Lecture 9,10: Beam formingTransmit diversity

Aliazam Abbasfar

Outline

Beam forming

Transmit Diversity

Space-time codes

Diversity gain – Power gainSNR = ‖h‖2 SNRavg = L SNRavg ‖h‖2/L

Diversity gain ‖h‖2/L E[ ‖h‖2/L ] 1 Less likely to fade deeply

Power gain : L SNRavg

Array gain

Antenna array Antenna arrays

Combining waves linearly (TX or RX) Radiation pattern Gain = maximum radiation pattern / isotropic radiation

intensity Main lobe and side lobes Beam width

3-db (Half power) beam-width

Changes angular radiation intensity Total radiated power is constant

Side lobes cannot be ignored

Phased array Combine phase-shifted signals

Beam formingChoice of coefficients makes the beam

Direction of the beam can changeAdaptive beam formingTracking

Coefficients can be chosen to null out interferenceAntenna pattern has nulls at certain

directions

Array transfer function φm= 2πD(m-1)sin()/

If Gm = exp(-jφm) , the beam point to the direction Unity gain and linear phase shift (linear phased array)

Array is a spatial filter Combine arriving signals with different weights Place nulls in the direction of interfering signals

M-antenna array can place M-1 nulls in the beam pattern

M

1m

jφm

M

1m

jφm

tj

m

mc

e G)A(θ

e Gx(t)eRer(t)

Some Array patterns

Frequency response of array Array response is a function of direction and frequency

(dependence) A phased array has a bandwidth

Nulls have limited bandwidth too

The bandwidth can be increased using delay lines instead of phase shifters Using filters in each branch

Multi-beam formingA single array can be used to form

different beams for two signalsSuperposition lawTX or RX

Adaptive beam formingThe direction of main lobe or the nulls can

change by changing the weights

Smart antenna changes the weights adaptively to track a target or minimize a cost function

Minimize mean-square-error (MSE) Use adaptive filter algorithms such as LMS and RLS

Transmit diversityIf channel response is known at the TX

SC, MRC, and EGC can be usedMRC is optimum, Why?

The same diversity gainLess power gain vs RX diversity

The total power should be divided among branches

Channel response is measured in the receiver

Sent to TX using a back channel Use downlink channel response in TDD systems

Space-time codes Can we achieve TX diversity if the channel response is not

known? YES Add temporal encoding

Repetition code + antenna muxing

Coded system

2

1

2

1

2

1

2

1

2

1

2

1

n

n

h

h

y

y

n

n

h

h

0

0

y

yx

x

x

2

1

2

1

2

1

2

1

2

1

2

1

2

1

21

21

2

1

n

n

)cos(h

)sin(h-

)sin(h

)cos(h

y

y

n

n

h

h

)cos()sin(0

0)sin()cos(

y

y

x

x

xx

xx

Alamouti scheme Very simple 2-antenna transmit diversity

No rate reduction

T1 T2 Antenna 1 : x1 x2* Antenna 2 : x2 -x1*

Encodes x1 and x2 into two orthogonal vectors Decouples data detections

SNR = ‖h‖2 SNR0/2 Similar to MRC (half power gain) Full diversity order (2)

Extension to more antennas (OSTBC) Real symbols ( for any MT) Lower rate ( ¾ for MT=3,4, ½ for any MT)

2

1

2

1*1

2*2

1*2

1

2

1

2

1*1

*2

21

2

1

n

n

h-

h

h

h

y

y

n

n

h

h

y

y

x

x

xx

xx

Pair-wise error probability (PEP)

P( XA XB | h) = Q (‖(XA-XB)h‖ /2n)

L

l ll

L

llhQ

U

UU

QQ

12

2

1

2

e

BABA

ABBABAe

4/SNR1

1

2

SNRP

~*)(*)(

~*

~

2

SNR)(*)(*

2

SNRP

hh

XX XX

h hh XX XXh

Space-time code designDiversity order (L)

Rank criterion : L = min [rank(XA-XB)] L <= N and MT

Shortest error event in coded systems

Full diversity order (L = MT)

Coding gain Determinant criterion Squared distance products in coded systems

])X(X*)X(Xdet[SNR

4

SNR

4P

BABAL

L

12e

L

l l

MIMO diversityIf there are MR antenna at the RX

P( XA XB | H) = Q (‖(XA-XB)H‖ /2n)

Diversity order = L MR <= MT MR Alamouti scheme = 2 MR

RR M

BABAL

LM

12e ])X(X*)X(Xdet[SNR

4

SNR

4P

L

l l

Ch. 7 GoldsmithCh. 3.3 Tse

Reading