Optimal Resource Allocation for OFDM Uplink Communication: A Primal-Dual Approach

Transcription

1 Optimal Resource Allocation for OFDM Uplink Communication: A Primal-Dual Approach Minghua Chen and Jianwei Huang The Chinese University of Hong Kong Acknowledgement: R. Agrawal, R. Berry, V. Subramanian J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 1 / 18

2 OFDM Systems Frequency band divided into several parallel orthogonal carriers/tones. High spectrum efficiency. Eliminate inter-symbol-interference (ICI) due to multi-path fading. Applications: WiMAX (802.16), Wi-Fi (802.11a/g), DSL, etc. J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 2 / 18

3 OFDM Systems Frequency band divided into several parallel orthogonal carriers/tones. High spectrum efficiency. Eliminate inter-symbol-interference (ICI) due to multi-path fading. Applications: WiMAX (802.16), Wi-Fi (802.11a/g), DSL, etc. How to perform distributed and efficient resource allocation? J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 2 / 18

4 Resource Allocation Assign each user a utility, U i ( ), depending on delay, throughput, etc. J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 3 / 18

5 Resource Allocation Assign each user a utility, U i ( ), depending on delay, throughput, etc. Scheduler maximizes first order change in total utility. max U(X(t)) r = max U i (X i (t))r i, r R(e) r R(e) i Motivated by channel-aware and gradient-based scheduling Myopic policy, requires no knowledge of channel or arrival statistics. J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 3 / 18

6 Resource Allocation Assign each user a utility, U i ( ), depending on delay, throughput, etc. Scheduler maximizes first order change in total utility. max U(X(t)) r = max U i (X i (t))r i, r R(e) r R(e) i Motivated by channel-aware and gradient-based scheduling Myopic policy, requires no knowledge of channel or arrival statistics. Resource allocation = weighted rate maximization Network : assigning carriers to users User : assign power over carriers J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 3 / 18

7 Achievable Rate Per Carrier If a carrier j is allocated to a single user i: r ij = log(1 + SNR) = log (1 + p ij e ij ) p ij = power user i allocates to carrier j. eij = received SNR/unit transmit power. J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 4 / 18

8 Achievable Rate Per Carrier If a carrier j is allocated to a single user i: r ij = log(1 + SNR) = log (1 + p ij e ij ) p ij = power user i allocates to carrier j. eij = received SNR/unit transmit power. By allowing time-sharing: ( ) r ij = x ij log 1 + p ij x ij e ij x ij [0, 1] = fraction of carrier j allocated to user i. J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 4 / 18

9 Achievable Rate Per Carrier If a carrier j is allocated to a single user i: r ij = log(1 + SNR) = log (1 + p ij e ij ) p ij = power user i allocates to carrier j. eij = received SNR/unit transmit power. By allowing time-sharing: ( ) r ij = x ij log 1 + p ij x ij e ij x ij [0, 1] = fraction of carrier j allocated to user i. Why time-sharing? Convexify the problem. Can be achieved in practice. Not affecting the optimal objective value with large number of carriers. J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 4 / 18

10 OFDMA rate region Rate region: R(e) = { r : r i = j ( x ij log 1 + p ) } ije ij, i x ij where (x, p) X such that (x, p) 0 Each user i: j p ij P i (uplink transmission power constraint) Each carrier j: i x ij 1 (channel allocation constraint) J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 5 / 18

11 OFDMA rate region Rate region: R(e) = { r : r i = j ( x ij log 1 + p ) } ije ij, i x ij where (x, p) X such that (x, p) 0 Each user i: j p ij P i (uplink transmission power constraint) Each carrier j: i x ij 1 (channel allocation constraint) Variation: sub-channelization (bundle carriers to reduce overhead) Interleaved ( standard mode) Adjacent (Band AMC mode) Random (e.g. frequency hopped) J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 5 / 18

12 Weighted Rate Maximization Problem max w r = r R(e) max (x,p) X w i i j ( x ij log 1 + p ) ije ij x ij (OPT) J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 6 / 18

13 Weighted Rate Maximization Problem max w r = r R(e) max (x,p) X w i i j ( x ij log 1 + p ) ije ij x ij (OPT) Concave maximization problem Non-strictly concave: typically many local/global optimal allocations J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 6 / 18

14 Weighted Rate Maximization Problem max w r = r R(e) max (x,p) X w i i j ( x ij log 1 + p ) ije ij x ij (OPT) Concave maximization problem Non-strictly concave: typically many local/global optimal allocations What is known: centralized optimal and (very good) heuristic algorithms [HSBA 07] What we will show: distributed optimal algorithm [This talk] J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 6 / 18

15 Centralized Optimal Algorithm [HSBA 07] Problem (OPT) is convex and satisfies Slater s condition No duality gap. Consider Lagrangian: L(x, p, λ, µ) := i + i w i j ( x ij log 1 + p ) ije ij x ij λ i ( P i j p ij ) + j µ j ( 1 i x ij ). Dual function: L(λ, µ) = max L(x, p, λ, µ) (x,p) X Optimal objective value to Problem (OPT): V = min L(λ, µ) (λ,µ) 0 J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 7 / 18

16 A Dual-Based Centralized Algorithm [HSBA 07] 1 For fixed λ, analytically solve for µ(λ), x(λ) and p(λ). 2 Multi-dimensional subgradient search of optimal λ. 3 Find optimal x and p by solving a system of linear equations (multiple solutions). J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 8 / 18

17 A Dual-Based Centralized Algorithm [HSBA 07] 1 For fixed λ, analytically solve for µ(λ), x(λ) and p(λ). 2 Multi-dimensional subgradient search of optimal λ. 3 Find optimal x and p by solving a system of linear equations (multiple solutions). Properties: Centralized computation (step 3) Slow convergence (step 2) High signaling overhead (steps 1, 2 and 3): all channel information and power constraints J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 8 / 18

18 Distributed Primal-Dual Algorithm Try to reach a saddle point of Lagrangian L(x, p, λ, µ): ẋ ij = k x ij L/ x ij = k x ij (f ij (x ij, p ij ) µ j ) + x ij, (mobile) ṗ ij = k p ij L/ p ij = k p ij (g ij(x ij, p ij ) λ i ) + p ij, (mobile) λ i = ki λ L/ λ i = ki λ p ij P i, (mobile) j λ i ( ) + µ j = k µ j L/ µ j = k µ j x ij 1, (base station) i µ j + J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 9 / 18

19 Distributed Primal-Dual Algorithm Advantages: Distributed and simple updates by mobiles and base station Low communication overhead: user feedback x ij s and base station announces µ j J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 10 / 18

20 Distributed Primal-Dual Algorithm Advantages: Distributed and simple updates by mobiles and base station Low communication overhead: user feedback x ij s and base station announces µ j Challenge: How to achieve global convergence with non-strictly concave objective functions Difficult in general Example: multi-path routing [Voice 06] J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 10 / 18

21 Convergence Result: Part I Theorem (Convergence to an Invariant Set) All trajectories of the primal-dual system converge to an invariant set V 0 globally and asymptotically. All optimal solutions of Problem (OPT) are contained in set V 0. J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 11 / 18

22 Convergence Result: Part I Theorem (Convergence to an Invariant Set) All trajectories of the primal-dual system converge to an invariant set V 0 globally and asymptotically. All optimal solutions of Problem (OPT) are contained in set V 0. Proof: constructing a proper Lyapunov function. Question: Will set V 0 contain non-optimal solutions? J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 11 / 18

23 Convergence Result: Part II Theorem (Convergence to a Global Optimal Solution) All trajectories of the primal-dual system globally converge to optimal solutions of Problem (OPT) under properly chosen stepsizes. J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 12 / 18

24 Convergence Result: Part II Theorem (Convergence to a Global Optimal Solution) All trajectories of the primal-dual system globally converge to optimal solutions of Problem (OPT) under properly chosen stepsizes. Set V 0 only contains optimal solutions. Stepsize choice is easy: e.g., not all k µ j s are the same. Proof: Over set V 0, the nonlinear system reduces to a linear one. The linear system in marginal stable. Set V0 contains only the optimal solution if (λ, µ) is completely observable from B[x T, p T ] T. J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 12 / 18

25 Relationship Invariant set Optimal solutions Complete Observability Invariant set Optimal solutions J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 13 / 18

26 Simulation Set-up Single cell, M = 4 users. e ij = (fixed location-based term) (frequency selective fast fading) Fixed term = empirical distribution. frequency selective term = block fading in time (2msec coh. time 250Hz Doppler); standard ref. mobile delay spread (6 taps, 1 µsec ). 5 MHz BW, 512 carriers. Adjacent channelization, 8 carriers/subchannel. Resource allocation over 20 OFDM symbols. Randomly generated weights in [0, 1]. J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 14 / 18

27 Convergence of Primal and Dual Variables 1 Dual variables λ 1.5 Dual variables µ Total power allocation of all users 4 Total channel allocation of all channels Numer of iterations Numer of iterations J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 15 / 18

28 Convergence of Total Weighted Rate 12 x 105 total weighted rate (bits/sec) Optimal solution Primal feasible solution Numer of iterations Achieve 90% within 500 iterations J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 16 / 18

29 Conclusions Topic: optimal resource allocation for uplink OFDM systems Algorithm: Primal-Dual algorithm Properties Distributed and simple updates Low communication overhead per iteration Future Work: Convergence with delay and asynchronous updates Faster convergence J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 17 / 18

30 Contact jwhuang J. Huang (CUHK) Primal-Dual Algorithm for OFDM Uplink 18 / 18