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MIMO Simulation Tutorial

-2-

연세대학교전기전자공학과

황규호[email protected] 2013-01-10



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


MU-MIMO Downlink Massive MIMO




1. ZF (Zero-Forcing) Beamformer 2. MMSE (Minimum Mean Squared Error) Beamformer




Point-to-point MIMO system

Rx scheme

Receiver : ZF, MMSE, ML, etc.

Tx scheme

Precoding(beamforming) : ZF BF, MMSE BF

DEMUX

MT symbols

Detection MUX

MT symbols

x1

x2

xMT

y1

y2

yMR

nMR

n1

n2

H

T M R M

R T M M

y Hx n 1

R M 1

T M 1

R M

MT : number of Tx antennas

MR : number of Rx antennas

x : data stream




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 r

h x h x n r

11 21 1 1 1 11

12 22 2 2 2 2

H

1h h x n s n

h h x n s n

x x

H H

x1

x2

r 1

r 2

h11

h12

h21

h22

1H1

2

s

s

21 : Normalized factor for transmit signalF

H x




Zero-Forcing (ZF) Beamforming

Received signal

11 1 21 2 1 1

12 1 22 2 2 2

h x h x n r h x h x n r

1 1 1 1 11

2 2 2 2 2

1 1s n s n r s n s n r

x

H H

21 : Normalized factor for transmit signalF

H x

x s n

1 1 1 1 12

1

1Tr Tr Tr T N

H H H

k k

H H UΣ V VΣ U Σ Σ

12

1

1diag , ,

T

T

N

N

k k

n n

x s n s

Noise enhancement by normalized factor




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

21

arg min ( ) E

WW HWx z x

12

2

T T n

x

W H HH I

,

Tr T

T

N W W

WW




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)

B E R

Comparison between receiver and precoder (2×2)

ML receiver

ZF receiver

MMSE receiver

ZF BF

MMSE BF




1. System modeling2. ZF, Block diagonalization




MU-MIMO BC system model (MISO case)

User 1

User 2

User K Base Station

1s

2s

K s

1h

2h

K h

Precoding

( )W

[ ] [ ] [ 1][ 1] [1 ] [ 1] 1,T T T T

K

k k k k k j k j j k K N K K N K K K N N j j k

y P s P s n

y H W P s n h w h w

1 1

1

| |

, ,

| |K

K K

P

P

h 0

H W w w P

h 0

─ ─

─ ─

1

whereK

k k T

k

P P

w

Tx power constraint

K : number of users

hk: channel vector of user k

wk: beamforming vector of user k

sk: data stream for user k




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 k N 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

K H

Precoding

( )sW

K s

User

1

Receiver

(G1)

User

1

Receiver

(G2)

User

K

Receiver

(GK )

1. All users have NR antennas

2. NT ≥ NRൈK

3. Full rank

Assumptions

1 1

1

| |

, ,

| |K

K K

P

P

H 0

H W W W P

H 0

─ ─

─ ─

1

whereK

k k T F k

P P

W

Tx power constraint




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 trace H H H

W H HH HH

1,

1K

k k k k k j j k k k

j j k y H W s H W s n s n Signal attenuation

Interference nulling




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

Hs Ws

CI: Channel InversionMUI: Multiuser Interference

[ ][ ]

0 for allT R R T

j k N N N N

j k

H W

1eff

H

2eff H

3eff

H





Step 1) Precoding matrix design for MUI elimination

In order to zero-interference, W j 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

T T T T T

j 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 R R R T T R

H

j j j j j

N K N N K N N N K N K N K N N N K

H U Σ V V

(1)1

(1)21

(1)2 32

1 1 2 3 4 (1)3 43

(0 )14

(0 )

2

| | | |

| | | |

H

H

H

H

H

H

v

v

H v0H u u u u

H v

v

v

─ ─

─ ─

─ ─

─ ─

─ ─

─ ─

(0 )null space of j

j j

HW 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

eff

j j j j j

N N N K

H W H V H

1:[ ] [ ]

, R

R R R

eff H H

j j j j j j j j N P N N N

H U Σ V W V G U

jW j

G

(0)

1:[ ] [ ] [ ]

ˆ R

T R T R

j j j j j N N N N P P N

W W W V V

Let P =NT −NR(K − 1)

[ ] R T

N N

(1) (0)

[ 1 ] [ 1 ] [ 1 ]1 1 [ 1 ]

R T R T T R R R T T R

H

j j j j j

N K N N K N N N K N 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 k

j j k

K

k k k k k k j j j k

j j k

eff

k k k k k

H H

k k k k k k k N

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 Tr k

K K k k

BD k T

k k

R P

P

Σ PI P

MUI nulling

ISI elimination




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 5

0

1

2

3

4

5

6

7

Number of users (K)

S u m r

a t e ( b p s / H z )

Sum rate performance

ZF

BD w/o WF

BD /w WF




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 5

0

1

2

3

4

5

6

7

8

Number of Rx antennas per UE (Mr)

S u m r

a t e ( b p s / H z )

Sum rate performance

ZF

BD w/o WF

BD /w WF




1. User Selection




Multiuser Diversity

In wireless communication, users experience different channel

conditions.

By using Efficient Scheduling, multiuser diversity can be achieved

User 2

Service

User 1

Service

User 3

Service

User 2

Service

<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/Hz

CH 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 rate도출maximum sum rate조합선택

CH 3

12bps/Hz




User 1

User 2

User 3

CH 2

10bps/Hz

CH 1

ZF-BF


User 1

User 2

User 3

12bps/Hz 15bps/Hz

11 bps/Hz


Heuristic search: 채널상태좋은사용자 1명선택추가적으로사용자추가하여 sum capacity계산및선택

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 3

9 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




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)

S u m R

a t e C a p a c i

t y

Round Robin

Heuristic Algorithm

Exhaustive Search System Parameters


Number of UE antennas (Mr) 2

Number of MSs (K) 2





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 1 H H H

M M M H y H Hx H n

. .

, =const.

1 a s H

M K M

H y x

Topic 4. Massive MIMO (1)MAC : Multiple Access Channel

1 M

M K

1K

1 M

2

1 1 2 1

2

22 1 2

2

1 2

1

H H

K

H H

K H

H H K K K

M M M

M M M M

M M M

h h h h h

hh h h h

H H

hh h h h

as M

. . 0a s

M

. . 1a s

M

By strong law of large numbers




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 N N

p

y H x n

2log det( ) H d N M N

pC

M I HH2

2

0

SNR d

d p p

N H H

d p

2log det( ) H d M M N

pC

M 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( ) H d N

pC

M I HH

21 1 2 1

21

2 1 21

2

1 2

| |1 1 1

| |

H H N

H

H H H

N

N H H

N N N

M M M

h h h h hh

h h hHH h h

h

h h h h h

1 21

where i i i

i M M

h h h

h

2 2

2 21

1 0

Var 1i i M i h h

h E h M M

h

1 H

N M

HH I

* * * 1 2

1 1 2 2

Gaussian Gaussian Gaussian

10

H

i j i j i j i j M M M

i j

g g gh h h h h h E g E h

M M M

h h

Conclusion




Capacity at Receiver (N>M)

For large N with IID complex-Gaussian channel H,

2log det( ) H d M

pC

M I H H

2

1 1 2 1

21

2 1 21

2

1 2

| |1 1 1

| |

H H

M H

H

H

M

H

M H H

M M M

N N

M M N M N

h h h h hh

h h hH H h h

h

h h h h h

1 21

whereT

i i i

i N N

h h h

h

2 2

2 21

1 0

Var 1i i N i h h

h E h N N

h

1 H

M

N N

M N M H H I

* * * 1 21 1 2 2

Gaussian Gaussian Gaussian

10

H

i j i j i j i j N N N

i j

g g gh h h h h h E g E h

N 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 0

log det 0 0 log (1 )

0 0 1

H d M N N N d N

d

d

d N N

pC p M

p

N p

p

I HH I I

2 2

2 2

log det log det( )

1 0 0

log det 0 0 log (1 )

0 0 1

H d d N M M M M

d

d

d

M M

p NpC

M M

Np

M Np

M M

Np

M

I H H I I

1 H

N

M

HH I

1 H

M

N N

M N M H H I

Independent with MLinearly increase as N

Increase as N with log shape




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

M

h

H

i j

M

h h

Converges to 1 as M increases

Converges to 0 as M increases




1. MU-MIMO Downlink Massive MIMO




Conventional linear precoding

Received signal after using linear precoding

MRT ZFBF

H W H 1

H H

W H HH

noise1,desired signalinterference

K

k d k k k d k i i

i i k

y p s p s

h w h w n

2

2

1,

SINR

1

d k k

k K

d k i

i i k

p

p

h w

h w 2log 1 SINR

k k R sum

1

K

k

k

R E R

Rate of user k Ergodic sum rate

MRT: Maximal Ratio Transmission

ZFBF: Zero-Forcing Beamforming

SINR of the kth user




Deterministic form of the SINRk & Rsum as M, K →∞

MRT

ZFBF

2

. .

2

1,

SINR as ,1

1

H d k k a s

mrt d k K

H d d k i

i i k

p

p M M K

p p K K

h h

h h

. .

1

1

SINR as ,tr

a s zf d

k d H

p M K p M K

K

H H

sum 2log 11

mrt d

d

p M R K

p K K

sum 2

1log 1

zf

d

M K R K p K

SINR: Signal-to-Interference-plus-Noise Ratio

CSI: Channel State Infromation

2 2 2 22

21,

1 1~ ,

1 2

K H H H H

k i k i k i M F

i i k

E M KM K

h h h h h h H

1 11/tr Diversity order of ZF-BF H M K

K

H H




하향링크 Massive MIMO에서의Transmit-MRC vs. ZF-BF

제한적인 user수에대하여안테나수에따른성능 (Pd = 1)

Cross point ☞

2~ log 12 1

MRC M R K

K

2

1~ log 1 ZF M K

R K K

• ZF-BF sum-rate

singlecross

1=2 1

2 1

M M K M K

K 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

S u m - r a t e

Massive MIMO (Single cell)

ZF-single-theory

ZF-single-

MRC-single-theory

MRC-single

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

# of BS antenna

S u m

- r a t e

Massive MIMO (Single cell)

ZF-single-theory

ZF-single-

MRC-single-theory

MRC-single

• Transmit MRC sum-rate




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 MRC이득이있음

2~ log 11

MRC SNR M R K

SNR K K

2

1~ log 1 ZF

SNR M K R K

K

0 0

1k P

SNR N N

Transmit-MRC ZF-BF

singlecross 1 K M K

SNR



40 60 80 100 120 140 160 180 200-6

-4

-2

0

2

4

6

8

# of BS antenna

S I N R ( d B

)

MRC-single-theory

MRC-single-ZF-single-theory

ZF-single-


SNR에따른하향링크 Massive MIMO에서의Transmit-MRC

vs. ZF-BF

M에따른수신 SINR비교

수신 SINR이 0dB를만족하는지점

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