NSF Workshop: Bridging the Gap between Wireless ...ece.vt.edu/thou/NSF_Workshop/Session1/Kumar.pdfMultiple-Antenna Channels,” IEEE Trans. Inform. Theory, May 2003. P Vijay Kumar

NSF Workshop: Bridging the Gap between Wireless Networking Technologies and

Advances at the Physical Layer

A DMT and Distributed Space-Time Coding Perspective

P.Vijay Kumar

Electrical-Engineering SystemsUniversity of Southern California

Los Angeles

at times on leave of absence at theIndian Institute of Science

Bangalore

August 27, 2007

P Vijay Kumar NSF Workshop: Bridging the Gap

Collaborators

I M. Anand

I P. Bhambhani

I S. Birenjith

I P. Elia

I S. Kannan

I K Raj Kumar

I Hsiao-feng Lu

I F. Oggier

I S. Pawar

I B. Sethuraman

I K. Vinodh


Overview

S D

min-cut=3

D

I Setting: quasi-static, flat-fading, coherent, channel modelwith the diversity-multiplexing gain tradeoff (DMT) asperformance measure

I present a powerful code-construction technique that cangenerate DMT-optimal codes for a variety of point-to-pointand cooperative communication networks


Thoughts on the Road Ahead

I relaxing or justifying the assumption of a synchronous network

I relaxing or justifying the assumption of a a coherent network

I what role should feedback play (if any) ?

I how does one match a space-time code to an error-correctingcode ?

I network coding or cooperative communication for wirelessnetworks (or both) ?

I extensions of current line of research:I determine the DMT of a general, single-source, single-sink

networkI extend to the case of multiple source-sink pairsI build up an toolbox of codes for various situationsI develop faster decoding algorithms


Point-to-Point MIMO Communication Setting

●

●●

TransmitAntenna Array

●

●

●

ReceiveAntennaArray

tn rn

I quasi-static channel - fixed for T channel usesI flat fadingI receiver knows fade coefficients {hij}I focus on large SNR behaviorI Channel model: given space-time code matrix X and noise

matrix W :

Y = HX + W , Wij ∈ CN (0, 1)


Diversity-Multiplexing Gain Tradeoff (DMT) nt = nr = 4

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

12

14

16

Multiplexing Gain rD

iver

sity

Gai

n d(

r)

I Spatial multiplexing gain r defined by:

Rate R = r log SNR bits/channel use .

I Diversity gain d(r) defined by:

limSNR→∞

Pe(r).= SNR−d(r).

L. Zheng & D. Tse, “Diversity and Multiplexing: A Fundamental Tradeoff in

Multiple-Antenna Channels,” IEEE Trans. Inform. Theory, May 2003.


DMT-Optimal Space-Time Code Construction

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

12

14

16

Multiplexing Gain r

Div

ersi

ty G

ain

d(r)

I H. Yao and G. W. Wornell, “Achieving the full MIMO diversity-multiplexingfrontier with rotation-based space time codes,” in Proc. Allerton Conference,Oct. 2003. (2 antenna case)

I H. El Gamal, G. Caire and M. O. Damen, “Lattice Coding and DecodingAchieve the Optimal DiversityMultiplexing Tradeoff of MIMO Channels,” IEEETrans.Inform Theory, June 2004. (lattice-based construction, not explicit)

I P. Elia, K.Raj Kumar, S. A. Pawar, PVK, Hsiao-feng Lu, “Explicit,Minimum-Delay Space-Time Codes Achieving the D-MG Tradeoff ”, IEEETrans.Inform Theory, Sep. 2006. (general, explicit and minimum-delay)


References on CDA Codes

I B. A. Sethuraman, B. Sundar Rajan, and V. Shashidhar, “Full-Diversity,High-Rate, Space–Time Block Codes From Division Algebras,” IEEE Trans.Info. Theory, vol. 49, no. 10, pp. 2596–2616, Oct. 2003.

I J. -C. Belfiore and G. Rekaya,“Quaternionic Lattices for Space-Time Coding,” inProc. IEEE Information Theory Workshop, Paris, France, Mar-Apr. 2003, pp.267270.

I J.-C. Belfiore, G. Rekaya and E.Viterbo, “The Golden code: a 2× 2 full-ratespace-time code with non-vanishing determinants,” IEEE Trans. Inform.Theory, vol.51, No. 4, April 2005, pp. 1432-1436.

I F. Oggier, G. Rekaya, J. C. Belfiore, E. Viterbo, “Perfect space-time blockcodes,” IEEE Trans. Inform. Theory, vol. 52, no. 9, pp. 3885–3902, Sep. 2006.

I P. Elia, B. Sethuraman, and PVK, “Perfect Space-Time Codes for Any Numberof Antennas,” to appear in the Nov. 2007 issue of the IEEE Trans. Info. Theory.


Properties of a CDA-Based Code

I DMT optimal for every statisticaldescription of the fading channel(approximate universality)

I zero error probability inno-outage region

I optimal codes for MIMO-OFDM

I Examples include perfect codessuch as the Golden code

I can be used to construct optimalcodes for the 1-bit ARQ channel

I enable construction of MIMOdiversity-embedded ST codes

●

●●

TransmitAntenna Array

●

●

●

ReceiveAntennaArray

1bit of noiseless feedback

tn rn

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

12

14

16

Spatial Multiplexing Gain r=R/log(SNR)

Div

ersi

ty G

ain

d(r)

L=1L=2L=4L=8L=20


References

I S. Tavildar and P. Viswanath, “Approximately universal codes over slow fadingchannels,” IEEE Trans. Info. Theory, July 2006.

I Sameer A. Pawar, K.Raj Kumar, Petros Elia, PVK and B.A.Sethuraman, “Minimum-Delay Space-Time Codes Achieving the DMD Tradeoff of theMIMO-ARQ Channel,” accepted for publication in the IEEE Trans. Inform.Theory.

I Petros Elia and PVK, “Approximately universal, explicit, DMT-optimalconstructions for the dynamic-decode-and-forward cooperative wireless relaynetwork,” Available at arXiv:0706.3502v1 [cs.DM] 24 Jun 2007.

I S. Diggawi, D. Tse, “Fundamental Limits of Diversity-Embedded Codes overFading Channels,” presented at ISIT 2005.


Cooperative Relay Communication

S D

Rn

R2gn

g2

hn

...

h2

g1

I quasi-static assumption

I half-duplex nodes

I flat fading

I {gi , hi} are channel-fading coefficients and are known to the receiver

I two classes of protocolsI amplify and forward (AF)I decode and forward (DF)


DMT of Several Protocols

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

DMT of Various Protocols (2 relays)

Multplexing Gain r

Div

ersi

ty G

ain

TDB

DDF

OAF, NAF

NSDF

SAF

OSDF


References

I A. Sendonaris, E. Erkip, and B. Aazhang, “User Cooperation Diversity–Part I:System Description,” IEEE Trans. Commun., vol. 51, no.11, pp. 1927–1938,Nov. 2003.

I J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative Diversity inWireless Networks: Efficient Protocols and Outage Behavior,” IEEE Trans. Info.Theory, vol. 50, no. 12, pp. 3062-3080, Dec. 2004.

I J. N. Laneman and G. W. Wornell, “Distributed Space-Time Coded Protocolsfor Exploiting Cooperative Diversity in Wireless Networks,” IEEE Trans. Inform.Theory, vol. 49, no. 10, pp. 2415-2525, Oct. 2003.

I K. Azarian, H. El Gamal, and P. Schniter, “On the AchievableDiversity-Multiplexing Tradeoff in Half-Duplex Cooperative Channels,” IEEETrans. Info. Theory, vol. 51, no. 12, pp. 4152–4172, Dec. 2005.

I Petros Elia, K. Vinodh, M. Anand and PVK, “D-MG Tradeoff and OptimalCodes for a Class of AF and DF Cooperative Communication Protocols,”submitted to the IEEE Trans. Inform. Theory, Jan. 2007.


References

I S. Yang and J.-C. Belfiore, “Optimal Space-Time Codes for the MIMOAmplify-and-Forward Cooperative Channel,” submitted to IEEE Trans. WirelessComm., Sept. 2005.

I S. Yang and J.-C. Belfiore, “Towards the Optimal Amplify-and-ForwardCooperative Diversity Scheme,” submitted to IEEE Trans. Wireless Comm.,Mar. 2006.

I M. Yuksel and E. Erkip, “Cooperative Wireless Systems: ADiversity-Multiplexing Tradeoff Perspective,” submitted to IEEE Trans. Info.Theory, Sept. 2006. Available: http://arxiv.org/pdf/cs.IT/0609122.


DMT-Optimal Codes under an AF Protocol

Channel model under an example AF protocol:

[yT

1| yT

2

]=

[g1 g2h2

] [xT

1 xT2

xT1 AT

2

]+ nT

can be rewritten as

Y = HX + W

[y

1y

2

]=

[g1I g2h2A

T

g1I

] [x1

x2

]+

[n1

n2

]Approximate universality property can be invoked to prove DMToptimality of CDA codes.S. Yang and J.-C. Belfiore, “Optimal Space-Time Codes for the MIMO

Amplify-and-Forward Cooperative Channel,” IEEE Trans. Inform. Th., Feb. 2007 .


DMT-Optimal Code Construction under a DF Protocol

Harder since code design has to be simultaneously optimal formultiple channels, but surprisingly do-able:

yT = H

26664

x t1(1)

x t2(1)

x t2(3)

x t3(1)

x t3(3)

37775 + wT

where

2664

x i (1)x i (2)x i (3)x1(4)

3775 = φi

0BB@

2664

`0 γσ(`3) γσ2(`2) γσ3(`1)`1 σ(`0) γσ2(`3) γσ3(`2)`2 σ(`1) σ2(`0) γσ3(`3)`3 σ(`2) σ2(`1) σ3(`0)

3775

1CCA .

I Petros Elia and PVK, “Approximately universal, explicit, DMT-optimalconstructions for the dynamic-decode-and-forward cooperative wireless relaynetwork,” Available at arXiv:0706.3502v1 [cs.DM] 24 Jun 2007.


Multi-hop Networks

S DD

Signalling schemes need to be able to deal with:

I broadcast nature of the network

I presence of interference

I half-duplex operation of nodes


Layered Networks

S D

I studied under clustering and non-clustering ( nodes in intermediate layers)situations

I DMT in the clustered case known and and optimal code under DF exists

I in the non-clustered case, DMT under AF protocol is known, optimal code yetto be found

I S. Borade L. Zheng and R. Gallager, “Amplify and forward in wireless relaynetworks: Rate, Diversity, and Network Size,” to appear in IEEE Trans. Inform.Theory, Oct. 2007.

I ShengYang and Jean-Claude Belfiore, “ of MIMO Multi-hop Relay Channels,”arXiv:0708.0386v1 [cs.IT] 2 Aug 2007.


K -path Multi-hop Networks

S D

min-cut=3

D

I min-cut yields the diversity gain d(0)I under full-duplex operation, if min-cut K occurs at either

source or destination then DMT of such a network is given by

d∗(r) = K (1− r) .

and is achievable (CDA codes for the parallel channel).


K Isolated Paths under Half-Duplex Operation

S D DS D

Here with few exceptions (figure on right), a DMT of K (1− r) canbe shown to be achieved.This is despite the possible presence of backflow:

S D

backflow

A

B C

A

B A C

CA

C

B


Half-Duplex Operation in Presence of Interference

I Certain types of interference canbe handled and permit a DMT ofK (1− r) to be achieved.

I Other types of interference appearto be harder to deal with.

I Layered K -path networks can alsobe handled.

S DD

S D

Problematicinterference

S D


Thanks!


Back-up Slides


Form of the Cyclic-Division-Algebra Code

Code matrices are of the form

X =

24 `0 γσ(`2) γσ2(`1)

`1 σ(`0) γσ2(`2)`2 σ(`1) σ2(`0)

35

● ● ●

●●●

●

●

●

●

●

● ● ●

● ●

where the `i are derived from

information-bearing symbolsz }| {�`0 `1 `2

�=

basis vectorsz }| {�γ1 γ2 γ3

�

⇑QAM message symbolsz }| {24 `0,1 `1,1 `2,1

`0,2 `1,2 `2,2

`0,3 `1,3 `2,3

35


Documents

NSF Workshop: Bridging the Gap between Wireless ...ece.vt.edu/thou/NSF_Workshop/Session1/Kumar.pdfMultiple-Antenna Channels,” IEEE Trans. Inform. Theory, May 2003. P Vijay Kumar