Download - MIMO Simulation Tutorial -2-
Contents Topic 1. MIMO Precoder
ZF (Zero-Forcing) Beamformer MMSE (Minimum Mean Squared Error) Beamformer
Topic 2. Multi-user MIMO (1) System modeling ZF, Block diagonalization
Topic 3. Multi-user MIMO (2) User selection
Max throughput, Round-Robin
Topic 4. Massive MIMO (1) Motivation of Massive MIMO Fundamental Overview of Massive MIMO
Topic 5. Massive MIMO (2) MU-MIMO Downlink Massive MIMO
Topic 1. MIMO Precoder
1. ZF (Zero-Forcing) Beamformer2. MMSE (Minimum Mean Squared Error) Beamformer
Topic 1. MIMO Precoder Point-to-point MIMO system
Rx scheme Receiver : ZF, MMSE, ML, etc.
Tx scheme Precoding(beamforming) : ZF BF, MMSE BF
DEMUXMT symbols
Detection MUXMT symbols
x1
x2
xMT
y1
y2
yMR
nMR
n1
n2
H
TM RM
R TM M
y Hx n 1RM 1TM 1RM
MT : number of Tx antennasMR : number of Rx antennas
x : data stream
Topic 1. MIMO Precoder Zero-Forcing (ZF) Beamforming
Assumption # of transmit antennas (NT) = # of receive antennas (NR) = 2
Received signal
11 1 21 2 1 1
12 1 22 2 2 2
h x h x n rh x h x n r
11 21 1 1 1 11
12 22 2 2 2 2
H
1h h x n s nh h x n s n
x x
H H
x1
x2
r1
r2
h11
h12
h21
h22
1H1
2
ss
21 : Normalized factor for transmit signal F
H x
Topic 1. MIMO Precoder Zero-Forcing (ZF) Beamforming
Received signal
11 1 21 2 1 1
12 1 22 2 2 2
h x h x n rh x h x n r
1 1 1 1 11
2 2 2 2 2
1 1s n s n rs n s n r
x
H H
21 : Normalized factor for transmit signal F
H x
x s n
1 1 1 1 12
1
1Tr Tr TrTN
H H H
k k
H H UΣ V VΣ U Σ Σ
121
1 diag , ,T
T
N
Nk k
n n
x s n s
Noise enhancement by normalized factor
Topic 1. MIMO Precoder Minimum Mean Square Error (MMSE) Beamforming
MMSE precoder
Precoder which can minimize mean square error Considering noise enhancement in ZF-BF
Optimal MMSE precoder
Considering power normalization
21arg min ( )E W
W HWx z x
12
2T T n
x
W H HH I
,
TrT
T
N W WWW
Topic 1. MIMO Precoder
System Parameters
Modulation QPSK
Number of Tx antennas 2
Number of Rx antennas 2
Transmit SNR 0~20dB
0 2 4 6 8 10 12 14 16 18 2010
-4
10-3
10-2
10-1
100
SNR(dB)
BE
R
Comparison between receiver and precoder (2×2)
ML receiverZF receiverMMSE receiverZF BFMMSE BF
Topic 2. Multi-user MIMO (1)
1. System modeling2. ZF, Block diagonalization
Topic 2. Multi-user MIMO (1) MU-MIMO BC system model (MISO case)
User 1
User 2
User KBase Station
1s
2s
Ks
1h
2h
Kh
Precoding( )W
[ ] [ ] [ 1][ 1] [1 ] [ 1] 1,T T T T
K
k k k k k j k j j kK N K KN K KK N N j j ky P s P s n
y H W P s n h w h w
1 1
1
| |, ,
| |K
K K
P
P
h 0H W w w P
h 0
─ ─
─ ─1
where K
k k Tk
P P
w
Tx power constraint
K: number of usershk: channel vector of user kwk: beamforming vector of user ksk: data stream for user k
Topic 2. Multi-user MIMO (1) MU-MIMO BC system model (MIMO case)
Received signal vector at user k after linear precoding
[ 1] [ ] [ ] [ 1] 1,R R T T R R
K
k k k k k j j kN N N N N N j j k
s s s s sy H W s n y H W s H W s n
Base Station
1s
2s
1H
2H
KH
Precoding( )sW
Ks
User 1Receiver
(G1)
User 1Receiver
(G2)
User KReceiver
(GK)
1. All users have NR antennas2. NT ≥ NR K3. Full rank
Assumptions
1 1
1
| |, ,
| |K
K K
P
P
H 0H W W W P
H 0
─ ─
─ ─1
where K
k k TFk
P P
W
Tx power constraint
Topic 2. Multi-user MIMO (1) Channel Inversion (zero-forcing)
Pseudo inverse of the channel prior to transmission Multi-user interference nulling
γ is scaling factor
Received signal at user k
If channel is ill-conditioned, i.e., one of the singular values of (HHH)-1 is very large, γ will be large, and the SNR at the Rx will be low
1 1CI 1 where traceH H H
W H HH HH
1,
1K
k k k k k j j k k kj j k
y H W s H W s n s n Signal attenuation
Interference nulling
Topic 2. Multi-user MIMO (1) Block Diagonalization
Concept of BD
Generalization of the CI for MU-MIMO with multiple antenna at Rx
Precoding matrix Ws is designed to suppress the MUI completely.
To eliminate all the MUI, the following constraint is imposed.
H1
H2
H3
Hs
W1 W2 W3
Ws
1 1H W
2 2H W
3 3H W
HsWs
CI: Channel InversionMUI: Multiuser Interference
[ ][ ]
0 for all T RR T
j kN NN N
j k
H W
1effH
2effH
3effH
Topic 2. Multi-user MIMO (1) Block Diagonalization
Step 1) Precoding matrix design for MUI elimination
In order to zero-interference, Wj should be in the null space of
The SVD of is given by
Example) When NT=6, NR=2, K=3, for j=1
0 for all j k j k H W
1 1 1
[ 1 ]
R T
TT T T Tj j j K
N K N
H H H H H jH
jH
(1) (0)
[ 1 ] [ 1 ] [ 1 ]1 1 [ 1 ]
R T R T T RR R T T R
H
j j j j jN K N N K N N N KN K N K N N N K
H U Σ V V
(1)1(1)21(1)
2 321 1 2 3 4 (1)
3 43(0)14(0)2
| | | |
| | | |
H
H
H
H
H
H
vv
H v0H u u u u
H vvv
─ ─
─ ─
─ ─
─ ─
─ ─
─ ─
(0)
null space of
j
j j
H
W V
CI: Channel InversionMUI: Multiuser Interference
Topic 2. Multi-user MIMO (1) Block Diagonalization
Step 2) Precoding matrix ( ) and receiver ( ) design for ISI elimination Effective channel matrix for user j
Optimal Tx/Rx scheme: SVD based Eigen Beamforming
Then, we obtain the final precoding matrix
(0)
[ 1 ]R T R
effj j j j j
N N N K
H W H V H
1:[ ] [ ]
, R
R R R
eff H Hj j j j j j j jN
P N N N
H U Σ V W V G U
jW jG
(0)
1:[ ] [ ] [ ]
ˆR
T R T R
j j j j j NN N N P P N
W W W V V
Let P=NT−NR(K − 1)
[ ]R TN N
(1) (0)
[ 1 ] [ 1 ] [ 1 ]1 1 [ 1 ]
R T R T T RR R T T R
H
j j j j jN K N N K N N N KN K N K N N N K
H U Σ V V
(0)
null space of
j
j j
H
W V
Topic 2. Multi-user MIMO (1) Block Diagonalization
Received signal after BD
Sum rate for BD
Optimal power loading matrix (Pk) can be obtained by the water-filling
1,
1,
1:
ˆ ˆ
R
K
k k k k k k k k j j kj j k
K
k k k k k k j j j kj j k
effk k k k k
H Hk k k k k k kN
k k k
z G y G H W s H W s n
G H W W s H W W s n
G H W s n
U U Σ V V s n
Σ s n
2
2 21 1
max log det subject to Trk
K Kk k
BD k Tk k
R P
P
Σ PI P
MUI nulling
ISI elimination
Topic 2. Multi-user MIMO (1) Sum rates in terms of the number of users (K).
Sum rates of ZF and BD. Sum rate of BD without WF Sum rate of BD with WF
System Parameters
Number of BS antennas (Mt) 10
Number of MS antennas (Mr) 2
Number of MS candidates (N) 10
Transmit SNR 0dB
1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
3
4
5
6
7
Number of users (K)
Sum
rat
e (b
ps/H
z)
Sum rate performance
ZF
BD w/o WFBD /w WF
Topic 2. Multi-user MIMO (1) Sum rates in terms of the number of receive antennas per user (Mr).
Sum rates of ZF and BD. Sum rate of BD without WF Sum rate of BD with WF
System Parameters
Number of BS antennas (Mt) 10
Number of MSs (K) 2
Number of MS candidates (N) 10
Transmit SNR 0dB
1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
3
4
5
6
7
8
Number of Rx antennas per UE (Mr)
Sum
rat
e (b
ps/H
z)
Sum rate performance
ZF
BD w/o WFBD /w WF
Topic 3. Multi-user MIMO (2)
1. User Selection
Topic 3. Multi-user MIMO (2) Multiuser Diversity
In wireless communication, users experience different channel conditions.
By using Efficient Scheduling, multiuser diversity can be achieved
User 2Service
User 1Service
User 3Service
User 2Service
<Time>
<Channel Quality>User 1 User 2 User 3 Serviced quality
Best User Selection Each user measure CSI of overall channel, and feedback to BS
CSI: Channel State Information (SNR, C/I, data rate, etc.)
BS selects the best user set Exhaustive search: NCK
User 1
User 2
User 3
CH 2
10bps/HzCH 1
ZF-BF
User 1 User 2 User 3
User 1
User 2
User 3
12bps/Hz 15bps/Hz
12bps/Hz
15bps/Hz
11 bps/Hz 9 bps/Hz
9 bps/Hz 10 bps/Hz
Scheduling Algorithm
Best User Selection : 기지국에서모든조합의 sum rate 도출maximum sum rate 조합선택
CH 3
12bps/Hz
Topic 3. Multi-user MIMO (2)
User 1
User 2
User 3
CH 2
10bps/HzCH 1
ZF-BF
User 1 User 2 User 3
User 1
User 2
User 3
12bps/Hz 15bps/Hz
11 bps/Hz
Scheduling Algorithm
Heuristic search: 채널상태좋은사용자 1명선택추가적으로사용자추가하여 sum capacity 계산및선택
CH 3
Step 1
8 bps/Hz
12bps/Hz
15bps/Hz 9 bps/Hz
Topic 3. Multi-user MIMO (2)
9 bps/Hz
Heuristic user selection As # of users increases, best user selection is getting almost impossible So BS selects the best user first, and then find its co-users with
exhaustive search: N + (N-1)C(K-1)
User 1
User 2
User 3
CH 1
ZF-BF
User 1 User 2 User 3
User 1
User 2
User 3
Scheduling Algorithm
Round Robin: 임의로사용자조합선택
CH 39 bps/Hz
CH 2
9 bps/Hz 8 bps/Hz
10bps/Hz 12bps/Hz 15bps/Hz
12bps/Hz
15bps/Hz
11 bps/Hz
Topic 3. Multi-user MIMO (2)
Round robin user selection BS selects users randomly without considering CSI Guarantee the fairness between users
Topic 3. Multi-user MIMO (2) Performance comparison
Best user selection, Heuristic user selection, Round robin user selection Using ZF BF
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
Tx SNR (dB)
Sum
Rat
e C
apac
ity
Round Robin
Heuristic AlgorithmExhaustive Search System Parameters
Number of BS antennas (Mt) 4
Number of UE antennas (Mr) 2
Number of MSs (K) 2
Number of MS candidates (N) 3
Topic 4. Massive MIMO (1)
1. Motivation of Massive MIMO2. Fundamental Overview of Massive MIMO
Motivation of Massive MIMO Consider a MIMO MAC ( M : # of RX antennas, K : # of TX)
If the BS process its receive signal by matched filtering,
By the strong law of large numbers,
With an unlimited number of antennas− Uncorrelated interference and noise vanish− The matched filter is optimal− The transmit power can be made arbitrarily small
M K
y Hx n ,H n : i.i.d. with zero mean and unit variance
1 1 1H H H
M M M y H y H Hx H n
. ., =const.
1 a sHM KM H y x
Topic 4. Massive MIMO (1)MAC : Multiple Access Channel
1M
M K
1K 1M
21 1 2 1
222 1 2
21 2
1
H HK
H HK
H
H HKK K
M M M
M M MM
M M M
h h h h h
hh h h hH H
hh h h h
as M . . 0a s
M. . 1a s
MBy strong law of large numbers
Topic 4. Massive MIMO (1) Channel Model
# of TX antenna M, # of RX antennas N IID complex-Gaussian channel H, x, n with zero mean and unit variance is downlink transmission power Receiver has perfect knowledge of H
Received SNR/ Capacity at Receiver
1 11d N M M NN
p
y H x n
2log det( )HdN M N
pCM I HH2
2
0
SNR dd
pp
N
HH
dp
2log det( )HdM M N
pCM I H H
M > N
M < N
SNR : Signal to Noise Ratio
Topic 4. Massive MIMO (1)
Capacity at Receiver (M>N)
For large M with IID complex-Gaussian channel H,
2log det( )HdN
pCM
I HH2
1 1 2 121
2 1 21
21 2
| |1 1 1
| |
H HN
HH H H
N
NH H
N N N
M M M
h h h h hh
h h hHH h hh
h h h h h
1 2
1where i i i
i MM
h h h
h
2 22
21
1 0
Var 1i i
Mih h
h E hM M
h
1 HNM
HH I
* * * 1 21 1 2 2
Gaussian Gaussian Gaussian
1 0H
i j i j i j i j MM M
i j
g g gh h h h h h E g E hM M M
h h
Conclusion
Topic 4. Massive MIMO (1)
Capacity at Receiver (N>M)
For large N with IID complex-Gaussian channel H,
2log det( )HdM
pCM
I H H2
1 1 2 121
2 1 21
21 2
| |1 1 1
| |
H HMH
HH
MHM
H HM M M
N NM M N M N
h h h h hh
h h hH H h hh
h h h h h
1 2
1where
Ti i ii N
Nh h h
h
2 22
21
1 0
Var 1i i
Nih h
h E hN N
h
1 HM
N NM N M
H H I
* * * 1 21 1 2 2
Gaussian Gaussian Gaussian
1 0H
i j i j i j i j NN N
i j
g g gh h h h h h E g E hN N N
h h
Conclusion
Topic 4. Massive MIMO (1)
Point-to-point MIMO Large number of transmit antennas
Large number of receive antennas
2 2
2 2
log det( ) log det( )
1 0 0log det 0 0 log (1 )
0 0 1
HdM N N N d N
d
d
d N N
pC pMp
N pp
I HH I I
2 2
2 2
log det log det( )
1 0 0
log det 0 0 log (1 )
0 0 1
Hd dN M M M M
d
d
d
M M
p NpCM M
NpM NpM
MNpM
I H H I I
1 HNM
HH I
1 HM
N NM N M
H H I
Independent with MLinearly increase as N
Increase as N with log shape
Topic 4. Massive MIMO (1) Simulation result
M = 1 ~ 500, hi = M X 1 Real Gaussian Vector
0 50 100 150 200 250 300 350 400 450 500-0.5
0
0.5
1
1.5
2
M
2
i
Mh
Hi j
Mh h
Converges to 1 as M increases
Converges to 0 as M increases
Topic 5. Massive MIMO (2)
1. MU-MIMO Downlink Massive MIMO
Topic 5. Massive MIMO (2) Conventional linear precoding
Received signal after using linear precoding
MRT ZFBF
HW H 1H H W H HH
noise1,
desired signalinterference
K
k d k k k d k i ii i k
y p s p s
h w h w n
2
2
1,
SINR1
d k kk K
d k ii i k
p
p
h w
h w 2log 1 SINRk kR sum1
K
kk
R E R
Rate of user k Ergodic sum rate
MRT: Maximal Ratio TransmissionZFBF: Zero-Forcing Beamforming
SINR of the kth user
Topic 5. Massive MIMO (2) Deterministic form of the SINRk & Rsum as M, K →∞
MRT
ZFBF
2
. .
2
1,
SINR as , 11
Hdk k a s
mrt dk K
Hd dk i
i i k
pp M M K
p p K K
h h
h h
. .
1
1SINR as , tr
a szf dk d
H
p M Kp M KK
H H
sum 2log 11
mrt d
d
p MR Kp K K
sum 21log 1zf
dM KR K p
K
SINR: Signal-to-Interference-plus-Noise RatioCSI: Channel State Infromation
2 2 2 222
1,
1 1 ~ , 1 2
KH H H H
k i k i k i M Fi i k
E M KMK
h h h h h h H
1 11/tr Diversity order of ZF-BFH M KK
H H
Topic 5. Massive MIMO (2) 하향링크 Massive MIMO에서의Transmit-MRC vs. ZF-BF
제한적인 user 수에대하여안테나수에따른성능 (Pd = 1)
Cross point ☞
2~ log 12 1
MRC MR KK
21~ log 1ZF M KR K
K
• ZF-BF sum-rate
singlecross
1 =2 12 1
M M K M KK K
K=5, Mcross=9 K=30, Mcross=59
30 40 50 60 70 80 90 1000
10
20
30
40
50
60
# of BS antenna
Sum
-rat
e
Massive MIMO (Single cell)
ZF-single-theoryZF-single-MRC-single-theoryMRC-single
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
25
# of BS antenna
Sum
-rat
e
Massive MIMO (Single cell)
ZF-single-theoryZF-single-
MRC-single-theoryMRC-single
• Transmit MRC sum-rate
Topic 5. Massive MIMO (2) SNR에따른하향링크 Massive MIMO에서의Transmit-MRC
vs. ZF-BF SNR(노이즈분산)을고려한 sum-rate
Single-cell
Cross-point
수신 SINR이 0dB를만족하는지점
결론 : Very low SINR 영역 (SINR < 0dB) 에서만 MRC 이득이있음
2~ log 11
MRC SNR MR KSNR K K
2
1~ log 1ZF SNR M K
R KK
0 0
1kPSNRN N
Transmit-MRC ZF-BF
singlecross 1 KM K
SNR
40 60 80 100 120 140 160 180 200-6
-4
-2
0
2
4
6
8
# of BS antenna
SIN
R(d
B)
MRC-single-theoryMRC-single-
ZF-single-theoryZF-single-
Topic 5. Massive MIMO (2) SNR에따른하향링크 Massive MIMO에서의Transmit-MRC
vs. ZF-BF
M에따른수신 SINR 비교
수신 SINR이 0dB를만족하는지점
Topic 5. Massive MIMO (2) 하향링크 Massive MIMO에서의Transmit-MRC vs. ZF-BF
BER 성능비교
20 40 60 80 100 120 140 160 180 20010
-5
10-4
10-3
10-2
BE
R
# of BS antenna
MRC
ZF-BF
K=30, SNR=0dB 일때, M에따른 BER 비교 K=30, M=40 일때, SNR에따른 BER 비교
0 1 2 3 4 5 6 7 8 9 1010
-6
10-5
10-4
10-3
10-2
BE
R
SNR
MRC
ZF-BF
MUI가충분히제거되지못함