Optimality of binary power-control in a single cell via majorization
aa r X i v : . [ c s . I T ] M a y Optimality of binary power-control in a singlecell via majorization
Hazer Inaltekin and Stephen V. Hanly Department of Electrical and Electronic Engineering, University of Melbourne, Australia. Department of Electrical and Computer Engineering, National University of Singapore, Singapore.
AbstractThis paper considers the optimum single cell power-control maximizing the aggregate (uplink)communication rate of the cell when there are peak power constraints at mobile users, and alow-complexity data decoder (without successive decoding) at the base station. It is shown, viathe theory of majorization, that the optimum power allocation is binary , which means links areeither “on” or “off”. By exploiting further structure of the optimum binary power allocation,a simple polynomial-time algorithm for finding the optimum transmission power allocationis proposed, together with a reduced complexity near-optimal heuristic algorithm. Sufficientconditions under which channel-state aware time-division-multiple-access (TDMA) maximizesthe aggregate communication rate are established. Finally, a numerical study is performed tocompare and contrast the performance achieved by the optimum binary power-control policywith other sub-optimum policies and the throughput capacity achievable via successive decoding.It is observed that two dominant modes of communication arise, wideband or TDMA, and thatsuccessive decoding achieves better sum-rates only under near perfect interference cancellationefficiency. Index Terms
Communication networks, power-control, sum-rate capacity, TDMA, successive decoding
This research was supported in part by the Australian Research Council, under Grant DP-11-0102729, and NUS startup grant263-000-572-133. The authors can be contacted at email addresses: [email protected] and [email protected], respectively.
I. I
NTRODUCTION
Next generation 4G wireless communication systems are required to support all-IP servicesincluding high data rate multimedia traffic [1], with bit rate targets as high as Gbit/s for lowmobility, and
Mbit/s for high mobility [2]. Transmission at such high rates is certainly achiev-able today on point-to-point links, using the great advances made in wireless communicationsover the past couple of decades. But in wireless networks, including 4G systems, interferencebetween links remains as a fundamental bottleneck that needs to be overcome [3]. Part of thechallenge arises from the broadcast nature of the shared wireless medium: transmission powerhas to be allocated to each link, but this allocation has knock-on effects on other links in thenetwork. Much progress has been made on this problem when target rates are specified foreach user and the objective is to minimize total transmit power in the network [4]. However,solving for optimum power allocations that maximize the total Shannon-theoretic sum-rate inthe presence of interfering links seems to be much harder: It is generally a non-linear , non-convex constrained optimization problem [5]. This motivates a search for structure leading tosimplifications in the power allocation problem for sum-rate maximization.In this paper, we focus on the optimum allocation of transmission powers to mobile terminals inorder to maximize the total communication sum-rate when a low-complexity single-user decoder(without successive decoding) is used at the base station. This is the conventional single cellmatched filter detection based uplink model: All mobiles are in the same cell and must all bedecoded at the same base station. Even though this optimization problem is non-convex, wesolve it by identifying an underlying Schur-convex structure in the objective sum-rate function.We show that the optimum power allocation is binary , i.e., a user either transmits with fullpower or does not transmit at all. By utilizing the binary structure of the sum-rate maximizingoptimum power allocation, we observe two dominant modes of communication: either the bestuser transmits with full power, which can be considered a channel quality based time-division-multiple-access (TDMA) mode, or all users transmit with full power, which can be considereda wideband (WB) mode. This result has implications for implementing joint power-control andscheduling, and helps to theoretically justify existing engineering approaches, such as code-division-multiple-access (CDMA), and scheduling based on channel quality.We also compare sum-rates achieved by the optimum power-control policy with throughput DRAFT capacity limits that can be achieved by successive decoding. Our results indicate that gains overthe simple optimum binary power-control due to advanced interference cancellation techniquescan be harvested only if the cancellation efficiency is near-perfect.II. R
ELATED WORK
In this paper, we are motivated by recent work on interference networks that shows that binarypower-control is often close to optimal when interference is treated as Gaussian noise, links havemaximum (peak) power constraints, and the objective is to maximize the sum-rate, even if itis not necessarily optimal in general [6]. “Binary” here just means that a link is either “on” or“off”, either at zero power, or maximum power, without taking any value in the continuum ofpossible values between and the peak power level.In addition to [6], some other works such as [7], [8] and [9] also motivate us to investigate theoptimality of binary power-control. Both [7] and [8] consider jointly optimal allocation of ratesand transmission powers in CDMA networks under alternative objectives such as maximization ofthe sum of signal-to-interference-plus-noise-ratios ( SINR ) [7] and the packet success probability[8]. Both approaches convert the problem into a convex optimization problem, and show that theoptimum power-control is indeed binary under such approximations. In [9], the authors provedthe optimality of an almost binary power-control strategy, up to one exceptional transmissionpower level in the continuum between and the peak power level, maximizing the total uplinkcommunication rate.The results reported in [6] as well as in other works raise the further question: When is“binary” power-control exactly optimal? It has been shown in very recent work [10] that binarypower-control is optimal when there is total symmetry amongst the links, i.e., all direct link gainshave one particular value, and all the cross-link gains have another particular value (possibly thesame value as the direct link gain, but not necessarily). One interesting feature of the result isthat it is as if the sum-rate function of the powers were either Schur-convex, or Schur-concave(even though it is neither), leading to the observed result that either all links should be “on”or just one link should be “on” at the optimal solution. A two-link Schur-convex/Schur-concavestructure is observed and used, but it does not generalize to more than two links.In the present paper, we study the sum-rate maximization problem for the classical multipleaccess channel, where all the links terminate in a common receiver node, but the link gains can DRAFT be arbitrary. In this setting, we show that the power-control problem can be solved quite easilyvia an underlying Schur-convex structure. In contrast to the symmetric network of interferinglinks, it is no longer necessarily an all-or-one result: It is possible for the chosen set of links thatare “on” to be larger than a singleton, but smaller than the set of all users, but it always consistsof users with the best channels. On the other hand, we will observe from numerical results thatthe dominant modes, in terms of probability, correspond to the all-on or one-on solutions.Majorization theory and Schur-convex/concave structures were also successfully utilized insome previous works, including [11], [12], [13] and [14], to answer important questions incommunications theory. This paper is another successful application of majorization theory toprove the optimality of binary power-control.In [11], the authors focus on the transceiver design for point-to-point multiple-input-multiple-output (MIMO) communication systems. By using extra degrees of freedoms provided by multi-ple transmitter and receiver antennas, and assuming either minimum mean-square error (MMSE)receiver or zero-forcing receiver, they show that the optimum linear precoder at the transmitteris the one diagonilazing the channels ( i.e., independent noise at all channels and no interferenceamong them) when the cost function to be minimized is Schur-concave (or, the objective functionto be maximized is Schur-convex). Their results do not directly apply to the our problem sincewe consider the sum-rate maximization in the presence of interfering links in this paper. In fact,we solve a special case of an open problem posed in [11] in chapter 5 on the optimum designof transceivers for the MIMO multiple-access channel.In [12], the authors focus on the design of capacity achieving spreading code sequences forthe CDMA multiple-access channel without fading. They allow multi-user detection for jointprocessing of users. Even though the performance figure of merit we are interested in this paperis also related to the information capacity, our problem set-up is different than the set-up in[12]. In this paper, we look at the capacity achieving transmission power allocations, rather thanthe optimum spreading code sequence design, for Fading Gaussian channels in the presenceof interfering links. For example, our objective sum-rate function is Schur-convex whereas itis Schur-concave in [12]. In [13], the same authors extend the analysis in [12] to the caseof colored noise. In [14], they analyze the user capacity , which is defined as the maximumnumber of users that can be admitted to the system by allocating spreading code sequencesand transmission powers optimally without violating minimum
SINR requirements, of CDMA
DRAFT systems. In this work, we focus on achieveable sum-rates rather than on user capacity.Our results are different from the corresponding classic results in [15]. In [15], the maximumShannon-theoretic sum-rate is considered, whereas in the present paper, we treat interferenceas pure Gaussian noise. Although our assumption simplifies the receiver, it complicates thepower optimization problem. We note that the capacity region of the Gaussian multiple-accesschannel is well understood, and it is known that all points of the boundary of the rate region canbe achieved by successive decoding [16]. The optimal power-control for the Fading Gaussianmultiple-access channel with channel state information at the transmitters is also well understood[17]. In the present paper, we arrive at the problem from a different angle, where our interestis in understanding the structure of power-control problems in which interference is treatedas Gaussian noise (very relevant for general interference networks), which excludes successivedecoding or other multi-user decoding techniques.From a practical perspective, treating interference as Gaussian noise is the approach takenin most existing systems, including cellular systems. Note that the uplink of a cell is indeed amultiple-access channel. Successive decoding is more complex to implement, and suffers fromerror propagation, which is mainly a problem if channels cannot be estimated very reliably. Wenote that Qualcomm has recently produced a chip for successive decoding [18], so we cannotbe sure that successive decoding will not be used in practice. Indeed, we believe it will be. Inthe present paper, we provide a comparison between the performance of the optimum binarypower-control scheme with that of successive decoding, under various assumptions about theefficiency of the cancellation process. We expect that, in practice, successive decoding will becombined with user scheduling, to reduce the potential for error propagation, and the presentpaper provides insight into the problem of combined power-control and user scheduling, as willbe shown. III. N
ETWORK M ODEL , M
AJORIZATION AND N OMENCLATURE
In this section, we will introduce the network model and some basic concepts from the theoryof majorization.
DRAFT
A. Network Model
We focus on the uplink communication scenario where n mobile users communicate with asingle base station. At time-slot t , the received signal at the base station is given by the basebanddiscrete-time Gaussian multiple-access channel as Y ( t ) = n X i =1 p h i ( t ) X i ( t ) + W ( t ) , where X i ( t ) and h i ( t ) are the transmitted signal and the channel fading coefficient of the i th user,respectively, and W ( t ) is white Gaussian noise with variance σ at the base station. We assumethat W ( t ) represents the cumulative effect of the thermal noise and other-cell interference at thebase station. Without loss of generality, we assume that all users are subject to the same peaktransmission power constraint of P , i.e., E [ | X i ( t ) | ] ≤ P for all t . We call a power allocationvector (at time-slot t ) P = ( P , · · · , P n ) ⊤ binary if P i is either P or for all i . The signal-to-noise-ratio (
SNR ) of the communication system under consideration is defined to be the ratio ρ = Pσ .In Section IV-A, we will solve the optimum power allocation problem for time-invariant (slowfading) channels characterized by a fixed channel vector h , i.e., h i ( t ) = h i for all t . Extensionsto time-varying (fast fading) channels are straightforward. B. Majorization and Nomenclature R m and R m + represent the set of m dimensional column vectors with real and real non-negative coordinates, respectively. For a vector x in R m , we denote its ordered coordinates by x (1) ≥ · · · ≥ x ( m ) , and diag ( x ) represents the diagonal matrix with entries of x at the diagonal.When we write (in boldface), we mean the vector of ones. For x and y in R m , we say x majorizes y and write it as x (cid:23) M y if we have P ki =1 x ( i ) ≥ P ki =1 y ( i ) when k = 1 , · · · , m − ,and P mi =1 x ( i ) = P mi =1 y ( i ) . If the users in the original rate maximization problem have different peak transmission power constraints given by the peakpower vector P = ( P , · · · , P n ) ⊤ , then solving the modified optimization problem having the uniform peak power constraint P and the fading processes that are scaled versions of the ones in the original problem by a factor of P i P , for all i ∈ { , · · · , n } ,will be enough to find the optimal transmission power allocation for the original problem. If there is a minimum transmission power P min requirement to maintain some level of control traffic in the network, then P is defined to be binary if P i is either P or P min for all i . DRAFT
A function g : R m R is said to be Schur-convex if x (cid:23) M y implies g ( x ) ≥ g ( y ) ; g issaid to be strictly Schur-convex if g is Schur-convex, and x (cid:23) M y implies g ( x ) > g ( y ) for all x and y which are not a permutation of each other. g is Schur-concave if − g is Schur-convex.Intuitively, a Schur-convex function increases when the dispersion among the components of itsargument increases.Schur-convex/concave functions frequently arise in mathematical analysis and engineeringapplications, e.g., [11], [12], [13], [14] and [19]. For example, every function that is convexand symmetric is also a Schur-convex function. Another important example of a Schur-convexfunction is a separable-convex function. A function g : I m R , where I ⊆ R is an interval,is said to be a separable-convex function if g is of the form g ( x ) = P mi =1 f ( x i ) , where f isa convex function on I . Then, any separable-convex function is also a Schur-convex function.(See [20] or [21].) IV. M AIN RESULTS
A. Optimality of Binary Power-control
In this section, we will prove the optimality of binary power-control for single cell communi-cation systems without successive decoding at the base station. We begin by assuming that thechannel is time-invariant and characterized by a fixed channel vector h ∈ R n + given at time .The vector h can be generated according to a probability distribution, but once it is generated,it is fixed and known by the base station. For this case, we drop the time index, and write thesum-rate per slot as R h ( P ) = 12 n X i =1 log h i P i σ + P nj =1 h j P j { j = i } ! , (1)where P = ( P , · · · , P n ) ⊤ is the vector of transmission powers. The base of the logarithmfunction in (1) is equal to the natural number e , and therefore communication rates in this paperare measured in terms of nats per time-slot.The sum-rate in (1) can be achieved using Gaussian input distributions and random codingarguments, and this is the focus of the present paper. In general, these rates are not optimal, andhigher rates in the multi-user capacity region are known to be achievable [22]. In fact, there isnothing inherently suboptimal about using Gaussian codebooks: The suboptimality of (1) comesfrom a failure to exploit the information content in the interference, which can be removed via DRAFT cancellation. Nevertheless, we will treat the interference as Gaussian noise in the present paper,and in this context the relevant achievable rates are given in (1).We are interested in solving the following non-convex optimization problem.maximize R h ( P ) subject to P (cid:22) P . (2)Even though R h ( P ) is a non-convex function of transmission powers, it is a strictly Schur-convexfunction of received powers at the base station, which will enable us to obtain the solutions forthe non-convex optimization problem in (2). Lemma 1:
Let D = N ni =1 [0 , h i P ] , x = diag ( P ) · h ( i.e., x changes as P changes), and write R h ( x ) as a function of x = ( x , · · · , x n ) ⊤ as R h ( x ) = 12 n X i =1 log x i σ + P nj =1 x j { j = i } ! . (3)Then, R h ( x ) is a strictly Schur-convex function of x on D . Proof:
Fix B ≥ , and define D B = { x ∈ R n : x ∈ D and P ni =1 x i = B } . On D B = ∅ , wecan write R h ( x ) as R h ( x ) = 12 n X i =1 log (cid:18) σ + Bσ + B − x i (cid:19) . We define g ( y ) = P ni =1 log (cid:16) σ + Bσ + B − y i (cid:17) on [0 , B ] n . Note that g ( y ) is a separable-convexfunction on [0 , B ] n since log (cid:16) σ + Bσ + B − y (cid:17) is a strictly convex function on [0 , B ] . Thus, we concludethat g ( y ) is strictly Schur-convex on [0 , B ] n . Since R h ≡ g on D B , we also conclude that R h isa strictly Schur-convex function on D B for any B ≥ such that D B = ∅ . Since D = S B ≥ D B ,this last observation further implies that R h is a strictly Schur-convex function on D .Note that x is in D if and only if P (cid:22) P . Therefore, maximizing R h ( x ) on D is equivalentto solving the optimization problem in (2). This observation together with the Schur-convexityof R h will be the key for characterizing the optimum power allocation vectors.The following are two simple facts about an optimum power allocation vector P ∗ solving(2). At P ∗ , there must exist at least one user transmitting with positive power, and if thereis only one user transmitting with positive power, this user must transmit with full power. Italso directly follows from the Schur-convexity of R h that if there are more than one userstransmitting with positive power, one of them must transmit with full power. Otherwise, we can This can also bee seen by using simple scaling arguments [6].
DRAFT majorize the received power vector x = diag ( P ∗ ) · h , and obtain a strictly better sum-rate byre-adjusting transmission powers without violating the transmission power constraint. The nexttheorem establishes the binary nature of P ∗ and its structural properties. Theorem 1:
Any P ∗ solving the problem (2) is a binary power allocation vector at which theusers transmitting with full power correspond to the ones having the best channel gains. Proof: : See Appendix A.We now address the issue of uniqueness. Let P ( h ) = ( P ( h ) , · · · , P n ( h )) ⊤ be any optimalbinary power allocation. Note that this definition extends the model to allow fading, and we canconsider P ( h ) as providing a power control policy, adaptive to changing channel conditions.Then the following theorem provides uniqueness. Theorem 2:
Any optimal power-control policy P ∗ ( h ) assigns the channel to the best usersfor almost all fading states. If the stationary distribution of the fading process is absolutelycontinuous, then P ∗ ( h ) is unique up to a set of measure zero. Proof:
See See Appendix B.We note that the set of optimum power allocation vectors solving (2) is not necessarily asingleton. However, Theorem 2 establishes uniqueness if the channel state vector is generatedby an absolutely continuous distribution, which is a valid assumption for most practical systems.Therefore, when we refer to an optimum power allocation vector or power-control policy for therest of the paper, we will use P ∗ -notation without any ambiguity.Finally, it is important to consider what the constraint in (2) means in the case of a fadingchannel. We can interpret this constraint as a peak power constraint. If P were an averagepower constraint on the powers modulating Gaussian codebooks [17], then we would replace theconstraint that P ( h ) (cid:22) P for all h ∈ R n + with the less onerous constraint that E [ P ( h )] (cid:22) P .The reason for interest in peak power constraints is that in practice it is necessary to operatewithin the linear range of a power amplifier, and this may preclude bursts of power that may berequired if only the average power is constrained. B. Polynomial-time Algorithm for Finding P ∗ In this section, we provide a polynomial-time algorithm for finding the optimum powerallocation vector P ∗ ( h ) for a given channel state vector h . One of the consequences of thestructure of the optimum power-control policy established above is that it is piecewise constant: DRAFT0
There exists a partition of the fading state space into n − regions upon each of which theoptimum power-control policy is constant: P ∗ ( h ) = X S⊆{ , ··· ,n } , S6 = ∅ P S { h ∈D S } , where P S = ( P , · · · , P n ) ⊤ is a transmission power profile such that P i = P { i ∈S} , and the D S is the region on which only the users in S transmit with full power, and the rest are notscheduled for transmission. Even though it is possible to give exact characterizations of theseoptimum power-control regions when there are only a few users (e.g., see the two-user examplein Section V), it becomes prohibitively complex to determine them when there are many users.On the other hand, the structure of the optimum binary power allocation established aboveallows us to construct a simple, polynomial-time algorithm to compute the optimum powerprofile for any realized fading state and any number of users in the cell, which can be hard-coded into a scheduler circuit, without the need for any explicit characterization of the optimumpower-control regions. The suggested algorithm takes a fading state h as an input, computes thesum-rates R k ( h ) at which the best k , ≤ k ≤ n , users transmit with full power, and returnsthe optimum sum-rate maximizing transmission power profile at which only the best k ∗ usersare scheduled for transmission with full power. The pseudocode for this simple polynomial-timealgorithm is shown below. V. W HEN IS
TDMA
OPTIMAL ?In this section, we will establish the conditions under which the channel-state aware TDMApolicy, in which the channel is allocated to the best user, is optimal for maximizing sum-rate insingle cell wireless communication systems. Optimality of this TDMA policy was established(under symmetric fading distributions) in previous works such as [17] and [15] when evensuccessive decoding for interference cancellation is allowed, and users are subject to an average power constraint. On the other hand, as Theorems 1 and 2 suggest, this TDMA policy isnot always optimal in the communication scenario considered in this paper where successivedecoding is not allowed, and users are subject to peak power constraints. The following two-userexample further illustrates this point quantitively.
Example 1:
When there are two users in the system, the sum-rate maximizing power allocation P ∗ ( h ) is either ( P, ⊤ , (0 , P ) ⊤ , or ( P, P ) ⊤ for any given fading state h = ( h , h ) ⊤ by DRAFT1
Algorithm 1
Algorithm for computing optimum power allocation
Input:
Fading state h ∈ R n Output:
Max. sum-rate R h ( P ∗ ) and opt. power profile P ∗ ∈ R n + Initialization: R ( h ) := log (cid:0) ρh (1) (cid:1) , k ∗ := 1 , R h ( P ∗ ) := R ( h ) for k = 2 to n do R k ( h ) = P ki =1 log (cid:18) h ( i ) ρ − + P kj =1 h ( k ) { j = i } (cid:19) if R k ( h ) > R h ( P ∗ ) then R h ( P ∗ ) = R k ( h ) , k ∗ = k end ifend forreturn (i) Max. sum rate: R h ( P ∗ ) . (ii) P ∗ : allocate TX power P to the best k ∗ users, andzero to the rest.Theorem 2. Writing down the aggregate communication rate expressions for all three casesseparately, and comparing them, one can derive the following conditions for the optimal powerallocation for the two-user communication scenario: P ∗ ( h ) ⊤ = ( P, ⊤ if h > ρ − √ h ρ and h ≥ h (0 , P ) ⊤ if h > ρ − √ h ρ and h > h ( P, P ) ⊤ if h ≤ ρ − √ h ρ and h ≤ ρ − √ h ρ . (4)These three optimum power allocation regions are illustrated in Fig. 1. For any fading state h lying inside the shaded region in Fig. 1, the TDMA policy becomes suboptimal, and the sum-rateis maximized by allocating the full transmission power to both users. This situation occurs whenboth users experience similar and severe channel conditions, i.e., h i ≤ ρ − √ , i = 1 , . Onthe other hand, if the channel conditions experienced by users are relatively different from eachother, or any of them is good enough, i.e., h i > ρ − √ , then the TDMA policy maximizes thesum-rate.Note that the shaded region on which the TDMA policy is suboptimal shrinks to a point inthe high SNR regime when ρ grows to infinity. Therefore, in the high SNR regime, we see onemode of communication with very high probability: Only the best user transmits with full power.On the other hand, in the low
SNR regime where ρ goes to zero, the shaded region grows and DRAFT2
PSfrag replacements h h h hhh h ρ − √ ρ − √ ρ − ρ − P ∗ ( h ) = ( P, P ) ⊤ P ∗ ( h ) = (0 , P ) ⊤ P ∗ ( h ) = ( P, ⊤ h = ρ − √ h ρh = ρ − √ h ρ h = h Fig. 1. Optimum power allocation regions for the two-user communication scenario. For fading states lying in the shaded area,the TDMA policy is not optimal, and the sum-rate is maximized when both users transmit with full power. covers the whole positive orthant in the R -plane. Therefore, in the low SNR regime, we againsee only one mode of communication with very high probability: All users transmit with fullpower.When there are more than just two users, and for moderate
SNR values, other modes ofcommunication in which the best k , < k < n , users transmit with full power can arise.Roughly speaking, the present discussion implies that the performance loss arising from the useof the TDMA policy for scheduling the best user critically depends on the relative strength ofthe peak transmission power with respect to the total noise power, including the backgroundnoise and other-cell interference, present in the system. These observations will be the guidingprinciples for the proof of the optimality of the TDMA policy in the next theorem, and theywill be further supported through numerical results in Section VI.Figure 1 also illustrates why P ∗ is unique when the fading process has a continuous distri-bution. When h lies on the boundary where any two of these three regions intersect, there aremore than one power profile maximizing the sum-rate. For example, all three power profiles (0 , P ) ⊤ , ( P, ⊤ and ( P, P ) ⊤ perform equally well for sum-rate maximization at the point h = (cid:16) ρ − √ , ρ − √ (cid:17) ⊤ . However, the probability of such a pathological case happening DRAFT3 is zero, and P ∗ can be almost surely uniquely determined if the joint stationary distribution ofthe fading process is absolutely continuous. Theorem 3:
For all n ≥ , if h (1) ≥ (e − ρ − for a fading state h , then the channel-stateaware TDMA policy in which the channel is assigned to the user with the best channel statemaximizes the sum-rate at this fading state. Proof:
See Appendix C.VI. N
UMERICAL R ESULTS AND D ISCUSSIONS
A. Optimal modes: WB and TDMA
In spite of the relative simplicity of Algorithm 1, we note that its worst case complexityis O ( n ) when there are n users, due to the ordering of the channel states of users and thesummations involved. In this section, we examine the sum-rate performance of the heuristicallyderived scheme that simply takes the best of two choices: Either all users on at full power, whichwe call the wideband strategy (WB), or, exactly one user on at full power (the best user), whichwe call the TDMA strategy. To test out how well this suboptimal strategy works, we use thefollowing simulation model.We consider a circular cell centered at the base station and having radius [unit distance](usually in kilometers). We focus on low, moderate and high density networks, and vary the SNR parameter between − dB and dB to identify the performance of the power-controlledsingle cell communication systems for a broad spectrum of network parameters. The users areuniformly distributed over the network domain with node density λ [nodes per unit area]. Thefading model includes both slow-fading, modeled by means of the bounded path-loss function x α for α > [23], and Rayleigh fast-fading, modeled by means of independent unit exponentialrandom variables. All simulations are performed in C over at least independent networkrealizations to obtain average aggregate communication rate figures.We begin by examining the empirical distribution of k ∗ , the number of users scheduled in anyfading state by Algorithm 1 (the optimal algorithm). In Figs. 2 and 3, we show the empiricaldistribution obtained for k ∗ over independent network realizations when ( λ ≈ ) and The same conclusions continue to hold for different cell sizes, different path-loss models including the unbounded path-lossmodel and generalized fading models including log-normal shadowing and other possible random factors.
DRAFT4 −6 −5 −4 −3 −2 −1 No. of Users Scheduled for Transmission P r o b . D e n s i t y o f k * SNR = −10dBSNR = 0dBSNR = 10dB
Fig. 2. Empirical probability density function of the optimumnumber of users scheduled for transmission. ( λ ≈ ) −7 −6 −5 −4 −3 −2 −1 No. of Users Scheduled for Transmission P r o b . D e n s i t y o f k * SNR = −10dBSNR = 0dBSNR = 10dB
Fig. 3. Empirical probability density function of the optimumnumber of users scheduled for transmission. ( λ ≈ ) ( λ ≈ ) users are uniformly distributed over the network domain for SNR values − dB, dB and dB. Similar conclusions continue to hold for different values of node density and the SNR parameter.In all cases, even though other modes of communication are quite possible, TDMA and WBmodes predominantly arise. The reason for such behavior is that when the channel state ofthe best user is good enough, we schedule just this user to maximize the communication rate;otherwise, the channels of the remaining users are also in deep fades, creating a domino effectand all users are scheduled together to maximize the communication rate. Similar observationswere also made in [10], and proven to hold for the symmetric network of interfering links.Similarly, here, we can prove that scheduling a single user becomes certain as we scale up thenode density. To see why this is so, consider first a model with a fixed number, n , of users, thatwe place uniformly at random in the cell. Since we have an i.i.d. model for the user locations, wecan let F ( h ) be the cumulative distribution function of the channel of a randomly selected user.Then the probability that all the users fail the condition of Theorem 3 is F n ((e − ρ − ) whichdecays exponentially in n , irrespective of the SNR. Thus, for a large number of users we willalmost certainly just schedule the best user, although the number of users required to observe DRAFT5 −30 −20 −10 0 10 20 3010 −3 −2 −1 SNR Parameter (dB) Su m − r a t e [ N a t s p er S l o t] Opt. Bin. Power−controlHeuristic
Fig. 4. Comparison of sum-rates achieved by the optimum bi-nary power-control and the heuristic algorithm choosing eitherthe TDMA mode or WB mode for transmission. ( λ = 0 . ) −30 −20 −10 0 10 20 3010 −3 −2 −1 SNR Parameter (dB) Su m − r a t e [ N a t s p er S l o t] Opt. Bin. Power−controlHeuristic
Fig. 5. Comparison of sum-rates achieved by the optimumbinary power-control and the heuristic algorithm choosingeither the TDMA mode or WB mode for transmission. ( λ = 1 ) this phenomena will be larger for lower SNR. It is a straightforward extension from this fixed n model to the above numerical model, where the probability becomes E (cid:2) F N ((e − ρ − ) (cid:3) ,where N is the Poisson number of users with intensity λ , and one can show that this also decaysexponentially in λ . This phenomena is illustrated in Figure 3 where only the best user is selectedat SNR = 10 dB.In Figs. 4, 5, 6 and 7, we compare the sum-rates achieved by the heuristic algorithm thatsimply chooses the best of the two extreme modes (WB or TDMA) with the rates achievedby the optimum binary power-control policy. As illustrated in these figures, the performanceachieved by the heuristic algorithm almost perfectly tracks the performance achieved by theoptimum power-control, and therefore it can be implemented to maximize communication rates insingle cell communication systems for all practical purposes without any noticeable performancedegradation. Especially, for systems with large numbers of users, the proposed heuristic algorithmwill run an order of magnitude faster than Algorithm 1. We also note that the knee of the sum-rate curves (more apparent for high density networks) at which they become non-differentiablecorresponds to a phase transition from the WB mode to the TDMA mode for scheduling users[10]. DRAFT6 −30 −20 −10 0 10 20 3010 −2 −1 SNR Parameter (dB) Su m − r a t e [ N a t s p er S l o t] Opt. Bin. Power−controlHeuristic
Fig. 6. Comparison of sum-rates achieved by the optimumbinary power-control and the heuristic algorithm choosingeither the TDMA mode or WB mode for transmission. ( λ = 5 ) −30 −20 −10 0 10 20 3010 −2 −1 SNR Parameter (dB) Su m − r a t e [ N a t s p er S l o t] Opt. Bin. Power−controlHeuristic
Fig. 7. Comparison of sum-rates achieved by the optimum bi-nary power-control and the heuristic algorithm choosing eitherthe TDMA mode or WB mode for transmission. ( λ = 10 ) B. Benefits from successive decoding
In this section, we compare the aggregate communication rate achieved by the optimumbinary power-control policy with the throughput capacity limits that can be achieved throughsuccessive decoding. When the receiver is capable of successively decoding the received signalswith cancellation efficiency β ∈ [0 , , which represents the amount of cancelled signal power,the throughput capacity can be given by C SIC ( β ) = 12 E h " n X i =1 log h ( i ) ρ − + P nj =1 h ( j ) { j = i } − β P i − j =1 h ( j ) ! . (5)In (5), we used the usual decoding order in which the strongest users are decoded first andsubtracted from the composite signal (see [18], [24] and [25]). Note that we obtain the classicalthroughput capacity equation C SIC (1) = E h [log (1 + ρ P ni =1 h i )] if the interference can becancelled perfectly ( β = 1 ) [17]. Thus, there is no need for user scheduling when consideringsuccessive decoding under peak power constraints, and perfect channel state information at thebase station. However, in practical implementations, β is usually bounded away from one due toimperfect channel and signal estimations. In these cases, it may pay to do some user selection, butin the numerical results below, we assume that all users are scheduled for successive interferencecancellation, as in (5). DRAFT7 −30 −20 −10 0 10 20 3010 −3 −2 −1 SNR Parameter (dB) Su m − r a t e [ N a t s p er S l o t] Opt. Bin. Power−controlSuccessive Decoding ( β = 0.5)Successive Decoding ( β = 0.7)Successive Decoding ( β = 0.9)Successive Decoding ( β = 1) Fig. 8. Comparison of the sum-rate achieved by the opti-mum binary power-control and the throughput capacity limitsachieved by successive decoding. ( λ = 0 . ) −30 −20 −10 0 10 20 3010 −3 −2 −1 SNR Parameter (dB) Su m − r a t e [ N a t s p er S l o t] Opt. Bin. Power−controlSuccessive Decoding ( β = 0.5)Successive Decoding ( β = 0.7)Successive Decoding ( β = 0.9)Successive Decoding ( β = 1) Fig. 9. Comparison of the sum-rate achieved by the opti-mum binary power-control and the throughput capacity limitsachieved by successive decoding. ( λ = 1 ) In Figs. 8, 9, 10 and 11, we depict the sum-rates achieved by the optimum power-control policyand the throughput capacity limits achieved through successive decoding. As it must, the perfectsuccessive signal decoding capability increases the rates of communication that can be achievedin single cell communication systems. In particular, for high density networks with moderate
SNR values, the performance increase achieved by the perfect successive decoding can be asmuch as two times the average sum-rate achieved by the optimal binary power-control treatingall signals as noise. On the other hand, if the interference cancellation is not perfect and someresidual signal power remains after each cancellation step, the sum-rate achieved by successivedecoding saturates as
SNR increases, and the optimum binary power-control can achieve highercommunication rates. Therefore, practical successive interference cancellations at the chip level(e.g., QUALCOMM CSM6850) require near-perfect cancellation efficiency to harvest potentialgains due to complex successive decoding process.In its favour, successive decoding does provide more fairness to users, as it enables all usersto transmit and achieve sustainable data rates simultaneously. It is particularly well suited to themultiple cell context, as discussed in the conclusions section of [26], but we do not investigatethat scenario in the present paper. Nor do we consider the impact of average power constraints,which may be very important in practice [17].
DRAFT8 −30 −20 −10 0 10 20 3010 −2 −1 SNR Parameter (dB) Su m − r a t e [ N a t s p er S l o t] Opt. Bin. Power−controlSuccessive Decoding ( β = 0.5)Successive Decoding ( β = 0.7)Successive Decoding ( β = 0.9)Successive Decoding ( β = 1) Fig. 10. Comparison of the sum-rate achieved by theoptimum binary power-control and the throughput capacitylimits achieved by successive decoding. ( λ = 5 ) −30 −20 −10 0 10 20 3010 −2 −1 SNR Parameter (dB) Su m − r a t e [ N a t s p er S l o t] Opt. Bin. Power−controlSuccessive Decoding ( β = 0.5)Successive Decoding ( β = 0.7)Successive Decoding ( β = 0.9)Successive Decoding ( β = 1) Fig. 11. Comparison of the sum-rate achieved by theoptimum binary power-control and the throughput capacitylimits achieved by successive decoding. ( λ = 10 ) VII. C
ONCLUSIONS
This paper exploits the Schur-convexity property of the sum-rate function of received powers,to show that binary power-control is optimal for the multiple-access channel, when interferenceis treated as Gaussian noise, and there are peak power constraints on the users. If the fadingdistribution is absolutely continuous, then the optimum binary power-control policy is unique.We provide an algorithm to find the optimum power allocation, as a function of the channel state,that is polynomial in the number of users in the cell. However, we also present numerical resultsfor a realistically dimensioned single cell system which suggest that there is essentially no lossin restricting attention to the best of two possible allocations in each channel state: (i) The bestuser transmits at peak power with other users switched off, as in channel-state aware TDMA,(ii) all users transmit simultaneously at peak power. This drastically reduces the complexity ofthe power allocation problem. Finally, we compared all such schemes with successive decoding.Our main conclusions regarding successive decoding are that as far as sum-rate maximizationis concerned, successive decoding can gain up to about a factor of 2 over the optimal binarypower-control scheme for the single cell model considered in the present paper, provided thatthe interference cancellation is perfect, and the SNR is moderate (not high or low). However,at high or low SNR, the gain is much less than that, and if the cancellation efficiency is less
DRAFT9 than 1 ( i.e., some small fraction of the interference remains) then the optimum binary power-control approach is superior, as it is not interference limited. It must be noted that this analysispertains to only a single cell system, and to sum-rate maximization under peak power constraints.With multiple cells, and different objectives (such as maximization of logarithmic utilities) theconclusions are likely to be very different.A
PPENDIX AP ROOF OF T HEOREM P ∗ , there cannot be two different users i and j with < P ∗ i < P and < P ∗ j < P . To obtain a contradiction, suppose there exist such two users. Let x = diag ( P ∗ ) · h , x i = h i P ∗ i and x j = h j P ∗ j . Since P ∗ is a solution for (2), we have R h ( x ) ≥ R h ( y ) for all y ∈ D = N ni =1 [0 , h i P ] .Without loss of generality, assume x i ≥ x j . But now, we can re-adjust transmission powerlevels to achieve < y i = x i + ǫ ≤ h i P and ≤ y j = x j − ǫ < h j P for some ǫ ≥ smallenough. Then, the received power vector y formed as y i = x i + ǫ , y j = x j − ǫ and y k = x k for k = i, j , belongs to D and majorizes x . By Lemma 1, R h ( y ) > R h ( x ) , which produces acontradiction. As a result, if P ∗ is a solution for (2), there can be at most one exceptional userwith transmission power c in (0 , P ) . Others either transmit with full power, or do not transmitat all.We will now show that this exceptional case does not happen. Suppose c ∈ (0 , P ) . Let m bethe index of the user with power c , and S be the subset of users transmitting with full power.Let H = P i ∈S h i . Then, R h ( x ) on N i ∈ S [0 , h i P ] N [0 , h m P ] can be written as R h ( x ) = 12 X i ∈S log x i σ + x m + P j ∈S x j { j = i } ! + 12 log x m σ + P j ∈ S x j ! = 12 X i ∈S log h i ρ − + H + ch m P − h i ! + 12 log ch m P ρ − + H ! . y (cid:23) M x if and only if there exists a doubly-stochastic matrix A such that x = Ay . We can construct A as follows. For k = i, j , let A k,l = { l = k } , l ∈ { , · · · , n } . Let A i,l = a { l = i } , A i,l = (1 − a ) { l = j } , A j,l = (1 − a ) { l = i } and A j,l = a { l = j } , l ∈ { , · · · , n } . To find a , we solve for a = x i − x j + 2 ǫ x j − ǫx j − x i − ǫ x i + ǫ − x i x j , which produces a = x i − x j + ǫx i − x j +2 ǫ . DRAFT0
We define the following function on [0 , h m ] . g ( x ) = 12 X i ∈S log (cid:18) h i ρ − + H − h i + x (cid:19) + 12 log (cid:18) xρ − + H (cid:19) , whose derivative with respect to x is g ′ ( x ) = 12 1 ρ − + H + x − X i ∈ S h i ρ − + H − h i + x ! .g has to be maximized at x = ch m P because P ∗ solves (2). Since f ( x ) = 1 − P i ∈ S h i ρ − + H − h i + x is a strictly increasing function of x , we have g ′ ( x ) > for x > if f (0) ≥ . Thus, g ( h m ) >g (cid:0) ch m P (cid:1) , which is a contradiction. If f ( h m ) ≤ , we have g ′ ( x ) < for x < h m . Thus, g (0) >g (cid:0) ch m P (cid:1) , which is a contradiction. Similarly, if f ( h m ) > and f (0) < , we have g (cid:0) ch m P (cid:1) < max { g (0) , g ( h m ) } , which is another contradiction. As a result, c must be either zero or P , whichproves that P ∗ is binary, and it strictly dominates any non-binary power allocation vector.To see why the users with the best channel states transmit with full power, assume that h i > h j , P ∗ i = 0 and P ∗ j = P . We can achieve the same aggregate communication rate by setting thetransmission power of the i th user to P h j h i < P and that of the j th user to zero. However,such a transmission power allocation can be strictly dominated by a binary transmission powerallocation as proven above. Therefore, users transmitting with full power correspond to the oneswith the best channel states when transmission powers are allocated according to P ∗ .A PPENDIX BP ROOF OF T HEOREM P ∗ ( h ) directly follows from Theorem 1 and some measure theoreticarguments. Therefore, we focus on the uniqueness of P ∗ ( h ) . We define the sum-rate at a fadingstate h when the best k users transmit with full power as R k ( h ) = 12 k X i =1 log h ( i ) ρ − + P kj =1 h ( j ) { j = i } ! . We want to show that S = { h ∈ R n : ∃ k, m such that k = m and R k ( h ) = R m ( h ) } has prob-ability zero with respect to the stationary distribution of the fading process. To this end, itis enough to show that S has zero volume since the stationary fading distribution is absolutelycontinuous. Suppose not. Then, we can find m > k such that S k,m = { h ∈ R n : R k ( h ) = R m ( h ) } has positive volume. First, let m = k + 1 . This means that we can find a point y ∈ S k,k +1 and DRAFT1 a small R k +1 -ball B ( y , ǫ ) ⊆ S k,k +1 centered around y . This implies that as a function of itslargest ( k + 1) th component (keeping other coordinates constant at y ( i ) , ≤ i ≤ k ), R k +1 ( h ) is constant over (cid:0) y ( k +1) − ǫ, y ( k +1) + ǫ (cid:1) . One can show that this cannot happen by taking thepartial derivative of R k +1 ( h ) with respect to h ( k +1) .Similarly, if m ≥ k + 2 , we can find a point y ∈ S k,m and a small R m -ball B ( y , ǫ ) ⊆ S k,m centered around y such that R m ( h ) is constant over this ball as a function of its largest ( k + j ) th , j = 1 , · · · , m − k , components. However, by following the same steps in Lemma 1, it is nothard to show that R m ( h ) is a strictly Schur-convex function as a function of the largest m elements of h . Therefore, R m ( h ) cannot be constant over B ( y , ǫ ) as a function of its largest ( k + j ) th , j = 1 , · · · , m − k , components since we can obtain a different h from a given h ,both in B ( y , ǫ ) , such that h (cid:23) M h by only perturbing the largest ( k + j ) th , j = 1 , · · · , m − k ,components. A PPENDIX CP ROOF OF T HEOREM h , we derive another fading state g = h (1) by making the channelconditions of all users the same and equal to h (1) . For these two fading states, we have R g ( P ∗ ) ≥ R h ( P ∗ ) , (6)since any set of received powers that can be achieved under h can be achieved under g . Now,note that if P ∗ schedules only one user for transmission with full power at g , then it schedulesonly the best user for transmission with full power at h since the maximum sum-rate at g formsan achievable upper bound for the maximum sum-rate at h for this case.By using the structural properties of P ∗ established in Theorem 1, we can write R g ( P ∗ ) as R g ( P ∗ ) = 12 k ∗ X i =1 log (cid:18) h (1) ρ − + ( k ∗ − h (1) (cid:19) = 12 k ∗ log (cid:18) ρh (1) k ∗ − ρh (1) (cid:19) for some optimal k ∗ ∈ { , · · · , n } . Our aim is to find a condition on h (1) under which we canshow that k ∗ = 1 .A similar problem was addressed in [10] but for a different model: the symmetric network ofinterfering links. This is a model in which there are n links, each with a different receiver node, DRAFT2 and each link interferes with all the others. The symmetry refers to the fact that the direct linkgain is unity for all links, and the cross-link gain is √ ǫ between any pair of links. See figure 1in [10] for an illustration of this model. In [10] the received power is denoted by P max but ifwe replace that by ρh (1) then the sum-rate in this model, with n links on, is given by R n ( ǫ ) = n log (cid:18) ρh (1) ǫ ( n − ρh (1) (cid:19) . Note that if ǫ = 1 then this gives the same rate as n links on in the model of the present appendix,under fading state g , and indeed the symmetric network model degenerates into, effectively, asymmetric multiple access model in the special case ǫ = 1 .We can use results from [10], Section IV B, to obtain the condition on h (1) that we need.Section IV B examines the special case of binary power control in which a link is either on atfull power or switched right off. 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