Post on 25-Apr-2020
transcript
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