Outlines Introduction and Motivation Dynamic Fair Channel Allocation for Wideband Systems Department of Mobile Communications Eurecom Institute Sophia Antipolis 19/10/2006
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
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)
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??
5/36 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?
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?
Single Antenna Systems Part I Single Antenna Systems
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
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
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)1 2............. PK 1 (H) PM K (H) 3 7 5 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
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 8 2 3+ >< 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)
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.
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 2 2 3 MX MX R m k (n) = R k, E 4 Pk m (n) 5 = P k k K m=1 m=1
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
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
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
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
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.
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=128 2 1.5 M=K=2 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 PDF of the minimum allocated channel gain using Max-Min Allocation for different values of M.
Single Antenna Systems 19/36 System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations 10 0 10 1 10 2 CDF 10 3 K=M=2 K=M=32 10 4 K=M=128 30 25 20 15 10 5 0 5 10 15 SNR (db) CDF of the minimum allocated channel gain using Max-Min Allocation for different values of M.
Single Antenna Systems 20/36 System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations 2.5 2 Averaged rate per sub channel (bits/dim) 1.5 1 0.5 Ergodic sum rate max Optimal Delay Limited Orthogonal Delay Limited Max min 0 0 10 20 30 40 50 60 70 Number of Sub channels Averaged throughput over Rayleigh fading at 0-dB with fair and unfair allocations.
Single Antenna Systems 21/36 System and Channel Models Optimal Dynamic Channel Allocations Orthogonal Dynamic Channel Allocations 1.8 1.7 Independent frequency channel gains Correlated frequency channel gains 1.6 Spectral efficiency 1.5 1.4 1.3 1.2 1.1 1 0 20 40 60 80 100 120 140 160 180 200 Frequency bandwidth (MHz) Spectral efficiency variation with system bandwidth for fixed number of users and sub-carriers with the max-min allocation policy. /36
Multiple Antenna Transmission Part II Multiple Antenna Systems
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
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
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
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
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
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
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)
Multiple Antenna Transmission System Model Allocation strategies 30/36 SE as a function of M, SNR=0dB 7 4.5 6 4 3.5 5 SE (bps/hz) 4 SE (bps/hz) 3 2.5 3 2 ZF DPC Opportunistic Beamforming Max Min Allocation Delay Limited Equal Rate 2 1.5 ZF DPC Opportunistic Beamforming Max Min Allocation Delay Limited Equal Rate 1 0 20 40 60 80 100 120 140 Number of Sub channels 1 0 20 40 60 80 100 120 140 Number of Sub channels 4 Transmit Antennas 2 Transmit Antennas
Multiple Antenna Transmission System Model Allocation strategies 31/36 SE as a function of M, SNR=0dB 4 3 3.5 2.5 3 SE (bps/hz) 2.5 2 SE (bits/dim/sub channel) 2 1.5 1.5 1 SM, Nt=4 SM, Nt=2 Singla Tx antenna STC, Nt=2 STC, Nt=4 0.5 0 10 20 30 40 50 60 70 Number of Sub channels 1 0.5 2 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) 10 20 30 40 50 60 70 Nb of Sub channels SM and STC comparison SM-Max-Min: Channel Spacing effect
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
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
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. 2816-2831, 1998. 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. 1332-1340, Sept. 1995.
Conclusions Thank you for your attention issam.toufik@eurecom.fr