Lecture 9,10: Beam forming Transmit diversity Aliazam Abbasfar.
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Transcript of Lecture 9,10: Beam forming Transmit diversity Aliazam Abbasfar.
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
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