Diversity

Wha Sook Jeon Mobile Computing and Communications Lab.

1

Introduction (1)

The idea behind diversity is to send the same data over independent fading paths

Macro-diversity − Diversity to mitigate the effects of shadowing − is generally implemented by combining signals received by several

base stations or access points − requires coordination among the different base stations, which is

implemented as a part of networking protocols in infrastructure-based wireless networks

2

Introduction (2)

Micro-diversity − Diversity techniques that mitigate the effect of multipath fading − Space diversity: by using multiple transmit or receive antennas − Angle (or directional) diversity: with smart antennas which are

antenna array with adjustable phase at each antenna element − Frequency diversity: by transmitting the same narrowband signal at

different carrier frequencies − Path diversity: spread spectrum with RAKE receiver − Time diversity: by transmitting the same date at different time

(coding or interleaving)

3

Scope of This Chapter

We focus on space diversity Receiver Diversity

− Combining Techniques ■ Selection Combining ■ Threshold Combining ■ Maximal Ratio Combining ■ Equal Gain Combining

Transmitter Diversity − Channel known at transmitter − Channel unknown at transmitter

■ Space Time Transmit Diversity (STTD)

Receiver Diversity

5

System model for Receiver Diversity (1)

ijii ea θα −=

Co-phasing: Removal of phase through multiplication by

BNa

BEra

M

ii

M

isii

01

2

2

1

∑

∑

=

=Σ

×

=γ

Identical noise PSD N0/2 on each branch and pulse shaping such that BTs=1

6

System model for Receiver Diversity (2)

1=ir

0

1 00

2

1 0

01

2

2

1

1 NME

NN

NE

BNa

BEras

M

i

M

i

s

M

ii

M

isii

=

=×

=

∑

∑

∑

∑

=

=

=

=Σγ

00 1 NNra ii ==

Example (no fading) - -

7

Diversity Gain

With fading, the combining of multiple independent fading path leads to a more favorable distribution for

Performance of a diversity system − Average symbol error probability

■ where is a symbol error probability in AWGN channel with SNR

− Outage probability ■

Diversity Gain − Performance advantage in as a result of diversity

combining outs PP and

γγγ γ dpPP ss )()(0 Σ∫∞

=

)(γsP γ

γγγγγ

γ dppPout ∫ Σ=≤= Σ

0

00 )()(

∑γ

8

Selection Combining (1) The combiner outputs the signal on the branch with the highest SNR Cumulative distribution function (cdf) of

− For M-branch diversity with uncorrelated Rayleigh fading amplitude,

− On ith branch:

− Outage probability of the selection combiner for target

■

− pdf of : differentiating relative to

■

− Average SNR of combiner output:

■

∑γ

∏=

Σ <=<=<=Σ

M

iiM pPpP

121 )()],...,,(max[)()( γγγγγγγγγγ

,1)( iiepi

iγγ

γγ −= iePout

γγγ 01)( 0−−=

0γM

M

iout eepP i ]1[)1()()( 00

100

γγγγγγγ −−

=Σ −=−=<= ∏

0γ∑γ )( 0γoutP

γγγγγ γγ −−−−=

ΣeeMp M 1]1[)(

∑∫∫=

∞ −−−∞

Σ =−==Σ

M

i

M

ideeMdp

10

1

0

1 ]1[ )( γγγ

γγγγγ γγγγγ

The average SNR for all

branches are the same

9

Selection Combining (2)

Outage Probability in Rayleigh fading

( )Mout eP γγ 01 −−=

10

Selection Combining (3)

Average Pb of BPSK in Rayleigh fading

γγγ γ dpQ bb )()2(Σ∫

11

Threshold Combining (1)

The combiner scans each branch in sequential order and outputs the first signal whose SNR is above a given threshold

Co-phasing is not required because only one branch output is used at a time

Tγ

Switch-and-stay combining (SSC) − Once a branch is chosen,

the combiner outputs that signal as long as the SNR on that branch remains the desired threshold.

two branches

12

Threshold Combining (2) Cdf of , the SNR of the combiner output with two branches:

For Rayleigh fading of each branch with −

− Outage probability for a given

− Probability density function ■

− Average symbol (bit) error probability for DPSK:

■

∑γ

)()( : 000 γγγ γ∑= PPout

γ

≥+≤≤<

=Σ )()()(

, )()()(

T1

T

21

21

γγγγγγγγγγγ

γγγ

γγγ PPp

PPP

TT

T

.,

21 1

)(T

T)(

)(

γγγγ

γγγγγγ

γγγγγγγ

γ ≥<

+−+−−

=+−−

+−−−

Σ T

TT

eeeee

P

T

T )1)(2( )1)(1(

)(γγγγ

γγ

γγγγγ

γγγγ

γ ≥<

−−

=−−

−−

Σ eeee

PT

T

)1()1(2

1)(21

0

γγγγγγ

γ

γγγ TTT eeedpePb

−−−−∞+−

+==

Σ∫Average symbol error proba. for DPSK on AWGN

13

Maximal Ratio Combining (1)

Combiner Output SNR

−

■ Envelope of combiner output:

■ Total noise PSD: −

The goal is to choose the ai to maximize

− when −

022 Nra ii =

sM

i ii Erar ∑==

1

∑==

M

i itot NaN1 0

2 22

( )∑

∑=

=Σ == M

i i

M

i sii

tot a

EraNN

r

12

2

1

0

2 1γ

( )∑∑

∑∑

===

=Σ =≤=

M

ii

M

i

siM

i i

M

i sii

NEr

a

EraN 11 0

2

12

21

0

1 γγ ( ) ∑ ∑∑ = ==≤

M

i

M

i iiM

i ii rara1 1

222

1 since

Σγ

( )∑∑

∑∑

===

=Σ ===

M

ii

M

i

siM

i i

M

i sii

NEr

a

EraN 11 0

2

12

2

1

0

1 γγ

14

Maximal Ratio Combining (2)

Distribution of − Assume i.i.d Rayleigh fading on each branch with the same average SNR − pdf of : M-stage Erlang distribution with mean and variance

■

− Outage probability for a given

■

− Average symbol (bit) error probability for BPSK modulation

■

Σγ

0γ

γΣγ γM 2γM

0 )!1(

)(1

≥−

=−−

Σγ

γγγ

γγ

γ Mep M

M

∫ ∑=

−−

Σ −−==<=

Σ

0

01

10

0 )!1()(1)()(

γ γγγ

γγγγγγM

k

k

out kedppP

)1( where

21

12

1)()2(1

00

γγ

γγγ γ

+=Γ

Γ+

+−

Γ−

== ∑∫−

=

∞

Σ

mM

m

M

bm

mMdpQP

15

Maximal Ratio Combining (3)

Outage Probability in Rayleigh fading

16

Maximal Ratio Combining (4)

Average Pb of BPSK in Rayleigh fading

17

Equal Gain Combining Simple technique which co-phases the signal on each branch and then

combines them with equal weighting, Combiner output SNR , assuming the same noise PSD N0/2 in each branch

−

For i.i.d. Rayleigh fading with two branches having average branch SNR

− Cdf of :

− Outage Probability:

− Pdf of :

− Average bit error rate for BPSK

γ

iji e θα −=

( )21

0

1 ∑=Σ =M

i si ErMN

γ

Σγ

Σγ

Σγ

( ){ }γγγπγγ γγγγγ 2211)( 2 QeeP −−−= −−Σ

)( 0γγ∑= PPout

−

−−= −−

Σ γγ

γγ

γγγπ

γγ γγγγ

γ221 1

411)( 2 Qeep

( )

+−−== −∞

∫ Σ

2

01115.0)()2( γγγγ γ dpQPb

Transmit Diversity

19

Channel Known at Transmitter A transmit diversity system with M transmit antennas and one receive antenna

is considered

We assume that the path gain of the ith antenna is known at transmitter.

The signal is multiplied by and then sent through the ith antenna.

Because the symbol energy Es in the transmitted signal s(t) is a constant,

Received signal: The weights ai to achieve the maximum SNR:

The resulting SNR:

− When the channel gains are known at transmitter, the transmit diversity is

similar to the receiver diversity with MRC − If all antennas has the same gain ri = r, − There is an array gain of M corresponding to an M-fold increase in SNR

over a single antenna transmitting with full power

ijii ea θα −=

ijier θ

∑ ==

M

i ia1

2 1

∑==

M

i ii tsratr1

)()(

∑=

=M

i i

ii

r

ra1

2

∑∑==

Σ ==M

ii

M

ii

s rNE

11

2

0

γγ

02 NEMr s=Σγ

20

Channel Unknown at Transmitter-Alamouti Scheme

The transmitter no longer knows the channel gain − If the transmit energy is divided equally among antenna, no performance

advantage is obtained Alamouti Scheme

− This scheme is designed for a digital communication system with two antennas

− The scheme to combine both space and time diversity (STTD)

*211

*22

22

21

*211

*22

*221

*11

22

21

*221

*11

)(

)(

nhnhshhrhrhz

nhnhshhrhrhz

−++=−=

+++=+=

STTD encoder

STTD decoder

s1 s2

s1 -s2*

s2 s1*

h1

h2

r1 r2 z1 z2

2*12

*212

122111

nshshrnshshr++−=

++=

21

STTD-Alamouti Scheme

Channel estimation with known data (x1, x2)

Diversity gain of 2

Array gain of 1 − The symbols s1 and s2 are transmitted simultaneously with energy Es/2. − The received SNR for zi

12*212

22

2112

*212

22*111

22

2122

*111

)(ˆ)(ˆ

xnxnhxxxrxrh

xnxnhxxxrxrh

−++=−=

−++=−=

222

22

12

112

22

11

~)(

~)(

nshhz

nshhz

++=

++=

( )0

222

21

2 NEhh s

i ×+

=γ