Multi-User Communication: Capacity, Duality, and Cooperation. Nihar Jindal Stanford University Dept. of Electrical Engineering

Transcription

1 Multi-User Communication: Capacity, Duality, and Cooperation Niar Jindal Stanford University Dept. of Electrical Engineering February 3, 004

2 Wireless Communication Vision Cellular Networks Wireless LAN s Wireless Broadband Ad-Hoc Networks Distributed Sensing Many more

3 Callenges in Wireless Communication Multi-user interference Time-varying cannels Cross-layer integration Hig data rate requirements Limited power budget Cost constraints

4 Outline Cannel Capacity Multi-user Cannels Multi-user MIMO Cannels Cooperation in Ad-Hoc Networks

5 Cannel Capacity Set of all simultaneously acievable data rates {R, R,,R K } R R R Motivation: Fundamental system limits Motivate practical transmission strategies R

6 Cannel Capacity: Network erspective Application Transport Network ysical/mac Capacity limits performance of pysical/mac layer How large of a multi-user bit pipe do I ave? Can consider iger-layer issues (sceduling, routing, etc.) using tese limits No delay constraints No complexity constraints

7 Cross-Layer Design Application Transport Network ysical/mac Transmission of multimedia over wireless Adapt video rate (quality) and routing to cannel conditions Reallocate resources (power, rate) at base station in response to network conditions Use pat diversity (network coding) for robustness

8 Single Cell Model Base Station Downlink (Broadcast) Uplink (Multiple-Access) Mobiles Downlink: Base as independent information to send to eac mobile (unicast messages) Uplink: Eac mobile as independent information to send to te base station

9 System Model Gaussian BC and MAC: same cannel gains and noise power ( n) ( n) z ( n ) x ( n ) ( ) x z(n) x + y ( n ) K (n) + y(n) x(n) () K (n) z K (n) x K (n) x x + y K (n) ( K ) Multiple-Access Cannel (MAC) Broadcast Cannel (BC)

10 Comparison of MAC & BC Differences: ower constraints Similarities: Gaussian inputs optimal Successive decoding optimal Capacity regions of BC & MAC known, but no relationsip between te two

11 BC Capacity Region = 5, = = C BC ( ;, )

12 Capacity Region Relationsip = 5, = Blue = BC Red = MAC =, = = C (, ;, ) CBC ( ;, ) MAC +

13 Capacity Region Relationsip = 5, = Blue = BC Red = MAC =, = =.5, =0.5 = C (, ;, ) CBC ( ;, ) MAC +

14 Duality: MAC to BC Blue = BC Red = MAC C BC ( ;, ) = Υ C MAC (, ;, ) 0

15 Sum ower Constraint MAC Union of MAC s = MAC wit sum power constraint ower pooled between MAC transmitters Transmitters ave independent messages Uplink Downlink Same capacity region!

16 Cannel Scaling Find MAC in terms of dual BC Scale cannel gain by, power by α α α α + + α α + + MAC BC MAC capacity unaffected by scaling

17 Scaled Cannel Capacity Scaled MAC subset of scaled BC 0 ), ; ( ), ;, ( > + α α α α α C C BC MAC Unscaled MAC capacity region is a subset of every scaled BC capacity region 0 ), ; ( ), ;, ( > + α α α C C BC MAC

18 Duality: BC to MAC Blue = Scaled BC Red = MAC Ι 0 ), ; ( ), ;, ( > + = α α α C C BC MAC α = α > 0 α >

19 Duality: Constant Cannels BC in terms of MAC MAC in terms of BC Υ MAC BC C C = 0 ), ;, ( ), ; ( Ι 0 ), ; ( ), ;, ( > + = α α α C C BC MAC

20 Multi-User MIMO Cannels X y BASE X M y K Cellular systems Future wireless LAN s? Wireless broadband (80.6)

21 Single-User MIMO x x x 3 y y y 3 ) (0, ~, I N n n n x x y y n Hx y m n mn m n m σ + = + = Μ Μ Λ Μ Ο Μ Λ Μ

22 Dual MIMO MAC-BC Same cannel gains and noise power on uplink/downlink (similar to a TDD system) n H + y x T H n x DOWNLINK/BC ULINK/MAC + y y H + x T H n y + = Hx + n, y = Hx n T T + y = H x + H x n

23 MIMO MAC & BC MIMO MAC Successive decoding optimal Capacity region known MIMO BC Capacity region not known Use dirty paper coding (DC) to acieve lower bound (acievable region) to capacity region Duality between MIMO MAC capacity region and MIMO BC acievable region

24 Optimal Transmission for BC Scalar BC Transmit sum of codewords (x and x ) Successive decoding (stronger user) MIMO BC Cannot use successive decoding Cannot always order users -TX antenna, -RX antenna example H =[ 0.5], H =[0.5 ] re-cancel interference at transmitter X = X +X H = H =5

25 Dirty aper Coding If additive interference known at TX but not RX, capacity same as if no interference Original result by Costa -D modulo sceme: Tomlinson-Harasima precoding (cannel equalization) N-D modulo sceme: Erez, Samai, Zamir Y=X+S+N X: Transmitted signal Y: Received signal S: Interference (known at TX but not at RX) N: Noise

26 Dirty aper Coding for BC Coding sceme (Caire & Samai): Coose codeword for user Treat tis codeword as interference to user ick signal for User using pre-coding Receiver experiences no interference: T = log(det(i + H Σ H )) R Signal for Receiver interferes wit Receiver : R det(i + H Σ ( = log det(i + H + Σ T ΣH Encoding order can be switced ) H ) T )

27 C DC Acievable Rate Region DC () = Υ (R R (R R det(i + H( Σ = log det(i + H = log = log ( T det(i + H )) ΣH ( T det(i + H Σ H )) det(i + H ( Σ = log det(i + H + Σ) H T Σ H ) Tr ( Σ +Σ ) ) H ) Union over all covariance matrices wic meet power constraint Equal to actual capacity region for scalar case Larger tan successive decoding region ),, + Σ Σ H T ( Σ, Σ ) T T ), ) )

28 MIMO MAC-BC Duality Teorem: Acievable BC region equals union of dual MAC capacity regions C DC ( ) = Υ 0 C MAC (, ) roof metod: For every MAC transmission strategy, sow tere exists an equivalent (i.e. same rates, same power) BC transmission strategy, vice versa

29 MIMO MAC-BC Duality Dual MAC Capacity Regions DC Region = Lower Bound C DC ( ) = Υ 0 C MAC (, )

30 Sum Rate Capacity Sum Rate Capacity: Maximum trougput (i.e. total data rate) acievable Sum rate capacity unknown for almost all nondegraded broadcast cannels Used dual caracterization to sow DC region acieves sum-rate capacity for general MIMO BC (arbitrary # of antennas, users) First sown by Caire & Samai for users, single receive antennas

31 Sato Upper Bound Definition: H y y = Hx + n H =, y = H y n x H H + n y Joint receiver wit correlated noise ( ) Σ z + y BC capacity only depends on marginals p(y x), p(y x), not on joint p(y,y x) BC sum rate capacity Cooperative capacity Get equality for worst-case noise correlation

32 MIMO BC Sum Rate Capacity Sato Upper Bound DC Region = Lower Bound roof uses Lagrangian duality

33 Sceduling & MIMO BC Coose point on region boundary according to function of queue lengts Use dual MAC caracterization to find boundary of entire region

34 Ad-Hoc/Sensor Networks No wired infrastructure Node to node communication May be underlying pysical penomenon

35 Ad-Hoc Network System Model TX RX G > G > TX RX TX as a message for RX, TX as a different message for RX Wat if TX and TX cooperate? Wat if RX and RX cooperate? Total system-wide power constraint Different gains for TX-TX cannel vs TX-RX cannel

36 Transmitter Cooperation TX RX TX RX Transmitters excange messages Use fraction of system power on separate frequency band Cooperatively transmit (-antenna BC) Upper bounded by broadcast cannel capacity

37 Receiver Cooperation TX RX TX RX Eac receiver amplifies-and-forwards own signal to oter receiver Use fraction of system power Similar to aving a second antenna Upper bounded by multiple-access capacity region

38 Receiver and Transmitter Cooperation TX RX TX RX Combines bot cooperation tecniques Upper bounded by MIMO capacity transmit antenna, receive antenna system

39 Capacity Region: G=0 db TX coop No coop RX coop TX cooperation rates significantly larger

40 Numerical Results: G=0 db MIMO Bound MAC/BC Bound TX coop No coop RX coop TX coop and TX/RX coop are identical

41 Numerical Results: G=0 db MIMO Bound MAC/BC Bound TX coop No coop RX coop TX coop acieves BC upper bound

42 Rate vs. Cannel Gain Acievable rates vs. ower (G) As G increases, approac upper bounds

43 Cooperation Insigts TX cooperation alone performs well RX cooperation quite poor TX and RX cooperation only useful wen G is very large Rate increases of 50-00% feasible via cooperation

44 Conclusions Developed duality teory for MAC and broadcast cannels Establised extension of duality for MIMO cannels Found sum rate capacity of MIMO broadcast cannels Sown cooperation can significantly improve data rates in ad-oc networks

45 Future Work Duality generalization Multi-user MIMO cannels wit imperfect cannel knowledge Sensor/ad-oc networks: cooperation, transmission of correlated data, finite energy systems Cross-layer resource allocation for wireless systems