Competition and Cooperation in Multiuser Communication Environments

Transcription

1 Competition and Cooperation in Multiuser Communication Environments Wei Yu Electrical Engineering Department Stanford University April, 2002 Wei Yu, Stanford University

2 Introduction A multiuser communication environment is a competitive environment. z 1 x 1 x 2 Multiuser z 2 y 1 y 2 z m 1 x n 1 Channel z m y m 1 x n y m What is the role of competition? What is the value of cooperation? Wei Yu, Stanford University 1

3 Multi-Antenna Wireless Communication Examples: broadcast wireless access (802.11), wireless local loop (WLL) User 1 User 2 User N Base-Station Cooperation among multiple antennas within the same user Competition among the users Wei Yu, Stanford University 2

4 Digital Subscriber Lines (DSL), Ethernet DSL and Ethernet environments are interference-limited. FEXT User 1 User 2 central office downstream upstream NEXT User N Explore the benefit of cooperation. Manage the competition. Wei Yu, Stanford University 3

5 Goal To characterize channel capacity, optimum spectrum, and coding for Multiple Access Broadcast Interference assuming Gaussian noise. To illustrate the value of cooperation in these scenarios. Wei Yu, Stanford University 4

6 Gaussian Vector Channel Capacity: C =maxi(x; Y). Z n W 2 nc X n Y n Ŵ (Y n ) H Optimum Spectrum: Water-filling maximize 1 2 log HS xxh T + S zz S zz subject to tr(s xx ) P, S xx 0. Wei Yu, Stanford University 5

7 Multiple Access Channel No transmitter coordination. Only receiver coordination. z 1 x 1 z 2 y x 2 Capacity? Optimum Spectrum? Coding? Wei Yu, Stanford University 6

8 Capacity Region for Multiple Access Channel W 1 2 nr 1 W 2 2 nr 2 X n 1 (W 1 ) X n 2 (W 2 ) H 1 H 2 Z n Y n Ŵ 1 (Y n ) Ŵ 2 (Y n ) Capacity region: R 1 I(X 1 ; Y X 2 ); R 2 I(X 2 ; Y X 1 ); R 1 + R 2 I(X 1, X 2 ; Y). Ahlswede ( 71), Liao ( 72), Cover-Wyner ( 73) Wei Yu, Stanford University 7

9 Coding for Multiple Access Channel Superposition coding and successive decoding achieves I(X 1, X 2 ; Y): [ x1 x 2 ] H z Feedforward Decision [ ˆx1 ˆx 2 ] Feedback Implementation: Generalized Decision-Feedback Equalizer (GDFE). Cioffi, Forney ( 97), Varanasi, Guess ( 97) Wei Yu, Stanford University 8

10 Capacity Pentagon R 1 I(X 1 ; Y X 2 ) I(X 1 ; Y ) I(X 2 ; Y ) I(X 2 ; Y X 1 ) R 2 Fix an input distribution p(x 1 )p(x 2 ), the capacity region is a pentagon. Wei Yu, Stanford University 9

11 Vector Multiple Access Capacity Region R 1 Maximum R 1 + R 2 max(r 1 + R 2 ) max I(X 1, X 2 ; Y) over all p(x 1 )p(x 2 ). R 2 Wei Yu, Stanford University 10

12 Uplink Power Control in Wireless Systems Successive decoding achieves capacity in multiple access channels. User 1 User 2 User N Base-Station If channel state is known at the transmitter... What is the optimal power allocation? Wei Yu, Stanford University 11

13 Sum Capacity Maximization X 1 N(0,S 1 ) X 2 N(0,S 2 ) H 1 H 2 Z N(0,S zz ) Y maximize 1 2 log H 1S 1 H1 T + H 2 S 2 H2 T + S zz S zz subject to tr(s i ) P i, i =1, 2 S i 0, i =1, 2 Maximizing a concave objective with convex constraints. Wei Yu, Stanford University 12

14 Competitive Optimum Optimum S 1 is a water-filling covariance against S 2. C(S 1,S 2 ) Optimum S 2 is a water-filling covariance against S 1. (S 1,S 2) can be reached by eachuser iteratively waterfilling against eachother. (S 1,S 2) S 2 S 1 Multiple Access Channel Sum Capacity = Competitive Optimum Wei Yu, Stanford University 13

15 Iterative Water-filling Theorem 1. The iterative water-filling process, where each user water-fills against the combined interference and noise, converges to the sum capacity of a Gaussian vector multiple access channel. R 1 R 1 (S (0) 1,S(0) 2 ) (S (1) 1,S(0) 2 ) (S 1,S 2 ) R 2 R 2 S (0) 1 S (0) 2 S (1) 1 S (1) 2 (S 1,S 2) Wei Yu, Stanford University 14

16 Related Works Multiple access channel with ISI: Cheng and Verdu ( 93). Multiple access fading channel: Single-antenna: Knopp and Humblet ( 95), Hanly and Tse, ( 98). Multi-antenna (asymptotic): Viswanath, Tse, Anantharam ( 00) Multi-pathfading channels: Medard ( 00) CDMA channels: Viswanath and Anantharam ( 99), Yates, Rose ( 00) Iterative water-filling is a generalization for vector multi-access channels Wei Yu, Stanford University 15

17 Multi-user Diversity in Wireless Systems Solves the power control problem for multi-antenna fading channels: User 1 User 2 User N Base-Station Single receive antenna: one user should transmit at the same time. Multiple receive antennas: multiple users transmit at the same time. Wei Yu, Stanford University 16

18 Results So Far Vector Channel Multiple Access Broadcast C =maxi(x; Y) C =maxi(x 1, X 2 ; Y) Water-filling Iterative Water-filling Vector Coding GDFE?????? Wei Yu, Stanford University 17

19 Broadcast Channel Coordination at transmitter. No receiver coordination. z 1 x z 2 y 1 y 2 Capacity? Optimum Spectrum? Coding? Wei Yu, Stanford University 18

20 Broadcast Channel Capacity Introduced by Cover ( 72) Superposition coding: Cover ( 72). Degraded broadcast channel: Bergman ( 74), Gallager ( 74) Coding using binning: Marton ( 79), El Gamal, van der Meulen ( 81) Sum and product channels: El Gamal ( 80) Gaussian vector channel, 2 2 case: Caire, Shamai ( 00) General capacity region is a well-known open problem. We focus on a non-degraded Gaussian vector broadcast channel. Simultaneous and independent work was done by Vishwanath, Jindal, Goldsmith, and Viswanath, Tse. Wei Yu, Stanford University 19

21 Gaussian Vector Broadcast Channel Z 1 X 1 N(0,S 1 ) X H 1 Z 2 Y 1 X 2 N(0,S 2 ) H 2 Y 2 Superposition coding gives: R 1 = I(X 1 ; Y 1 )= 1 2 log H 1S 1 H T 1 + H 1 S 2 H T 1 + S z1 z 1 H 1 S 2 H T 1 + S z 1 z 1 R 2 = I(X 2 ; Y 2 )= 1 2 log H 2S 2 H T 2 + H 2 S 1 H T 2 + S z2 z 2 H 2 S 1 H T 2 + S z 2 z 2 Wei Yu, Stanford University 20

22 Writing on Dirty Paper Gaussian Channel... with Transmitter Side Information Z N(0,Szz) S N(0,Sss) Z N(0,Szz) X Y X Y C = 1 2 log S xx + S zz S zz C = 1 2 log S xx + S zz S zz Capacities are the same if S is known non-causally at the transmitter. Based on Gel fand and Pinsker ( 80), Heegard and El Gamal ( 83). Gaussian scalar channel: Costa ( 81). Vector channel: Yu, et al ( 01). Generalizations: Cohen, Lapidoth ( 01), Erez, Zamir, Shamai ( 01). Wei Yu, Stanford University 21

23 New Achievable Region Z n 1 W 1 2 nr 1 X n 1 (W 1,X n 2 ) X n H 1 Z n 2 Y n 1 Ŵ 1 (Y n 1 ) W 2 2 nr 2 X n 2 (W 2) H 2 Y n 2 Ŵ 2 (Y n 2 ) R 1 = I(X 1 ; Y 1 X 2 ) = 1 2 log H 1S 1 H T 1 + S z1 z 1 S z1 z 1 R 2 = I(X 2 ; Y 2 ) = 1 2 log H 2S 2 H T 2 + H 2 S 1 H T 2 + S z2 z 2 H 2 S 1 H T 2 + S z 2 z 2 Wei Yu, Stanford University 22

24 Converse Broadcast capacity does not depend on noise correlation: Sato ( 78). z 1 z 1 z 1 x 1 x 2 z 2 y 1 x 1 y 1 x 1 y 1 = y 2 y 2 y 2 x 2 z 2 z 2 x 2 }{{} { p(z1 )=p(z if 1) p(z 2 )=p(z 2), not necessarily p(z 1,z 2 )=p(z 1,z 2). Thus, sum-capacity C min S zz max S xx I(X; Y). Wei Yu, Stanford University 23

25 Least Favorable Noise Fix Gaussian input S xx : 1 minimize 2 log HS xxh T + S zz S zz [ ] Sz1 z subject to S zz = 1 S z2 z 2 S zz 0 Minimizing a convex function over convex constraints. Optimality condition: S 1 zz (HS xx H T + S zz ) 1 = if S zz > 0 at minimum. [ Ψ1 0 0 Ψ 2 ]. Wei Yu, Stanford University 24

26 GDFE Revisited [ x1 x 2 ] H z Feedforward Decision [ ˆx1 ˆx 2 ] Feedback Least Favorable Noise Feedforward filter is diagonal! Decision-feedback may be moved to the transmitter by precoding. R =min S zz I(X; Y) (i.e. with least favorable noise) is achievable. Wei Yu, Stanford University 25

27 Gaussian Broadcast Channel Sum Capacity Achievability: C max S xx min S zz I(X; Y). Converse (Sato): C min S zz (Diggavi, Cover 98): min S zz max S xx max S xx I(X; Y). I(X; Y) =max S xx min S zz I(X; Y). Theorem 2. Gaussian vector broadcast channel sum capacity is: C =max S xx whenever S zz > 0 at the saddle-point. 1 min S zz 2 log HS xxh T + S zz S zz Wei Yu, Stanford University 26

28 Gaussian Mutual Information Game Z N(0,S zz ) X N(0,S xx ) H Y Strategy Objective Signal Player {S xx :trace(s xx ) P } max I(X; Y) { [ ] } Fictitious Sz1 z S Noise Player zz : S zz = 1 0 min I(X; Y) S z2 z 2 Competitive equilibrium exists. Wei Yu, Stanford University 27

29 Saddle-Point is the Broadcast Capacity R(S xx,s zz ) The optimum S xx is a waterfilling covariance against S zz. The optimum Szz is a leastfavorable noise for Sxx. S xx (S xx,s zz) S zz Broadcast Channel Sum Capacity = Competitive Equilibrium Wei Yu, Stanford University 28

30 The Value of Cooperation z 1 z 1 z 1 x 1 y 1 x 1 y 1 x 1 y 1 z 2 z 2 z 2 x 2 y 2 x 2 y 2 x 2 y 2 max I(X; X + Z) maxi(x; X + Z) minmax I(X; X + Z) S xx S xx S zz S xx s.t. trace(s xx ) P s.t. S xx = [ ] S1 0 0 S 2 trace(s i ) P i, s.t. S zz = [ ] Sz1 z 1 S z2 z 2 trace(s xx ) P Wei Yu, Stanford University 29

31 Application: Multi-line Transmission in DSL Z User 1 User 2 central office User N Withcoordination, crosstalk can be pre-subtracted. Practical pre-subtraction: Tomlinson precoding. Optimal pre-subtraction: Dirty-paper precoding. Wei Yu, Stanford University 30

32 Performance: Vector DSL/Ethernet 20 VDSL lines Large improvement for short loops Courtesy: George Ginis Data rate (Mbps) Downstream No vectoring with noise A No vectoring with noise F Vectoring with noise A Vectoring with noise F Loop length (ft) Wei Yu, Stanford University 31

33 Results So Far Multiple Access Broadcast Interference C =maxi(x 1, X 2 ; Y) C =maxmini(x; Y)?? Iterative Water-filling Minimax?? GDFE GDFE Precoder?? Wei Yu, Stanford University 32

34 Interference Channel No transmitter coordination. No receiver coordination. z 1 x 1 y 1 z 2 x 2 y 2 Capacity is a difficult open problem. Wei Yu, Stanford University 33

35 DSL Interference Environment User 1 User 2 User 3 upstream central office Near-far problem: The closer user emits too much interference. Power back-off is necessary. Current system imposes a maximum power-spectral-density limit. Wei Yu, Stanford University 34

36 Power Control Problem Find an optimum (P 1 (f),p 2 (f)) to maximize: R 1 = R 2 = W 0 W 0 log log ( 1+ ( 1+ H 11 (f) 2 ) P 1 (f) N 1 (f)+ H 21 (f) 2 df, P 2 (f) H 22 (f) 2 ) P 2 (f) N 2 (f)+ H 12 (f) 2 df. P 1 (f) s.t. W 0 P 1 (f)df P 1, W 0 P 2 (f)df P 2 Finding the global optimum is computationally difficult. Wei Yu, Stanford University 35

37 Competitive Environment User 1 User 2 User 3 upstream central office Eachuser maximizes its own data rate regarding other users as noise. Non-cooperative game. Wei Yu, Stanford University 36

38 Iterative Water-filling N 1 (f) P H (f) P 2 N 2 (f) H 2 22 (f) P 2 P 1 f f P (0) 1 (0) (1) (1) (f) P (f) P (f) P (f) 2 Theorem 3. Under a mild condition, the two-user Gaussian interference game has a competitive equilibrium. The equilibrium is unique, and it can be reached by iterative water-filling. 1 2 Wei Yu, Stanford University 37

39 Distributed Power Control for DSL P 1 P 2 Iterative Water-filling Water-filling P 1 =P 1 Yes R 1 target? R 2 target? Yes P 2 =P 2 P 1 =P 1 + No No P 2 =P 2 + Control (P 1 (f),p 2 (f)) by setting (P 1, P 2 ). Wei Yu, Stanford University 38

40 Performance Rate Region 4 VDSL lines at 3000ft rates: 6.7Mbps 4 VDSL lines at 1000ft 10Mbps 21Mbps. 3000ft loops (Mbps) Iter. Water-filling Existing Method ft loops (Mbps) Competitive optimal points are much better than existing methods. Wei Yu, Stanford University 39

41 Conclusion Multiple Access Iterative water-filling achieves sum capacity. Broadcast Sum Capacity is a saddle-point of a mutual information game. Interference Competitive optimum is a desirable operating point. Wei Yu, Stanford University 40

42 Future Work Network Information Theory Multi-antenna/Multi-line Signal Processing Multiuser system design: physical layer vs network layer Applications to broadband access networks: Wireless Local Area Networks High-speed Ethernet Digital Subscriber Lines Computer Interconnects Wei Yu, Stanford University 41