MIMO Wireless Communications · MIMO Wireless Communications • Single Input Single Output (SISO)...
Transcript of MIMO Wireless Communications · MIMO Wireless Communications • Single Input Single Output (SISO)...
MIMO Wireless Communications:
An IntroductionAn Introduction
Dr. Rakhesh Singh Kshetrimayum
MIMO Wireless Communications
• Single Input Single Output (SISO)
Tx Rx
• What will happen if Tx and Rx employs multiple antennas?
Ant 1 Ant 1
Fig. 1 SISO system
MIMO Wireless Communications
• Multiple Input Multiple Output (MIMO)?Ant 1
Ant 1
Ant 2
. .
Tx Rx
Ant 2
Ant NT
Ant NR
.
.
.
.
.
.
Fig. 2 NT × NR MIMO system
channel
MIMO Wireless Communications
• What are advantages of MIMO?
• Capacity (C) for Single Input Single Output (SISO) system
( )SNRBWC += 1log2
• Data rate increase when
– Bandwidth (BW) and
– Signal to noise (SNR) power increase
Inherent problems:
• BW is precious, almost and always fixed for different applications
( )SNRBWC += 1log2
MIMO Wireless Communications
• Signal power increase
– battery lifetime decrease
– creates higher interference
– needs expensive RF amplifier – needs expensive RF amplifier
• In MIMO, spectral efficiency increase
– Without increasing BW and
– Signal power
MIMO Wireless Communications
• But, how?
• Basically two gains for MIMO systems:
– (i) MUX/rate gain
– Capacity behavior ( )SNRrR log≈– Capacity behavior
( )( )SNR
SNRR
SNRr
2log
lim
∞→=
( )SNRrR 2log≈
MIMO Wireless Communications• For instance
• How much is the achievable rate gain?Ant 1
Ant 1
Ant 2
Tx RxAnt 2
Ant 3
Ant 3
Fig. 3 Achievable rate gain with 3 × 3 MIMO system
P/SS/P
r=3
( )SNRR 2log3≈
MIMO Wireless Communications
(ii) Diversity gain
• Error probability behavior (minimize it)
• Slope of symbol error rate/ bit error rate (SER/BER curve
increases)
( ) d
e SNRSNRP−≈
increases)
• Behavior of error probability w.r.t. average transmit
power in log-log scale
• for asymptotically high power
( ){ }( )
2
2
loglim
log
eP SNR
dSNR SNR
= −→ ∞
MIMO Wireless Communications• How much is diversity gain?
Ant 1Ant 1
Ant 2
( ) 9−≈ SNRSNRPe
• For SISO (Rayleigh fading case), d=1,
Tx Rx
Ant 2
Ant 3Ant 3
Fig. 4 Diversity gain of 3 × 3 MIMO system
d=9
( ) 1−≈ SNRSNRPe
MIMO Wireless Communications
• How much is rate and diversity gain?Ant 1
Ant 1
Ant 2
r=1
Tx Rx
Ant 2
Ant 3
Ant 3
Fig. 5 Rate and diversity gain of 3 ×
3 MIMO system (Case I)
d=4
( )SNRR 2log≈
( ) 4−≈ SNRSNRPe
MIMO Wireless Communications
• How much is rate and diversity gain?Ant 1
Ant 1
Ant 2r=2
Tx Rx
Ant 2
Ant 3
Ant 3
Fig. 6 Rate and diversity gain of
3 × 3 MIMO system (Case II)
d=1
( )SNRR 2log2≈
( ) 1−≈ SNRSNRPe
MIMO Wireless Communications
• Need for a proper design for MIMO systems
• As r decrease, d increase
• As r increase, d decrease• As r increase, d decrease
• Diversity-multiplexing trade-off [1]
( )( ) ( )RTRTopt NNrrNrNd ,min0, ≤≤−−=
MIMO Wireless Communications
• Narrowband MIMO System ModelAnt 1
Ant 1
h12
h11
x1
y1
Tx Rx
Ant 2
Ant 2
Fig. 7 Narrowband MIMO system model for 2 × 2
MIMO system
h21
h22x2 y2
MIMO Wireless Communications
12121111 nxhxhy ++=At receiving antenna 1 (receives mixture of signals 1 &2)
At receiving antenna 2 (receives mixture of signals 1 &2,
a major problem in MIMO detection)a major problem in MIMO detection)
22221212 nxhxhy ++=
In matrix form, y=Hx+n
+
=
2
1
2
1
2221
1211
2
1
n
n
x
x
hh
hh
y
y
MIMO Wireless Communications
• For NT × NR MIMO system
+
=
T
N
N
n
n
x
x
hhh
hhh
y
y
L
L
2
1
2
1
22221
11211
2
1
+
=
RTTRRR
T
R NNNNNN
N
N n
n
x
x
hhh
hhh
y
y
MM
L
MOOM
L
M
22
21
222212
Channel matrixReceived signal
vector
Transmitted
signal vector
Noise
signal
vector
MIMO Wireless Communications
• In order to have performance analysis of
MIMO channel
– we need Analytical MIMO channel models
• Analytical MIMO channel model• Analytical MIMO channel model
– i.i.d. MIMO channel model
– Separately correlated MIMO channel model
– Uncorrelated keyhole MIMO channel model
MIMO Wireless Communications
• i.i.d. MIMO channel model (each element of
the channel matrix H is complex random
variable)
==+= ,2,1;,2,1; TR
imag
ij
real
ijij NjNijhhh LL
( ) ( )
×
−
×
=⇒
2
12
exp
2
12
1
2
1,0~
2/
/
/
imagreal
ijimagreal
ij
imagreal
ij
TRijijij
hhp
Nh
π
MIMO Wireless Communications
( )
Complex Gaussian distribution
�Joint distribution of real and imaginary part
� They are assumed independent
�PDF can be multiplied
( )
( ) ( )( ) ( )( )
( ) ( ) ( )( )( )
( ) ( )2
22
22
exp1
exp1
exp1
exp1
1,0~
ijij
imag
ij
real
ijij
imag
ij
real
ijij
Cij
hhp
hhhp
hhhp
Nh
−=⇒
+−=⇒
−−=⇒
π
π
ππ
MIMO Wireless Communications
( ) { }, ,
2 2
, ,
, 1, 1
1 1exp exp
R T R T
R T
N N N N
i j i jN Ni ji j
p h hπ π ==
= − = − ∑∏H H
( ) ( )( )1 H
For i.i.d. case,
( ) ( )( )1exp
R T
H
N Np Trace
π= −H H HH
“etr” is the abbreviation for “exponential trace”.
( ) { }HNNetrp TR HHHH −= −π
MIMO Wireless Communications
( )*
11 12 1 11 21 1
21 22 2 12 22 2
1 2 1 2
T R
T R
H
N N
N N
N N N N N N N N
trace
h h h h h h
h h h h h htrace
h h h h h h
=
HH
L L
L L
M O O M M O O M
L L1 2 1 2
22 2
11 12 1
22 2
21 22 2
2
1
R R R T T T R T
T
T
R
N N N N N N N N
N
N
N N
h h h h h h
h h h
h h htrace
h h
+ + +
+ + +=
+
L L
L L L L
L L L L
M O O M
L L L2 2
2
,2
,
, 1
R R T
R T
N N
N N
i j
i j
h
h
=
+ +
= ∑
L
MIMO Wireless Communications
• MIMO channel parallel decomposition
• To see how much is the capacity increase in MIMO systems
yx~ xy~
Fig. 8 Transmit precoding and receiver shaping (needs CSIR and CSIT)
nHxy +=
y
xVx ~=
x~ x
yUyH=~
y
MIMO Wireless Communications
• From SVD of the channel matrix H, we have,
( ) ( )nxVΣUUnHxUyUy +=+== HHHH~
HVUH ∑=
( ) ( )nxVΣUUnHxUyUy +=+==
( )nxVVΣUUy +=⇒ ~~ HH
nxΣy ~~~ +=⇒
MIMO Wireless Communications
• Component-wise
n
n
x
x
y
y~
~
~
~
00000
00000~
~
2
1
2
1
2
1
2
1
σ
σ
+
=
R
H
T
HH
R
H
N
R
N
RR
N
R
n
n
n
x
x
x
y
y
y
~
~
~
~
000000
00000
00000
00000
00000
~
~
2222
M
M
M
M
O
O
M
M
σ
σ
MIMO Wireless Communications
• Parallel RH Gaussian channels
nxy
nxy~~~
~~~
2222
1111
+=
+=
MOM
σ
σ
• Capacity?
• increases RH fold
RR
HHHH
NN
RRRR
ny
nxy
~~
~~~
=
+=
MOM
MOM
σ
MIMO Wireless Communications
• Rate Gain √
• Diversity gain ×
Space-time codes
Implemented at the transmitter side, needs CSIR Implemented at the transmitter side, needs CSIR
and block fading
• Why space-time codes?
( ) dGc
eSG
cP ≈
MIMO Wireless Communications
• where S is the SNR
• c is a scaling constant specific to the
– modulation employed and
– the nature of the channel– the nature of the channel
• Gc≥ 1 denotes the coding gain and
• Gd is the diversity order of the system
MIMO Wireless Communications
• Diversity gain/order determines the
– negative slope of an error rate curve plotted vs
SNR on a log-log scale
( )
• Space-time coded scheme with diversity order
Gd has
– an error probability at high SNR behaving as
( )2 2 2 2log log log loge d c d
P c G G G S≈ − −
( ) dG
eP S−
≈
MIMO Wireless Communications
If there is some coding gain, then
– average probability of error will be of the form
• If there were no array or power gain then
– the probability of error expression will be of the
form
( )
1d
e G
c
PG S
≈
( )
1d
e G
c
PG S
≈
MIMO Wireless Communications
• The coding gain determines the
– horizontal shift of uncoded system error rate
curve to the
• space time coded error rate curve • space time coded error rate curve
• plotted on a log-log scale obtained for the same
diversity order
Fig. 9 Illustration of diversity and coding gains
• BER curves are usually waterfall type but
– we have shown straight lines for illustration purpose only
MIMO Wireless Communications
• Alamouti Space-Time Codes
Fig. 10 A block diagram of Alamouti space-
time encoder
( )1 2 1 2, 0H
= =s s s s
r
r%
sFig. 11 Alamouti’s space-time decoding
MIMO Wireless Communications
• Received signal vector
• Received signals in two time intervals
= +r Hs n
• the output of the combiner
1 1 2 1 1
* *
2 2 1 2 2
r s s h n
r s s h n
= + −
−
+=
−=
*211
*2
*221
*1
*2
1
1*2
2*1
2
1
~
~
rhrh
rhrh
r
r
hh
hh
r
r
H= +r H Hs n% %
MIMO Wireless Communications
• For 2×1 MIMO system, two signals are picked up by
– the receiving antenna at the receiver
( ) ( ) 22
2
2
2
1211
2
2
2
11~~;~~ nshhrnshhr ++=++=
– the receiving antenna at the receiver
• Two signals are completely decoupled after the combining operation
– for Alamouti Space Time Codes
• Simplifies greatly the detection strategy in comparison
– to conventional MIMO detection
MIMO Wireless Communications
• Applying MLD
• For 0≤t≤T, we have
( )2 2args r h h s= − +% { }M
• For T≤t≤2T, we have,
( )2 2
1 1 1 2
argˆ
min{ }m
s r h h sm
= − +%
( )2 2
2 2 1 2
argˆ
min{ }ms r h h s
m= − +%
{ }M
kkm ss1=∈
MIMO Wireless Communications
MIMO detection
• Detect signals jointly
– since many signals are transmitted from the transmitter to the receiver
• Maximum likelihood (ML) detection outputs the • Maximum likelihood (ML) detection outputs the vector which
• minimizes the Euclidean distance between
– the received vector and
– all possible combinations of the transmitted symbol vectors 2min
argˆ Hsrx
s −=
MIMO Wireless Communications
• Consider 2 × 2 MIMO system
=
=
=
=
2
1
2
1
2221
1211
2
1;;;
n
n
s
s
hh
hh
r
rnsHr
2222212
+
=
⇒
+=
2
1
2
1
2221
1211
2
1
n
n
s
s
hh
hh
r
r
nHsr
MIMO Wireless Communications
• At the detector,
– we want to detect s1 and s2 at time t,
– but there exist interference between these two
signalssignals
• Assume that sk are modulated in M-ary
constellation i.e., sk є {s1,s2,…,sM}
2222121212121111 ; nshshrnshshr ++=++=
MIMO Wireless Communications
• We need to find the minimum metric of the
Euclidean distance
{ }( ) ( )
+−++−
∈
2
22212
2
12111,,2,1,
min jijiMji
shshrshshrL
• For instance, 16-QAM, (s1,s2) are (1 of 16
symbols, 1 of 16 symbols) implies 16×16 pairs
{ }
{ }( ) ( ) ( ) ( )
+−++++−++
∈
∈
2
22212222121
2
12111212111,,2,1,
,,2,1,
min jijiMji
Mji
shshnshshshshnshshL
L
MIMO Wireless Communications
• Metric calculations of 256 are required
• For 3×3 MIMO system,
– NT=NR=3, metric calculations of 163=4096 are
requiredrequired
• For 5×5 MIMO system,
– NT=NR=5, metric calculations of 165 =10,48,576
are required
• which is obviously impractical
MIMO Wireless Communications
• ML detectors are optimal but impractical
• Low complexity suboptimal detectors like
– Zero forcing (ZF) and
– Minimum mean square error (MMSE) – Minimum mean square error (MMSE)
• are preferable
• In linear detector,
• a linear preprocessor (W) is first applied
– to the received signal vector
rWsH=ˆ
MIMO Wireless Communications
• For instance in ZF
• H+ is the Moore Penrose pseudo-inverse of H
nHsrHrWs++ +=== H
ZFˆ
• H is the Moore Penrose pseudo-inverse of H
• Performance is not so good
• Consider a 2×2 MIMO system with s є S={-3,-
1,1,3}
( ) HHHZF HHHHW
1−+ ==
MIMO Wireless Communications
• The channel matrix is given by H=[2 0.5; 1 2;]
• Suppose the received signal vector is r=[1 0.9]T
• The ZF detector’s output is given by
• Thus the hard decision of for s becomes [1 1]T
– which is different from the ML decision of [1 -1]T
• ZF less complex than MLD, but poor performance
[ ]T2.05.0ˆ == +rHs
MIMO Wireless Communications
Antenna selection [2]
• Reduce hardware complexity
• A subset of total available antennas selected
– Based on capacity maximization – Based on capacity maximization
– Based on maximum received SNR
– Done at transmitter, at receiver or both
MIMO Wireless Communications• Transmit Antenna Selection/Maximal Ratio
Combining (TAS/MRC) [3]Ant 1
Ant 1
Ant 2Select
antenna 2,
then use
Tx Rx
Ant 2
Ant 3Ant 3
Fig. 12 3 × 3 TAS/MRC MIMO system
then use
only this 3
path
MIMO Wireless Communications
• How to select antenna?
• By maximizing the received SNR
• For Rayleigh fading channel,
– C are i.i.d. Chi square distributed with the probability
2
,
1
arg max{ | | }rN
i i j
j
I C h=
= =∑
– Ci are i.i.d. Chi square distributed with the probability
density function (PDF) and cumulative distribution
function (CDF) as
1
1
0
1( ) , 0
( 1) !
( ) 1 , 0!
r
r
N x
r
N ix
i
p x x e xN
xP x e x
i
− −
−−
=
= ≥−
= − ≥∑
MIMO Wireless Communications
• Order statistics
• PDF of maximum received SNR C(Nt) such that
C(1)≤ C(2) ≤…≤ C(Nt) can be given as [4]
1N −= 1
( )
11 1
0
( ) [ ( )] ( )
(1 ) ·( 1)! !
t
t
r
t r
N
N t
N iN Nx xt
it
p x N P x p x
N xe x e
N i
−
−− −− −
=
=
= −−
∑
MIMO Wireless Communications
• Assume binary phase shift keying (BPSK) for
TAS/MRC MIMO system
• Instantaneous SNR at the MRC receiver
2
• The average BER
2
0
( 2 ) ( )bb b b
P Q p dγγ γ γ∞
= ∫
Conditional error probability (CEP) for BPSK
( )t
r
t N
N
j
jNb Ch γγγ == ∑=
2
1
,
MIMO Wireless Communications
• Thus, closed form expression for BER Rayleigh fading is
( )
( )
( )( )( )
( )11
2
0 0
11
, 1 ! 11 ! 12 1
rrt
r
k tN t
k NN
tt r rN
k tr
N
kNP a N k N t
N kk
γ
γ
+−−
= =
− − = × + − × − − + ++
∑ ∑
• where, is the coefficient of in the expansion of
•
1( )
0
12 1
1
r
jN t
rj
j
N t j
j k
γ
γ
+ −−
=
+ − + × × + + + ∑
( , )t r
a N k 2tz
( )
2
1
0
2 1
!
r
ki
N
i
z
k
i
−
=
+
∑
MIMO Wireless Communications
• η − µ distribution proposed by M. Yacoub in 2000 (can analyze non LOS propagation)
• Can model many distributions
• Some of the special cases of η − µ distribution are
– Rayleigh distribution for η → 1, µ = 0.5, – Rayleigh distribution for η → 1, µ = 0.5,
– one sided Gaussian distribution for η → 1, µ = 0.25,
– Nakagami-m distribution for η → 1, µ = m/2 and
– Hoyt distribution for η → q2, µ = 0.5
• BER analysis for TAS/MRC over η − µ fading channel can be carried out in the similar way
MIMO Wireless Communications
Fig. 13: BER performance of (2, 1; 1) TAS/MRC system over η
− µ fading channels for η = 1 with BPSK modulation
MIMO Wireless Communications
Spatial modulation
• Two information bearing units
– Transmit antenna index : estimated at receiver
– A symbol from signal constellation : transmitted from antenna corresponding to transmit antenna indexantenna corresponding to transmit antenna index
–Advantages
• Higher capacity
• Reduced hardware complexity
• Avoidance of transmit antenna synchronization
• How?
MIMO Wireless Communications
MIMO Wireless Communications
Fig. 14 SM system model [5]
MIMO Wireless Communications
• k bit information blocks
• Sub blocks of m bits and n bits
• n bits spatially modulated
• m bits modulated using digital modulation • m bits modulated using digital modulation
schemes
• m bits are transmitted physically, effectively similar to
transmitting k=m+n bits
• Restriction on the number of transmit antennas
• Integer exponent of 2
MIMO Wireless Communications
SM Receiver
• MRC diversity scheme
• Two step detection of transmitted symbol
• Antenna index estimation ((h )Hy is noise except for • Antenna index estimation ((hj)Hy is noise except for
actual transmit antenna)
• ML symbol detection, only one transmit antenna [6]
ˆ arg maxj
j =
H
j
2
j
h y
h
{ }*ˆ
2
ˆˆ Re2min
arg q
H
jqjq xxj
yhhx −=
MIMO Wireless Communications
• Assume two estimation processes are independent
– Transmit antenna index estimate
– Estimation of the transmit symbol– Estimation of the transmit symbol
• Notation
– Pa is probability that the antenna index estimation is incorrect
– Ps is probability is that the transmitted symbol estimation is incorrect
MIMO Wireless Communications
• Probability of correct estimation
• Probability of error
( )( )sac PPP −−= 11
• Assume QAM
• CEP
( )( ) sasasace PPPPPPPP −+=−−−=−= 1111
( ) ( )2( )eP aQ b cQ bγ γ γ= −
MIMO Wireless Communications
• Table 2 MODULATION PARAMETERS FOR
VARIOUS MODULATION SCHEMES [8]
MIMO Wireless Communications
• Q-function as sum of exponentials [7]
• CEP becomes
( )22 2
321 1
12 4
xx
Q x e e−−
≈ +
• CEP becomes
• where a=2, b=1, and c=1 for 4-QAM
2 4 7
3 3 62( )12 4 144 16 24
,b b bb
b
e
a a c c cP e e e e e
γ γ γγγγ
− − −−−≅ + − − −
MIMO Wireless Communications
• the integral of fits the definition of MGF [9] of
the received SNR, MGF(γ).
( ) ( ) ( )dxxpxMGF X∫∞
−= exp γγ
• Probability of error in symbol detection
( )1 1 2 1 1 1 4 1 7
6 2 2 3 4 36 4 3 6 6s
MGF MGF MGFP MGF MGFγ γ γ γ
γ
+ − + +
≈
∫0
MIMO Wireless Communications
• Probability of error in transmit antenna index
estimation
• where
( )
+
≈=
34
1
412
1 γγγ MGFMGFQP effa
• where
• MGF for Rice fading case
2
ˆ2
eff
γγ = −j j
h h
( ) 1
1 1exp
Rice
K KMGF
K K
γγ
γ γ
+= −
+ + + +
MIMO Wireless Communications
• Fig. 15 SER vs SNR (dB) for 2×2 SM MIMO system considering Rician fading
MIMO Wireless Communications
• SM with antenna selection [10]
• Some of the selected antennas for SM may be
completely down
• SM systems combined with antenna selection• SM systems combined with antenna selection
Antennas selected at transmitter
SM applied over selected antennas with
better links
• CSI assumed to be available at transmitter
MIMO Wireless Communications
• SM with antenna selectionAnt 1
Ant 1
Ant 2Ant 2
Apply SM
on
selected
antennas
Tx Rx
Ant 4
Ant 4
Fig. 16 4 × 4 TAS SM MIMO system
Ant 3 Ant 3
antennas
which
give the
best links
MIMO Wireless Communications
• Order statistics
• Antenna Selection Parameter
2rN
∑
( )( )
( )
2
,
1
1
1
0
, 0
1 , 0!
r
r
j
r
Aj
N
j i j
i
N x
A
r
N ix
i
A h
x ef x x
N
xF x e x
i
=
− −
−−
=
=
= ≥Γ
= − ≥
∑
∑
MIMO Wireless Communications
• Ajs are arranged in ascending order
• S antennas corresponding to highest Ajs are
selected
• PDF of A(r) such that A ≤ A ≤…≤ A can be • PDF of A(r) such that A(1)≤ A(2) ≤…≤ A(Nt) can be
given as
• where r = Nt-S+1
( )( )
( )( ){ } ( ){ } ( )
111
, 1
t
j j jr
r N r
A A A A
t
f x F x F x f xB r N r
− −
= −− +
MIMO Wireless Communications
• The PDF of received SNR can be given as
• where and C (j,N ) is
( )( ) ( ) ( )
( ) ( ) ( )1
11
0 0
11 11 ,
1 , 1
t
tr
N i Mj x N i jN tr
t r
i r j tt r t
if x C j N x e
jN r N B i N iγ
−− − + ++ −
= = =
− = −
− + Γ − + ∑∑∑
• where and Ct(j,Nr) is
coefficient of xt in
( ) ( )1r tM N N i j= − − +
0
1
!
tr
N i jN i
i
x
i
− +
=
− ∑
MIMO Wireless Communications
• Outage probability
( ) ( )
( )
r
2 1, P
1 1t
R
out r
N
P R Aγγ
γ
−= <
= ∑
• where
( )( ) ( ) ( )
( ) ( )( )( )
( )( )
1
0 0
1 1,
1 , 1
, 111 ,
1
t
r
N
out
i rt r t
i Mj r th t
t r N tj t
t
P RN r N B i N i
N t N i jiC j N
j N i j
γ
γ γ
=
−
+= =
=− + Γ − +
+ − + +− × −
− + +
∑
∑ ∑
2 1R
thγ
γ
−=
MIMO Wireless Communications
Fig. 17: Outage Probability Vs. SNR curve for TAS SM MIMO
systems with antenna selection (2 bits/s/Hz)
MIMO Wireless Communications
• References:
1. L. Zheng and D. N. Tse, “Diversity and multiplexing: A fundamental
trade-off in multiple antenna channels,” IEEE Trans. Information
Theory, pp. 1073-96, May 2003.
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