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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
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Jan. 21, 2005 2
JWITC’05
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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Jan. 21, 2005 4
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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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Jan. 21, 2005 5
JWITC’05
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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Jan. 21, 2005 6
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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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Jan. 21, 2005 7
JWITC’05
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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Jan. 21, 2005 8
JWITC’05
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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Jan. 21, 2005 9
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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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Jan. 21, 2005 10
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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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Jan. 21, 2005 12
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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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Jan. 21, 2005 13
JWITC’05
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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Jan. 21, 2005 14
JWITC’05
Equal Rate Allocation & Size Proportional Rate Allocation
Optimized over all possible partitions under a given G
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Jan. 21, 2005 15
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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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Jan. 21, 2005 16
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Optimal Rate Allocation
m=n=8G=4
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Jan. 21, 2005 17
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Optimal Rate Allocation
m=n=8
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Jan. 21, 2005 18
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Optimal Rate Allocation
m=n=8
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Jan. 21, 2005 19
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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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Jan. 21, 2005 20
JWITC’05
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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Jan. 21, 2005 21
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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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Jan. 21, 2005 22
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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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Jan. 21, 2005 23
JWITC’05
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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Jan. 21, 2005 24
JWITC’05
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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Jan. 21, 2005 25
JWITC’05
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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Jan. 21, 2005 26
JWITC’05
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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Jan. 21, 2005 27
JWITC’05
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!
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Jan. 21, 2005 28
JWITC’05
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!
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