Multiuser OFDMA Resource Allocation Algorithms ... - Semantic Scholar

Multiuser OFDMA Resource Allocation Algorithms for In-Home Power-Line Communications Hao Zou, Sumanth Jagannathan, John M. Cioffi Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA {haozou, sumanthj, cioffi}@stanford.edu

Abstract— This paper reformulates the multi-user OFDMA resource allocation problems in broadcast and multiple-access channels by defining an equivalent interference channel. The resulting problems are then solved by employing recentlydeveloped low-complexity algorithms for spectrum optimization in the interference channel. The re-formulation averts the highcomplexity integer programming problem that otherwise results from an OFDMA system that allocates each user to separate subchannels. Simulation results indicate that the proposed algorithm achieves near-optimal performance and is suitable for PLC resource allocation applications.

I. I NTRODUCTION Power-Line communications (PLC) is an emerging technology for in-home video streaming and networking. Homeplug AV is the most current standard for in-home PLC and supports a typical data rate of 70-100 Mbps, and up to 200 Mbps [1]. However, since all devices transmit data on the same power cord, the bus topology of a PLC network limits the data rate when multiple users share the home PLC channel. The Homeplug AV standard targets mainly high-definition television (HDTV) applications. Since one channel of HDTV stream requires a data rate of 24 Mbps, even with the most recent Homeplug AV standard, only 80% of the outlets can support a data rate of 55 Mbps or better, equivalent to two HDTV channels [2]. PLC faces fierce competition from other wireless (e.g. 802.11n) and wire-line (e.g. Ethernet, cable modem, etc.) alternatives for home networking in terms of data rate for video-streaming performance. As a result, persistent improvements of PLC systems on data rate and Quality-ofService (QoS) are pivotal to its success. Dynamic resource allocation strategies have been proposed for wireless as well as wire-line networks to combat the rate-loss caused by interference in a multi-user scenario. A successful example is the level-2 dynamic spectrum management (DSM) algorithms proposed for digital subscriber-lines (DSL)[3-7]. PLC has many features similar to DSL; consequently, its multi-user resource allocation strategy can benefit from existing research results for DSL networks. However, while formulating the resource allocation problem for PLC, the similarities and differences in the topology between PLC and DSL need to be examined in detail. Firstly, in PLC, all users transmit their signals on the same wire, while DSL users typically transmit on separate twisted-pairs. Secondly, because of the bus topology in PLC, a subchannel is best assigned to an individual user and not shared by different users, i.e., a frequency-division multiple-access (FDMA) restriction is

usually employed. Notwithstanding these differences, PLC and DSL both have a central controller suitable for dynamic allocation of resources. In DSL, the central controller is the spectrum management center (SMC), while in PLC, it is called the central coordinator (CCo) [8]. Taking into account these constraints, this paper formulates the PLC resources allocation problem as a maximum weighted-sum-rate optimization problem with an FDMA constraint, where different users occupy different subchannels. A few resource allocation algorithms have been proposed for in-home PLC systems for the centrally coordinated topology [9], [10]. The proposed optimization algorithm in [10] maximizes sum data rate with a rate-ratio constraint to achieve fairness. However, the optimality of the algorithms are hard to analyze. Alternatively, weighted sum-rate maximization with FDMA constraints has been studied well in DSL and may be adopted in PLC networks [11], [12]. For example, reference [11] introduces a fast solution to the resource-allocation problem with a constant-energy assumption, while reference [12] applies a dual-decomposition method to the multi-user FDMA broadcast problem and shows that the solution is near-optimal when the number of tones goes to infinity. The rest of the paper is organized as follows: Section II provides an overview of the system model for PLC and formulates the PLC resource-allocation problem as a weighted-sum-rate problem with FDMA constraints. Section III introduces an equivalent interference reformulation to the FDMA problem and recasts it as a convex optimization problem. Section IV describes the constant-energy and the dual-decomposition approaches to the resource-allocation problem of [11], [12]. Section V shows performance comparisons of the algorithms with PLC channel and noise data. Section VI concludes the paper. II. PLC S YSTEM M ODEL AND R ESOURCE A LLOCATION P ROBLEM The topology of the PLC network considered in this paper follows [8]. The in-home PLC network has a Central Coordinator (CCo). The CCo connects externally to the internet service provider (ISP) through a DSL or fiber-optic broadband connection, and coordinates the data transfer within the home. Figure 1 shows a typical home network that follows a star topology. PLC devices in this star topology communicate by establishing bidirectional links with the CCo. In this paper, persistent resource allocation strategies for the downlink

978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.

Broadband Link to Internet Service Providers (ISPs) (DSL, Fiber, Cable etc.)

No polynomial-time efficient solution currently exists for this nonlinear 0-1 integer programming problem, justifying a re-formulation of problem (1) to reduce its computational complexity.

Room 1 PLC Modem 1 Power Cord

PLC Central Coordinator (CCo), or Home Gateway.

PLC Modem 2

Room K Room 2 PLC Modem 3

...

PLC Modem L

IPTV

PLC Modem 4

Fig. 1.

The star topology of In-Home PLC network

broadcast (BC) channel from the CCo to L users are first considered. The BC problem can be formulated as: maximize:

subject to:

N L l=1 n=1 L

ωl log2 (1 + Il,n pl,n gl,n )

l=1 n=1

Il,n = 1, ∀n ∈ {1, · · · N }

l=1

Il,n = 0 or 1, ∀l ∈ {1, · · · L, }, ∀n ∈ {1, · · · N } N L

III. R E - FORMULATION TO A C ONVEX O PTIMIZATION P ROBLEM The non-linear integer problem (1) can be reformulated into an interference-channel problem by introducing a large crosstalk channel gain G between users. In a network affected by severe crosstalk, the penalty of sharing subchannels is so high that the optimal solution must be FDMA. Such an Equivalent-Interference re-formulation of (1) can be shown as: N L p g l,n l,n ) ωl log2 (1 + maximize: 1 + j=l pj,n G l=1 n=1 (2) N L subject to: pl,n ≤ Pmax .

pl,n ≤ Pmax ,

l=1 n=1

(1) where L is the number of users in the PLC network; N is the total number of OFDM tones; ωl is the fairness weight assigned to user l; pl,n is the power allocated to Tone n of user l; Pmax is the total power constraint; gl,n is channel response squared over noise on tone n of user l, with the SNR-gap Γ [13] normalized within it. The system optimizes over the choice of on/off energies indicated by Il,n for user l on tone n. In fairness weight ωl for user l can be adjusted to achieve fairness in resource allocation. The fairness weight ωl can be viewed economically as a “monetary budget” for user l to purchase the bandwidth resource [14]. The channel condition can be viewed as a price factor for bandwidth resources in the PLC system. With a large ωl , user l can purchase or exchange more bandwidth in the PLC network even if the price (channel condition) is not favorable to the user. A larger purchasing power ωl can be incrementally granted to user l when needed. The rate-ratio across all users can thus be managed equally well without an explicit rate-ratio constraint. Moreover, in case an individual user suffers from an extremely poor channel condition, the whole system will not be affected as much as the case when a pre-assigned rate-ratio is fixed among the users, which in this case equals to assigning an infinite ωl to the user with extremely poor channel condition and will cause the data rates of all users to be near zero. The above optimization problem is a nonlinear 0-1 integer programming problem. Exhaustive search of allocation of the N subchannels to the users would require O(N L ) operations.

For sufficiently large G, it can be proved that the optimal solution of (2) is equivalent to (1): Lemma 1: For sufficiently large G, the optimal solution to the problem in (2) is FDMA, where each subchannel is assigned to at most one user. Proof: This lemma conforms to [15]. An intuitive proof is given here: Assume p∗l,n is an optimal but nonFDMA solution to (2), i.e. ∃n ∈ {1, 2, · · · N }, l1 , l2 , · · · lm ∈ {1, 2, · · · L}, m > 1, such that pl1 ,n , pl2 ,n , · · · , plm ,n = 0. For sufficiently large crosstalk coefficient G → ∞, pl,n gl,n 1+ j=l pj,n G → 0, ∀l ∈ {l1 , l2 , · · · lm }. As a result the sum data rate for these m users on tone n is: p g l,n l,n ) = 0, (3) log2 (1 + 1 + j=l pj,n G l∈{l1 ,l2 ,···lm }

i.e., the allocation of power on tone n for more than one user yields zero data rate improvement. Consequently any allocation of all the power to a user with the best channel condition on tone n, say user l1 , will not violate the power constraint but will yield a non-zero and higher weightedsum-rate, contradicting the assumption that p∗l,n is the optimal solution to (2). Theorem 1: Solving Problem 1 is equivalent to solving Problem 2. Proof: Let p∗l,n be the optimal solution to (2). Let pl,n be the optimal solution to (1). First, by Lemma 1, p∗l,n satisfies the FDMA rule. When the FDMA rule is satisfied, the objective functions of (2) and (1) are equivalent, i.e. : N L

p g l,n l,n ) j=l pj,n G

(4)

ωl log2 (1 + Jl,n pl,n gl,n ),

(5)

ωl log2 (1 +

l=1 n=1 N L

=

1+

l=1 n=1

L for some Jl,n satisfying Jl,n = 0 or 1 and l=1 Jl,n = 1 Since pl,n is an feasible solution to (2), p∗l,n achieves a greater


or equal value in 5 and is at least optimal for 1. Conversely, since p∗l,n is a feasible solution for (1), pl,n achieves a greater or equal value in (4) than p∗l,n does and is at least optimal for (2). Therefore, solving (1) is equivalent to solving (2). In practice, pn,l may be a small non-zero number as the crosstalk G increases and can be rounded to zero with negligible data rate loss over the network. The reformulated optimization problem (2) is a special case of the Spectrum Balancing problem in DSL Dynamic Spectrum Management (DSM) when all crosstalk coefficients are equal [7]. The Spectrum Balancing problem in DSM is non-convex but can be solved efficiently by Sequential Convex Programming (SCP), which iteratively optimizes a localized convex approximation to the non-convex objective function [16]. The SCALE algorithm proposed in [4] is an example of SCP with the following approximation function: log2 (1 + z) ≈ α log2 (z) + β,

(6)

For example, since the precision of the results depends on both the accuracy of (6) and (10), the precision of (10) can be set low for the first several iterations and can then be increased gradually with an increase in the precision (6). The speed of the new algorithm may also be increased by heuristically choosing a good initial power assignment. IV. C ONSTANT-E NERGY AND D UAL -D ECOMPOSITION M ETHODS An approximation to the optimal energy allocation of (1) is for a user to transmit equal energy across all the subchannels assigned by the CCo. Such a constant-energy optimization relaxation to problem (1) can be stated as follows: N L

maximize:

subject to:

l=1 n=1 L

Pmax ωl log2 (1 + Il,n gl,n L N l=1

j=1 Il,j

)

Il,n = 1, ∀n ∈ {1, · · · N }

l=1

where z0 α= 1 + z0

(7)

β = log2 (1 + z0 ) −

z0 log2 (z0 ). 1 + z0

(8)

Approximation (6) is tight when z = z0 . By (6), a local approximation to (2) can be obtained as: maximize:

N L l=1 n=1

subject to:

N L

ωl (αl,n log2 (

1+

p g l,n l,n ) + βl,n ) j=l pj,n G

pl,n ≤ Pmax ,

Il,n = 0 or 1, ∀l ∈ {1, · · · L, }, ∀n ∈ {1, · · · N }. (11) Hoo et al. presented a brute-force search based solution to the constant-energy optimization problem with a complexity of O(N 2 max(L, log2 N )) [11]. The equal-energy assignment of power performs well in high SNR regions but poorly in low SNR regions. Another method to solve the integer program (1) is to solve its dual problem [12]. Let P = {pl,n }. The Lagrangian dual L(λ, P ) of (1) can be formulated as: L(λ, P ) =

l=1 n=1

l=1 n=1

(9) where αl,n and βl,n are calculated from (7) and (8) with z0 replaced by the SNR of user l on tone n. Defining p˜l,n = ln(pl,n ), (9) becomes: maximize:

N L l=1 n=1

ep˜j,n G))}

j=l

subject to:

N L

ωl log2 (1 + pl,n gl,n )

+ λ(Pmax −

L N

.

(12)

pl,n )

l=1 n=1

Denoting the dual function g(λ) as: g(λ) = min L(λ, P ),

ωl {αln (log2 (gl,n ) + p˜l,n + βln

− log2 (1 +

N L

P

(13)

the dual problem of (1) can thus be formulated as: (10)

ep˜l,n ≤ Pmax .

maximize: g(λ) . subject to: λ ≥ 0

(14)

A bisection update of λ is applied in [12] to search for the optimal solution of (14).

l=1 n=1

The SCP solution to the non-convex problem (2) is obtained by iteratively solving its convex approximation (10) and updating αl,n and βl,n to tighten the approximation [4]. With about 10 steps of inner-loop convex optimization of (10) and outerloop tightening of (6), a near-optimal solution to (2) can be obtained. Standard convex optimization methods such as the interior point method, the subgradient method or the ellipsoid method can be applied to solve the convex approximation in (10). In addition to standard algorithms, several approaches exist for further acceleration of the execution speed of the algorithm.

V. S IMULATION R ESULTS AND D ISCUSSION Simulations of the constant-energy, dual-decomposition and the equivalent-interference algorithms are performed. The channel transfer functions are obtained from the Homeplug Alliance in-home PLC measurements [17]. The frequency-band from 2 to 30 MHz is divided into 573 OFDM subchannels, each having a bandwidth of 49 kHz. The PSD template for the noise generated by a typical home appliance is obtained from [18]. Random perturbations were added to the template to simulate the noise PSD of five different appliances. The effect


−80

Weight

SNR

−85

(1.0, 1.0, 1.0, 1.0) (1, 1.2, 1.5, 0.5) (1.5, 0.7, 1.0, 1.3)

HIGH LOW HIGH LOW HIGH LOW

Noise PSD (dBm/Hz)

−90 −95 −100

Constant Energy Dual Decomposition Virtual Inteference

6 Data rate of user 3 in Mbps

−115 −120 2

5

Fig. 2.

10 15 20 Frequency (MHz)

25

30

A typical Noise PSD (User 1)

of these noises at the receiver is modeled by passing their PSDs through the transfer function of the channel between the noise generating appliance and the receiver [19]. Five different channel transfer functions were chosen from the measurement database for this purpose. Figure 2 shows a typical home PLC Noise PSD. Figure 3 shows a typical home PLC channel response. The transmitter power is set to 10 dBm. A −30

5 4 3 2 1 0 0

Fig. 4.

2

4 6 Data rate of user 2 in Mbps

8

Rate region for User 2 and User 3 (Low SNR)

convergence rate for the equivalent interference method with the bounding inequality (6). Usually, over 90% of the optimal capacity is achieved within 10 iterations. The FDMA rule is also satisfied with high precision within 10 iterations. Figure 7 gives a plot of the weighted-sum-rate for weight vector (1.5, 0.7, 1.0, 1.3) vs. the interference strength k , where k is defined as the equivalent inference G over the mean of normalized channel response gl,n , i.e.:

−40 −50 Channel Gain (dB)

Equivalent Interference 158.00 22.055 156.19 13.516 206.25 28.912

W EIGHTED -S UM -R ATE (M BPS )

7

−110

−60 −70 −80

k=

−90 −100 5

Fig. 3.


25

30

1 NL

G L N l=1

(15)

n=1 gl,n

When the equivalent interference G is tens of dB below the mean of gl,n , the rate performance loss is about 3% for the PLC channel and noise data simulated. As G increases to the

A typical PLC Channel (User 1)

downstream channel with the CCo broadcasting to four users is considered. FDMA restriction is satisfied in the results and the power assignment for each user follows a water-filling pattern. Table I gives a comparison of the weighted-sum-rate in Mbps for the constant-energy, dual-decomposition, and equivalentinterference methods. In addition to the low SNR case where the PLC noise is generated as described above, results for a high SNR case with a low flat AWGN noise of −130 dBm is also given. Figure 4 shows the two-user rate region for User 2 and User 3 in the low SNR case. At high SNR, all the three methods return near-optimal solutions. At low SNR, the constant energy method has about 30% performance loss while the dual-decomposition and equivalent-interference methods still return near optimal solutions. Figure 5 shows the power assignment for the low SNR case with the weight vector ω = (1, 1.2, 1.5, 0.5). Figure 6 shows the outer-loop

−50 −60 −70 −80 PSD (dBm/Hz)

−110 0

Dual Decomposition 158.00 22.055 158.20 13.848 206.47 28.917

TABLE I

−105

−125 0

Constant Energy 157.83 15.872 155.93 9.663 205.96 20.716

−90 −100 −110 −120 −130 −140 −150 0

User 1 User 2 User 3 User 4 5


Fig. 5.

25

30

Power assignment


same magnitude of the mean of gl,n , the rate loss is negligible.

Weighted Sum Rate (%)

100

80

60

allocation by reformulating the weighted-sum-rate OFDMA problem into an equivalent-interference channel problem. Simulation results show that the proposed algorithm significantly outperforms the constant-energy optimization algorithm in scenarios when the appliance noise causes the SNR to become very low. It achieves equally well near-optimal solutions as the dual-decomposition algorithm in both low and high SNR situations, making it suitable as a persistent resource allocation algorithm for QoS demanding applications in PLC networks.

40

ACKNOWLEDGEMENTS This work was supported by France Télécom R&D.

20

0 0

R EFERENCES 5

10

Fig. 6.

15 20 Iteration Steps

25

30

Convergence Rate vs. Iteration

31

Weighted Sum Rate (Mbps)

30

29

28

27

26

25 −40

−30

−20

Fig. 7.

−10 0 k (dB)

10

20

30

Rate vs. Crosstalk

The equivalent interference algorithm also gives a straightforward solution to the uplink MAC channel problem where L users transmit data to the CCo with per user power constraints. While inner-loop convex problem of the equivalent interference method can still be solved efficiently with additional linear constraints, the complexity of the constant energy becomes exponential with L power constraints. The bisection updates of the dual variables λ in the dual-decomposition method will also leads to exponential complexity in this case. More sophisticated dual-variable updating techniques such as the Ellipsoid method need to be exploited for the dual decomposition method on the MAC channel. VI. C ONCLUSIONS Contention-free persistent resource allocation is preferable for QoS demanding applications such as HDTV. The Homeplug AV standard uses periodic Time-Division MultipleAccess (TDMA) as its persistent resource allocation strategy [20]. However, the performance of TDMA degrades rapidly when multiple QoS conncetions are requested. A multi-user OFDMA algorithm was proposed for in-home PLC resource

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