EA-452 Chapter 7

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Chapter 7

Equalization, Diversity

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Topics:

1) Review of relevant concepts

2) Inter Symbol Interference

3) Definition Equalization , Diversity andChannel Coding

4) Fundamentals of Equalization

5) A Generic Adaptive Equalizer

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Review of Relevant Concepts

Fading:

1) Flat Fading

2) Frequency Selective Fading

3) Other Multi-path Concerns

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Flat Fading

Flat Fading is caused by absorbers betweenthe two antennae and is countered by

antenna placement and transmit power level.

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Frequency Selective Fading

Frequency selective fading is

caused by reflectors between the

transmitter and receiver creating

multi-path effects.

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Effects of Frequency

Selective Fading

• The dips or fades in the response due to reflection

cause cancellation of certain frequencies at the

Receiver.

• Reflections off near-by objects (e.g. ground,

buildings, trees, etc) can lead to multi-path

signals of similar signal power to the direct

signal.• This can result in deep nulls in the received

signal power due to destructive interference.

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When the waves of multi-path signals are

out of phase, reduction of the signal

strength at the receiver can occur.

Figure Explaining Multi-path

Fading

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Other Multi-path Concerns

Apart from creating frequency selective

fading,Multi-path can also cause inter-symbol

interference.

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Coherence Bandwidth

Coherence Bandwidth is a statistical measure of

the range of frequencies over which the channelcan be considered flat that is a channel which

passes all spectral components with

approximately equal gain and linear phase. It is the

range of frequencies over which the two frequencycomponents have a strong potential for amplitude

correlation.

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Definition: Delay spread (frequency-selective

fading), in which a bit arrives at the receiver at

different times because of the different paths taken,

causing bits to run into each other and thus cause

inter-symbol interference (ISI). This limits theusable digital signaling rate far a given rate. In

other words, if the modulation bandwidth exceeds

the coherence bandwidth of the radio channel Inter

Symbol Interference or ISI occurs. This causessignificant error in high bit rate systems.

Inter Symbol Interference(ISI)

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Inter Symbol Interference

Representation

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Equalization : It is the used for compensatingInter Symbol Interference. An Equalizer within a

receiver compensates for the average range of

expected channel amplitude and delay

characteristics. Equalizers must be adaptive as the

channel is generally unknown and time varying.

Definition Equalization, Diversity & Channel

Coding

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Diversity: It is the technique used to compensate for

fading channel impairments. It is implemented by

using two or more receiving antennas. While

Equalization is used to counter the effects of ISI,

Diversity is usually employed to reduce the depthand duration of the fades experienced by a receiver

in a flat fading channel. These techniques can be

employed at both base station and mobile receivers.

Spatial Diversity is the most widely used diversitytechnique.

Diversity

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Spatial Diversity Technique- A

Brief Description

In this technique multiple antennas are strategically

spaced and connected to common receiving system.While one antenna sees a signal null, one of the

other antennas may see a signal peak, and the

receiver is able to select the antenna with the best

signal at any time. The CDMA systems use Rakereceivers which provide improvement through time

diversity.

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• Unlike Equalization, Diversity requires notraining overhead as a transmitter doesn’t

require one.

• It provides significant link improvementwith little added cost.

• It exploits random nature of wave

propagation by finding independent (

uncorrelated) signal paths for

communication.

Diversity Techniques- Highlights

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• It is a very simple concept where in one path undergoes a deep fade and another

independent path may have a strong signal.

• As there is more than one path to selectfrom, both the instantaneous and average

SNRs at the receiver may be improved,

often as much as 20-30 dB

Diversity Techniques- Highlights

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Channel coding: It is the technique which

improves mobile communication link performance

by adding redundant data bits in the transmittedmessage. In this technique, the base band portion

of the transmitter, a channel coder maps a digital

message sequence into another specific containing

greater number of bits than originally contained in

the message. The coded message is then modulated

for transmission in the wireless channel.

Channel Coding

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Channel Coding is used by the receiver to detect

or correct some or all of the errors introduced

by the channel in a particular sequence of

message bits. The added coding bits lower these

the raw data transmission rate through the

channel. There are two types of codes: Block

codes and convolutional codes.

Channel Coding (Cont’d)

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ISI has been recognized as the major obstacle to high

speed data transmission over mobile radio channels.

Equalization is a technique used to combat inter symbol interference. As the mobile fading channels

are random and time varying, equalizers must track

the time varying characteristics of the mobile

channel, and thus are called adaptive equalizers.

Fundamentals of Equalization

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Operating modes of adaptive

equalizer

1) Training mode

2) Tracking Mode

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Training Mode

Initially, a known, fixed length training sequence

is sent by the transmitter so that the receiver’s

equalizer may average to a proper setting. Thetraining sequence is a pseudo random signal or a

fixed, prescribed bit pattern. Immediately

following the training sequence, the user data is

sent.

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Utility of training sequence

The training sequence is designed to permit

an equalizer at the receiver to acquire the

proper filter coefficients in the worst possible

channel conditions. Therefore when the

training sequence is finished. Therefore filter

coefficients are near their optimal values for

reception of user data. An adaptive equalizer

at the receiver uses a recursive algorithm to

evaluate the channel and estimate filter

coefficients to compensate for the channel.

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Tracking mode

When the data of the users are received, the

adaptive algorithm of the equalizer tracks the

changing channel. As a result of this, theadaptive equalizer continuously changes the

filter characteristics over time. Equalizers are

widely used in TDMA Systems.

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Example 7.3

• Consider the design of a Digital carrier cellular system

Given

• F=900 Mhz, mobile velocity = 80 km/hour

• Symbol rate = 24.3 Kbps

Determine

The maximum doppler shift

The coherence time of the channelMax # of symbols that can be transmitted withoutupdating the equalizer

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Solution Example 7.3The max doppler shift = velocity/wavelength

Wavelength = 3*10^8/900*10^6 = 0.33 metersDoppler shift = 80 *1000/36000/0.33 = 66.67 Hz

Coherence time (using the geometric mean)

= 3/(4* doppler shift) *sqrt (1/Pi) = 6.3 4millisecs

Symbols transmitted without updating the equalizer =

6.34*24.3*1000 = 154 symbols

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)()()(*)( t t ht f t g eq

The goal of equalization is to satisfy the

above equation. In the frequency domain theabove equation can be expressed as

1)(*)( f F f H eq

Where Heq(f) and F(f) are Fourier transforms of

heq(t) and f(t) respectively.

Working of an Adaptive Equalizer (Cont’d)

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Block Diagram of Adaptive Equalizer

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A Generic Adaptive Equalizer

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Diversity

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Topics

DiversitySpace Diversity

Selection Diversity

Scanning Diversity

Maximum Ratio Combining

Equal Gain Combining

Polarization Diversity

Frequency DiversityTime Diversity

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DIVERSITY

A diversity scheme is a method that is used to develop

information from several signals transmitted over

independent fading paths.

It exploits the random nature of radio propagation by

finding independent (uncorrelated) signal paths for

communication.

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Diversity Technique

Macroscopic diversity Microscopic diversity

Diversity

Objective:

Combining the multiple signals in such a fashion so as toreduce the effects of excessive deep fades.

Types:

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MACROSCOPIC DIVERSITY

• Prevents Large Scale fading.

• Large Scale fading is caused by

shadowing due to variation in boththe terrain profile and the nature of the surroundings.

Large Scale fading is log normallydistributed signal.

• This fading is prevented byselecting an antenna which is notshadowed when others are, thisallows increase in the signal-to-noise ratio.

MICROSCOPIC DIVERSITY

• Prevents Small Scale fading.

• Small Scale fading is caused by

multiple reflections from thesurroundings. It is characterizedby deep and rapid amplitudefluctuations which occur as themobile moves over distances of afew wavelength.

• This fading is prevented byselecting an antenna which givesa strong signal that mitigates thissmall signal fading effect.

Types Of DiversityDiversity Technique

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Space Diversity

Principle :

A method of transmission or reception, or

both, in which the effects of fading are

minimized by the simultaneous use of two or

more physically separated antennas, ideally

separated by one half or more wavelengths.

Space Diversity

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Space Diversity

Signals received from spatially separated antennas on the mobilewould have essentially uncorrelated envelopes for antenna

separations of one half wavelength or more.

Generalized block diagram of

space diversity.

Selection Diversity Scanning Diversity Maximal Ratio Combining Equal Gain Combining

Space Diversity

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Selection Diversity

Principle :

Selecting the best signal among all thesignals received from different braches at

the receiving end.

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Derivation of Selection Diversity

Improvement

Consider M independent Rayleigh fading channels available at receiver

Diversity branch

Assumptions:

• Each branch has the same average SNR

• Instantaneous SNR for each branch = γ i

Selection Diversity

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Selection Diversity

The Signal-to-Noise ratio is defined as follows

Eb , N0 are constants

Where

E b - Average Carrier Energy

N 0 - Noise power spectral density ά - A random variable used to represent

amplitude values of the fading channel with

respect to E b /N 0

The instantaneous SNR ( γ i ) can be defined as

2

0

N

E SNR b

Instantaneous signal power per branch

Mean noise power per branchγ

i =

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Selection Diversity (cont’d)

For Rayleigh fading channels, α has a Rayleigh

distribution and so α 2 and consequently γ i have a

chi-square distribution with two degrees of freedom.

The probability density function for such a channel is

The pdf for a single branch that has SNR less thansome threshold γ is

0 exp1

)( i

ii p

ed d p P ii

iiir 1)exp(1

)()(00

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Selection Diversity (cont’d)The probability that all M independent diversity

branches receive signals which are less than a

threshold γ is

If a signal branch achieves SNR > γ then the

probability that SNR > γ for one or more branches is

)(1)(

M

,......1

M M r P e P

M

M ir e P P

11)(-1)(

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Cumulative distribution curves for output signals from selection

diversity for various values of M

The percentage of total

time interval during

which a signal is

below any given level

is called “ outage rate”

at that level.

When M = 1

γ/ Г = 1

10 Log(γ/ Г ) = 0

Selection Diversity

%3636.0

11

1

11

e

e P r

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Determination of average signal to noise

ratio

• Find the pdf of the fading signal

• Compute the derivative of PM(γ )

The mean SNR is

Where

1)(dd)(

1-M

ee M P p M M

dxee Mx p

x x

M

0

1-M

0

1)d(

x

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Evaluating this equation the average SNR improvement using

selection can be found

M

k k 1

1

Selection Diversity

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Selection Diversity Example

Assuming four branch diversity is used, where each branch

receives an independent Rayleigh fading signal. If the average

SNR is 20 dB, determine the probability that the SNR will drop

below 10 dB. Compare this with the case of a single receiver

without diversity.

γ = 10 dB

Г = 20 dB

γ/ Г = 0.1

With Selection Diversity

Without Diversity

.00008201dB)10(41.0

4

e P

.09501dB)10( 11.0

1 e P

Selection Diversity

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Conclusion

• Selection diversity offers an average improvement in thelink margin without requiring additional transmitter power or sophisticated receiver circuitry.

• Selection diversity is easy to implement because all that

is needed is a side monitoring station and an antennaswitch at the receiver.

• However it is not an optimal diversity technique becauseit does not use all of the possible branchessimultaneously.

• In practice the SNR is measured as (S+N)/N , since it isdifficult to measure SNR.

Selection Diversity

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Feedback or Scanning Diversity

Principle :

Scanning all the signals in a fixed

sequence until the one with SNR more

than a predetermined threshold is

identified.

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Scanning Diversity Explanation

Consider M independent Rayleigh fading

channels available at receiver.

Scanning Diversity

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Conclusion

• This method is very simple to implement, requiring only

one receiver.

• The resulting fading statistics are somewhat inferior to

those obtained by the other methods.

Scanning Diversity

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Maximal Ratio Combining

Principle :

Combining all the signals in a co-

phased and weighted manner so as to

have the highest achievable SNR at the

receiver at all times.


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Derivation of Maximum Ratio Combining

Improvement

Consider M branches which are maximal ratio combined in a co-

phased and weighted manner in order to achieve high SNR


M i l R ti C bi i

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Assumptions:

• The voltage signal γi from each of the M diversity

branches are co-phased to provide coherent

voltage addition and are individually weighted to

provide optimal SNR.

• Each branch has gain Gi

• Each branch has same average noise power N


M i l R ti C bi i

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Resulting signal envelope applied to the detector is

Assuming that all amplifiers have additivenoise at their input and that the noise isuncorrelated between different amplifiers.

The total noise power NT applied to the detector is the weighted sumof the noise in each branch.

Which results in a SNR applied to the detector γM

Using Chebychev’s inequality γM is maximized when


M

i

ii M r Gr 1

M

i

iT G N N 1

2

T

M M

N

r

2

2

N r

G ii

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The Maximized value is

Now

We have The E fieldreceived envelop

The received signal envelope for a fading mobile radio signal can bemodeled from two independent Gaussian random variables Tc andTs each having zero mean and equal variance σ2 .

M

i

i

M

i

i

N

r

N r

M N

r

N i

i

11

2

21

2

21

2

2

2

222

2

1

2

1 scii T T

N r

N

i sc z r t r t T t T t E )()()()( 22

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Hence γM is a chi-square distribution of 2M Gaussian random

variable with variance

The resulting pdf for γM is

The probability that γM is less than some SNR threshold γ is

22

2 N

0for !)1(

)(1

M M

M

M M

M

e p

M

M

k

k

M M M r k

ed p P 1

1

0!1

1)(


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Determination of average signal to noise

ratio

The before equation is the probability distribution for maximal ratio

combining.

Hence the mean SNR is

M M

i

M

i

i M

11

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Principle :

Combining all the signals in a co-phased

manner with unity weights for all signal

levels so as to have the highest

achievable SNR at the receiver at all

times.

E l G i C bi i

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• In certain cases it is not convenient to provide for thevariable weighting capability.

• This allows the receiver to exploit signals that are

simultaneously received on each branch.

• The probability of producing an acceptable signal from a

number of unacceptable inputs is still retained.

• The performance is marginally inferior to maximal ratiocombining and superior to selection Diversity.

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Principle :

Polarization diversity relies on

the decorrelation of the tworeceive ports to achieve diversity gain. The two receiver ports must

remain cross-polarized.


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Effective Diversity is obtained with a Correlation

Coefficient below 0.7

In order to keep the correlation at this level

• space diversity at a base station requires antenna

spacing of up to 20 wavelengths for the broadside

case, and even more for the inline case.

• Polarization diversity at a base station does not

require antenna spacing.

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Polarization Diversity(cont’d)

– At the base station, space diversity is considerablyless practical than at the mobile because thenarrow angle of incident fields requires largeantenna spacing.

– The comparatively high cost of using spacediversity at the base station prompts theconsideration of using orthogonal polarization.

– Polarization diversity provides two diversitybranches and allows the antenna elements to beconsidered.

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In the early days of cellular radio, all subscriber units were

mounted in vehicles or used vertical whip antennas. Today,

however, over half of the subscriber units are portable. This

means that most subscribers are no longer using vertical

polarization due to hand-tilting when the portable cellular phone

is used. This recent phenomenon has sparkled interest inpolarization diversity at the base station.

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Theoretical Model for Polarization Diversity

Assuming that a signal is transmitted from a mobile stationwith vertical( or horizontal) polarization, and is received by apolarization diversity antenna with two branches at the basestation.

The measured horizontal and vertical polarization paths

between a mobile and a base station are uncorrelated.The decorrelation for the signal in each polarization is caused bymultiple reflections in the channel between the mobile and basestation antenna.

This results in signals of different amplitudes and phasereflections.

In reality there is some dependence of the received polarizationon the transmitted polarization.

Theoretical Model for Polarization

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Diversity(Cont’d)

v1 , v2 -Two antenna elements

Which make ±α angle

(polarization angle) with the Y-

axis.

A mobile station is located in

the direction of offset angle β

from the main beam direction

of the diversity antenna.

Theoretical Model for Polari ationPolarization Diversity

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Diversity(Cont’d)

Some of the vertically polarized signals transmitted from themobile station are converted to the horizontally polarized signal,because of multipath propagation.

Horizontally Polarized Component

Vertically Polarized Component

These signals x and y are received at β= 0

Assuming that r 1and r 2 have independent Rayleigh distribution,

Φ1 and Φ2 have independent, uniform distribution.No cross coupling between the diversity antenna elements isassumed.


)cos( 11 t r x

)cos( 22 t r y

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Correlation Coefficient ρ The received signal values at v1 and v2 are

Substituting x and y

Where

coscossin1 y x coscossin2 y x

t br ar t br ar sin)sinsin(cos)coscos( 221122111 t br ar t br ar sin)sinsin(cos)coscos( 221122112

cos

cossin

b

a


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(Cont’d)

Amplitudes for v1 and v2 are

Therefore,

The correlation coefficient is defined as follows

2

2211

2

22111 )sinsin()coscos( br ar br ar R

2

2211

2

22112 )sinsin()coscos( br ar br ar R

)cos(2 2121

22

2

22

11 abr r br ar R

)cos(2 2121

22

2

22

12 abr r br ar R

22

1

22

1

22

2

22

1

2

2

2

1

2

2

2

1

)()(

.

R R R R

R R R R


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(cont’d)

Assumptions

2

2

22

2

22

1

2

1 Rbr ar R

)cos(2)cos(2. 221

22

2

22

1221

22

2

22

1

2

2

2

1 abr r br ar abr r br ar R R

24

2

24

1

2

2

2

1 . br ar R R

2221

22

2

22

1

22

1 )cos(2)( abr r br ar R

22

2

222

2

2

1

44

2

44

1 4

R

bar r br ar


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Substituting the values

Because r 1 and r 2 follow Rayleigh distribution

Let then ρ will be

Where Г is the cross polarization discrimination of the propagation path betweena mobile and a base station.

222

2

2

1

42

2

2

4

2

42

2

1

4

1

222

2

2

1

42

2

2

4

2

42

2

1

4

1

2

2

bar r br r ar r

bar r br r ar r

22

2

4

2

22

1

4

1

4

4

r r

r r

2

1

2

2

r

r

2

22

222

22

22

costancostan

baba

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Polarization Diversity :Average Signal loss L

The average level of vertically polarized signal level

is

We have seen before that

Hence

The signal loss is

2

0 R

2

2

2

0 r R

22

222

221

21 Rbr ar R

2

2

2

1

r

r

22

2

2

2

1 bar

r L


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Three factors determine ρ and L

Polarization angle α

Offset angle β

Cross polarization discriminator Г

When Г= 0

At α = 500 and β= 330

ρ = 0

When Г= 0 dB =1

At α = 450 and β= 900

2

22

222

22

22

costan

costan

ba

ba

221ba L

dB L 345cos90cos45sin1

1 222

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Frequency Diversity

Principle :

The same information signal is

transmitted and received simultaneously on two

or more independent fading carrier frequencies.

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Frequency Diversity

– The rational behind this technique is thatfrequencies separated by more than thecoherence bandwidth of the channel will notexperience the same fade.

– The probability of simultaneous fade will be theproduct of the individual fading probabilities.

– This is often employed in microwave LOS linkswhich carry several channels in a frequencydivision multiplex mode(FDM).

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Frequency Diversity

– This technique not only requires spare bandwidth, butalso requires that there be as many receivers as there

are channels used for the frequency diversity.

However, for critical traffic, the expense may be

justified.

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Time Diversity

Principle :

The signals representing the

same information are sent over the same

channel at different times.

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Time Diversity

– Time Diversity repeatedly transmits

information at time spacing that exceeds the

coherence time of the channel.

– Multiple repetitions of the signal will be

received with multiple fading conditions,thereby providing for diversity.

– A modern implementation of time diversity

involves the use of RAKE receiver for spread

spectrum CDMA, where multipath channelprovides redundancy in the transmitted

message.

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Polorization Diversity

Selection Diversity Scanning Diversity Maximal Ratio Combining Equal Gain Combining

Space Diversity Frequency Diversity Time Diversity

Diversity

References

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References

• Wireless Communications- Theodore S. Rappaport.

• Mobile Communication Engineers Theory and application

– William C.Y.Lee.

• Cox, D.C., “Antenna Diversity Performance in Mitigating the effects of Portable

Radiotelephone Orientation and Multipath Propagation,”

IEEE Transactions on Communications,vol.62, No.9, pp.2695-2712, November 1983.

• Jakes, W. C., “ A Comparison of specific space Diversity Technique for

Reduction of Fast Fading in UHF Mobile Radio Systems,”

IEEE Transactions on Vehicular Technology, Vol. VT-20, No.4, pp.81-93,

November 1971.

• Lemieux, J. F., Tanany, M., and Hafez, H.M., “ Experimental Evaluation of Space/Frequency/Polarization Diversity in the Indoor Wireless Channel,”

IEEE Transactions on Vehicular Technology, Vol. 40, No.3, pp.569-574, August

1993.

References

7/30/2019 EA-452 Chapter 7


• Rappaport, T.S., and Hawbaker, D.A, “Wide band MicrowavePropagation Parameters Using Circular Frequency Reuse Efficiency for the Reverse Channel ,”

IEEE Transactions on Vehicular Technology, Vol. 40, No.2, pp.231-242, February 1992.

• Vaughan , R., “ Polarization Diversity in Mobile Communications,”

IEEE Transactions on Vehicular Technology, Vol. 39, No.3, pp.177-186, August 1990.

• Kozono , S., “ Base Station Polarization Diversity Reception for MobileRadio,”

IEEE Transactions on Vehicular Technology, Vol. VT-33, No.4, pp.301-306, November 1985.

• Lee, W.C.Y, “ Polarization Diversity System for Mobile Radio,” IEEE Transactions on Communications, Vol. 20, pp.912-922, October 1972.

EA-452 Chapter 7

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