Dynamic Fair Channel Allocation for Wideband Systems

Transcription

1 Outlines Introduction and Motivation Dynamic Fair Channel Allocation for Wideband Systems Department of Mobile Communications Eurecom Institute Sophia Antipolis 19/10/2006

2 Outline of Part I Outlines Introduction and Motivation 2 Single Antenna Systems System and Channel Models Optimal Dynamic Channel Allocations Wideband MAC Ergodic Capacity Wideband MAC Delay Limited Capacity Orthogonal Dynamic Channel Allocations System Representation Delay Limited Rates Max-Min Allocation

3 Outline of Part II Outlines Introduction and Motivation 3 Multiple Antenna Transmission System Model Allocation strategies Spatial Multiplexing Max-Min Allocation (SM-Max-Min) Space Time Coding Max-Min allocation (STC-Max-Min) Delay Limited Rate Allocation (DLR)

4 Outlines Introduction and Motivation 4/36 Multi-user Diversity H Frequency User1 User2 user1 user2 user1 user2 f 2 user2 user1 user2 f 1 Time Multiuser Diversity High spectral efficiency Maximum Fairness High spectral efficiency Penalty??

5 5/36 Fairness Outlines Introduction and Motivation Soft QoS Example: Internet browsing,... Prop. fairness, Max throughput,... achieve multi-user diversity. Hard QoS Example: VoIP, Real Time Video,... Is it possible to exploit MUD for this?

6 5/36 Fairness Outlines Introduction and Motivation Soft QoS Example: Internet browsing,... Prop. fairness, Max throughput,... achieve multi-user diversity. Hard QoS Example: VoIP, Real Time Video,... Is it possible to exploit MUD for this?

7 Single Antenna Systems Part I Single Antenna Systems

8 Single Antenna Systems 7/36 System Setting System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations Multiple Access Channel(MAC) Single Antenna users: K = {1,..., K } M parallel sub-channels Block Fading

9 Single Antenna Systems 8/36 Signal and Channel Settings System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations The n-th block received signal on sub-channel m is given by Y m (n) = KX k=1 q P m k (n)hm k (n)x m k (n) + Z m (n) H : the set of possible joint fading states. Assume that H is bounded The channel state at block fading n can be represented by the channel gain matrix 2 H(n) = 6 4 H 1 1 (n) H2 1 (n) HM 1 (n) H2 1 (n) HK 1 (n) HM K (n) 3 7 5

10 Single Antenna Systems 9/36 Signal and Channel Settings System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations We call power allocation 2 P (H) = 6 4 P 1 1 (H) P2 1 (H) PM 1 (H) P2 1 (H) PK 1 (H) PM K (H) Notation simplicity: P (n) = P (H (n)) Feasible power allocation: power allocation that satisfies 2 3 MX E 4 Pk m (n) 5 P k m=1

11 Single Antenna Systems System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations 10/36 Wideband MAC Ergodic Capacity Wideband MAC ergodic capacity 8 2 [ < C ERG h P i : R : X! MX R k E 4 Pk S 3 log 1 + Pm k (n) Hm k (n) 2 = 5, S K9 N E m Pm P k S m=1 0 ; k k k=1,...,k Maximum ergodic sum rate is achieved by >< Pk m (n) = 4 1 λk N 0 H >: k m(n) 2 5 if 0 otherwise H m k (n) 2 λ k λ k H m k (n) 2 where [x] + = max(0, x)

12 Single Antenna Systems System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations 11/36 Wideband MAC Delay Limited Capacity Wideband delay limited capacity is defined as C d (P) [ h P i P:E m Pm P k k k=1,...,k 8 \ < : R : X! 9 MX Pk S R k log 1 + Pm k Hm k 2 =, S K N H H k S m=1 0 ; Flat fading (Hanly and Tse, 1998) polymatroid structure = Explicit characterization. Not valid for wideband only an implicit Lagrangian characterization is possible.

13 Single Antenna Systems System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations 12/36 Wideband MAC Delay Limited Capacity Theorem For a given power constraint vector P, the boundary surface of C d (P) is the set of rate vectors R such that there exist λ R K + and for each block time n, there exist a power allocation matrix P(.), a rate allocation matrix R(.) and a rate reward vector α (n) R K +, where for each sub-channel m, (R m (n), P m (n)) is a solution to the optimization problem and subject to max r,p KX α k (n)r k λ k p k k=1 X Pk S r k log 1 + p! k Hm k (n) 2 σ k S MX MX R m k (n) = R k, E 4 Pk m (n) 5 = P k k K m=1 m=1

14 Single Antenna Systems System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations 13/36 Wideband MAC Delay Limited Capacity Define the marginal utility functions for user k as u m k (z) α k (n) λ k z + N 0 H k m(n), = The optimal rate allocation R m k (n) = Z = The optimal power allocation P m k (n) = 1 H m k (n) 2 0 k = 1,..., K 1 n N 0 + z I u m k (z) > um k (z), k k and u m o k (z) > 0 dz Z 0 I nuk m (z) > um k (z), k k and uk m o (z) > 0 dz where I{.} is the indicator function

15 Single Antenna Systems 14/36 Graph Representation System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations Users Sub-channels User 1 W 1,2 W 1,1 Sub-channel 1 W 1,K W 2,1 User 2 W 2,K W 2,2 Sub-channel 2 W K,1 User K W K,2 W K,K Sub-channel K Graph representation of the system

16 Single Antenna Systems 15/36 Graph Representation System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations Users Sub-channels User 1 Sub-channel 1 User 2 Sub-channel 2 User K Allocation Example Sub-channel K

17 Single Antenna Systems 16/36 Delay Limited rates System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations Mathematical formulation of the optimization subject to β m k kx k=1 MX m=1 min β m k,pm k K M k=1 m=1 β m k Pm k = {0, 1}, for all k, m β m k β m k = 1, for all m «log 1 + Pm k.hk m = Rk, for all k N 0

18 Single Antenna Systems 17/36 Max-Min Allocation System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations Finding the permutation π = arg max π Π min k=0,...,k 1 Hπ k = Guarantees that at any given time instant the minimum channel gain allocated is the best possible among all allocations = Maximizes the minimum of all user rates when equal and fixed power.

19 Single Antenna Systems 18/36 System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations 2.5 x 10 3 M=K=32 M=K= M=K= PDF of the minimum allocated channel gain using Max-Min Allocation for different values of M.

20 Single Antenna Systems 19/36 System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations CDF 10 3 K=M=2 K=M= K=M= SNR (db) CDF of the minimum allocated channel gain using Max-Min Allocation for different values of M.

21 Single Antenna Systems 20/36 System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations Averaged rate per sub channel (bits/dim) Ergodic sum rate max Optimal Delay Limited Orthogonal Delay Limited Max min Number of Sub channels Averaged throughput over Rayleigh fading at 0-dB with fair and unfair allocations.

22 Single Antenna Systems 21/36 System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations Independent frequency channel gains Correlated frequency channel gains 1.6 Spectral efficiency Frequency bandwidth (MHz) Spectral efficiency variation with system bandwidth for fixed number of users and sub-carriers with the max-min allocation policy. /36

23 Multiple Antenna Transmission Part II Multiple Antenna Systems

24 Multiple Antenna Transmission System Model Allocation strategies Broadcast Channel(BC) Single Antenna users: K = {1,..., K } M parallel sub-channels of bandwidth W Block Fading

25 Multiple Antenna Transmission System Model Allocation strategies 24/36 Spatial Multiplexing Max-Min Allocation (SM-Max-Min) The SINR value of user k on channel m, if it is assigned antenna n t γ k (m, n t ) = P N t H k,m [n t ] 2 N 0 + N t n t n t P N t H k,m [n t ] 2 Max-Min allocation: Permatation π s.t. π = arg max π Π min k=0,...,k 1 γπ k

26 Multiple Antenna Transmission System Model Allocation strategies 25/36 Space Time Coding Max-Min allocation (STC-Max-Min) For a fair comparison of the STC with SM = W is divided to N t adjacent sub-bands of bandwidth W = W N t The SINR value for user k on sub-band m is given by γ k,m = P Nt N t n t =1 P N t H k,m [n t ] 2 N 0 Max-Min allocation: Permatation π s.t. π = arg max π Π min k=0,...,k 1 γπ k

27 Multiple Antenna Transmission System Model Allocation strategies 26/36 Delay Limited Rate Allocation (DLR) Target rate vector to acheive with no delay in the transmission The optimal solution is difficult to perform Sub-optimal solution in two steps: Sub-channel and antenna allocation Power adaptation

28 Multiple Antenna Transmission System Model Allocation strategies 27/36 Delay Limited Rate Allocation (DLR) Target rate vector to acheive with no delay in the transmission The optimal solution is difficult to perform Sub-optimal solution in two steps: Sub-channel and antenna allocation Allocation according to SM-Max-Min Power adaptation

29 Multiple Antenna Transmission System Model Allocation strategies 28/36 Delay Limited Rate Allocation (DLR) {k i } i=1...nt the set of users scheduled in a sub-channel m The rate constraint can be expressed as γ k i is the target SINR P ki = X j i A the N t N t matrix given by 8 >< A i,j >: H ki [j] 2 N 0 γ P kj H ki [i] 2 γ k k + i i H ki [i] 2 (1) 0 j = i H ki [j] 2 H ki [i] 2 γ k i j i B the N t 1 vector such that B i = N 0 H ki [i] 2 γ k i

30 Multiple Antenna Transmission System Model Allocation strategies 29/36 Delay Limited Rate Allocation (DLR) Equation (1) can then written as (I A) P = B (2) A is a nonnegative, primitive matrix, so that Perron-Frobenius theory guarantees the existence of a dominant, positive eigenvalue r. Theorem (Hanly, 1995) The carrier to interference equation (2) has a positive solution if and only if r < 1. If r < 1 then there is a unique solution P given by P = (I A) 1 B (3)

31 Multiple Antenna Transmission System Model Allocation strategies 30/36 SE as a function of M, SNR=0dB SE (bps/hz) 4 SE (bps/hz) ZF DPC Opportunistic Beamforming Max Min Allocation Delay Limited Equal Rate ZF DPC Opportunistic Beamforming Max Min Allocation Delay Limited Equal Rate Number of Sub channels Number of Sub channels 4 Transmit Antennas 2 Transmit Antennas

32 Multiple Antenna Transmission System Model Allocation strategies 31/36 SE as a function of M, SNR=0dB SE (bps/hz) SE (bits/dim/sub channel) SM, Nt=4 SM, Nt=2 Singla Tx antenna STC, Nt=2 STC, Nt= Number of Sub channels Tx antennas (independent frequency channel gains) 2 Tx antennas (correlated frequency channel gains B=20MHz) 2 Tx antennas (correlated frequency channel gains B=5MHz) 1 Tx antenna (independent frequency channel gains) 1 Tx antenna (correlated frequency channel gains B=20MHz) 1 Tx antenna (correlated frequency channel gains B=5MHz) Nb of Sub channels SM and STC comparison SM-Max-Min: Channel Spacing effect

33 Conclusions 32/36 Different dynamic Hard Fairness Allocation strategies are proposed Optimal and Orthogonal delay limited rates Ergodic sum rates DL & Hard Fairness = High Spectral efficiency penalty

34 Conclusions 33/36 Different dynamic Hard Fairness Allocation strategies are proposed Optimal and Orthogonal delay limited rates Ergodic sum rates DL & Hard Fairness = High Spectral efficiency penalty

35 Conclusions Hanly Stephen V. and David N. C. Tse Multi-access fading channels: Part II: Delay limited capacities. IEEE Trans. on Info. Theory, vol. 44, pp , Hanly Stephen An algorithm for combined cell-site selection and power control to maximize cellular spread spectrum capacity IEEE J. Select. Areas Commun., Vol. 13, pp , Sept

36 Conclusions Thank you for your attention