mimo simulation tutorial -2-

MIMO Simulation Tutorial-2-

연세대학교전기전자공학과

황규호 [email protected]

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


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


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


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 MSs (K) 2


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


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


User 1

User 2

User 3

CH 2

10bps/HzCH 1

ZF-BF


User 1

User 2

User 3

12bps/Hz 15bps/Hz

11 bps/Hz


Heuristic search: 채널상태좋은사용자 1명선택추가적으로사용자추가하여 sum capacity 계산및선택

CH 3

Step 1

8 bps/Hz

12bps/Hz

15bps/Hz 9 bps/Hz


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


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


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 UE antennas (Mr) 2

Number of MSs (K) 2


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


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


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


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


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

mimo simulation tutorial -2-

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