MIMO Multihop Relay Channel from the Diversity Perspective www.comelec.enst.fr/~syang Ecole Nationale Supérieure des Télécommunications (ENST) 46, rue Barrault, 75013 Paris France May 3 rd , 2007 CNRS-LSS, Supélec Sheng Yang Joint work with Prof. Jean-Claude Belfiore
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MIMO Multihop Relay Channel from the Diversity Perspective
www.comelec.enst.fr/~syangEcole Nationale Supérieure des Télécommunications (ENST)
46, rue Barrault, 75013 ParisFrance
May 3rd, 2007CNRS-LSS, Supélec
Sheng YangJoint work with Prof. Jean-Claude Belfiore
LAN
How to cooperate to optimize the diversity ??
Outline
• System Model and Assumptions
• Part I : Amplify-and-Forward
• Part II : Achieving the Upper Bound via Partition
• Conclusions
3
Outline
• System Model and Assumptions
• Part I : Amplify-and-Forward
• Part II : Achieving the Upper Bound via Partition
• Conclusions
System Model and Assumptions
• N-hop channel, N+1 virtual nodes equipped with multiple antennas.
• Rayleigh slow fading, i.i.d.
• Node # i can only hear node # i-1.
• All relays work in full-duplex mode (generalization to half-duplex mode trivially).
• Channel state information (CSI) at receiver only.
source destination...
5
... .........relays
n0 n1 nN-1 nN
• N MIMO subchannels
• Transmitted signal is function of received signal
• The relaying functions depends on the relaying strategy.
• Same transmit power constraint is imposed for all nodes.
6
channel 1 channel N
Diversity-Multiplexing Tradeoff[Zheng and Tse, IT 03]
Definition
A channel is said to have a diversity-multiplexing tradeoff (DMT) if for each multiplexing gain ,
and .
7
Definition
Two channels are said to be equivalent if they have the same DMT.
Outline
• System Model and Assumptions
• Part I : Amplify-and-Forward
• Part II : Achieving the Upper Bound via Partition
• Conclusions
• Each antenna simply normalizes the received signal to the same power level
• The end-to-end MIMO channel is
9
Amplify-and-Forward (AF)
channel 1 channel N
.
.
Rayleigh Product Channel
Lemma
The AF multihop channel is equivalent to the Rayleigh product (RP) channel defined by
with
.
We identify the AF multihop channel with the RP channel.
10
DMT of the Rayleigh Product Channel
Theorem
The DMT of the (n0,n1,...,nN) RP channel is a piecewise-linear function connecting the points (k, d(k)), where
11
• For N=1, the DMT is
• For N=2, the DMT is
.
.
.
with being ordered values of (n0,n1,...,nN).
Properties of the DMT (I)
12
• Only depends on the ordered version of (n0,n1,...,nN).
• The diversity bottleneck of the RP channel is an Rayleigh channel, since
.
Properties of the DMT (II)
13
The DMT equivalence class is uniquely represented by the minimal form.
Theorem (Reduction)
The (n0,n1,...,nN) channel is equivalent to the channel if and only if
( )
• In particular, iff .
Definition
is said to be the minimal (horizontal) form if is the minimum integer such that is satisfied. is the order of the RP channel.
( )
Intuition from the DMT (I)
14
Rayleigh MIMO channel
If k is the “network flow”, then d(k) is the minimum “cost” to limit the network flow to k. In particular, d(0) is the “disconnection cost”.
... ... ... ...
......
......
Intuition from the DMT (II)
15
Rayleigh product channel
Example : (2,2,2)
16
basis change
diversity 3!!
diversity 4
canonical basis
Example : The (5,...,5) Channel
17
0
5
10
15
20
25
0 1 2 3 4 5
multiplexing gain
dive
rsity
gai
n
Outline
• System Model and Assumptions
• Part I : Amplify-and-Forward
• Part II : Achieving the Upper Bound via Partition
• Conclusions
Upper Bound on the Diversity
19
• Proved using the cut set bound or data processing theorem.
• The diversity bottleneck of the upper bound is one of the subchannels. Generally larger than the AF diversity.
Theorem
The DMT of the (n0,n1,...,nN) multihop channel with any relaying strategy is upper-bounded by
.
UB AFdiversity
Serial Partition
• The all-DF scheme partitions the multihop channel into N serial subchannels.
• In general, K serial AF subchannels by K-1 intermediate decoding nodes D1, ... ,DK-1, and the DMT is
• Smaller partition size means lower complexity.
The DMT upper bound is achieved when all relaying clusters have full antenna cooperation and the decode-and-forward (DF) scheme is used.
.
20
Example : (3, 1, 4, 2)
21
diversity 2
diversity 3
diversity 33 4 8
3 2
3 8
all DF
partial DF
partial DF
22
Out
age
Prob
abili
ty
Received Eb/N0
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All AF All DF(3,1)+(1,4,2)(3,1,4)+(4,2)
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Example : (3, 1, 4, 2)
Distributed Schemes
23
• In most cases, no intermediate decoding is possible.
• Distributed space-time processing is needed.
Example (1, n, 1)
...• Relays do the coding [Jing and Hassibi, TWC06]: linearly
process the received signal in different ways and forward.
• Source does the coding [Elia et al., Allerton05]: only one relay is used at a time, n independent parallel channels in the time domain.
Parallel Partition
24
Example (2, 2, 2)
Lemma
There exist exactly independent paths in an (n0,n1,...,nN) multihop channel.
Theorem
parallel AF subchannels are enough to achieve the maximum diversity.
1
1
1
1
Option I
2
2
Option II
Flip-and-Forward
25
Both maximum diversity gain and multiplexing gain are achieved in a completely distributed way !!
+
-
Theorem
By creating parallel AF subchannels, the Flip-and-Forward (FF) scheme achieves the maximum diversity. And the DMT is lower-bounded by that of the AF scheme.
-
26
Flip-and-Forward : Generalization
27
+
- --
Flip-and-Forward : Generalization
Non-Independent Partition
28
• Independent partition is not a necessary condition for maximum diversity.
• The size of non-independent partition is smaller than the independent one.
Partition size is two instead of four!!
Example (2, 2, 2, 3)
Coding the Multihop Channel
29
• With the DF scheme, the source and the intermediate decoding nodes use DMT-achieving codes, e.g., the Perfect codes [Oggier et al. IT06, Elia et al. IT06].
• With the FF scheme, the relays flip (and amplify) the received signal in a coordinated way to create K parallel AF subchannels . The source transmits an M x L matrix in the subchannel , i.e.,
The DMT-achieving ST codes with minimum delay is proposed in [Yang and Belfiore, ISIT06] with L=M. Hence the total coding delay is .
.
30
Sym
bol E
rror
Rat
e
Received Eb/N0
Performance : AFF vs. AF
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
0 3 6 9 12 15 18 21 24 27 30
Symbol Error Rate
Eb/N0(dB)
2222, AFF vs AF
(2,2,2,2), AFF(2,2,2,2), AF(2,4,3), AFF(2,4,3), AF
(2,2,2,2) AFF (2,2,2,2) AF (2,4,3) AFF (2,4,3) AF
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1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
0 3 6 9 12 15 18 21 24 27 30
Symbol Error Rate
Eb/N0(dB)
2222, AFF vs AF
(2,2,2,2), AFF(2,2,2,2), AF(2,4,3), AFF(2,4,3), AF
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
0 3 6 9 12 15 18 21 24 27 30
Symbol Error Rate
Eb/N0(dB)
2222, AFF vs AF
(2,2,2,2), AFF(2,2,2,2), AF(2,4,3), AFF(2,4,3), AF
Outline
• System Model and Assumptions
• Part I : Amplify-and-Forward
• Part II : Achieving the Upper Bound via Partition
• Conclusions
31
Conclusions
• Both distributed and non-distributed relaying schemes are proposed for the MIMO multihop channel.
• Maximum diversity gain is achieved via partition : serial or parallel.
• Serial partition is intermediate decoding, when and where to decode.
• From parallel partition to flip-and-forward, completely distributed, minimum relaying delay and complexity.
• DMT-achieving coding schemes available.
Thank you for your patience!!32
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