Standard and Quasi-Standard Stochastic Power Control Algorithms

Transcription

1 Standard and Quasi-Standard Stochastic Power Control Algorithms Jie Luo, Sennur Ulukus, Anthony Ephremides IEEE Transaction on Information Theory, vol. 51, no. 7, July 2005 Presented by Hamed Farhadi Communication Theory Laboratory Royal Institute of Technology (KTH) Stockholm, Sweden Sep 27, 2011 Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12

2 Outline Power control in wireless networks Standard power control Standard stochastic power control Quasi-standard stochastic power control Convergence of stochastic power control algorithms Example Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12

3 Power Control in Wireless Networks Uplink scenario: Quality of service requirement: SINR i = Optimization problem: min s.t. i SINR i γ i K j=1 p j h ii p i j i h ijp j +σ 2 γ i Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12

4 Classification of Power Control Algorithms Central/distributed power control: Central: high complexity, global CSI Distributed: low complexity, local CSI, adaptive to channel variations Deterministic/stochastic power control: Deterministic: perfect knowledge of interference power Stochastic: noisy observations of interference power Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12

5 Standard Power Control SINR requirements can be represented as p I(p) I(p) is a standard function if p 0 Positivity: I(p) 0 Monotonicity: if p p, then I(p) I(p ) Scalability: ηi(p) I(ηp), η > 1 Standard power control algorithm p(n+1) = (1 α)p(n)+αi(p(n)), 0 < α 1 Converges to p, p = I(p ) if the problem is feasible. p is the feasible power vector with smallest elements. Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12

6 Standard Stochastic Power Control Ĩ(p,θ) is a random estimate of I(p) Standard stochastic interference function satisfies p 0: Mean condition: E[Ĩ(p,θ) p] = I(p) I(p) is a deterministic standard interference function. Lipschitz condition: K 1 > 0, p 1,p 2, I(p 1 ) I(p 2 ) 2 K 1 p 1 p 2 2 Growing condition (constraint on estimation noise): K 2 > 0,E[ Ĩ(p,θ) I(p) 2 p] K 2 (1+ p 2 ) Standard stochastic power control algorithm: p(n+1) = (1 α(n))p(n)+α(n)ĩ(p(n),θ(n)) Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12

7 Quasi-standard Stochastic Power Control Quasi-standard stochastic interference function satisfies: Mean condition: E[Ĩ(p,θ) p] = I(p)+g(p) I(p) is a deterministic standard interference function, and g(p) is a bias term. Bias condition: K 3 > 0,0 β(n) 1; g(p) K 3 β(n)(1+ p(n) ) Lipschitz condition: K 1 > 0, p 1,p 2 ; I(p 1 ) I(p 2 ) 2 K 1 p 1 p 2 2 Growing condition: K 2 > 0; E[ Ĩ(p,θ) I(p) g(p) 2 p] K 2 (1+ p 2 ) Standard stochastic power control algorithm: p(n+1) = (1 α(n))p(n)+α(n)ĩ(p(n)) Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12

8 Deterministic and Stochastic Standard Power Control Deterministic standard power control: p(n +1) = p(n) α(n)(p(n) I(p(n))) Stochastic standard power control: p(n+1) = p(n) α(n)(p(n) I(p(n)))+α(n)(Ĩ(p(n)) I(p(n))) }{{} estimation noise Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12

9 Probability One Convergence If n=0 α(n) =, n=0 α(n)2 <, then the standard stochastic power control algorithm converges to p, given by p = I(p ), with probability one. If n=0 α(n) =, n=0 α(n)2 <, n=0α(n)β(n) < then the quasi-standard stochastic power control algorithm converges to p, given by p = I(p ), with probability one. Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12

10 Convergence in Probability If α 0, N > 0, such that α(n) α,β(n) α, n N then ε > 0, K 8 (ε) > 0, such that in the standard and quasi-standard stochastic power control algorithm: lim suppr( p(n) p ε) K 8 α n Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12

11 Example Uplink of CDMA system with K users and M base stations Chip match filter output: z i = K j=1 pj h ij b j s j +v i Output of linear filter: y i = K j=1 pj h ij b j (c T i s j )+c T iv i Fixed base station assignment with match filter receiver: c i = s i I i (p) = γi h ii ( j i p jh ij (s T i s j ) 2 +σ 2 ) Received signal power: E[y 2 i ] = j p jh ij (s T i s j ) 2 +σ 2 I i (p) = γi h ii E[y 2 i ] γ ip i The stochastic estimate of I i (p) is: Ĩ i = γi h ii y 2 i γ i p i Ĩ is a standard stochastic interference function Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12

12 Thank you! Standard and Quasi-Standard Stochastic Power Control Algorithms Sep 27, / 12