Jan. 21, 2005 1

JWITC’05

Towards a Better Diversity-Multiplexing Tradeoff in MIMO Systems

Lin DAIThe Hong Kong University of Science and Technology

Joint work with Prof. K. B. Letaief and Miss S. Sfar

Jan. 21, 2005 2

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MIMO Systems

Tx Rx

H

Tx Rx

H

multiple replicasDiversityGain

H: Only one dimension!

Tx Rx

MIMOChannel

H

...

...

...

• Diversity gain √

• Multiplexing gain √

Channel dimensions increase!

Transmit diversity

Receivediversity

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Diversity Gain

rN≤ Receive diversitydp SNR−∼Error probability

tN≤ Transmit diversity

t rN N≤ ⋅ MIMO systems

How to achieve the maximum diversity gain?

• Space-Time Trellis Codes (STTC)[Tarokh’98]

• Space-Time Block Codes (STBC)[Alamouti’98]

• ……

STC

Recb ....

. .. .

. .. .. .. .

. .. .

Diversity-based Schemes

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Multiplexing Gain

{ }min , logt rC N N SNR∼Capacity

number of degrees of freedom

• Bell Lab Layered Space-Time(BLAST) [Foschini’98]

• ……

LST Recb2

b1

bm

. . .

Tx Rx......

Spatial Multiplexing

Multiplexing-based Schemes

Data rate increases!

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Diversity-Multiplexing Tradeoff

(0, mn)

(1, (m-1)(n-1))

(2, (m-2)(n-2))

(r, (m-r)(n-r))

(min(m,n), 0)

Spatial Multiplexing Gain: r=R/logSNR

Div

ersi

ty G

ain:

d*(

r)

MIMO systems can provide both diversity gain and spatial multiplexing gain.

For a system with m transmit and n receive antennas, the optimal diversity-multiplexing tradeoff curve is given by [Zheng&Tse’03]:

*( ) ( )( )d r m r n r= − −

V-BLAST

STBC

• Alamouti’s scheme (m=2, n=1)

• D-BLAST (MMSE, ignoring the overhead)

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How to Achieve the Optimal Tradeoff?

• Explicit code construction– [Yao’03]: a structured coding scheme for two-transmit

two-receive antenna systems with code duration two – [Ma’02]: full-rate-full-diversity codes based on LCF

coding and ML decoding – [Gamal’04]: LAttice Space-Time (LAST) codes under

generalized minimum Euclidean distance lattice decoding

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How to Achieve the Optimal Tradeoff?

• Signal Processing

– [Tarokh’99]: combined array processing and space-time coding

• [Prasad’01]: LSTBC

• [Kim’03]: variable-rate STBC

• [Tao’04]: optimized receiver designa good tradeoff

between two gainsGroup Transmission and Detection

low complexity

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Group Transmission and Detection

S/P

STTC1

STTC2

STTCG

1 2[ , ,...]b b=B

1b

2b

Gb

1ts

2ts

Gts

1m

2m

Gm Tx

Tx

Tx............

......

n Rx

... ...

GZF-Decoding

Group Detectionduring Ti time slots

STTC DecodingIU ˆ

ib

1,...,i G=For each group ,iG

• MLD inside the group• interference cancellation among the groups

Quasi-static, flat, Rayleigh

combination of LST and STC

better diversity gain

better multiplexing gain Group Zero Forcing

(GZF)Group Successive

Interference Cancellation (GSIC)LST STC

tradeoff

a good tradeoff

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Our Contributions

• Derive the optimal diversity-multiplexing tradeoff of a group detector.

• Propose two novel group-transmission-group-detection based schemes where much better diversity-multiplexing tradeoff can be achieved compared to the existent schemes.

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Optimal Tradeoff Function of a Group Detector

Theorem 1: When the block length , where , the diversity-multiplexing tradeoff of an m-transmit-n-receive system with GZF (GSIC) is given by

where

| | | | 1x xL n≥ − + −G G 1,...,arg max | |x ii G==G G

{ },1,... 1( ) min ( ) ;i i iG

outi G id r d r r r

==

⎧ ⎫= =⎨ ⎬⎩ ⎭

∑G G G

( )( ), ( )i i i iout i id r n r r= − − −G G G GG G

group size of the i-th group

the number of groups

allocated rate of the i-th group

irG

iG• The tradeoff function clearly depends on the multiplexing rate , as

well as the particular partition which is specified by G and .• 3 rate allocation schemes are examined: equal rate allocation, size

proportional rate allocation, optimal rate allocation.

(3, 2, 2, 1)(2, 2, 2, 2)(4, 1, 2, 1)

…

partition

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Equal Rate Allocation

/i

r r G=G

{ },{ , }1

( ) min ( ) ( | | )(| | )i

GZF p G GZFEqRate G Gi G

r rd r d r n mG G=

= = − + − −G G G

( , ) ( )( )GZFEqRatem r m rd G r n mG G G G⎢ ⎥ ⎢ ⎥= − + − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Step 1: optimized over all possible partitions with G fixed

Maximum achievable multiplexing rate is

GG× G

Step 2: optimized over all possible G

G =2

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Size Proportional Rate Allocation

| | /i i

r r m=G G

,{ , }( ) | | ( | | | |)(1 )GZF p GPropRate G G Gr rd r n mm m

= − + − −G G G

Step 1: optimized over all possible partitions with G fixed

,{ , }( ) ( )(1 )GZF p GPropRatem m r m rd r n mG G m G m⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − + − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Maximum achievable multiplexing rate is m

Step 2: optimized over all possible G

G =2

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Equal Rate Allocation & Size Proportional Rate Allocation

m=n=8G=3

Optimized over all possible partitions

equal rate

size proportional rate allocation

equal rate

size proportional rate allocation

GZF

GSIC

GZF

GSIC

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Equal Rate Allocation & Size Proportional Rate Allocation

Optimized over all possible partitions under a given G

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Optimal Rate Allocation

maximize

{ }1

( ) min ( )i i

GZF GZF

i Gd r d r

== G G

subject to

1; [0,| |], 1,...,

i i

G

ii

r r r i G=

⎧ ⎫= ∈ ∀ =⎨ ⎬

⎩ ⎭∑ G G G

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Optimal Rate Allocation

m=n=8G=4

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Optimal Rate Allocation

m=n=8

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Optimal Rate Allocation

m=n=8

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Remarks• Optimal rate allocation > Size proportional rate allocation

> Equal rate allocation

• With optimal rate allocation, GZF performs closely to GSIC and sometimes even outperforms it.

• Both GZF and GSIC can efficiently bridge the gap between the optimal performance and BLAST via decreasing G.

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Proposed Scheme 1: GLST (Group Layered Space-Time)

S/P

STBC1

STBC2

STBCG

1 2[ , ,...]b b=B

1b

2b

Gb

1ts

2ts

Gts

1m

2m

Gm Tx

Tx

Tx............

......

n Rx... ...

GZF-Decoding

Group Detectionduring Ti time slots

STBC DecodingIU ˆ

ib

1,...,i G=For each group ,iG

Decoding-GZF

n Rx... ...

Collectthe

receivesignalsover T

time slots

Linearspace-timeprocessing

IIu GroupDetection

IIr

... ...

1b̂

2b̂

ˆGb

The equivalent channel becomes Tnxm.

For each time slot, group detector is applied to an nxm channel.

Better performance can be achieved by group detection since the receive dimensions increase from n to Tn. GLST should have a better

tradeoff than LSTBC.

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Proposed Scheme 2: QoGST (Quasi-Orthogonal Group Space-time)

S/P

STBC1

STBC2

STBCG

1 2[ , ,...]b b=B

1b

2b

Gb

1ts

2ts

Gts

1m

2m

Gm Tx

Tx

Tx............

......

Decoding-GZF

n Rx... ...

Collectthe

receivesignalsover T

time slots

Linearspace-timeprocessing

IIu GroupDetection

IIr

... ...

1b̂

2b̂

ˆGb

* *1 1,..., ,...,T T⎡ ⎤ ⎡ ⎤= + ⎣ ⎦⎣ ⎦X A b A b B b B b

independent to each other

S/Pinter-groupSTBC

1 2[ , ,...]b b=B

1b

2b

bGb

1ts

2ts

mGts

1m

2m

mGm Tx

Tx

Tx............

......

inter-groupSTBC

• inter-group interference can be better suppressed thanks to the orthogonal nature of STBC• interleaving gain can be achieved

• without suppressing the inter-group interference• no interleaving gain

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Design Example of Inter-group STBCConsider a case that the bit stream is divided into G=2 groups and transmitted by m=4 transmit antennas over T=2 time slots.

*1,1 1,2

*1,2 1,1

*2,1 2,2

*2,2 2,1

GLST

b bb bb bb b

⎡ ⎤−⎢ ⎥⎢ ⎥= ⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

X

m

t

*1,1 2,1

*1,2 2,2

*2,1 1,1

*2,2 1,2

QoGST

b bb bb bb b

⎡ ⎤−⎢ ⎥−⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

X

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Tradeoff Comparison: LSTBC, GLST & QoGST• LSTBC

• GLST

• QoGST

( ) ( )(1 / )LSTBC m md r g n m g Tr K= − + −

_ _( ) ( ) ( )lower GLST GLST upper GLSTd r d r d r≤ ≤

( )1 /mg n Tr K−( )( )/ /bx Tr G g Tr G− −

_ _( ) ( ) ( )lower QoGST QoGST upper QoGSTd r d r d r≤ ≤

( / )( / )m b bG n Tr G g Tr G− −( )( )/ /m b bG n m g Tr G g Tr G− + − −

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Tradeoff Comparison: LSTBC, GLST & QoGSTm=K=6, n=4, 2 groups

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

Spatial Multiplexing Gain: r

Div

ersi

ty G

ain:

d

dLSTBC(r)lower bound of GLST, dlow er-GLST(r)upper bound of GLST, dupper-GLST(r)lower bound of QoGST, dlow er-QoGST(r)upper bound of QoGST, dupper-QoGST(r)

(m, n) optimal, d*(r)

QoGST: g=3, Gm=Gb=2, T=2

GLST: G=2, gm=gb=3, T=4

LSTBC: G=2, gm=gb=3, T=4

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Tradeoff Comparison: LSTBC, GLST & QoGSTm=K=6, n=4

QoGST: g=2, Gm=Gb=3, T=4

GLST: G=3, gm=gb=2, T=2

, 3 groups

LSTBC cannot work in this case!

• LSTBC requires

• GLST (QoGST) requires 1mTn m g≥ − +

1mn m g≥ − +

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

Spatial Multiplexing Gain: r

Div

ersi

ty G

ain:

d

GLST

GLST

QoGST

QoGST

GLST: G=2, gm=gb=3, T=4

QoGST: g=3, Gm=Gb=2, T=2

GLST: G=3, gm=gb=2, T=2

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FER Comparison: LSTBC, GLST & QoGST

5 10 15 20 2510

-4

10-3

10-2

10-1

100

SNR (dB)

FER

GLST

QoGST

LSTBC

K=m=6, n=4QoGST: Gm=Gb =3, g=2GLST: gm=gb=3, G=2

K=m=6, n=4LSTBC: gm=gb=3, G=2

K=m=6, n=4QoGST: Gm=Gb =2, g=3GLST: gm=gb=2, G=3

K=m=4, n=2QoGST: Gm=Gb =2, g=2GLST: gm=gb =2, G=2

3dB1 dB 4dB

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FER Comparison: LSTBC, GLST & QoGST

5 10 15 20 2510

-4

10-3

10-2

10-1

100

SNR (dB)

FER

GLST

QoGST

LSTBC

K=m=6, n=4QoGST: Gm=Gb =3, g=2GLST: gm=gb=3, G=2

K=m=6, n=4LSTBC: gm=gb=3, G=2

K=m=6, n=4QoGST: Gm=Gb =2, g=3GLST: gm=gb=2, G=3

K=m=4, n=2QoGST: Gm=Gb =2, g=2GLST: gm=gb =2, G=2

3dB1 dB 4dB

LSTBC cannot work in these cases!

12dB

11dB

QoGST and GLST have better diversity-multiplexing tradeoff than LSTBC!

Jan. 21, 2005 28

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Thank you!

Towards a Better Diversity-Multiplexing Tradeoff in MIMO SystemsMIMO SystemsDiversity GainMultiplexing GainDiversity-Multiplexing TradeoffHow to Achieve the Optimal Tradeoff?How to Achieve the Optimal Tradeoff?Group Transmission and DetectionOur ContributionsOptimal Tradeoff Function of a Group DetectorEqual Rate AllocationSize Proportional Rate AllocationEqual Rate Allocation & Size Proportional Rate AllocationEqual Rate Allocation & Size Proportional Rate AllocationOptimal Rate AllocationOptimal Rate AllocationOptimal Rate AllocationOptimal Rate AllocationRemarksProposed Scheme 1: GLST (Group Layered Space-Time)Proposed Scheme 2: QoGST (Quasi-Orthogonal Group Space-time)Design Example of Inter-group STBCTradeoff Comparison: LSTBC, GLST & QoGSTTradeoff Comparison: LSTBC, GLST & QoGSTTradeoff Comparison: LSTBC, GLST & QoGSTFER Comparison: LSTBC, GLST & QoGSTFER Comparison: LSTBC, GLST & QoGSTThank you!