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Diversity techniquesfor

flat fading channels

BER vs. SNR in a flat fading channel

Different kinds of diversity techniques

Selection diversity performance Ma imum Ratio !om"ining performance

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BER vs. SNR in a flat fading channel

#n a flat fading channel $or narro%"and system&' the !#R$channel impulse response& reduces to a single impulse

scaled "y a time(varying comple coefficient.)he received $equivalent lo%pass& signal is of the form

*roakis' +rd Ed. ,-(+

( ) ( ) ( ) ( ) ( ) j t r t a t e s t n t = +

e assume that the phase changes /slo%ly0 and can "eperfectly tracked

12 important for coherent detection

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BER vs. SNR $cont.&

e assume3

the time(variant comple channel coefficient changesslo%ly $12 constant during a sym"ol interval&

the channel coefficient magnitude $1 attenuation

factor& a is a Rayleigh distri"uted random varia"lecoherent detection of a "inary *S4 signal $assumingideal phase synchroni5ation&

6et us define instantaneous SNR and average SNR 3

{ }2 20 0 0b ba E N E a E N = =

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BER vs. SNR $cont.&

Since

using

%e get

( ){ }

{ }2 22

20,

a E aa p a e a

E a

=

( ) ( ) p a

pd da

=

( ) 00

1 0 . p e

=

Rayleigh distri"ution

E ponential distri"ution

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BER vs. SNR $cont.&

)he average "it error pro"a"ility is

%here the "it error pro"a"ility for a certain value of a is

e thus get

#mportant formulafor o"taining

statistical average

#mportant formulafor o"taining

statistical average( ) ( )0

e e P P p d

=

( ) ( ) ( )2 02 2 .e b P Q a E N Q = =

( ) 0 00 00

1 12 1 .

2 1e P Q e d

= = +

7(*S4

7(*S4

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BER vs. SNR $cont.&

8ppro imation for large values of average SNR is o"tainedin the follo%ing %ay. 9irst' %e %rite

0

0 0

1 1 11 1 1

2 1 2 1e P

= = + + +

)hen' %e use

%hich leads to

1 1 2 x x+ = + +K

0 01 4 .e P for large

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SNR

BER

9requency(selective channel$no equali5ation&

9lat fading channel

8 :Nchannel

$no fading&

9requency(selective channel$equali5ation or Rake receiver&

/BER floor0

BER vs. SNR $cont.&

01 4e P

( )e P =

means a straight line in log;log scale

0( ) =

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BER vs. SNR' summary

Modulation

7(*S4

D*S4

7(9S4$coh.&

7(9S4$non(c.&

( )e P e P 0( )e P for large

( )2Q

2e

( )Q

2 2e

0

0

11

2 1

+ 01 4

01 2

01 2

01

( )01 2 2 +

0

0

11

2 2

+

( )01 2 +

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Better performance through diversity

Diversity the receiver is provided %ith multiple copiesof the transmitted signal. )he multiple signal copiesshould e perience uncorrelated fading in the channel.

#n this case the pro"a"ility that all signal copies fade

simultaneously is reduced dramatically %ith respect tothe pro"a"ility that a single copy e periences a fade.

8s a rough rule3

0

1e L

P

is proportional to

BERBER 8verage SNR8verage SNR

Diversity ofL3th order

Diversity ofL3th order

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Different kinds of diversity methods

Space diversity: Several receiving antennas spaced sufficiently far apart$spatial separation should "e sufficently large to reducecorrelation "et%een diversity "ranches' e.g. 2 ,< &.

Time diversity: )ransmission of same signal sequence at different times$time separation should "e larger than the coherencetime of the channel&.

Frequency diversity: )ransmission of same signal at different frequencies$frequency separation should "e larger than thecoherence "and%idth of the channel&.

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Diversity methods $cont.&

Polarization diversity: =nly t%o diversity "ranches are availa"le. Not %idelyused.

Multipath diversity:Signal replicas received at different delays$R84E receiver in !DM8&

Signal replicas received via different angles ofarrival $directional antennas at the receiver&

Equali5ation in a )DM;)DM8 system providessimilar performance as multipath diversity.

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Selection diversity vs. signal com"ining

Selection diversity: Signal %ith "est quality is selected.

Equal Gain om!ining "EG #Signal copies are com"ined coherently3

Ma$imum %atio om!ining "M% & !est S'% is achieved#Signal copies are %eighted and com"ined coherently3

1 1

i i L L j j

EGC i ii i

Z a e e a

= == =

2

1 1

i i L L

j j MRC i i i

i i

Z a e a e a

= == =

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Selection diversity performance

e assume3$a & uncorrelated fading in diversity "ranches$! & fading in i 3th "ranch is Rayleigh distri"uted$c & 12 SNR is e ponentially distri"uted3

( ) 00

1, 0 .ii i p e

=

*ro"a"ility that SNR in "ranch i is less than threshold y 3

( ) ( ) 00

1 . y

yi i i P y p d e

< = = !D9!D9

*D9*D9

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Selection diversity $cont.&

*ro"a"ility that SNR in every "ranch $i.e. all L "ranches&is less than threshold y 3

( ) ( ) 01 20

, , ... , 1 .

L y L y

L i i P y p d e

< = =

( ) ( ) ( ) ( )1 2 1 2, , , . L L p p p p =K K

Note3 this is true only if the fading in different "ranches isindependent $and thus uncorrelated& and %e can %rite

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Differentiating the cdf $cumulative distri"ution function&%ith respect to y gives the pdf

( )0

01

0

1 y

L y e p y L e

=

%hich can "e inserted into the e pression for average "iterror pro"a"ility

( ) ( )0

.e e P P y p y dy

=

)he mathematics is unfortunately quite tedious ...

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> "ut as a general rule' for large it can "e sho%n that

0

1e L P

is proportional to

regardless of modulation scheme $7(*S4' D*S4' 7(9S4&.

0

)he largest diversity gain is o"tained %hen moving fromL 1 , to L 1 7. )he relative increase in diversity gain

"ecomes smaller and smaller %hen L is further increased.)his "ehaviour is typical for all diversity techniques.

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SNR

BER

9lat fading channel' Rayleigh fading'

L 1 ,8 :Nchannel

$no fading&

( )e P =

0( ) =

BER vs. SNR $diversity effect&

L 1 7L 1 - L 1 +

9or a quantitative picture $relatedto Ma imum Ratio !om"ining&'

see *roakis' +rd Ed.' 9ig. ,-(-(7

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MR! performance

Rayleigh fading 12 SNR in i 3th diversity "ranch is

( )2 2 20 0

b bi i i i

E E a x y

N N = = +

:aussian distri"utedquadrature components

:aussian distri"utedquadrature componentsRayleigh distri"uted magnitudeRayleigh distri"uted magnitude

#n case of L uncorrelated "ranches %ith same fadingstatistics' the MR! output SNR is

( ) ( )2 2 2 2 2 2 21 2 1 10 0

b b L L L

E E a a a x y x y

N N = + + = + + +K K

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MR! performance $cont.&

)he pdf of follo%s the chi(square distri!ution %ith 7 L degrees of freedom

( )( ) ( )

1 1

0 0 1 !o o

L L

L L p e e

L L

= =

Reduces to e ponential pdf %hen L 1 ,Reduces to e ponential pdf %hen L 1 ,

1

0

11 12 2

L k L

ek

L k P k

=

+ + =

9or 7(*S4' the average BER is ( ) ( )0

e e P P p d

=

( ) ( )2e P Q =( )0 01 = +

:amma function 9actorial

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9or large values of average SNR this e pression can "eappro imated "y

0

2 11

4

L

e

L P

L

=

%hich again is according to the general rule

0

1.e L P is proportional to

*roakis' +rd Ed.,-(-(,

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)he second term in the BER e pression does notincrease dramatically %ith L3

( )( )2 1 2 1 !

1 1! 1 !

L L L L L L = = =

3 2

10 3

35 4

L

L

L

= == =

= =

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BER vs. SNR for MR!' summary

Modulation

7(*S4

D*S4

7(9S4$coh.&

7(9S4$non(c.&

( )e P 0( )e P for large

( )2Q

( )Q

0

2 11 L

e

L P

Lk =

0 9or large

4k =2k =

2k =1k =

*roakis +rd Ed.,-(-(,

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hy is MR! optimum peformance?

6et us investigate the performance of a signal com"iningmethod in general using ar"itrary %eightingcoefficients .

Signal magnitude and noise energy;"it at the output of the

com"ining circuit3

1

L

i ii

Z g a=

= 201

L

t ii

N N g =

= SNR after com"ining3

( )2

2

20

b i ib

t i

E g a Z E N N g

= =

i g

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8pplying the Sch%ar5 inequality

it can "e easily sho%n that in case of equality %e musthave %hich in fact is the definition of MR!.

)hus for MR! the follo%ing important rule applies $therule also applies to S#R 1 Signal(to(#nterference Ratio&3

( ) 2 2 2i i i i g a g a

i i g a=

1

Li

i

=

= =utput SNR or S#R 1 sum of"ranch SNR or S#R values

hy is MR! optimum peformance? $cont.&

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Matched filter 1 @full(scale@ MR!

6et us consider a single sym"ol in a narro%"and system$%ithout #S#&. #f the sampled sym"ol %aveform "eforematched filtering consists of L)* samples

the impulse response of the matched filter also consistsof L)* samples

, 0,1, 2, ,k r k L= K

*k L k h r =

and the output from the matched filter is

2

0 0 0

L L L

k L k L k L k k k k k

Z h r r r r = = =

= = = MR! AMR! A

Definition of matched filterDefinition of matched filter

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Matched filter 1 MR! $cont.&

0r Lr )) )) ))

*0 Lh r = 1h 1 Lh Lh

2

0

L

k k

Z r =

=

)he discrete(time $sampled& matched filter can "epresented as a transversal 9#R filter3

Z

12 MR! including all L ,values of k r

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7_227_2005