1

Diversity & EqualizerDiversity & Equalizer

Ref.: S. Saunders, Antennas & Propagation, ch.15-16.

R. B. Wu

Diversity Channel Model� In N independently fading channels, if outrage probability

is p then probability of losing communications is pN. For example, p=10%, N=3, then pN =0.1%.

2

R. B. Wu

� Two criteria� Fading in individual branches has low cross-correlation.� Mean power available from each branch is almost equal.

� Correlation coefficients between two branches

� Mean power in channel i

� Design for good diversity: to obtain channels with low correlation coefficients and high mean power.

Criteria for Useful Branches

( )( )[ ]21

*2211

12 σσµαµαρ −−= E

[ ]2

2i

iE

Pα

=

R. B. Wu

Space Diversity

( )

∫

∑∑

=

=

=

=

=

=

−=

π

θ

φ

θθθρ

α

α

θφ

2

012

12

11

)sinexp()()(

scatterers eduncorrelat Assuming

,scatterers of presenceIn sinexp

antennasbetween difference phase

djkdpd

er

r

jkd

s i

s

n

ij

i

n

i i

Environments with significant scatterers widely spread around the antenna will result in good space diversity.

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R. B. Wu

Mobile Station Space Diversity

=

λπρ

θdJd

p2)(

)( ddistribute uniform Assuming

012

A reasonable d for horizontal antenna spacing is ~0.5λ. Due to correlation & mutual coupling, it can even be ~0.1λ.

For vertical separation, wave spreads in small extent. The spacing should be larger than the horizontal separation.

d/λ

R. B. Wu

Base Station Space Diversity� Spacings required are much greater than mobile case,

particularly when mobile is not from broadside direction.� Horizontal space diversity is commonly applied, while

vertical spacing is rarely used.

−

⋅

= θ

λπθ

λπρ 2

2

0012 sin431cos2)(

rrdJ

rrdJd ss

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R. B. Wu

Polarization Diversity� Reflection & diffraction are polarization sensitive.

� Two polarizations are uncorrelated, but cross-polarized component has considerably lower power than the co-polar.

� Cross-polar ratio Γ=E[|Ev|2]/E[|Eh|2] ~ 6dB in macrocell or ~7.4dB in microcell in 900MHz band & ρ ~ 0.1.

� Polarization diversity are almost as high as space diversity in areas with reasonable scattering, while in open areas where scattering is small, space diversity is more reliable.

� Mixed scheme with antennas spatially separated & differently polarized ! ),()( hrραρρ ×=

R. B. Wu

Other Diversity Methods� Time diversity

� Transmitting same signals multiple times, spaced apart in time sufficiently that channel fading is decorrelated.

� Seldom used in practice since it reduces system capacity and introduces a transmission delay.

� The principle is applied to improving efficiency in coded modulation scheme.

� Frequency diversity� Two frequency components spaced wider than the coherence

bandwidth experience uncorrelated fading in wideband channels.

� The principle is implicitly employed in some forms of equaliser.

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R. B. Wu

Selection Combining

Mean power increase required to decrease fade probability by one decade is now (10/N), compared to 10dB as N=1.

NsN

sNrsepP

Γ≈−==< Γ− γγγγγ γ )1(),,,(

:channelRayleigh of branchest independen NFor

fade21 L

Γ<<sγ

R. B. Wu

Diversity Gain in Selection Combining� Diversity gain is reduced if the power in two branches is

unequal.� Significant benefit is obtained even for large power rations.

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R. B. Wu

Switched Combining� �Switch & stay� policy with only one receiver.� The threshold has to be set in relation to the mean power

on each branch.

R. B. Wu

Equal-Gain Combining

)()(

)()(21

2211

2121

2211

θθ

θθθθ

jj

jjjj

enenrrsenesrenesry

−−

−−

+++=

+++=

instantaneous SNR: ( ) ( )

( ) gain 3dB ), (if 22

4

211212121

221

2

21212

2121 21

γγγγγγγ

γ θθ

==++=

+=

++= −−

Njj

c PrrenenErr

Improvement over selection & switching combining

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R. B. Wu

Maximum Ratio CombiningChoosing the branch weights to

achieve maximum SNR

( )

+=

+=

=

∑∑

∑

==

=

N

iii

N

ii

N

N

iiii

N

ii

nP

wny

Pw

11

2

1

*

1

;

αα

α

α

overall SNR:

∑∑∑∑∑=====

=

=

=

N

ii

N

iiN

N

ii

N

iii

N

iic rPrnEr

11

22

1

22

1

*21

2

1

221 γαγ

R. B. Wu

Effect of Branch Correlation

Fading probability for two correlated Rayleigh branches

BER for BPSK when using max. ratio combining in Rayleigh channel

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R. B. Wu

Comparison of Combining Methods

� For two uncorrelated, equal mean power Rayleigh channels� Easy to achieve gains equivalent to > 10dB power savings.

R. B. Wu

Introduction to Equalizers

b=[b0, b1, �, bm-1]: a sequence of m binary bitsDiscrete-time system model

kj

jkjkk

D

Dj jkjkkkk

nuhhuy

uhsnsy

++=

=+=

∑∑

≠−

−= −

00

;

InterSysmbol Interference

)( 0 kTtyyk +=

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R. B. Wu

First Nyquist Criterion

kkTh δ=)(Requirement for zero ISI

THHn T

na =+≡⇔ ∑∞

−∞=)()( 2πωω

( )∫ ∑

∑ ∫∫−∞

−∞=

∞

−∞=

+ −∞

∞−

−

⋅+=

===

T kTjn T

n

n

Tn

Tn

kTjkTjk

deH

deHdeHkThh

/2

02

/)1(2

/2

)(21

)(21)(

21)(

π ωπ

π

π

ωω

ωωπ

ωωπ

ωωπ

Constant)( =⋅=∴ ∑∞

−∞=kTkj

ka ehTH ωω

Total folded spectrumSpectral density

fπω 2=T/2π T/4πT/2π−T/4π− 0

R. B. Wu

Zero-forcing Equalizer

TTHC

ycu

a

M

Mi ikik

πωωω ≤=⇒

= ∑ −= −

for )()( ISI zero

;�

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R. B. Wu

Least Mean Squares Equalizer� Noise enhancement problem by zero-forcing equalizer

� LMS equalizer

[ ]

[ ] [ ][ ]T

Mkkk

kyuH

yy

yuyy

M

Mi ikikkk

yyyk

ukEkkE

J

ycuEuuEJ

21

*

1

22

,,,)(

;)( ,)()(

; minimize

;�

−−

−

−= −

=

==

=⇒

−=−= ∑

Ly

yryyR

rRc

EQ

R. B. Wu

Adaptive Equalizer

Training sequence

Iterative LMS algorithm

kkkkkkk uuee � ;*1 −=+=+ ycc µ

parameter size step :µ

Peramble

MidambleBurst duration

Training sequenceInformation bits

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R. B. Wu

DS-CDMA System

R. B. Wu

Rake Receiver

Max

imum

ratio

co

mbi

ning

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R. B. Wu

Conclusions� Diversity is powerful to deal with signal fading, if the

branches has low cross-correlation and similar power� Gains are achieved at the expense of extra hardware, e.g.,

extra antennas and receivers. � Optimum approach depends on mechanism & geometry of

the multipath scattering which produces fading.

� Equalizer structures can overcome the negative effects of wideband channel and improve the performance.

� Appropriate choice is compromise between computational complexity & performance.