1. ECE559VV – Fall07 Course Project Presented by Guanfeng Liang Distributed Power Control and Spectrum Sharing in Wireless Networks

2. Outline Background Power control Spectrum sharing Conclusion

3. Background Interference is the key factor that limits the performance of wireless networks To handle interference, can optimize by means of Frequency allocation: Power control: Or, jointly - spectrum sharing: f f f

4. Power Control N users, M base stations, single channel, uplink P j - transmit power of user j h kj - gain from user j to BS k z k – variance of independent noise at BS k

5. General Interference Constraints Fixed Assignment: BS a j is assigned to user j Minimum Power Assignment: each user is assigned to the BS that maximizes its SIR Limited Diversity: BS’s in K j are assigned to user j

6. Standard Interference function Definition: Interference function I(p) is standard if for all p≥0, the following properties are satisfied. Positivity - I(p) ≥0 Monotonicity - If p ≥ p’, then I(p) ≥ I(p’). Scalability – For all a>1, aI(p)>I(ap). I FA, I MPA, I LD are standard. For standard interference functions, minimized total power can be achieved by updating p(t+1)=I(p(t)) in a distributed fashion, asynchronously. (Yates’95)

7. Spectrum Sharing Power is uniformly allocated across bandwidth W Transmission rate is not considered What should we do if power is allowed to be allocated unevenly? Can “rate” optimality be achieved in a distributed manner?

8. Settings M fixed 1-to-1 user-BS assignments Noise profile at each BS: N i (f) Random Gaussian codebooks – interference looks like Gaussian noise

9. Rate Region Pareto Optimal Point

10. Optimization Problem Global utility optimization maximization U(R 1,…,R M ) reflects the fairness issue Sum rate: U sum (R 1,…,R M ) = R 1 +…+R M Proportional fairness: U PF (R 1,…,R M ) = log(R 1 )+…+log(R M ) In general, U is component-wise monotonically increasing => optimal allocation must occur on the boundary R *

11. Examples

12. Infinite Dimension Theorem 1: Any point in the achievable rate region R can be obtained with M power allocations that are piecewise constant in the intervals [0,w 1 ), [w 1,w 2 ),…,[w 2M-1,W], for some choice of {w i } i=1. 2M-1. Theorem 2: Let (R 1,…,R M ) be a Pareto efficient rate vector achieved with power allocations {p i (f)} i=1,…,M. If h i,j h j,i >h i,i h j,j then p i (f)p j (f)=0 for all f [0,W].

13. Non-Cooperative Scenarios Non-convex capacity expression -> rate region not easy to compute Another approach: view the interference channel as a non- cooperative game among the competing users -> competitive optimal Assumptions: Selfish users user i tries to maximize U i (R i ) -> maximize R i

14. Gaussian Interference Game(GIG) Each user tries to maximize its own rate, assuming other users’ power allocation are known. Well-known Water-filling power allocation

15. Iterative Water-filling (Yu’02)

16. Equilibrium Theorem 3: Under a mild condition, the GIG has a competitive equilibrium. The equilibrium is unique, and it can be reached by iterative water-filling. Nash Equilibrium

17. Is the Equilibrium Optimal? NO! Example: h 1,1 =h 2,2 =1, h 1,2 =h 2,1 =1/4, W=1, N 1 =N 2 =1, P 1 =P 2 =P Water-filling -> flat power allocation: Orthogonal power allocation

18. Repeated Game Utility of user i : Decision made based on complete history Advantage: much richer set of N.E., hence have more flexibility in obtaining a fair and efficient resource allocation

19. Equilibriums of a Repeated Game Fact: frequency-flat power allocations is a N.E. of the repeated game with AWGN. Theorem 4: The rate R i FS achieved by frequency-flat power spread is the reservation utility of player i in the GIG. Result: If the desired operating point (R 1,…,R M ) is component-wise greater than (R 1 FS,…,R M FS ), there is no performance loss due to lack of cooperation. (Tse’07)

20. Results

21. Summary Performance optimization of wireless networks 1-D: power = power control Distributed power control with constant power allocation 2-D: power + frequency = spectrum sharing One shot GIG – iterative water-filling Repeated game 3-D: power + frequency + time Cognitive radio

22. Thank you and Questions?