Multiuser Transmit Optimization for Multicarrier Broadcast Channels ...
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Multiuser Transmit Optimization for Multicarrier Broadcast Channels: Asymptotic FDMA Capacity Region and Algorithms Louise M. C. Hoo, Bijit Halder, Member, IEEE, José Tellado, and John M. Cioffi, Fellow, IEEE
Abstract—We derive optimal and suboptimal multiuser transmit-optimization methods for a multicarrier broadcast channel with intersymbol interference under the frequency-division multiple-access (FDMA) restriction. The general FDMA-based multicarrier broadcast problem is formulated as a maximum weighted rate-sum problem. Given each user’s subchannel assignment, the optimal transmit strategy is achieved by multilevel waterfilling. Unfortunately, the problem of finding the optimal subchannel assignments is combinatorial. However, by relaxing the FDMA restriction, we obtain a convex reformulation that allows for efficient computation of the optimal solution, and therefore, a characterization of the FDMA capacity region for a broadcast channel. If all users share the same transmission medium, we prove that the optimal frequency partitioning among the users has an ordered structure that can be exploited to significantly reduce the computational complexity. To make multiuser transmit-optimization schemes practical for applications with relatively fast time-varying user data-rate requirements or priorities, further reduction in computational complexity is necessary. This is achieved by restricting the energy distribution to be constant across the used subchannels. Simulations indicate the low-complexity constant-energy methods presented are very robust, and suffer from negligible performance loss. Index Terms—Broadcast channel, constant-energy optimization, frequency-division multiple access (FDMA), multilevel waterfilling, multiuser loading, multiuser transmit optimization, orthogonal frequency-division multiplexing (OFDM).
I. INTRODUCTION
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N MOST communications networks, multiple users or multiple services requested by the same user must share a limited pool of resources, such as transmit power (energy), bandwidth (spectrum), and transmit duration. These users or services typically experience different channel characteristics, and will desire different data rates and demand different qualities of services (QoS) that include target error rates, latencies, and availability. Moreover, the transmission medium and the desired transmission
Paper approved by Y. Li, the Editor for Wireless Communications Theory of the IEEE Communications Society. Manuscript received December 17, 2002; revised November 15, 2003. This paper was presented in part at the IEEE Wireless Communications and Networking Conference, Chicago, IL, September 2000, and in part at the IEEE International Conference on Communications, Helsinki, Finland, June 2001. L. M. C. Hoo is with Broadcom Corporation, Sunnyvale, CA 94085 USA (e-mail: [email protected]). B. Halder was with Telicos Corporation, Santa Clara, CA 95050 USA (e-mail: [email protected]). J. Tellado is with Teranetics Corporation, Santa Clara, CA 95054 USA (e-mail: [email protected]). J. M. Cioffi is with the STAR Lab, Stanford University, Stanford, CA 94305 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.829570
Fig. 1. An L-user broadcast channel.
requirements may vary over time. Using a feedback channel to communicate back to the transmitter the channel information estimated at the receiver of each active user, multiuser transmit optimization strives to achieve a higher multiuser rate or spectral efficiency by optimizing the allocation of available bandwidth and transmit power among the users in a manner that balances system throughput and fairness. Multiuser transmit optimization is practical for use in wireline systems, fixed wireless systems, or in low-mobility environments where the channels vary slowly. If transmit reoptimization is done at a slower rate than the symbol rate, the overhead in the feedback channel can be low. The channel we will focus on is the broadcast channel. The broadcast channel consists of a sender with a transmit power and bandwidth budget sending independent information simultaneously to multiple receivers. Some examples of the broadcast channel are the downlink transmission of a wireless communication system and the downstream transmission of a wireline network. Fig. 1 illustrates a broadcast channel of one sender and receivers. We solve the multiuser transmit-optimization problem for multicarrier modulation systems under different performance criteria for the broadcast channel, using the same circular channel approach employed in [1]–[3]. This approach, which results in the discrete Fourier transform (DFT) basis functions as eigenfunctions of any finite impulse response (FIR) channel, is adopted by any multicarrier system that uses discrete multitone (DMT) or orthogonal frequency-division multiplexing
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HOO et al.: MULTIUSER TRANSMIT OPTIMIZATION FOR MULTICARRIER BROADCAST CHANNELS
Fig. 2. Multicarrier broadcast channel with intersymbol interference (ISI) viewed as parallel Gaussian broadcast channels.
N
(OFDM). The orthogonal decomposition of each user’s channel frequency response into parallel independent additive white Gaussian noise (AWGN) subchannels, using the DFT, allows any multicarrier broadcast channel to be equivalently viewed parallel broadcast channels, one for each frequency, as as illustrated in Fig. 2. For each frequency/subchannel , the model for the broadcast channel is given by (1) where is the magnitude of the channel frequency response is a zero-mean Gaussian noise variable with variof user , , is the input, and is the output for user . The ance variables , , are the energies and bits allocated to subchannel . In general, superposition coding, together with successive interference cancellation, is necessary for achieving multiuser capacity [2], [3]. This implies that some subchannels need to be shared among different users, thereby making decoding a highly complex problem. In most practical systems, the improvement in capacity is not sufficient to justify the increase in complexity. To simplify the transmitter and receiver implementations, the focus here is on multiuser transmit optimization based on the frequency-division multiple-access (FDMA) scheme. In an FDMA scheme, orthogonality between the users is achieved by having the users occupy different sets of subchannels. An FDMA scheme is a natural choice for a multicarrier system, since it already has an inherent FDMA structure in place. Furthermore, an FDMA scheme has been shown to be the capacity-achieving scheme in some special cases [2], [4]. Given each user’s subchannel assignment, the maximum multiuser rate is achieved by performing separate waterfilling
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for each user in the multiple-access channel case, and joint waterfilling for each user in the broadcast channel case. Effectively, the nonoverlapping nature of the subchannel assignments decouples the multiuser system into a system of independent users. However, this multiuser rate can be far below the multiuser capacity if the subchannel assignments are chosen arbitrarily, often resulting in many inefficiently used subchannels. Unfortunately, due to the combinatorial nature of the FDMA restriction, finding the optimal subchannel assignments requires an exhaustive search. Hence, the exponential complexity of the optimal FDMA multiuser transmit-optimization methods makes them impractical. In [5]–[7], low-complexity suboptimal algorithms were proposed to closely approximate the optimal FDMA strategy for supporting several services with different QoS under various performance criteria in a multiservice OFDM system. Later, in [4] and [8], these algorithms were proven to be asymptotically optimal by relaxing the FDMA restriction to permit time sharing of each subchannel. This relaxation technique was also used in [9] and [10]. In [9], it was applied to find the optimal subchannel allocation, which minimizes the total transmission power while satisfying each user’s minimum rate constraint in a multiuser OFDM system. In [10], it was applied to characterize the FDMA capacity region for a Gaussian multiple-access channel with intersymbol interference (ISI). To make multiuser transmit-optimization schemes practical for applications with relatively fast time-varying user data-rate requirements or priorities, further reduction in computational complexity is achieved by restricting the energy distribution to be constant across the used subchannels [4], [11]–[13]. The organization of this paper is as follows. Section II formulates the Gaussian broadcast channel with the FDMA restriction and shows that this is a combinatorial problem. Section III describes the convex relaxation technique used to reduce this exponential complexity to a polynomial complexity. Section IV proves that the optimal frequency partition follows an ordered structure for some special cases. In Section V, to achieve further reduction in complexity, we place a constant-energy restriction over the used subchannels, thereby eliminating energy optimization. Computationally efficient FDMA-based multiuser transmit-optimization methods are dereived. Simulation results presented in Section VI show that these low-complexity methods are very robust, and suffer from negligible performance loss, even at low signal-to-noise ratios (SNRs), where the constant-energy approximation is less accurate. Section VII concludes the paper. II. FDMA FORMULATION By approximating each channel transfer function with a piecewise-constant model, the whole frequency spectrum is bins or subchannels,1 with each bin discretized into having a flat channel response. For the th frequency bin, let and be the energy allocated to user , and the ratio of the magnitude squared of the channel response and the noise power spectral density for user . Then, we formulate the problem of an -user transmit optimization under 1Frequency
bin and subchannel will be used interchangeably.
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the FDMA restriction for a broadcast channel as a weighted rate-sum formulation
vision multiple-access (TDMA) optimization problem, as follows:
Total energy constraint s.t. FDMA restriction (2) , the weight assigned to user , is constrained , is the user index set, is the frequency index set, and is the subchannel assignment for user . As , this piecewise-constant approximation becomes the optimal continuous FDMA formulation.2 of each Given the weights and the channel responses user, the objective in (2) is to find the energy distribution that maximizes the weighted sum of each user’s achievable rate subject to a total transmit-energy constraint and an FDMA restriction. Maximizing the weighted rate-sum for all possible ’s traces out the boundary of the capacity region. For any fixed subchannel assignments , the optimal FDMA solution to (2) is achieved by multilevel waterfilling [4], [13], whose underlying equations are where by
(3)
(4) where
if if
However, finding the optimal subchannel assignments among the users requires searches. Hence, the computation of the optimal FDMA solution has exponential complexity. In general, the number of used subchannels needs to be optimized until all energies are positive. (Refer to [4] for more details.) For the special case when all users have equal weights, an FDMA scheme among the users is indeed a capacity-achieving denoting the achievable rate scheme. Specifically, with for the broadcast channel is for user , the rate-sum maximized by assigning each subchannel to the user with the . In other words, best gain-to-noise-power ratio given by the transmitter should transmit information only to the user with the best reception for each subchannel, and the optimal energy distribution follows the waterfilling solution. III. FDMA–TDMA FORMULATION: A CONVEX RELAXATION If time sharing is allowed between users for each frequency bin, formulation (2) can then be recast as an FDMA-time-di2Any
N is accurate for a multicarrier formulation.
(5) is the time-sharing factor for user The additional variable in the th frequency bin. In other words, user occupies the th frequency bin for a fraction of the time. In this case, , . Another equivalent interpretation of for multicarrier transmission is that is the amount of bandwidth allocated to user in the th subchannel. This is equivalent to further partitioning of bandwidth in each subchannel, effectively creating a multicarrier system with a total number increases, the bin width of subchannels greater than . As decreases, which implies that the increase in the aggregate rate contributed by time sharing among the users decreases, as well. , no time sharing occurs, and the opIn the limit, as timum FDMA-TDMA solution is the optimum FDMA solution. Therefore, for sufficiently large , an FDMA strategy that assigns every subchannel to the user with the largest time-sharing factor will result in negligible performance loss relative to the optimum FDMA solution. This relaxation technique was also used in [9] and [10]. Observation 3.1: The FDMA-TDMA formulation of (5) is convex,sinceitisthemaximizationofaconcavecostfunctionover a convex constraint set. First, the cost function is a sum of functions of the form , and are some positive constants. Since any positive where linear combination of concave functions is concave, it suffices is concave in . to prove that This is achieved by proving that the Hessian for is positive semidefinite over the positive quadrant of , . Next, all the constraint functions are linear in , , and thus give convex sets. Since the intersection of convex sets is convex [14], the constraint set in (5) is convex. Since the FDMA-TDMA optimization problem is convex, a local maximum or minimum is also the global maximum or minimum. This optimal solution can be computed very efficiently (with polynomial complexity) using interior-point methods [14], thereby allowing us to characterize the asymptotic FDMA capacity region for the Gaussian broadcast channel with ISI. There are two other practical convex multiuser formulations for the broadcast channel. In the minimum total-energy formulation, the optimization criterion is to minimize the total power used by the sender, while satisfying the constraints that each receiver achieves a certain fixed rate. In the mixed formulation, the optimization criterion is to maximize a weighted sum of rates for a subset of users, while satisfying the fixed-rate constraints for the rest.
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Observation 3.2: The maximum-weighted rate-sum formulation of (5), the minimum total-energy formulation and the mixed formulation are equivalent. That is, given a problem in one formulation, we can construct a new problem recast as either of the other two formulations, such that the optimal energy allocation and optimal rates for all users are identical for all three formulations. IV. ORDERED FREQUENCY PARTITION If the channel-to-noise responses for all the users are simply , , scaled versions of each other, i.e., we will show in this section that the optimal frequency partition among the users for the FDMA-TDMA multiuser-optimization problem for the broadcast channel, as described in Section III, follows a specific ordering. Multiservice-optimization problems are practical examples where such conditions on the channel-to-noise responses are true. Multiservice optimization refers to the allocation of resources (energy and frequency) for services that simultaneously occupy the same transmission medium. These services are either fixed rate, e.g. voice, or variable rate, e.g. internet data, multimedia, and they require different noise immunities as characterized by the parameter . is In the context of multicarrier transmission, the parameter commonly known as the SNR gap, which is a function of the coding scheme of choice, the target probability of symbol error, and the desired noise margin. The multiservice problem for multicarrier systems was first studied in [5]–[7]. The fast loading algorithms proposed therein are based on assigning the subcarriers to the services in order of their respective gaps. Specifically, the service with the highest gap is assigned subcarriers with the highest SNRs, and the service with the smallest gap is allocated subcarriers with the worst SNRs. Therefore, the search for the optimum subcarrier allocation has been reduced from an exhaustive search to a polynomial search. These low-complexity algorithms were found to be applicable for solving the optimal frequency-duplex problem in very-high-speed digital subscriber line (VDSL) systems [10]. In [5] and [6], assumptions of high SNR and equal-energy distribution are made to justify such an ordered subcarrier assignment. In the following, we will present a rigorous proof for the optimality of such an ordered frequency partition without making these assumptions. Observation 4.1: For clarity, the focus first is on a two-user . To simplify the derivation of the proof, we scenario will substitute “ln” for “log” from now on. Under the assump, , the two-user minimum total-ention that ergy formulation is
s.t.
(6) Theorem 4.1: Without loss of generality, assume that the channel-gain-to-noise ratios, , , are distinct and are
Fig. 3. Optimal frequency partition for the three-user multiservice 0 0 and sorted in a descending FDMA-TDMA problem with 0 order. , , and are the waterfilling levels.
K K
Fig. 4.
K
>
>
g
Illustration for Observation 4.2.
sorted in a descending order. Then, the optimal frequency partition for formulation (6) is a two-band assignment with sharing of the boundary subchannel only, if at all. Furthermore, assume . Then, there exists , without loss of generality that , such that user 1 occupies subchannels 1 to where , and user 2 occupies subchannels to . Moreover, time sharing occurs, if at all, only at the boundary subchannel between user 1 and user 2. Proof: See the Appendix. Fig. 3 illustrates the optimal solution for the three-user multiservice problem. -user multiservice case makes use of The proof for the induction and the following observation. be the optimal Observation 4.2: (See Fig. 4.) Let . solution to the -user multiservice problem, where Also, assume that there are no shared subchannels. (This can be achieved by appropriately subdividing the shared subchannels and resolving the problem with the redefined larger set of subchannels.) Let be the set of subchannels allocated to the th user in accordance with the optimal solution. Now, define a -user problem by removing the th user from reduced , and its the set of users, as well as its energy set of subchannels from the set of all subchannels. Then, the optimal subchannel assignment and energy allocation for -user problem is identical to the -user the reduced users. problem, for the remaining Proposition 4.1: If for , then under the sorted channel assumption, the optimal solution to the -user multiservice problem must have an -band structure. Moreover, if , then the set of subchannels assigned to each user is in the
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Illustration for proof of Theorem 4.2.
following order. There exist
, where such that user 1 occupies , user occupies subchannels subchannels 1 to to , and user occupies subchannels to for , . Theorem 4.2: If Proposition 4.1 holds for some , then . it must hold for Proof: We present a proof by contradiction. In particular, -user problem we show that if the optimal solution to an does not have an -band structure, then we can construct a reduced -user problem with an optimal solution that does -user not have an -band structure. Assume that for an -band problem, the optimal solution does not have an structure. Then, there must be at least one user, say user , with more than one band of subchannels assigned. In other words, is composed of at the set of subchannels assigned to user least two bands, with a nonempty set of subchannels between those bands. Let be a user that is assigned one of the sub, we can select a channels from the set . Since user that is different from and . Let be such a user and be the set of subchannels assigned to . Now we formulate a reduced -user problem by removing from the set of users and from the set of subchannels. From Observation 4.2, we conclude that the optimal subchannel assignment for the reduced problem remains the same for users and , i.e., user uses some subchannels that are between subchannels assigned to user . Thus, the solution to the reduced -user problem does not have an -band structure, thereby contradicting our original assumption. A simple illustration of the steps for the proof of is shown in Fig. 5. The ordering contradiction for structure can be proven using a similar argument. Theorem 4.3: If for , then under the sorted channel assumption, the optimal solution to the -user problem, formulated either as maximum-weighted rate-sum, minimum total-energy, or mixed, has an -band structure, and with time sharing only at the boundary subchannels, , then user 1 if at all. Moreover, if is assigned the best subchannels first, followed by user 2, and , where so forth. Specifically, there exist such that user 1 occupies subchannels 1 to , user occupies subchannels to , and user occupies subchannels to for . Moreover, sharing occurs, if at all, at the boundary between user and user . subchannel Proof: The proof follows from Observation 3.2 and Theorems 4.1 and 4.2. V. CONSTANT-ENERGY OPTIMIZATION Although optimal multiuser loading algorithms have been proposed in [8]–[10] to solve various forms of the
FDMA-TDMA optimization problem, in general, the computed time-sharing factors are irrational, and thus, lead to large latencies in practice. Moreover, the computational complexity of these algorithms are not suited for applications with relatively fast time-varying user data-rate requirements or priorities. To achieve further reduction in computational complexity, this section presents a new approach that restricts the energy distribution to be constant across the used subchannels. The process of determining the optimal set of useful subchannels and then assigning them to the best users is called bandwidth optimization [4], [13]. We derive the optimal and one suboptimal constant-energy multiuser loading methods with bandwidth optimization for the broadcast channel. The constant-energy approach is motivated by two observa, tions of (4). If user experiences a flat channel, i.e., the optimal energy allocation is which is constant for all subchannels allocated to user . Second, at , which implies that the optimal enhigh SNRs, ergy distribution is constant. For frequency-selective channels, a constant transmitenergy distribution has been shown to be almost as good as the waterfilling energy distribution, as long as there is bandwidth optimization for the single-user case [15], [16]. For the broadcast channel, the weighted rate-sum maximization problem under joint constant-energy and FDMA restrictions becomes
s.t. or (7) where is the total transmit-energy budget. The set condenotes tains the indexes of all the used subchannels, and indicates that user the cardinality of . The variable is assigned subchannel , and indicates otherwise. The first two contraints in (7) represent the FDMA restriction, which implies that each subchannel is assigned to one user, at most. Associated with each user is the parameter . Given the set of used subchannels, , the cost function in (7) can be rewritten as follows: (8) which implies that the optimal strategy is to assign each subchannel in to the user that achieves the maximum weighted rate with the same power. Optimal Algorithm: to Let be the number of best subchannels to use. For , do 1) 2) 3) sum of biggest is the weighted aggregate rate from using best subchannels.
HOO et al.: MULTIUSER TRANSMIT OPTIMIZATION FOR MULTICARRIER BROADCAST CHANNELS
The maximum weighted aggregate rate is
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.
For each iteration, step 2 requires comparisons and log operations, while step 3 requires operations if using the divide-and-conquer-based sorting algorithms. . Hence, the worst-case complexity is Instead of searching all sets of best subchannels to use, different stopping criteria can be employed. One of them is presented here. Suboptimal Algorithm: 1) Initialize the iteration index and the set of useful . Thus, the number of subchannels . useful subchannels is 2) Initialize the set that contains subchannel assignment for , . user , . 3) Set the weighted aggregate rate . 4) Set 5) Assign to user where . 6) Compute , stop. 7) If 8) Else, remove subchannel
Fig. 6.
High SNR scenario: average SNR of 20 dB.
. . from
where .
9) , and . Return to 4. Essentially, the suboptimal algorithm outlined starts off with using all subchannels, and then removing the subchannel from that contributes the least weighted rate until no further increase in the weighted rate-sum occurs. This algorithm has a operations, but should only reworst-case complexity of operations. quire The constant-energy formulation simplifies the multiuser transmit-optimization problem tremendously but with negligible performance loss. Although slightly better performance can be achieved by relaxing the restriction that each user be allocated the same amount of energy, this approach will involve set assignments, making the problem combinatorial. Previously known constant-energy multiuser loading algorithms can be found in [11] and [12]. In [11], the optimization criterion is to minimize the total power used, while satisfying the constraint that each user is allotted a fixed number of subchannels. In [12], the optimization criterion is to maximize the minimum achievable rate among all users. However, none of these methods optimize over the set of useful subchannels, unlike the proposed algorithms.
VI. ALGORITHM PERFORMANCE ANALYSIS The performances of four multiuser transmit-optimization algorithms for the broadcast channel are evaluated in this section. These algorithms are: 1) FDMA-Ex (Section II): the optimal FDMA method of exponential complexity; 2) FDMA-TDMA (Section III): the optimal FDMA-TDMA method of polynomial complexity; 3) FDMA-CE (Section V): the suboptimal constant-energy FDMA method;
4) FDMA-Fixed: a method that preassigns fixed sets of subchannels to each user without bandwidth optimization, i.e., without using channel knowledge of each user. Specifically, the total number of subchannels is evenly divided among all users, if possible, with any remaining subchannels assigned, one at a time, to the user with the largest weight first, and so forth. The ordering among the users is arbitrarily fixed. The downlink transmission (broadcast channel) of a three-user multicarrier system is considered. It is assumed that all users experience independent channels where each channel frequency response corresponds to a frequency-selective Rayleigh fading channel, characterized by an exponential power delay profile with a normalized delay spread of three. The formulation used is the weighted rate-sum formulation. Hence, the performance measure is the achievable averaged weighted rate-sum normalized by the total number of dimensions. (A complex subchannel on the performance of has two dimensions.) The effect of each algorithm is investigated. For each , 20 independent channel realizations are generated for each user. At dB, , and an average SNR of 20 dB, the results in Fig. 6 show that FDMA-CE suffers from negligible rate loss with respect to FDMA-Ex. The reason for this is that the constant-energy approximation is very good for high SNR cases. On the other hand, FDMA-Fixed has a rate loss of 18%. At an average SNR of 5 dB, the results in Fig. 7 show that FDMA-CE suffers from an average rate loss of 0.4%, while FDMA-Fixed incurs an average rate loss of 31%. The perfor, mance comparisons were not carried out beyond because it would require a prohibitive effort to compute the optimal FDMA solution, which has exponential complexity. The main reason that FDMA-CE works well for low-SNR cases is because of multiuser diversity. Since the users experience mutually independent channel frequency responses, it is likely that one or more users will find each subchannel favorable. Effectively, the multiuser FDMA-CE algorithm selects the user with the best subchannel gain for each subchannel, creating a new channel frequency response that is relatively flatter,
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VII. CONCLUSION The computation of the FDMA capacity region for broadcast channels with ISI requires exponential complexity. By relaxing the FDMA restriction, we obtain a convex reformulation that allows for efficient computation of the optimal solution, and therefore, a characterization of the asymptotic FDMA capacity region. We also prove that the optimal frequency partition has an ordered structure under certain conditions. By enforcing a constant-energy restriction, further reduction in computational complexity is achieved. We present the optimal and one suboptimal constant-energy method with bandwidth optimization. Simulation results show that these methods suffer from negligible performance loss, even at low SNRs, where the constant-energy approximation is not valid. APPENDIX PROOF OF THEOREM 4.1
Fig. 7. Low SNR scenario: average SNR of 5 dB.
The proof involves the application of the Karush–Kuhn–Tucker (KKT) optimality conditions. For the and derivation of the proof, we will assume that . We set up the Lagrangian function as follows:
(9) Fig. 8. Average SNR of 10 dB.
thereby making the constant-transmit-energy assumption accurate even for low-SNR cases. Since the FDMA-TDMA (with energy or power optimization) formulation is a convex relaxation to the FDMA formulation, we can use the optimal FDMA-TDMA solution, called FDMA-TDMA in Fig. 8, to upper bound the respective performance loss of FDMA-CE and FDMA-Fixed relative to subchannels. Two conclusions can be FDMA-Ex for drawn from Fig. 8. The first is that the additional complexity incurred for power optimization gives only a slight rate improvement (0.13%) over the much simpler constant-energy scheme. The second is that a considerable increase in spectral efficiency, 26% in this case, can be achieved from bandwidth optimization using channel knowledge at the transmitter. In general, the magnitude of the gain in spectral efficiency will depend on the degree of correlation among the users’ channel frequency responses. For instance, if the channel frequency responses are all flat, then there is no gain from performing bandwidth optimization.
Setting the partial derivatives of , and to zero, we get
with respect to
,
,
(10)
(11) and (12) Rewriting (12), we get (13)
HOO et al.: MULTIUSER TRANSMIT OPTIMIZATION FOR MULTICARRIER BROADCAST CHANNELS
Equation (13) is the classical waterfilling equation, and and are the optimal waterfilling levels for users 1 and 2, respectively. Equation (13) asserts that there exist positive constants and such that, for each , (13) hold true when and . If both , are negative, then subchannel is not used by either user and should be excluded from the opis negative, then subchannel timization process. If is not used by user 1 (2) and should be excluded from the optimization set for user 1 (2). Substituting (13) for and , and combining (10) and (11), we obtain (14) where
, and , . Since the optimization problem is convex, (13) and (14), as well as the fixed-rate constraints, the total subchannel-width constraint in , , formulation (6), and the nonnegativity constraints on , are the necessary and sufficient optimality conditions. and If the th subchannel is shared, then , which, in turn, implies that and due to the complementary slackness condition [14]. As a result, the right-hand side of (14) becomes zero. In the following, we show that there is at most one value of , or equivalently, one , that , thereby proving that there is sharing of at satisfies most one subchannel. We consider two distinct cases. Case 1) . For this case, for , , and . Therefore, is a strictly increasing function of when . Hence, there cannot be more than one subchannel for a constant . In [4], it that satisfies was proven that when , gives a suboptimal solution as the total energy required to satisfy all fixed-rate constraints decreases with any change to the shared subchannel. . Case 2) , , from (13), For , we see that the relations have to hold true. In other words, . Using for , we can the fact that , further determine that . Since and for and , is a strictly convex function with a minimum at . Hence, and or , where and . Then, there cannot be more than one subchannel for which such that . On the other hand, if , there is no feasible solution satisfying all specified constraints. (Refer to [4] for more details and illustrations.)
Fig. 9.
C
0 f (1=g
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The next step is to prove that the optimal frequency partition among the users is a two-band partition with user 1 assigned the best subchannels first, followed by user 2. (This order is true , which is assumed here.) Again, we refer only when to (14), which is the KKT optimality condition that has to be satisfied for each subchannel . We consider three possible cases. and , then subchannel is shared. 1) If and , which means that subchannel 2) If is assigned to user 1, then and , due to the complementary slackness condition. Thus, from (14), . and , which means that subchannel 3) If is assigned to user 2, then and , due to the complementary slackness condition. Thus, from (14), . Since is the optimal solution when [4], from Case 2, is convex. Therefore, , is concave, as shown in Fig. 9, and crosses the zero axis at most once. With ’s sorted in a dethat satisfy scending order, the subchannels with the largest are assigned only to user 1, and the subchannel whose gain-to-noise ratio satisfies is time shared between both users, and the subchannels with that satisfy are assigned only the smallest to user 2. In other words, if subchannel is the subchannel should that is time shared, then the user with the higher subchannels, while the other user solely occupy the best subchannels, giving rise to solely occupies the worst a two-band frequency partitioning structure with time sharing only of the boundary subchannel.3 ACKNOWLEDGMENT The first author, L. M. C. Hoo, would like to thank W. Yu for helpful discussions on the proof of Theorem 4.1. REFERENCES [1] W. Hirt and J. L. Massey, “Capacity of the discrete-time Gaussian channel with intersymbol interference,” IEEE Trans. Inform. Theory, vol. 34, pp. 380–388, May 1988. [2] R. S. Cheng and S. Verdú, “Gaussian multiaccess channels with ISI: Capacity region and multiuser waterfilling,” IEEE Trans. Inform. Theory, vol. 39, pp. 773–785, May 1993. 3In [10], the same approach was used to prove a two-band partition for a two-user Gaussian multiple-access channel if both users have the same channel characteristics.
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[3] A. J. Goldsmith and M. Effros, “The capacity region of broadcast channels with intersymbol interference and colored Gaussian noise,” IEEE Trans. Inform. Theory, vol. 47, pp. 219–240, Jan. 2001. [4] L. M. C. Hoo, “Multiuser transmit optimization for multicarrier modulation systems,” Ph.D. dissertation, Stanford Univ., Stanford, CA, Dec. 2000. [5] L. M. C. Hoo, J. Tellado, and J. M. Cioffi, “Dual QoS loading algorithms for multicarrier systems offering different CBR services,” in Proc. IEEE PIMRC, vol. 1, Boston, MA, Sept. 1998, pp. 278–282. [6] , “Dual QoS loading algorithms for DMT systems offering CBR and VBR services,” in Proc. IEEE GlobeCom, vol. 1, Sydney, Australia, Nov. 1998, pp. 25–30. , “Discrete dual QoS loading algorithms for multicarrier systems,” [7] in Proc. IEEE Int. Conf. Communications, Vancouver, BC, Canada, June 1999, pp. 796–800. [8] L. M. C. Hoo, B. Halder, J. Tellado, and J. M. Cioffi, “Asymptotic FDMA capacity region for broadcast channels with ISI,” in Proc. IEEE Int. Conf. Communications, vol. 6, Helsinki, Finland, June 2001, pp. 1648–1653. [9] C. Y. Wong and R. S. Cheng, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Select. Areas Commun., vol. 17, pp. 1747–1757, Oct. 1999. [10] W. Yu and J. Cioffi, “FDMA capacity of Gaussian multiple-access channels with ISI,” IEEE Trans. Commun., vol. 50, pp. 102–111, Jan. 2002. [11] C. Y. Wong, C. Y. Tsui, R. S. Cheng, and K. B. Letaief, “A real-time subcarrier allocation scheme for multiple-access downlink OFDM transmission,” in Proc. IEEE Vehicular Technology Conf., vol. 2, Amsterdam, The Netherlands, Sept. 1999, pp. 1124–1128. [12] W. Rhee and J. M. Cioffi, “Increase in capacity of multiuser OFDM system using dynamic subchannel allocation,” in Proc. IEEE Vehicular Technology Conf., vol. 2, Tokyo, Japan, May 2000, pp. 1085–1089. [13] L. M. C. Hoo, J. Tellado, and J. M. Cioffi, “FDMA-based multiuser transmit optimization for broadcast channels,” in Proc. IEEE Wireless Communication Networking Conf., vol. 2, Chicago, IL, Sept. 2000, pp. 597–602. [14] J.-B. Hiriart-Urruty and C. Lemarâechal, Convex Analysis and Minimization Algorithms. Berlin, Germany: Springer-Verlag, 1993. [15] P. S. Chow, “Bandwidth optimized digital transmission techniques for spectrally shaped channels,” Ph.D. dissertation, Stanford Univ., Stanford, CA, May 1993. [16] B. Schein and M. Trott, “Suboptimal power spectra for colored Gaussian channels,” in Proc. IEEE Int. Symp. Information Theory, 1997, p. 340.
Louise M. C. Hoo received the B.S.E.E. degree (with distinction) from Cornell University, Ithaca, NY, in 1994, and the M.S.E.E. and Ph.D. degrees from Stanford University, Stanford, CA, in 1996 and 2000, respectively. She is currently a Senior Staff Scientist with Broadcom Corporation, Home and Wireless Networking Division, Sunnyvale, CA. She has been working on physical layer algorithm design, and development and testing of the IEEE 802.11-based wireless LAN systems. She has also held internship positions at Amati Communications Corporation (now part of Texas Instruments), CA, Hewlett-Packard Laboratories, CA, and AT&T Bell Laboratories, NJ. She has over ten IEEE publications.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 6, JUNE 2004
Bijit Halder (M’98) received the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 1998. Since 1998, he has been working in the industry and is involved in transceiver design for DSL and Ethernet. His research interests include development of low-complexity signal-processing algorithms and efficient systems design for communications.
José Tellado received the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA in 1994 and 1999, respectively. He is currently the Director of Systems Design at Teranetics, Santa Clara, CA, which he cofounded in 2003 to develop very-high-speed Ethernet IC solutions. Prior to Teranetics, he was the Modem Architect at Intel’s Broadband Wireless Access group. Before that, in addition to being an Employee–Founder and the Director of Technology at Iospan Wireless, San Diego, CA, he was also the Chief Designer of the innovative MIMO-OFDM AirBurst™ technology that was acquired by Intel. At Stanford University, he proposed new advanced methods for multicarrier modulation (DMT/OFDM) in the areas of peak-to-average-power ratio reduction and multiuser transmit optimization. He has also consulted for Apple Computers Advanced Technology Wireless Group and Globalstar in the areas of WLAN and low-orbit satellite cellular systems, respectively. He is the author of over 16 IEEE publications, 10 standards contributions, 15 patents, and the book Multicarrier Modulation With Low PAR: Applications to DSL and Wireless (Norwell, MA: Kluwer, 2000).
John M. Cioffi (S’77–M’78–SM’90–F’96) received the B.S.E.E. degree in 1978 from the University of Illinois at Urbana-Champaign and the Ph.D.E.E. degree in 1984 from Stanford University, Stanford, CA. He was with Bell Laboratories from 1978 to 1984, and IBM Research from 1984 to 1986. Since 1986, he has been with the Electrical Engineering Department, Stanford University, where he is currently a Professor of Electrical Engineering. He founded Amati Com. Corporation in 1991 (purchased by TI in 1997) and was Officer/Director from 1991 to 1997. He is currently on the Boards of Directors of Marvell, Teknovus, ASSIA, and Teranetics. He is on the Advisory Boards of Halisos Networks, Ikanos, and Portview Ventures. He is a member of the U.S. National Research Council’s CSTB. His specific interests are in the area of high-performance digital transmission. He has published over 250 papers and holds over 40 patents. Dr. Cioffi has received various awards, including Member, National Academy of Engineering (2001); IEEE Kobayashi Medal (2001); IEEE Millennium Medal (2000); Institute of Electrical Engineers J.J. Tomson Medal (2000); 1999 University of Illinois Outstanding Alumnus; 1991 IEEE Communications Magazine Best Paper Award; 1995 ANSI T1 Outstanding Achievement Award; and the National Science Foundation Presidential Investigator (1987–1992).
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 6, JUNE 2004
Multiuser Transmit Optimization for Multicarrier Broadcast Channels: Asymptotic FDMA Capacity Region and Algorithms Louise M. C. Hoo, Bijit Halder, Member, IEEE, José Tellado, and John M. Cioffi, Fellow, IEEE
Abstract—We derive optimal and suboptimal multiuser transmit-optimization methods for a multicarrier broadcast channel with intersymbol interference under the frequency-division multiple-access (FDMA) restriction. The general FDMA-based multicarrier broadcast problem is formulated as a maximum weighted rate-sum problem. Given each user’s subchannel assignment, the optimal transmit strategy is achieved by multilevel waterfilling. Unfortunately, the problem of finding the optimal subchannel assignments is combinatorial. However, by relaxing the FDMA restriction, we obtain a convex reformulation that allows for efficient computation of the optimal solution, and therefore, a characterization of the FDMA capacity region for a broadcast channel. If all users share the same transmission medium, we prove that the optimal frequency partitioning among the users has an ordered structure that can be exploited to significantly reduce the computational complexity. To make multiuser transmit-optimization schemes practical for applications with relatively fast time-varying user data-rate requirements or priorities, further reduction in computational complexity is necessary. This is achieved by restricting the energy distribution to be constant across the used subchannels. Simulations indicate the low-complexity constant-energy methods presented are very robust, and suffer from negligible performance loss. Index Terms—Broadcast channel, constant-energy optimization, frequency-division multiple access (FDMA), multilevel waterfilling, multiuser loading, multiuser transmit optimization, orthogonal frequency-division multiplexing (OFDM).
I. INTRODUCTION
I
N MOST communications networks, multiple users or multiple services requested by the same user must share a limited pool of resources, such as transmit power (energy), bandwidth (spectrum), and transmit duration. These users or services typically experience different channel characteristics, and will desire different data rates and demand different qualities of services (QoS) that include target error rates, latencies, and availability. Moreover, the transmission medium and the desired transmission
Paper approved by Y. Li, the Editor for Wireless Communications Theory of the IEEE Communications Society. Manuscript received December 17, 2002; revised November 15, 2003. This paper was presented in part at the IEEE Wireless Communications and Networking Conference, Chicago, IL, September 2000, and in part at the IEEE International Conference on Communications, Helsinki, Finland, June 2001. L. M. C. Hoo is with Broadcom Corporation, Sunnyvale, CA 94085 USA (e-mail: [email protected]). B. Halder was with Telicos Corporation, Santa Clara, CA 95050 USA (e-mail: [email protected]). J. Tellado is with Teranetics Corporation, Santa Clara, CA 95054 USA (e-mail: [email protected]). J. M. Cioffi is with the STAR Lab, Stanford University, Stanford, CA 94305 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.829570
Fig. 1. An L-user broadcast channel.
requirements may vary over time. Using a feedback channel to communicate back to the transmitter the channel information estimated at the receiver of each active user, multiuser transmit optimization strives to achieve a higher multiuser rate or spectral efficiency by optimizing the allocation of available bandwidth and transmit power among the users in a manner that balances system throughput and fairness. Multiuser transmit optimization is practical for use in wireline systems, fixed wireless systems, or in low-mobility environments where the channels vary slowly. If transmit reoptimization is done at a slower rate than the symbol rate, the overhead in the feedback channel can be low. The channel we will focus on is the broadcast channel. The broadcast channel consists of a sender with a transmit power and bandwidth budget sending independent information simultaneously to multiple receivers. Some examples of the broadcast channel are the downlink transmission of a wireless communication system and the downstream transmission of a wireline network. Fig. 1 illustrates a broadcast channel of one sender and receivers. We solve the multiuser transmit-optimization problem for multicarrier modulation systems under different performance criteria for the broadcast channel, using the same circular channel approach employed in [1]–[3]. This approach, which results in the discrete Fourier transform (DFT) basis functions as eigenfunctions of any finite impulse response (FIR) channel, is adopted by any multicarrier system that uses discrete multitone (DMT) or orthogonal frequency-division multiplexing
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Fig. 2. Multicarrier broadcast channel with intersymbol interference (ISI) viewed as parallel Gaussian broadcast channels.
N
(OFDM). The orthogonal decomposition of each user’s channel frequency response into parallel independent additive white Gaussian noise (AWGN) subchannels, using the DFT, allows any multicarrier broadcast channel to be equivalently viewed parallel broadcast channels, one for each frequency, as as illustrated in Fig. 2. For each frequency/subchannel , the model for the broadcast channel is given by (1) where is the magnitude of the channel frequency response is a zero-mean Gaussian noise variable with variof user , , is the input, and is the output for user . The ance variables , , are the energies and bits allocated to subchannel . In general, superposition coding, together with successive interference cancellation, is necessary for achieving multiuser capacity [2], [3]. This implies that some subchannels need to be shared among different users, thereby making decoding a highly complex problem. In most practical systems, the improvement in capacity is not sufficient to justify the increase in complexity. To simplify the transmitter and receiver implementations, the focus here is on multiuser transmit optimization based on the frequency-division multiple-access (FDMA) scheme. In an FDMA scheme, orthogonality between the users is achieved by having the users occupy different sets of subchannels. An FDMA scheme is a natural choice for a multicarrier system, since it already has an inherent FDMA structure in place. Furthermore, an FDMA scheme has been shown to be the capacity-achieving scheme in some special cases [2], [4]. Given each user’s subchannel assignment, the maximum multiuser rate is achieved by performing separate waterfilling
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for each user in the multiple-access channel case, and joint waterfilling for each user in the broadcast channel case. Effectively, the nonoverlapping nature of the subchannel assignments decouples the multiuser system into a system of independent users. However, this multiuser rate can be far below the multiuser capacity if the subchannel assignments are chosen arbitrarily, often resulting in many inefficiently used subchannels. Unfortunately, due to the combinatorial nature of the FDMA restriction, finding the optimal subchannel assignments requires an exhaustive search. Hence, the exponential complexity of the optimal FDMA multiuser transmit-optimization methods makes them impractical. In [5]–[7], low-complexity suboptimal algorithms were proposed to closely approximate the optimal FDMA strategy for supporting several services with different QoS under various performance criteria in a multiservice OFDM system. Later, in [4] and [8], these algorithms were proven to be asymptotically optimal by relaxing the FDMA restriction to permit time sharing of each subchannel. This relaxation technique was also used in [9] and [10]. In [9], it was applied to find the optimal subchannel allocation, which minimizes the total transmission power while satisfying each user’s minimum rate constraint in a multiuser OFDM system. In [10], it was applied to characterize the FDMA capacity region for a Gaussian multiple-access channel with intersymbol interference (ISI). To make multiuser transmit-optimization schemes practical for applications with relatively fast time-varying user data-rate requirements or priorities, further reduction in computational complexity is achieved by restricting the energy distribution to be constant across the used subchannels [4], [11]–[13]. The organization of this paper is as follows. Section II formulates the Gaussian broadcast channel with the FDMA restriction and shows that this is a combinatorial problem. Section III describes the convex relaxation technique used to reduce this exponential complexity to a polynomial complexity. Section IV proves that the optimal frequency partition follows an ordered structure for some special cases. In Section V, to achieve further reduction in complexity, we place a constant-energy restriction over the used subchannels, thereby eliminating energy optimization. Computationally efficient FDMA-based multiuser transmit-optimization methods are dereived. Simulation results presented in Section VI show that these low-complexity methods are very robust, and suffer from negligible performance loss, even at low signal-to-noise ratios (SNRs), where the constant-energy approximation is less accurate. Section VII concludes the paper. II. FDMA FORMULATION By approximating each channel transfer function with a piecewise-constant model, the whole frequency spectrum is bins or subchannels,1 with each bin discretized into having a flat channel response. For the th frequency bin, let and be the energy allocated to user , and the ratio of the magnitude squared of the channel response and the noise power spectral density for user . Then, we formulate the problem of an -user transmit optimization under 1Frequency
bin and subchannel will be used interchangeably.
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the FDMA restriction for a broadcast channel as a weighted rate-sum formulation
vision multiple-access (TDMA) optimization problem, as follows:
Total energy constraint s.t. FDMA restriction (2) , the weight assigned to user , is constrained , is the user index set, is the frequency index set, and is the subchannel assignment for user . As , this piecewise-constant approximation becomes the optimal continuous FDMA formulation.2 of each Given the weights and the channel responses user, the objective in (2) is to find the energy distribution that maximizes the weighted sum of each user’s achievable rate subject to a total transmit-energy constraint and an FDMA restriction. Maximizing the weighted rate-sum for all possible ’s traces out the boundary of the capacity region. For any fixed subchannel assignments , the optimal FDMA solution to (2) is achieved by multilevel waterfilling [4], [13], whose underlying equations are where by
(3)
(4) where
if if
However, finding the optimal subchannel assignments among the users requires searches. Hence, the computation of the optimal FDMA solution has exponential complexity. In general, the number of used subchannels needs to be optimized until all energies are positive. (Refer to [4] for more details.) For the special case when all users have equal weights, an FDMA scheme among the users is indeed a capacity-achieving denoting the achievable rate scheme. Specifically, with for the broadcast channel is for user , the rate-sum maximized by assigning each subchannel to the user with the . In other words, best gain-to-noise-power ratio given by the transmitter should transmit information only to the user with the best reception for each subchannel, and the optimal energy distribution follows the waterfilling solution. III. FDMA–TDMA FORMULATION: A CONVEX RELAXATION If time sharing is allowed between users for each frequency bin, formulation (2) can then be recast as an FDMA-time-di2Any
N is accurate for a multicarrier formulation.
(5) is the time-sharing factor for user The additional variable in the th frequency bin. In other words, user occupies the th frequency bin for a fraction of the time. In this case, , . Another equivalent interpretation of for multicarrier transmission is that is the amount of bandwidth allocated to user in the th subchannel. This is equivalent to further partitioning of bandwidth in each subchannel, effectively creating a multicarrier system with a total number increases, the bin width of subchannels greater than . As decreases, which implies that the increase in the aggregate rate contributed by time sharing among the users decreases, as well. , no time sharing occurs, and the opIn the limit, as timum FDMA-TDMA solution is the optimum FDMA solution. Therefore, for sufficiently large , an FDMA strategy that assigns every subchannel to the user with the largest time-sharing factor will result in negligible performance loss relative to the optimum FDMA solution. This relaxation technique was also used in [9] and [10]. Observation 3.1: The FDMA-TDMA formulation of (5) is convex,sinceitisthemaximizationofaconcavecostfunctionover a convex constraint set. First, the cost function is a sum of functions of the form , and are some positive constants. Since any positive where linear combination of concave functions is concave, it suffices is concave in . to prove that This is achieved by proving that the Hessian for is positive semidefinite over the positive quadrant of , . Next, all the constraint functions are linear in , , and thus give convex sets. Since the intersection of convex sets is convex [14], the constraint set in (5) is convex. Since the FDMA-TDMA optimization problem is convex, a local maximum or minimum is also the global maximum or minimum. This optimal solution can be computed very efficiently (with polynomial complexity) using interior-point methods [14], thereby allowing us to characterize the asymptotic FDMA capacity region for the Gaussian broadcast channel with ISI. There are two other practical convex multiuser formulations for the broadcast channel. In the minimum total-energy formulation, the optimization criterion is to minimize the total power used by the sender, while satisfying the constraints that each receiver achieves a certain fixed rate. In the mixed formulation, the optimization criterion is to maximize a weighted sum of rates for a subset of users, while satisfying the fixed-rate constraints for the rest.
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Observation 3.2: The maximum-weighted rate-sum formulation of (5), the minimum total-energy formulation and the mixed formulation are equivalent. That is, given a problem in one formulation, we can construct a new problem recast as either of the other two formulations, such that the optimal energy allocation and optimal rates for all users are identical for all three formulations. IV. ORDERED FREQUENCY PARTITION If the channel-to-noise responses for all the users are simply , , scaled versions of each other, i.e., we will show in this section that the optimal frequency partition among the users for the FDMA-TDMA multiuser-optimization problem for the broadcast channel, as described in Section III, follows a specific ordering. Multiservice-optimization problems are practical examples where such conditions on the channel-to-noise responses are true. Multiservice optimization refers to the allocation of resources (energy and frequency) for services that simultaneously occupy the same transmission medium. These services are either fixed rate, e.g. voice, or variable rate, e.g. internet data, multimedia, and they require different noise immunities as characterized by the parameter . is In the context of multicarrier transmission, the parameter commonly known as the SNR gap, which is a function of the coding scheme of choice, the target probability of symbol error, and the desired noise margin. The multiservice problem for multicarrier systems was first studied in [5]–[7]. The fast loading algorithms proposed therein are based on assigning the subcarriers to the services in order of their respective gaps. Specifically, the service with the highest gap is assigned subcarriers with the highest SNRs, and the service with the smallest gap is allocated subcarriers with the worst SNRs. Therefore, the search for the optimum subcarrier allocation has been reduced from an exhaustive search to a polynomial search. These low-complexity algorithms were found to be applicable for solving the optimal frequency-duplex problem in very-high-speed digital subscriber line (VDSL) systems [10]. In [5] and [6], assumptions of high SNR and equal-energy distribution are made to justify such an ordered subcarrier assignment. In the following, we will present a rigorous proof for the optimality of such an ordered frequency partition without making these assumptions. Observation 4.1: For clarity, the focus first is on a two-user . To simplify the derivation of the proof, we scenario will substitute “ln” for “log” from now on. Under the assump, , the two-user minimum total-ention that ergy formulation is
s.t.
(6) Theorem 4.1: Without loss of generality, assume that the channel-gain-to-noise ratios, , , are distinct and are
Fig. 3. Optimal frequency partition for the three-user multiservice 0 0 and sorted in a descending FDMA-TDMA problem with 0 order. , , and are the waterfilling levels.
K K
Fig. 4.
K
>
>
g
Illustration for Observation 4.2.
sorted in a descending order. Then, the optimal frequency partition for formulation (6) is a two-band assignment with sharing of the boundary subchannel only, if at all. Furthermore, assume . Then, there exists , without loss of generality that , such that user 1 occupies subchannels 1 to where , and user 2 occupies subchannels to . Moreover, time sharing occurs, if at all, only at the boundary subchannel between user 1 and user 2. Proof: See the Appendix. Fig. 3 illustrates the optimal solution for the three-user multiservice problem. -user multiservice case makes use of The proof for the induction and the following observation. be the optimal Observation 4.2: (See Fig. 4.) Let . solution to the -user multiservice problem, where Also, assume that there are no shared subchannels. (This can be achieved by appropriately subdividing the shared subchannels and resolving the problem with the redefined larger set of subchannels.) Let be the set of subchannels allocated to the th user in accordance with the optimal solution. Now, define a -user problem by removing the th user from reduced , and its the set of users, as well as its energy set of subchannels from the set of all subchannels. Then, the optimal subchannel assignment and energy allocation for -user problem is identical to the -user the reduced users. problem, for the remaining Proposition 4.1: If for , then under the sorted channel assumption, the optimal solution to the -user multiservice problem must have an -band structure. Moreover, if , then the set of subchannels assigned to each user is in the
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Fig. 5.
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Illustration for proof of Theorem 4.2.
following order. There exist
, where such that user 1 occupies , user occupies subchannels subchannels 1 to to , and user occupies subchannels to for , . Theorem 4.2: If Proposition 4.1 holds for some , then . it must hold for Proof: We present a proof by contradiction. In particular, -user problem we show that if the optimal solution to an does not have an -band structure, then we can construct a reduced -user problem with an optimal solution that does -user not have an -band structure. Assume that for an -band problem, the optimal solution does not have an structure. Then, there must be at least one user, say user , with more than one band of subchannels assigned. In other words, is composed of at the set of subchannels assigned to user least two bands, with a nonempty set of subchannels between those bands. Let be a user that is assigned one of the sub, we can select a channels from the set . Since user that is different from and . Let be such a user and be the set of subchannels assigned to . Now we formulate a reduced -user problem by removing from the set of users and from the set of subchannels. From Observation 4.2, we conclude that the optimal subchannel assignment for the reduced problem remains the same for users and , i.e., user uses some subchannels that are between subchannels assigned to user . Thus, the solution to the reduced -user problem does not have an -band structure, thereby contradicting our original assumption. A simple illustration of the steps for the proof of is shown in Fig. 5. The ordering contradiction for structure can be proven using a similar argument. Theorem 4.3: If for , then under the sorted channel assumption, the optimal solution to the -user problem, formulated either as maximum-weighted rate-sum, minimum total-energy, or mixed, has an -band structure, and with time sharing only at the boundary subchannels, , then user 1 if at all. Moreover, if is assigned the best subchannels first, followed by user 2, and , where so forth. Specifically, there exist such that user 1 occupies subchannels 1 to , user occupies subchannels to , and user occupies subchannels to for . Moreover, sharing occurs, if at all, at the boundary between user and user . subchannel Proof: The proof follows from Observation 3.2 and Theorems 4.1 and 4.2. V. CONSTANT-ENERGY OPTIMIZATION Although optimal multiuser loading algorithms have been proposed in [8]–[10] to solve various forms of the
FDMA-TDMA optimization problem, in general, the computed time-sharing factors are irrational, and thus, lead to large latencies in practice. Moreover, the computational complexity of these algorithms are not suited for applications with relatively fast time-varying user data-rate requirements or priorities. To achieve further reduction in computational complexity, this section presents a new approach that restricts the energy distribution to be constant across the used subchannels. The process of determining the optimal set of useful subchannels and then assigning them to the best users is called bandwidth optimization [4], [13]. We derive the optimal and one suboptimal constant-energy multiuser loading methods with bandwidth optimization for the broadcast channel. The constant-energy approach is motivated by two observa, tions of (4). If user experiences a flat channel, i.e., the optimal energy allocation is which is constant for all subchannels allocated to user . Second, at , which implies that the optimal enhigh SNRs, ergy distribution is constant. For frequency-selective channels, a constant transmitenergy distribution has been shown to be almost as good as the waterfilling energy distribution, as long as there is bandwidth optimization for the single-user case [15], [16]. For the broadcast channel, the weighted rate-sum maximization problem under joint constant-energy and FDMA restrictions becomes
s.t. or (7) where is the total transmit-energy budget. The set condenotes tains the indexes of all the used subchannels, and indicates that user the cardinality of . The variable is assigned subchannel , and indicates otherwise. The first two contraints in (7) represent the FDMA restriction, which implies that each subchannel is assigned to one user, at most. Associated with each user is the parameter . Given the set of used subchannels, , the cost function in (7) can be rewritten as follows: (8) which implies that the optimal strategy is to assign each subchannel in to the user that achieves the maximum weighted rate with the same power. Optimal Algorithm: to Let be the number of best subchannels to use. For , do 1) 2) 3) sum of biggest is the weighted aggregate rate from using best subchannels.
HOO et al.: MULTIUSER TRANSMIT OPTIMIZATION FOR MULTICARRIER BROADCAST CHANNELS
The maximum weighted aggregate rate is
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.
For each iteration, step 2 requires comparisons and log operations, while step 3 requires operations if using the divide-and-conquer-based sorting algorithms. . Hence, the worst-case complexity is Instead of searching all sets of best subchannels to use, different stopping criteria can be employed. One of them is presented here. Suboptimal Algorithm: 1) Initialize the iteration index and the set of useful . Thus, the number of subchannels . useful subchannels is 2) Initialize the set that contains subchannel assignment for , . user , . 3) Set the weighted aggregate rate . 4) Set 5) Assign to user where . 6) Compute , stop. 7) If 8) Else, remove subchannel
Fig. 6.
High SNR scenario: average SNR of 20 dB.
. . from
where .
9) , and . Return to 4. Essentially, the suboptimal algorithm outlined starts off with using all subchannels, and then removing the subchannel from that contributes the least weighted rate until no further increase in the weighted rate-sum occurs. This algorithm has a operations, but should only reworst-case complexity of operations. quire The constant-energy formulation simplifies the multiuser transmit-optimization problem tremendously but with negligible performance loss. Although slightly better performance can be achieved by relaxing the restriction that each user be allocated the same amount of energy, this approach will involve set assignments, making the problem combinatorial. Previously known constant-energy multiuser loading algorithms can be found in [11] and [12]. In [11], the optimization criterion is to minimize the total power used, while satisfying the constraint that each user is allotted a fixed number of subchannels. In [12], the optimization criterion is to maximize the minimum achievable rate among all users. However, none of these methods optimize over the set of useful subchannels, unlike the proposed algorithms.
VI. ALGORITHM PERFORMANCE ANALYSIS The performances of four multiuser transmit-optimization algorithms for the broadcast channel are evaluated in this section. These algorithms are: 1) FDMA-Ex (Section II): the optimal FDMA method of exponential complexity; 2) FDMA-TDMA (Section III): the optimal FDMA-TDMA method of polynomial complexity; 3) FDMA-CE (Section V): the suboptimal constant-energy FDMA method;
4) FDMA-Fixed: a method that preassigns fixed sets of subchannels to each user without bandwidth optimization, i.e., without using channel knowledge of each user. Specifically, the total number of subchannels is evenly divided among all users, if possible, with any remaining subchannels assigned, one at a time, to the user with the largest weight first, and so forth. The ordering among the users is arbitrarily fixed. The downlink transmission (broadcast channel) of a three-user multicarrier system is considered. It is assumed that all users experience independent channels where each channel frequency response corresponds to a frequency-selective Rayleigh fading channel, characterized by an exponential power delay profile with a normalized delay spread of three. The formulation used is the weighted rate-sum formulation. Hence, the performance measure is the achievable averaged weighted rate-sum normalized by the total number of dimensions. (A complex subchannel on the performance of has two dimensions.) The effect of each algorithm is investigated. For each , 20 independent channel realizations are generated for each user. At dB, , and an average SNR of 20 dB, the results in Fig. 6 show that FDMA-CE suffers from negligible rate loss with respect to FDMA-Ex. The reason for this is that the constant-energy approximation is very good for high SNR cases. On the other hand, FDMA-Fixed has a rate loss of 18%. At an average SNR of 5 dB, the results in Fig. 7 show that FDMA-CE suffers from an average rate loss of 0.4%, while FDMA-Fixed incurs an average rate loss of 31%. The perfor, mance comparisons were not carried out beyond because it would require a prohibitive effort to compute the optimal FDMA solution, which has exponential complexity. The main reason that FDMA-CE works well for low-SNR cases is because of multiuser diversity. Since the users experience mutually independent channel frequency responses, it is likely that one or more users will find each subchannel favorable. Effectively, the multiuser FDMA-CE algorithm selects the user with the best subchannel gain for each subchannel, creating a new channel frequency response that is relatively flatter,
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VII. CONCLUSION The computation of the FDMA capacity region for broadcast channels with ISI requires exponential complexity. By relaxing the FDMA restriction, we obtain a convex reformulation that allows for efficient computation of the optimal solution, and therefore, a characterization of the asymptotic FDMA capacity region. We also prove that the optimal frequency partition has an ordered structure under certain conditions. By enforcing a constant-energy restriction, further reduction in computational complexity is achieved. We present the optimal and one suboptimal constant-energy method with bandwidth optimization. Simulation results show that these methods suffer from negligible performance loss, even at low SNRs, where the constant-energy approximation is not valid. APPENDIX PROOF OF THEOREM 4.1
Fig. 7. Low SNR scenario: average SNR of 5 dB.
The proof involves the application of the Karush–Kuhn–Tucker (KKT) optimality conditions. For the and derivation of the proof, we will assume that . We set up the Lagrangian function as follows:
(9) Fig. 8. Average SNR of 10 dB.
thereby making the constant-transmit-energy assumption accurate even for low-SNR cases. Since the FDMA-TDMA (with energy or power optimization) formulation is a convex relaxation to the FDMA formulation, we can use the optimal FDMA-TDMA solution, called FDMA-TDMA in Fig. 8, to upper bound the respective performance loss of FDMA-CE and FDMA-Fixed relative to subchannels. Two conclusions can be FDMA-Ex for drawn from Fig. 8. The first is that the additional complexity incurred for power optimization gives only a slight rate improvement (0.13%) over the much simpler constant-energy scheme. The second is that a considerable increase in spectral efficiency, 26% in this case, can be achieved from bandwidth optimization using channel knowledge at the transmitter. In general, the magnitude of the gain in spectral efficiency will depend on the degree of correlation among the users’ channel frequency responses. For instance, if the channel frequency responses are all flat, then there is no gain from performing bandwidth optimization.
Setting the partial derivatives of , and to zero, we get
with respect to
,
,
(10)
(11) and (12) Rewriting (12), we get (13)
HOO et al.: MULTIUSER TRANSMIT OPTIMIZATION FOR MULTICARRIER BROADCAST CHANNELS
Equation (13) is the classical waterfilling equation, and and are the optimal waterfilling levels for users 1 and 2, respectively. Equation (13) asserts that there exist positive constants and such that, for each , (13) hold true when and . If both , are negative, then subchannel is not used by either user and should be excluded from the opis negative, then subchannel timization process. If is not used by user 1 (2) and should be excluded from the optimization set for user 1 (2). Substituting (13) for and , and combining (10) and (11), we obtain (14) where
, and , . Since the optimization problem is convex, (13) and (14), as well as the fixed-rate constraints, the total subchannel-width constraint in , , formulation (6), and the nonnegativity constraints on , are the necessary and sufficient optimality conditions. and If the th subchannel is shared, then , which, in turn, implies that and due to the complementary slackness condition [14]. As a result, the right-hand side of (14) becomes zero. In the following, we show that there is at most one value of , or equivalently, one , that , thereby proving that there is sharing of at satisfies most one subchannel. We consider two distinct cases. Case 1) . For this case, for , , and . Therefore, is a strictly increasing function of when . Hence, there cannot be more than one subchannel for a constant . In [4], it that satisfies was proven that when , gives a suboptimal solution as the total energy required to satisfy all fixed-rate constraints decreases with any change to the shared subchannel. . Case 2) , , from (13), For , we see that the relations have to hold true. In other words, . Using for , we can the fact that , further determine that . Since and for and , is a strictly convex function with a minimum at . Hence, and or , where and . Then, there cannot be more than one subchannel for which such that . On the other hand, if , there is no feasible solution satisfying all specified constraints. (Refer to [4] for more details and illustrations.)
Fig. 9.
C
0 f (1=g
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).
The next step is to prove that the optimal frequency partition among the users is a two-band partition with user 1 assigned the best subchannels first, followed by user 2. (This order is true , which is assumed here.) Again, we refer only when to (14), which is the KKT optimality condition that has to be satisfied for each subchannel . We consider three possible cases. and , then subchannel is shared. 1) If and , which means that subchannel 2) If is assigned to user 1, then and , due to the complementary slackness condition. Thus, from (14), . and , which means that subchannel 3) If is assigned to user 2, then and , due to the complementary slackness condition. Thus, from (14), . Since is the optimal solution when [4], from Case 2, is convex. Therefore, , is concave, as shown in Fig. 9, and crosses the zero axis at most once. With ’s sorted in a dethat satisfy scending order, the subchannels with the largest are assigned only to user 1, and the subchannel whose gain-to-noise ratio satisfies is time shared between both users, and the subchannels with that satisfy are assigned only the smallest to user 2. In other words, if subchannel is the subchannel should that is time shared, then the user with the higher subchannels, while the other user solely occupy the best subchannels, giving rise to solely occupies the worst a two-band frequency partitioning structure with time sharing only of the boundary subchannel.3 ACKNOWLEDGMENT The first author, L. M. C. Hoo, would like to thank W. Yu for helpful discussions on the proof of Theorem 4.1. REFERENCES [1] W. Hirt and J. L. Massey, “Capacity of the discrete-time Gaussian channel with intersymbol interference,” IEEE Trans. Inform. Theory, vol. 34, pp. 380–388, May 1988. [2] R. S. Cheng and S. Verdú, “Gaussian multiaccess channels with ISI: Capacity region and multiuser waterfilling,” IEEE Trans. Inform. Theory, vol. 39, pp. 773–785, May 1993. 3In [10], the same approach was used to prove a two-band partition for a two-user Gaussian multiple-access channel if both users have the same channel characteristics.
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[3] A. J. Goldsmith and M. Effros, “The capacity region of broadcast channels with intersymbol interference and colored Gaussian noise,” IEEE Trans. Inform. Theory, vol. 47, pp. 219–240, Jan. 2001. [4] L. M. C. Hoo, “Multiuser transmit optimization for multicarrier modulation systems,” Ph.D. dissertation, Stanford Univ., Stanford, CA, Dec. 2000. [5] L. M. C. Hoo, J. Tellado, and J. M. Cioffi, “Dual QoS loading algorithms for multicarrier systems offering different CBR services,” in Proc. IEEE PIMRC, vol. 1, Boston, MA, Sept. 1998, pp. 278–282. [6] , “Dual QoS loading algorithms for DMT systems offering CBR and VBR services,” in Proc. IEEE GlobeCom, vol. 1, Sydney, Australia, Nov. 1998, pp. 25–30. , “Discrete dual QoS loading algorithms for multicarrier systems,” [7] in Proc. IEEE Int. Conf. Communications, Vancouver, BC, Canada, June 1999, pp. 796–800. [8] L. M. C. Hoo, B. Halder, J. Tellado, and J. M. Cioffi, “Asymptotic FDMA capacity region for broadcast channels with ISI,” in Proc. IEEE Int. Conf. Communications, vol. 6, Helsinki, Finland, June 2001, pp. 1648–1653. [9] C. Y. Wong and R. S. Cheng, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Select. Areas Commun., vol. 17, pp. 1747–1757, Oct. 1999. [10] W. Yu and J. Cioffi, “FDMA capacity of Gaussian multiple-access channels with ISI,” IEEE Trans. Commun., vol. 50, pp. 102–111, Jan. 2002. [11] C. Y. Wong, C. Y. Tsui, R. S. Cheng, and K. B. Letaief, “A real-time subcarrier allocation scheme for multiple-access downlink OFDM transmission,” in Proc. IEEE Vehicular Technology Conf., vol. 2, Amsterdam, The Netherlands, Sept. 1999, pp. 1124–1128. [12] W. Rhee and J. M. Cioffi, “Increase in capacity of multiuser OFDM system using dynamic subchannel allocation,” in Proc. IEEE Vehicular Technology Conf., vol. 2, Tokyo, Japan, May 2000, pp. 1085–1089. [13] L. M. C. Hoo, J. Tellado, and J. M. Cioffi, “FDMA-based multiuser transmit optimization for broadcast channels,” in Proc. IEEE Wireless Communication Networking Conf., vol. 2, Chicago, IL, Sept. 2000, pp. 597–602. [14] J.-B. Hiriart-Urruty and C. Lemarâechal, Convex Analysis and Minimization Algorithms. Berlin, Germany: Springer-Verlag, 1993. [15] P. S. Chow, “Bandwidth optimized digital transmission techniques for spectrally shaped channels,” Ph.D. dissertation, Stanford Univ., Stanford, CA, May 1993. [16] B. Schein and M. Trott, “Suboptimal power spectra for colored Gaussian channels,” in Proc. IEEE Int. Symp. Information Theory, 1997, p. 340.
Louise M. C. Hoo received the B.S.E.E. degree (with distinction) from Cornell University, Ithaca, NY, in 1994, and the M.S.E.E. and Ph.D. degrees from Stanford University, Stanford, CA, in 1996 and 2000, respectively. She is currently a Senior Staff Scientist with Broadcom Corporation, Home and Wireless Networking Division, Sunnyvale, CA. She has been working on physical layer algorithm design, and development and testing of the IEEE 802.11-based wireless LAN systems. She has also held internship positions at Amati Communications Corporation (now part of Texas Instruments), CA, Hewlett-Packard Laboratories, CA, and AT&T Bell Laboratories, NJ. She has over ten IEEE publications.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 6, JUNE 2004
Bijit Halder (M’98) received the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 1998. Since 1998, he has been working in the industry and is involved in transceiver design for DSL and Ethernet. His research interests include development of low-complexity signal-processing algorithms and efficient systems design for communications.
José Tellado received the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA in 1994 and 1999, respectively. He is currently the Director of Systems Design at Teranetics, Santa Clara, CA, which he cofounded in 2003 to develop very-high-speed Ethernet IC solutions. Prior to Teranetics, he was the Modem Architect at Intel’s Broadband Wireless Access group. Before that, in addition to being an Employee–Founder and the Director of Technology at Iospan Wireless, San Diego, CA, he was also the Chief Designer of the innovative MIMO-OFDM AirBurst™ technology that was acquired by Intel. At Stanford University, he proposed new advanced methods for multicarrier modulation (DMT/OFDM) in the areas of peak-to-average-power ratio reduction and multiuser transmit optimization. He has also consulted for Apple Computers Advanced Technology Wireless Group and Globalstar in the areas of WLAN and low-orbit satellite cellular systems, respectively. He is the author of over 16 IEEE publications, 10 standards contributions, 15 patents, and the book Multicarrier Modulation With Low PAR: Applications to DSL and Wireless (Norwell, MA: Kluwer, 2000).
John M. Cioffi (S’77–M’78–SM’90–F’96) received the B.S.E.E. degree in 1978 from the University of Illinois at Urbana-Champaign and the Ph.D.E.E. degree in 1984 from Stanford University, Stanford, CA. He was with Bell Laboratories from 1978 to 1984, and IBM Research from 1984 to 1986. Since 1986, he has been with the Electrical Engineering Department, Stanford University, where he is currently a Professor of Electrical Engineering. He founded Amati Com. Corporation in 1991 (purchased by TI in 1997) and was Officer/Director from 1991 to 1997. He is currently on the Boards of Directors of Marvell, Teknovus, ASSIA, and Teranetics. He is on the Advisory Boards of Halisos Networks, Ikanos, and Portview Ventures. He is a member of the U.S. National Research Council’s CSTB. His specific interests are in the area of high-performance digital transmission. He has published over 250 papers and holds over 40 patents. Dr. Cioffi has received various awards, including Member, National Academy of Engineering (2001); IEEE Kobayashi Medal (2001); IEEE Millennium Medal (2000); Institute of Electrical Engineers J.J. Tomson Medal (2000); 1999 University of Illinois Outstanding Alumnus; 1991 IEEE Communications Magazine Best Paper Award; 1995 ANSI T1 Outstanding Achievement Award; and the National Science Foundation Presidential Investigator (1987–1992).
