Resource Allocation in OFDM Systems Jianwei Huang Princeton University Joint work with R. Berry, M. Honig, M. Chiang V. Subramanian, R. Agrawal, R. Cendrillon, M. Moonen Sponsors: NSF, Motorola, Alcatel WINLAB Seminar, April 2006 J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 1 / 46
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 (Princeton Univ.) OFDM Resource Allocation April 2006 2 / 46
Resource Allocation in OFDM Systems OFDMA (Orthogonal Frequency Division Multiple Access) systems ◮ Each user is assigned a subset of carriers. ◮ No interference among users. ◮ Need to determine user-carrier association and power allocation. Interference limited OFDM systems ◮ Each user can transmit over all carriers. ◮ Interference among active users in the same carrier. ◮ Need to determine power allocation to mitigate interference. J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 3 / 46
Resource Allocation in OFDM Systems Scheduling & Resource Allocation in WiMax Networks fa ding users base station tim e (OFDMA) OFDM R 2 T 1 T 2 5 km CO CO CO CO CP CP C C P P R 1 2 km 4 km T 4 RT1 RT1 RT1 RT1 CP CP CP CP T 3 3 km 3.5 km RT RT RT2 RT2 2 2 CP CP CP CP (Interf-limited) R 3 4 km 3 km RT3 RT3 RT3 RT3 C CP C CP P P R 4 Power Control in Wireless Spectrum Management in Ad Hoc Networks DSL Networks J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 4 / 46
Model Summary Part I II III Motivation WiMAX Ad Hoc DSL Infrastructure Ad Hoc Infrastructure Network Wireless Wireless Wired No Interference Interference Interference Objective Scheduling & Power Allocation Power Allocation Power Allocation J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 5 / 46
Main References WiMAX : J. Huang, V. Subramanian, R. Agrawal, and R. Berry, “Downlink Scheduling and Resource Allocation for OFDM Systems,” CISS 2006 Ad Hoc : J. Huang, R. Berry and M. Honig, “Distributed Interference Compensation for Wireless Networks,” to appear in IEEE Journal on Selected Areas in Communications , May 2006 DSL : J. Huang, R. Cendrillon, M. Chiang, and M. Moonen, “Autonomous spectrum balancing (ASB) for digital subscriber lines,” submitted to ISIT 2006 More related publications can be found at www.princeton.edu/ ∼ jianweih J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 6 / 46
Part I: WiMAX Network Scheduling & Resource Allocation in WiMax Networks fa ding users base station tim e (OFDMA) OFDM R 2 T 1 T 2 5 km CO CO CO CO CP CP C C P P R 1 2 km 4 km T 4 RT1 RT1 RT1 RT1 CP CP CP CP T 3 3 km 3.5 km RT RT RT2 RT2 2 2 CP CP CP CP (Interf-limited) R 3 4 km 3 km RT3 RT3 RT3 RT3 C CP C CP P P R 4 Power Control in Wireless Spectrum Management in Ad Hoc Networks DSL Networks J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 7 / 46
WiMAX Network fading users base station time Based on 802.16 and provide MAN broadband connectivity. ◮ Cover a distance of up to 5Kms and shared data speed up to 70Mbps. Defines a scheduling based MAC protocol. J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 8 / 46
Gradient-based Scheduling We consider downlink channel aware scheduling. Many approaches accomplish this via gradient-based scheduling. ◮ Assign each user a utility, U i ( · ), depending on delay, throughput, etc. ◮ Scheduler maximizes choose rate r = ( r 1 , . . . , r N ) T from the rate region R ( e ) to solve: � r ∈R ( e ) ∇ U ( X ( t )) · r = max max w i r i , r ∈R ( e ) i J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 9 / 46
Gradient-based Scheduling We consider downlink channel aware scheduling. Many approaches accomplish this via gradient-based scheduling. ◮ Assign each user a utility, U i ( · ), depending on delay, throughput, etc. ◮ Scheduler maximizes choose rate r = ( r 1 , . . . , r N ) T from the rate region R ( e ) to solve: � r ∈R ( e ) ∇ U ( X ( t )) · r = max max w i r i , r ∈R ( e ) i ◮ Myopic policy, requires no knowledge of channel or arrival statistics. ◮ Depends on the utility functions, could lead to various allocation rules ⋆ Proportional fair, maximum rate, stabilizing policies, etc. J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 9 / 46
OFDMA Rate Region � � � � 1 + p ij e ij � R ( e ) = r : r i = x ij log , ( x , p ) ∈ X , x ij j where ◮ x ij = time fraction of tone j allocated to user i . ◮ p ij = power allocated to user i on tone j . ◮ e ij = received SNR/unit power (i.e., channel condition). J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 10 / 46
OFDMA Rate Region � � � � 1 + p ij e ij � R ( e ) = r : r i = x ij log , ( x , p ) ∈ X , x ij j where ◮ x ij = time fraction of tone j allocated to user i . ◮ p ij = power allocated to user i on tone j . ◮ e ij = received SNR/unit power (i.e., channel condition). ◮ Feasible region � � � � X := ( x , p ) ≥ 0 : x ij ∈ { 0 , 1 } , ∀ i , j , x ij ≤ 1 , ∀ j , p ij ≤ P . i ij J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 10 / 46
Joint Scheduling and Resource Allocation Problem Solve at every scheduling interval: Problem: 1A-INT � 1 + p ij e ij � � � max V ( x , p ) := w i x ij log x ij ( x , p ) ∈X i j Technical Challenges : ◮ Integer constraints on x ij . ◮ Want to obtain low-complexity fast algorithms. J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 11 / 46
Problem with Relaxed Integer Constraints We consider the following problem with relaxed integer constraints. Problem: 1B-RELAX � 1 + p ij e ij � � � max V ( x , p ) := w i x ij log x ij ( x , p ) ∈ ˜ X i j by replacing x ij ∈ { 0 , 1 } in X with 0 ≤ x ij ≤ 1 in ˜ X . J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 12 / 46
Problem with Relaxed Integer Constraints We consider the following problem with relaxed integer constraints. Problem: 1B-RELAX � 1 + p ij e ij � � � max V ( x , p ) := w i x ij log x ij ( x , p ) ∈ ˜ X i j by replacing x ij ∈ { 0 , 1 } in X with 0 ≤ x ij ≤ 1 in ˜ X . We will show ◮ Typically, optimal solution of Problem 1B-RELAX is also optimal for Problem 1A-INT . ◮ If not, a near optimal solution of Problem 1A-INT can be found with almost no additional complexity. J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 12 / 46
Solve Problem 1B-RELAX Convex problem that satisfies Slater’s condition. ◮ No duality gap. Consider Lagrangian : � 1 + p ij e ij � � � L ( x , p , λ, µ ) := x ij log w i x ij i j � � � � � � � + λ P − p ij + µ j 1 − x ij . i , j j i Optimizing over x , p and µ , find close form solution of L ( λ ). Dual function L ( λ ) is convex. ◮ Find the optimal λ ∗ by 1-D iterative search. J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 13 / 46
Optimal Primal Variables Given λ ∗ , µ ∗ , let ( x ∗ , p ∗ ) = arg max ( x , p ) ∈X L ( x , p , λ ∗ , µ ∗ ) . which lead to � 1 , µ ij ( λ ∗ ) = max i µ ij ( λ ∗ ) x ∗ ij = µ ij ( λ ∗ ) < max i µ ij ( λ ∗ ) 0 , ◮ Requires a simple sort of users per tone j . J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 14 / 46
Optimal Primal Variables Given λ ∗ , µ ∗ , let ( x ∗ , p ∗ ) = arg max ( x , p ) ∈X L ( x , p , λ ∗ , µ ∗ ) . which lead to � 1 , µ ij ( λ ∗ ) = max i µ ij ( λ ∗ ) x ∗ ij = µ ij ( λ ∗ ) < max i µ ij ( λ ∗ ) 0 , ◮ Requires a simple sort of users per tone j . In most cases, no tie occurs on any tone j . ◮ Only one user per tone ⇒ optimal solution for Problem 1A-INT . J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 14 / 46
Optimal Primal Variables Given λ ∗ , µ ∗ , let ( x ∗ , p ∗ ) = arg max ( x , p ) ∈X L ( x , p , λ ∗ , µ ∗ ) . which lead to � 1 , µ ij ( λ ∗ ) = max i µ ij ( λ ∗ ) x ∗ ij = µ ij ( λ ∗ ) < max i µ ij ( λ ∗ ) 0 , ◮ Requires a simple sort of users per tone j . In most cases, no tie occurs on any tone j . ◮ Only one user per tone ⇒ optimal solution for Problem 1A-INT . Occasionally, ties occur on some tones. ◮ Results in multiple users per tone ⇒ not primal feasible. ◮ Need to break the ties to find a feasible primal solution for Problem 1A-INT . J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 14 / 46
Break the Ties for Problem 1A-INT Break the ties: choose one user per tone. ◮ Each choice corresponds to one power allocation p ∗ . ij p ∗ Utilize subgradient of dual function L ( λ ): P − � ij . J. Huang (Princeton Univ.) OFDM Resource Allocation April 2006 15 / 46
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