Invoking:
 Stationary in time variable 𝑡 .
 And uncorrelated scattering in time variable delay 𝜏 .
 Thus, we may reformulate equation (8.41) to be;
 Where;
 𝜏1, 𝜏2; Δ𝑡 is the Definition for the Autocorrelation Function.
 𝜏1 is Propagation Delay of the 1st Path involved in the Calculation.
 𝜏2 is Propagation Delay of the 2nd Path involved in the Calculation.
 Δ𝑡 is Difference between the observation times 𝑡1 𝑎𝑛𝑑 𝑡2 .
 ℎ~
𝜏2, 𝑡 + ∆𝑡 is The Complex Impulse Response of the Channel and is called
“The Input Delay Spread Function of the Channel”.
 ℎ~∗ 𝜏1; 𝑡 is the Complex Conjugation of “The Input Delay Spread Function of
the Channel”.
 𝛿 𝜏1 − 𝜏2 is a Delta Function.
 𝜏1, Δ𝑡 is the Multipath Autocorrelation Profile of the Channel.
𝝉 𝟏, 𝝉 𝟐; 𝜟𝒕 = 𝑬 𝒉~∗ 𝝉 𝟏; 𝒕 𝒉∼ 𝝉 𝟐; 𝒕 + ∆𝒕
= 𝝉 𝟏; ∆𝒕 𝜹 𝝉 𝟏 − 𝝉 𝟐
(8.42)

MULTIPATH CHANNELS
 Using 𝜏 in place of 𝜏1 ,
 The remaining function in equation (8.42) is redefined as:
 Where;
 𝜏; Δ𝑡 is the Multipath Autocorrelation Profile of the Channel.
 𝜏 is time variable delay.
 Δ𝑡 is Difference between the observation times 𝑡1 𝑎𝑛𝑑 𝑡2 .
 ℎ~
𝜏; 𝑡 is The Complex Impulse Response of the Channel and is
called “The Input Delay Spread Function of the Channel”.
 ℎ∼∗ 𝜏; 𝑡 + Δ𝑡 is the Complex Conjugation of “The Input Delay Spread
𝒓 𝒉~ 𝝉; 𝜟𝒕 = 𝑬 𝒉∼ 𝝉; 𝒕 𝒉∼∗ 𝝉; 𝒕 + 𝚫𝒕 (8.43)

MULTIPATH CHANNELS
 The Autocorrelation Function is
shown in Fig 8.39 of 𝐻~ 𝑓; 𝑡 is
defined by;
 Where:
 𝑅 𝐻~ 𝑓1, 𝑡1; 𝑓2, 𝑡2 is The
Autocorrelation Function.
 𝑓1, 𝑓2 are Two Frequencies in
The Spectrum of a Transmitted
Signal.
 𝑡1 time at which the output of the
1st Path is observed.
 𝑡2 time at which the output of the
2nd Path is observed.
 𝐻~ 𝑓2; 𝑡2 is Time Varying
Transfer Function.
 𝐻~∗ 𝑓1; 𝑡1 is the Complex
Conjugation of Time Varying
Transfer Function.
𝑹 𝑯∼ 𝒇 𝟏, 𝒕 𝟏; 𝒇 𝟐, 𝒕 𝟐 = 𝑬 𝑯~∗ 𝒇 𝟏; 𝒕 𝟏 𝑯~ 𝒇 𝟐; 𝒕 𝟐 (8.44)
Fig 8.39 Convolution, Cross-
Correlation and Autocorrelation

MULTIPATH CHANNELS
 We find that The Autocorrelation Functions;
 𝑅 𝐻~ 𝑓1, 𝑡1; 𝑓2, 𝑡2 and 𝑅ℎ~ 𝜏1, 𝑡1; 𝜏2, 𝑡2 ,
 Are Related by a form of Two Dimensional Fourier Transformation,
 As Follows:
 Where;
 𝑅 𝐻∼ 𝑓1, 𝑡1; 𝑓2, 𝑡2 and 𝑅ℎ∼ 𝜏1, 𝑡1; 𝜏2, 𝑡2 are The Autocorrelation
Functions.
 𝑓1, 𝑓2 are Two Frequencies in The Spectrum of a Transmitted Signal.
 𝑡1 time at which the output of the 1st Path is observed.
 𝑡2 time at which the output of the 2nd Path is observed.
 𝜏1 is Propagation Delay of the 1st Path involved in the Calculation.
 𝜏2 is Propagation Delay of the 2nd Path involved in the Calculation.
𝑹 𝑯∼ 𝒇 𝟏, 𝒕 𝟏; 𝒇 𝟐, 𝒕 𝟐 =
−∞
∞
𝑹 𝒉~ 𝝉 𝟏, 𝒕 𝟏; 𝝉 𝟐, 𝒕 𝟐 𝒆𝒙𝒑 𝒋𝟐𝝅 𝒇 𝟏 𝝉 𝟏 − 𝒇 𝟐 𝝉 𝟐 𝒅𝝉 𝟏 𝒅𝝉 𝟐 (8.45)

MULTIPATH CHANNELS
 Invoking Stationary in The Time Domain,
 We may Reformulate Equation (8.44) as:
 Where;
 𝑅 𝐻~ 𝑓1, 𝑓2; ∆𝑡 is The Autocorrelation Function.
 𝑓1, 𝑓2 are Two Frequencies in The Spectrum of a Transmitted
Signal.
 ∆𝑡 is Difference between the observation times 𝑡1 𝑎𝑛𝑑 𝑡2 .
 𝐻~
𝑓2; 𝑡 + ∆𝑡 is Time Varying Transfer Function.
 𝐻~∗ 𝑓1; 𝑡 is the Complex Conjugation of Time Varying Transfer
Function.
𝑹 𝑯∼ 𝒇 𝟏, 𝒇 𝟐; ∆𝒕 = 𝑬 𝑯∼∗ 𝒇 𝟏; 𝒕 𝑯∼ 𝒇 𝟐; 𝒕 + ∆𝒕 (8.46)

MULTIPATH CHANNELS
 If we assume stationary in Frequency Domain;
 We may write;
 Where;
 𝑅 𝐻~ 𝑓, 𝑓 + ∆𝑓; ∆𝑡 is The Autocorrelation Function.
 𝑓 is any Frequency in The Spectrum of a Transmitted Signal.
 ∆𝑓 is Difference between 𝑓1 𝑎𝑛𝑑 𝑓2 .
 𝑟 𝐻~ ∆𝑓; ∆𝑡 is Spaced Frequency Spaced Time Correlation Function of the
Channel.
 𝐻~ 𝑓 + ∆𝑓; 𝑡 + ∆𝑡 is Time Varying Transfer Function.
 𝐻~∗ 𝑓; 𝑡 is the Complex Conjugation of Time Varying Transfer Function.
𝑹 𝑯~ 𝒇, 𝒇 + ∆𝒇; ∆𝒕 = 𝒓 𝑯~ ∆𝒇; ∆𝒕
= 𝑬 𝑯~∗ 𝒇; 𝒕 𝑯~ 𝒇 + ∆𝒇; 𝒕
(8.47)

MULTIPATH CHANNELS
 Spaced Frequency Spaced Time Correlation Function of the Channel;
 Can be expressed as follows;
 Where;
Channel.
 𝑟ℎ~ 𝜏; Δ𝑡 is the Multipath Autocorrelation Profile of the Channel.
𝒓 𝑯~ ∆𝒇; ∆𝒕 =
−∞
∞
𝒓 𝒉~ 𝝉; 𝚫𝒕 𝒆𝒙𝒑 −𝒋𝟐𝝅𝝉 𝚫𝒇 𝒅𝝉 (8.48)

MULTIPATH CHANNELS
 We introduce a function 𝑆 𝜏, 𝑉
as follows;
 Where;
 𝑆 𝜏, 𝑉 is Scattering Function.
 𝑉 is the Doppler Shift or the
Apparent Change in Frequency.
 𝑟ℎ∼ 𝜏; Δ𝑡 is the Multipath
Autocorrelation Profile of the
Channel.
 Δ𝑡 is Difference between the
observation times 𝑡1 𝑎𝑛𝑑 𝑡2 .
 Scattering Environment as shown
in Fig 8.40 is defined by;
 Angles of Arrival.
 Excess delays in each path.
 Power of each path.
𝑺 𝝉, 𝑽 =
−∞
∞
𝒓 𝒉~ 𝝉; 𝚫𝒕 𝒆𝒙𝒑 −𝒋𝟐𝝅𝑽 𝚫𝒕 𝒅 𝚫𝒕 (8.49)
Fig 8.40 Scattering Environment
[27].

MULTIPATH CHANNELS
 We may write the following equation:
 Where:
 𝑟ℎ~ 𝜏; Δ𝑡 is the Multipath Autocorrelation Profile of the Channel.
 𝑆 𝜏; 𝑉 is Scattering Function.
 𝑉 is the Doppler Shift or the Apparent Change in Frequency.
𝒓 𝒉∼ 𝝉; 𝚫𝒕 =
−∞
∞
𝑺 𝝉; 𝑽 𝒆𝒙𝒑 𝒋𝟐𝝅 ∆𝒕 𝒅𝑽 (8.50)

MULTIPATH CHANNELS
 𝑆 𝜏; 𝑉 may be define in terms of 𝑟 𝐻∼ ∆𝑓; ∆𝑡 by;
 Applying a Fourier Transform with respect to ∆𝑡 and,
 An Inverse Fourier Transform with respect to ∆𝑓 as follows;
 Where;
 𝑉 is the Doppler Shift or the Apparent Change in Frequency.
Channel.
𝑺 𝝉; 𝑽 =
−∞
∞
𝒓 𝑯~ ∆𝒇; ∆𝒕 𝒆𝒙𝒑 −𝒋𝟐𝝅𝑽 ∆𝒕 𝒆𝒙𝒑 𝒋𝟐𝝅𝝉 ∆𝒇 𝒅 ∆𝒕 𝒅 ∆𝒇 (8.51)

MULTIPATH CHANNELS
 Fig 8.41 displays the
relationships between;
 𝑟ℎ~ 𝜏; ∆𝑡 “Multipath
Autocorrelation Profile”,
 𝑟 𝐻∼ ∆𝑓; ∆𝑡 “Spaced
Frequency Spaced Time
Correlation Function of
the Channel”,
 𝑆 𝜏; 𝑉 “Scattering
Function”.
Fig 8.41 Functional Relationships between
“Multipath Autocorrelation Profile”, “Spaced-
Frequency Spaced-Time Correlation Function”
and “Scattering Function”

MULTIPATH CHANNELS
 The Function 𝑆 𝜏; 𝑉 is called the “Scattering Function”
of the channel,
 Consider the transmission of a single tone 𝑓′
.
 Then the Resulting Filter output is:
 Where;
 𝑠0~ 𝑡 is The Complex Envelope of The Resulting Filter
Output.
 𝑓′
is Single Tone of Frequency (Relative to the Carrier).
 𝐻~
𝑓′
; 𝑡 is Time Varying Transfer Function.
𝒔 𝟎~ 𝒕 = 𝒆𝒙𝒑 𝒋𝟐𝝅𝒇′ 𝒕 𝑯~ 𝒇′; 𝒕 (8.52)

MULTIPATH CHANNELS
 The Autocorrelation Function of 𝑠0~ 𝑡 is;
 Where;
 𝑠0~ 𝑡 is The Complex Envelope of The Resulting Filter
Output.
 𝑠0~∗ 𝑡 is the Complex Conjugation of The Complex
Envelope of The Resulting Filter Output.
 𝑓′ is Single Tone of Frequency (Relative to the Carrier).
 𝐻~
𝑓′
; 𝑡 + ∆𝑡 is Time Varying Transfer Function.
 𝐻~∗ 𝑓′; 𝑡 is the Complex Conjugation of Time Varying
Transfer Function.
 𝑟 𝐻~ 0; ∆𝑡 is Spaced Frequency Spaced Time Correlation
Function of the Channel.
𝑬 𝒔 𝟎~∗ 𝒕 𝒔 𝟎~ 𝒕 + ∆𝒕 = 𝒆𝒙𝒑 𝒋𝟐𝝅𝒇′
∆𝒕 𝑬 𝑯~∗
𝒇′
; 𝒕 𝑯~
𝒇′
; 𝒕 + ∆𝒕
= 𝒆𝒙𝒑 𝒋𝟐𝝅𝒇′
∆𝒕 𝒓 𝑯~ 𝟎; ∆𝒕
(8.53)

MULTIPATH CHANNELS
 Putting ∆𝑓 = 0 in Equation (8.48);
 Then using Equation (8.50) we may write;
 Where;
 𝑟 𝐻~ 0; ∆𝑡 is Spaced Frequency Spaced Time Correlation Function of the
Channel.
 𝑟ℎ~ 𝜏; ∆𝑡 is Multipath Autocorrelation Profile.
 −∞
∞
𝑆 𝜏; 𝑉 𝑑𝜏 is the Power Spectral Density of the Channel Output (Expressed
as a Function of Time Delay 𝜏 and Doppler Shift 𝑉 .
 We can say that “Scattering Function” provides a Statistical Measure of the
𝒓 𝑯~ 𝟎; ∆𝒕 =
−∞
∞
𝒓 𝒉~ 𝝉; ∆𝒕 𝒅𝝉
= −∞
∞
−∞
∞
𝑺 𝝉; 𝑽 𝒅𝝉 𝒆𝒙𝒑 𝒋𝟐𝝅𝑽 ∆𝒕 𝒅𝑽
(8.54)

MULTIPATH CHANNELS
 Putting ∆𝑡 = 0 in Equation (8.43),
 We may write;
 Where;
 𝑃ℎ~ 𝜏 is “The Delay Power
Spectrum or The Multipath Intensity
Profile of The Channel”.
 𝑟ℎ~ 𝜏; 0 is “The Multipath
Autocorrelation Profile of the
Channel”.
 𝐸 is Statistical Expectation
Operator.
 ℎ~
𝜏; 𝑡 is “The Complex Impulse
Response of the Channel” and is
called “The Input Delay Spread
𝑷 𝒉~ 𝝉 = 𝒓 𝒉~ 𝝉; 𝟎 = 𝑬 𝒉~
𝝉; 𝒕 𝟐
(8.55)

MULTIPATH CHANNELS
8.6.1- Delay Spread and Doppler
Spread.
 Putting ∆𝑡 = 0 in Equation (8.50),
 Then using the 1st line of Equation (8.55),
 We may write;
𝑷 𝒉~ 𝝉 =
−∞
∞
𝑺 𝝉; 𝑽 𝒅𝑽 (8.56)
 Where:
 𝑃ℎ~ 𝜏 is “The Delay Power Spectrum or
The Multipath Intensity Profile of The
Channel”.
 𝑉 Doppler Shift.
 Equation (8.56) means that;
 The Delay Power Spectrum may be
defined in terms of The Scattering
Function,

MULTIPATH CHANNELS
 Fig 8.42 depicts a typical plot,
 Of the Power Spectral Density,
 Versus Excess Delay.
 The Excess Delay is measured
(w.r.t) The Time Delay for the
shortest echo path.
 The Power is Measured in dBm.
 The “Threshold Level” is shown in
Fig 8.41 ,
 Defines The Power Level below
which,
 the Receiver Fails to operate
Satisfactory. Fig 8.42 Example of a Power Delay
Profile for a Mobile Radio Channel.

MULTIPATH CHANNELS
 The Average Delay is defined as,
 The 1st Central Moment,
 (The Mean of) 𝑷 𝒉~ 𝝉 as shown by:
𝝉 𝒂𝒗 = 𝟎
∞
𝝉 𝑷 𝒉~ 𝝉 𝒅𝝉
𝟎
∞
𝑷 𝒉~ 𝝉 𝒅𝝉
(8.57)
 Where;
 𝜏 𝑎𝑣 is “The Average Delay”.
 𝜏 is “Time variable delay”.
 𝑃ℎ~ 𝜏 is “The Delay Power
Spectrum or The Multipath
Intensity Profile of The
Channel”.

MULTIPATH CHANNELS
 The Delay Spread is defined as follows;
𝝈 𝝉 = 𝟎
∞
𝝉 − 𝝉 𝒂𝒗
𝟐 𝑷 𝒉~ 𝝉 𝒅𝝉
𝟎
∞
𝑷 𝒉~ 𝝉 𝒅𝝉
𝟏
𝟐
(8.58)
 Where:
 𝜎𝜏 is “The Delay Spread”.
 𝜏 𝑎𝑣 is “The Average Delay”.
 𝑃ℎ~ 𝜏 is “The Delay Power Spectrum
or The Multipath Intensity Profile of The
Channel”.
 Note;
 𝐵𝑐 =
1
𝜎 𝜏
 𝐵𝑐 is “Coherence Bandwidth of the
Channel”.
Fig 8.43 Delay Spread Concept [28].
 As shown at Fig 8.43, Due to Multipath,
each symbol transmitted is received
multiple times at Receiver, creates
“echo” that its duration is “Delay

MULTIPATH CHANNELS
 We first set ∆𝑓 = 0 ,
 Which corresponds to transmission
of single tone over the channel.
 Then we get 𝑟 𝐻~ 0; ∆𝑡 .
 Hence, the Fourier Transform of
𝑟 𝐻~ 0; ∆𝑡 as follows;
𝑺 𝑯~ 𝑽 =
−∞
∞
𝒓 𝑯~ 𝟎; ∆𝒕 𝒆𝒙𝒑 −𝒋𝟐𝝅𝑽 ∆𝒕 𝒅 ∆𝒕 (8.59)
 Where;
 𝑆 𝐻~ 𝑉 is The “Doppler Spectrum of the
Channel”.
 𝑟 𝐻~ 0; ∆𝑡 is “Spaced Frequency
Spaced Time Correlation Function of the
Channel”
 𝑉 is “Doppler Shift”.
 NOTE:
 𝑆 𝐻~ 𝑉 defines The
Power Spectrum of the
Channel output,
 Expressed as a Function of
the Doppler Shift 𝑉 as
shown in Fig 8.44.
Fig 8.44 Doppler Spectrum of a
Channel Example [29]

MULTIPATH CHANNELS
 Doppler Spectrum may also be
defined,
 In terms of the Scattering Function,
 As follows;
𝑺 𝑯~ 𝑽 =
−∞
∞
𝑺 𝝉; 𝑽 𝒅𝝉 (8.60)
 Where;
 𝑆 𝐻~ 𝑉 is The “Doppler Spectrum
of the Channel”.
 𝑆 𝜏; 𝑉 is “Scattering Function”.

MULTIPATH CHANNELS
 The Doppler Spread of the
Channel,
 May be defined as follows;
𝝈 𝑽 = −∞
∞
𝑽 𝟐
𝑺 𝑯~ 𝑽 𝒅𝑽
−∞
∞
𝑺 𝑯~ 𝑽 𝒅𝑽
𝟏
𝟐
(8.61)
 Where;
 𝜎 𝑉 is “Doppler Spread of the
Channel”.
 𝑆 𝐻~ 𝑉 is The “Doppler Spectrum of
the Channel”.
 𝜏 𝑐 is “The Coherence Time of the
Channel”
 𝜏 𝑐 is Reciprocal of the Doppler
Spread.
Fig 8.45 Doppler Spread in Multipath
[30]. As shown in Fig 8.45, Due to Multipath, a
single sinusoid by base station is
received as summation of 3 sinusoids
𝑓𝑐 + 𝑓𝑑1 , 𝑓𝑐 + 𝑓𝑑2 𝑎𝑛𝑑 𝑓𝑐 + 𝑓𝑑

MULTIPATH CHANNELS
 The Average Fade Rate
is,
 Related to Doppler
Spread,
 As follows;𝒇 𝒆 = 𝟏. 𝟒𝟕𝟓 𝝈 𝑽 𝑪𝒓𝒐𝒔𝒔𝒊𝒏𝒈𝒔 𝑷𝒆𝒓 𝑺𝒆𝒄𝒐𝒏𝒅 (8.62)
 Where:
 𝑓𝑒 is “The Fade Rate of The Channel”.
 𝜎 𝑉 is “Doppler Spread of the Channel”.
 NOTE:
 The Fade Rate provides a Measure of the
Rapidity of Fading of the Channel.
 Some typical values in a mobile
environment as follows;
 𝜎𝜏 = 20𝜇𝑠 .
 𝜎 𝑉 = 40 − 80 𝐻𝑧 .

MULTIPATH CHANNELS
 The particular form of fading experienced
by,
 A Multipath channel depends on whether,
 The channel is viewed in,
 Frequency Domain or Time Domain
 If the channel is viewed in Frequency
Domain,
 Then “The Channel’s Coherence
Bandwidth” 𝑩 𝒄 is concerned.
 𝑩 𝒄 is a measure of The Transmission
Bandwidth for which Signal Distortion
across the channel becomes noticeable.
 Frequency Flat Fading;
 Occurs when 𝑩 𝒄 is large compared to the
message bandwidth.
 Frequency Flat may be said Frequency
nonselective.

MULTIPATH CHANNELS
 Frequency Selective Multipath
Channel,
 When 𝑩 𝒄 is small compared to the
bandwidth of the transmitted signal.
 If the channel is viewed in Time
Domain,
 Then “The Channel’s Coherence Time”
𝝈 𝝉 is concerned.
 𝝈 𝝉 is a measure of the Transmitted
Signal Duration for which Signal
Distortion across the channel
becomes noticeable.
 Time Selective Fading;
 Occurs when 𝝈 𝝉 is small compared
to the Duration of the Received
Signal.
 Time Flat Fading;
 Occurs when 𝝈 𝝉 is large compared

MULTIPATH CHANNELS
 Figure 8.46 shows The Classification
of Multipath Channels as follows;
1) Flat-Flat Channel;
 Means no significant variation in both
Frequency and Time.
2) Frequency Flat Channel;
 Means no significant variation in
Frequency only.
3) Time-Flat Channel;
 Means no significant variation in Time
only.
4) Non-Flat Channel;
 Means there is significant variation in
both Frequency and Time.
 The forbidden area shaded in Fig
8.46,
 Follows from the inverse between
Bandwidth and Time Duration.
Fig 8.46 The Four Classes of Multipath
Channels.

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ve%20and%20destructive%20of%20multipath%20image&ved=0ahUKEwissOaJy5XYAhWKmLQKHYYjAWM4ZBAzCCEoHzAf&iact=mrc
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dingchannels-12965573995327-phpapp01%2F95%2Fintroduction-to-wireless-fading-channels-34-
728.jpg%3Fcb%3D1447407100&imgrefurl=https%3A%2F%2Fwww.slideshare.net%2Fnitin_jain_india%2Fintroducti
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channels&docid=8YzzW36pC212EM&tbnid=owEi1qNTRQ5miM%3A&vet=10ahUKEwi0x7DFmafYAhWRYlAKHUlo
DvMQMwg-
KAEwAQ..i&w=728&h=546&bih=602&biw=1242&q=Doppler%20Spread%20of%20the%20Channel%20image&ved=
0ahUKEwi0x7DFmafYAhWRYlAKHUloDvMQMwg-KAEwAQ&iact=mrc&uact=8

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