IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 17, NO. 10, OCTOBER 1999
`
`1747
`
`Multiuser OFDM with Adaptive
`Subcarrier, Bit, and Power Allocation
`
`Cheong Yui Wong, Roger S. Cheng, Member, IEEE,
`Khaled Ben Letaief, Senior Member, IEEE, and Ross D. Murch, Senior Member, IEEE
`
`Abstract—Multiuser orthogonal frequency division multiplex-
`ing (OFDM) with adaptive multiuser subcarrier allocation and
`adaptive modulation is considered. Assuming knowledge of the
`instantaneous channel gains for all users, we propose a mul-
`tiuser OFDM subcarrier, bit, and power allocation algorithm
`to minimize the total transmit power. This is done by assigning
`each user a set of subcarriers and by determining the number
`of bits and the transmit power level for each subcarrier. We
`obtain the performance of our proposed algorithm in a multiuser
`frequency selective fading environment for various time delay
`spread values and various numbers of users. The results show that
`our proposed algorithm outperforms multiuser OFDM systems
`with static time-division multiple access (TDMA) or frequency-
`division multiple access (FDMA) techniques which employ fixed
`and predetermined time-slot or subcarrier allocation schemes.
`We have also quantified the improvement in terms of the overall
`required transmit power, the bit-error rate (BER), or the area of
`coverage for a given outage probability.
`
`Index Terms—Adaptive modulation, frequency selective fading
`channel, multiaccess communication, multiuser channel, orthog-
`onal frequency division multiplexing (OFDM), resource manage-
`ment.
`
`I. INTRODUCTION
`
`particular, subcarriers with large channel gains employ higher
`order modulation to carry more bits/OFDM symbol, while
`subcarriers in deep fade carry one or even zero bits/symbol.
`Integrated design of forward error correcting code and adap-
`tive modulation has also been studied using BCH code and
`trellis coded modulation (TCM) in [8] and [9], respectively.
`Although both references considered only time-varying flat
`fading channels, the same coded adaptive modulation design
`can be easily applied to OFDM systems. As different subcar-
`riers experience different fades and transmit different numbers
`of bits, the transmit power levels must be changed accordingly.
`The problem of optimal power allocation has also been studied
`in [10].
`In this paper, we consider extending OFDM with adaptive
`modulation to multiuser frequency selective fading environ-
`ments. When OFDM with adaptive modulation is applied in a
`frequency selective fading channel, a significant portion of the
`subcarriers may not be used. These are typically subcarriers
`which experience deep fade and are not power efficient to
`carry any information bit. In multiuser systems using static
`time-division multiple access (TDMA) or frequency-division
`multiple access (FDMA) as multiaccess schemes, each user
`is allocated a predetermined time slot or frequency band to
`apply OFDM with adaptive modulation. Consequently, these
`unused subcarriers (as a result of adaptive modulation) within
`the allocated time slot or frequency band of a user are wasted
`and are not used by other users. However, the subcarriers
`which appear in deep fade to one user may not be in deep
`fade for other users. In fact, it is quite unlikely that a subcarrier
`will be in deep fade for all users, as the fading parameters for
`different users are mutually independent. This motivates us to
`consider an adaptive multiuser subcarrier allocation scheme
`where the subcarriers are assigned to the users based on
`instantaneous channel information. This approach will allow
`all
`the subcarriers to be used more effectively because a
`subcarrier will be left unused only if it appears to be in deep
`fade to all users.
`We consider a multiuser subcarrier, bit, and power allo-
`cation scheme where all users transmit in all the time slots.
`Our objective is to minimize the overall transmit power by
`allocating the subcarriers to the users and by determining
`the number of bits and the power level transmitted on each
`subcarrier based on the instantaneous fading characteristics of
`all users. In this paper, we formulate the multiuser subcarrier,
`bit, and power allocation problem and propose an iterative
`algorithm to perform the multiuser subcarrier allocation. Once
`0733–8716/99$10.00 ª
`
`RECENTLY, intense interest has focused on modulation
`
`techniques which can provide broadband transmission
`over wireless channels for applications including wireless mul-
`timedia, wireless Internet access, and future-generation mobile
`communication systems. One of the main requirements on
`the modulation technique is the ability to combat intersymbol
`interference (ISI), a major problem in wideband transmission
`over multipath fading channels. There are many methods pro-
`posed to combat the ISI, e.g., [1]–[3]. Multicarrier modulation
`techniques, including orthogonal frequency division multiplex
`(OFDM), (e.g., [4]) are among the more promising solutions
`to this problem.
`the transmitter knows the instantaneous
`Assuming that
`channel transfer functions of all users, many papers [5]–[7]
`have demonstrated that significant performance improvement
`can be achieved if adaptive modulation is used with OFDM. In
`
`Manuscript received October 15, 1998; revised March 27, 1999. This work
`is supported in part by the Hong Kong Telecomm Institute on Information
`Technology and the Hong Kong Research Grant Council.
`The authors are with the Department of Electrical and Electronic Engi-
`neering, Hong Kong University of Science and Technology, Clear Water
`Bay, Hong Kong (e-mail: eeyui@ust.hk; eecheng@ust.hk; eekhaled@ust.hk;
`andeermurch@ust.hk).
`Publisher Item Identifier S 0733-8716(99)07861-0.
`
`1999 IEEE
`
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`IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 17, NO. 10, OCTOBER 1999
`
`the subcarrier allocation is determined, the bit and power
`allocation algorithm can be applied to each user on its al-
`located subcarriers. We also compare the performance of our
`proposed solution to various other static subcarrier allocation
`schemes.
`The results of the work can be applied, for instance, to
`the downlink transmission in a time division duplex (TDD)
`wireless communication system to improve the downlink
`capacity. In such a system, the base station (BS) can estimate
`the instantaneous channel characteristics of all
`the BS-to-
`mobile links based on the received uplink transmissions. The
`multiuser subcarrier, bit, and power allocation can then be
`used. It is clear that there is a certain amount of transmission
`overhead as the BS has to inform the mobiles about their
`allocated subcarriers and the number of bits assigned to
`each subcarrier.1 However, this overhead can be relatively
`small, especially if the channels vary slowly (e.g.,
`in an
`indoor low mobility environment), and the assignment is done
`once every many OFDM symbols. To further reduce the
`overhead, we can assign a contiguous band of subcarriers with
`similar fading characteristics as a group, instead of assigning
`each individual subcarrier. In this paper, we will not focus
`on how the subcarrier allocation information is transmitted.
`Instead, we will focus on how— and by how much—this
`new strategy can reduce the required transmit power; or how
`and by how much this new scheme can improve the bit-
`error rate (BER) for a fixed transmit power. Alternately, we
`also consider how and by how much this new scheme can
`increase the area of coverage for a given transmit power and
`target BER.
`While the bit allocation algorithm can be viewed as a
`practical implementation of the water-pouring interpretation
`for achieving the Shannon capacity of an ISI channel [13], the
`multiuser subcarrier and bit allocation algorithm presented in
`this paper is the counterpart of the multiuser water-pouring
`solution given in [14]. In information theoretic studies, the
`usual approach is to maximize the capacity (or information
`rate) under the power constraint. In this study, we focus
`on deriving practical algorithms that can support real-time
`multimedia data whose bit rates are generally fixed by the
`compression algorithms. Hence, we assume a given set of user
`data rates and attempt to minimize the total transmit power
`under a fixed performance requirement.
`The organization of this paper is as follows. In Section II,
`we will first give the system model and formulate the minimum
`overall transmit power problem. The optimization problem
`seeks to minimize the overall transmit power using combined
`subcarrier, bit, and power allocation schemes for multiuser
`OFDM systems. The bit and power allocation algorithm for
`a single-user system is studied in Section III. In Section IV,
`we derive a lower bound to the minimum overall transmit
`
`1 Note that the power level used does not need to be transmitted to the
`receiver in such a TDD system. As the subcarrier gain is known to the
`transmitter, it can adjust the transmit power level to achieve a predetermined
`receiver power level based on the number of bits allocated to that subcarrier.
`However, in FDD systems, the transmit power levels determined by the
`receiver have to be sent back to the transmitter. In such systems, the additional
`performance gain achieved by power allocation may not justify the cost of
`sending the transmit power level information to the transmitter.
`
`power by relaxing some of the constraints in the origi-
`nal problem. We also derive a suboptimal subcarrier al-
`location algorithm. In Section V, we compare the perfor-
`mance between our proposed method and other static ap-
`proaches via Monte Carlo simulations. Finally, we conclude in
`Section VI.
`
`II. SYSTEM MODEL
`The configuration of our multiuser adaptive OFDM system
`is shown in Fig. 1. We assume that the system has
`users
`and the
`th user has a data rate equal to
`bit per OFDM
`symbol. In the transmitter, the serial data from the
`users
`are fed into the subcarrier and bit allocation block which
`allocates bits from different users to different subcarriers. We
`assume that each subcarrier has a bandwidth that is much
`smaller than the coherence bandwidth of the channel and that
`the instantaneous channel gains on all the subcarriers of all
`the users are known to the transmitter. Using the channel
`information, the transmitter applies the combined subcarrier,
`bit, and power allocation algorithm to assign different sub-
`carriers to different users and the number of bits/OFDM
`symbol to be transmitted on each subcarrier. Depending on
`the number of bits assigned to a subcarrier,
`the adaptive
`modulator will use a corresponding modulation scheme, and
`the transmit power level will be adjusted according to the
`combined subcarrier, bit, and power allocation algorithm. We
`define
`to be the number of bits of the
`th user that are
`assigned to the
`th subcarrier. As we do not allow more than
`one user to share a subcarrier, it follows that for each
`, if
`,
`for all
`. We also assume that
`the adaptive modulator allows
`to take values in the set
`where
`is the maximum number
`of information bits/OFDM symbol that can be transmitted by
`each subcarrier.
`the output of the modulators
`The complex symbols at
`are transformed into the time domain samples by inverse
`fast Fourier transform (IFFT). Cyclic extension of the time
`domain samples, known as the guard interval, is then added
`to ensure orthogonality between the subcarriers, provided that
`the maximum time dispersion is less than the guard interval.
`The transmit signal is then passed through different frequency
`selective fading channels to different users.
`We assume that the subcarrier and bit allocation information
`is sent to the receivers via a separate control channel. At
`the receiver, the guard interval is removed to eliminate the
`ISI, and the time samples of the
`th user are transformed
`by the FFT block into modulated symbols. The bit allo-
`cation information is used to configure the demodulators
`while the subcarrier allocation information is used to extract
`the demodulated bits from the subcarriers assigned to the
`th user.
`In the frequency selective fading channel, different subcar-
`riers will experience different channel gains. We denote by
`the magnitude of the channel gain (assuming coherent
`reception) of the
`th subcarrier as seen by the
`th user.
`We assume that the single-sided noise power spectral density
`(PSD) level
`is equal to unity (i.e.,
`), for all
`
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`
`1749
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`Fig. 1. Block diagram of a multiuser OFDM system with subcarrier, bit, and power allocation.
`
`subcarriers and is the same for all users. Furthermore, we
`the required received power (in energy per
`denote by
`symbol) in a subcarrier for reliable reception of
`information
`bits/symbol when the channel gain is equal to unity. Note that
`the function
`depends on , and this allows different users
`to have different quality-of-service (QoS) requirements and/or
`different coding and modulation schemes. In order to maintain
`the required QoS at the receiver, the transmit power, allocated
`to the
`th subcarrier by the
`th user must equal
`
`Mathematically, we can formulate the problem as
`
`and the minimization is subjected to the constraints
`
`For all
`
`and
`
`(1)
`
`For all
`if there exists
`
`with
`
`then
`
`(2)
`
`(3)
`
`(4)
`
`Using these transmit power levels, the receiver can demodulate
`the modulated symbols at the output of the FFT processor and
`achieve the desired QoS’s of all users.
`The goal of the combined subcarrier, bit, and power al-
`location algorithm is then to find the best assignment of
`so that the overall transmit power, the sum of
`over all subcarriers and all users, is minimized for given
`transmission rates of the users and given QoS requirements
`specified through
`,
`. In order to make the
`problem tractable, we further require that
`is a convex and
`increasing function with
`. This condition essentially
`means that no power is needed when no bit is transmitted and
`that the required additional power to transmit an additional
`bit increases with
`[i.e.,
`is increasing in
`]. Almost all popular coding and modulation schemes satisfy
`this condition.
`It is important to note that even though the problem is
`formulated to minimize the overall transmit power for given
`QoS requirements, the same solution can be applied to improve
`the QoS’s of the users for a given overall transmit power.
`The latter can simply be achieved by increasing the power
`proportionally for all the subcarriers, while using the same set
`of
`.
`
`Note that constraint (3) is the data rate requirement and
`constraint (4) ensures that each subcarrier can only be used by
`one user. Moreover,
`is the set of all
`possible values for
`means that the th user
`, and
`does not use the
`th subcarrier to transmit any information.
`
`III. BIT ALLOCATION ALGORITHM
`FOR SINGLE USER CHANNEL
`Before we try to solve the multiuser allocation problem, we
`will first derive the bit allocation algorithm for the single-user
`environment. The single-user problem not only gives better
`understanding of the issues involved, but also provides a bit
`allocation algorithm that we will use in our multiuser solution.
`We can rewrite the optimization problem in (2) for the
`single-user case as
`
`and the minimization is under the constraint
`
`(5)
`
`(6)
`
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`

`
`;
`
`where
`
`and
`
`have to satisfy
`
`;
`
`and
`
`for all
`
`for all
`
`(7)
`
`(8)
`
`(9)
`
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`IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 17, NO. 10, OCTOBER 1999
`
`, which denotes the
`
`Note that we have dropped the subscript
`user in all notations.
`As the power needed to transmit a certain number of bits in
`a subcarrier is independent of the numbers of bits allocated to
`other subcarriers, it turns out that a greedy approach is optimal.
`A greedy algorithm assigns bits to the subcarriers one bit at a
`time, and in each assignment, the subcarrier that requires the
`least additional power is selected. The bit allocation process
`will be completed when all
`bits are assigned. Several papers
`(e.g., [15] and [16]) have provided various algorithms for this
`problem, and the basic structure of most algorithms are similar
`and can be described as follows:
`
`the optimization problem in (2) is a
`that
`turns out
`It
`combinatorial optimization problem. To make the problem
`tractable, we consider a different but similar optimization
`problem. We relax the requirement
`to allow
`to
`be a real number within the interval
`. Moreover, in order
`to deal with constraint (4),
`variables,
`,
`with values within the interval [0, 1], are introduced to the
`cost function as sharing factors of the th subcarrier. The new
`optimization problem becomes
`
`Initialization:
`For all
`, let
`Bit Assignment Iterations:
`Repeat the following
`
`and
`
`times:
`;
`
`;
`
`End;
`Finish:
`
`is the final bit allocation solution.
`
`The initialization stage computes, for each subcarrier, the
`additional power needed to transmit an additional bit. For
`each bit assignment iteration, the subcarrier that needs the
`minimum additional power is assigned one more bit, and the
`new additional power for that subcarrier is updated. After
`iterations, the final bit assignment gives the optimal bit
`allocation for each subcarrier. It is important to note that the
`bit allocation is optimal only for the given function
`,
`which depends on the selected modulation scheme. Different
`modulation schemes will lead to different
`, different bit
`allocation, and possibly lower transmit power
`.
`The concept of this algorithm is fairly simple, and many
`similar algorithms based on the same principle have been
`obtained before. In particular,
`there exist faster and less
`complex algorithms which can speed up the bit allocation
`process significantly (e.g., [15] and [16]). In our simulations,
`we use the algorithm given in [16].
`
`IV. MULTIUSER SUBCARRIER AND BIT ALLOCATION
`We have observed that, in the single-user case, a greedy
`approach which assigns one bit at a time to the subcarrier
`that requires the least additional power gives the optimal
`allocation in the sense of minimizing the overall transmit
`power. Unfortunately, the problem becomes more difficult in
`the multiuser environment. As users cannot share the same
`subcarrier, allocating bits to a subcarrier essentially prevents
`other users from using that subcarrier. This dependency makes
`any greedy algorithm a nonoptimal solution. It turns out that
`the optimal solution may not assign any of a user’s bits to the
`best subcarrier seen by that user. This may happen when the
`best subcarrier of a user is also the best subcarrier of another
`user who happens to have no other good subcarriers. Hence,
`the multiuser subcarrier and bit allocation problem is much
`more complicated to solve than that of the single-user case.
`
`satisfying the constraints (3)
`For any valid set of
`and (4) in the original optimization problem, we can let
`
`if
`if
`
`,
`.
`
`(10)
`
`and the
`Then, it is easy to show that the same set of
`corresponding
`defined in (10) satisfy the constraints (8)
`and (9) in the new optimization problem. Moreover, with
`defined in (10), the new cost function in (7) is equal to the
`cost function in (2). Hence, the minimization problem in (7)
`is the same as the original optimization problem, except that
`the minimization is done over a larger set. Consequently, the
`minimum power obtained in (7)
`is a lower bound to the
`minimum power obtained in (2),
`.
`Another way to interpret
`the optimization in (7) is to
`consider
`as the time-sharing factor for the
`th user of
`the th subcarrier. For example, in every OFDM symbol (
`being a very large number), user
`uses the
`th subcarrier
`in
`symbols. Clearly, the average (over
`symbols)
`information data rate and the average transmit power has to
`be scaled by the same factor
`. Hence, we can consider
`(7) as the optimization problem when the users are allowed
`to time-share each subcarrier over a large number of OFDM
`symbols. However, most wireless communication channels are
`time varying, and the channels may not stay unchanged long
`enough for timesharing to be feasible. Hence, in this paper,
`we will continue to consider the original problem in (2) and
`use the optimization problem in (7) as a lower bound, even
`though it has its own physical interpretation.
`The modified optimization problem in (7) is more tractable.
`However, even though the function
`is convex in ,
`the terms in the cost function have the form
`, and
`as a function of
`,
`is not convex in
`. To
`proceed further, we let
`and rewrite the cost
`function in terms of
`. The constraint on
`
`and
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`WONG et al.: MULTIUSER OFDM
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`1751
`
`becomes
`
`, and it can be easily shown that
`is convex in
`within the triangular
`region specified by
`and
`. In particular,
`the Hessian evaluated at any point within this region is a
`positive semidefinite matrix. Hence, we can reformulate the
`optimization problem in (7) as a convex minimization problem
`over a convex set. That is
`
`where
`
`and
`
`have to satisfy
`
`and
`
`for all
`
`for all
`
`(11)
`
`(12)
`
`(13)
`
`These necessary conditions can be interpreted by the fact
`that if the minimum occurs within the constrained region [(0,
`1) for
`and (0,
`) for
`], then the derivative
`evaluated at the minimum point must be zero. On the other
`hand, if the optimal solution occurs at a boundary point, then
`the derivative must be positive along all directions pointing
`toward the interior of the constraint set. Then, (17) follows
`from considering the boundary point at
`.
`From (15) and (17), we can conclude that
`
`where
`
`if
`if
`if
`
`Moreover, from (16) and (17), it follows that
`
`if
`if
`
`(18)
`
`;
`
`.
`
`;
`
`(19)
`
`Using standard optimization techniques in [17], we obtain the
`Lagrangian
`
`where
`
`(14)
`
`are the Lagrangian multipliers for the
`and
`where
`constraints (12) and (13), respectively.
`After differentiating
`, re-
`and
`with respect to
`spectively, we obtain the necessary conditions for the optimal
`solution,
`and
`. Specifically, if
`, we have
`
`if
`if
`if
`
`and
`
`if
`if
`
`.
`
`(15)
`
`(16)
`
`On the other hand, if
`
`, then
`
`, and we have
`
`for all
`
`and
`
`(17)
`
`(20)
`Since constraint (13) must be satisfied, we find from (19)
`that for each
`, if
`for
`are all
`different, then only the user with the smallest
`can
`use that subcarrier. In other words, for the
`th subcarrier, if
`are different for all
`, then
`
`where
`
`for all
`
`(21)
`
`(22)
`
`Hence, it follows that for a fixed set of Lagrange multipliers
`,
`, we can use them to determine
`for each
`using (22). The
`and
`obtained will then form an
`optimal solution for the optimization problem; however, the
`individual rate constraint (12) may not be satisfied.
`In order to find the set of
`such that the individual
`rate constraints are satisfied, we have obtained an iterative
`searching algorithm. Starting with some small values for all
`, this iterative procedure increases one of the
`until the
`data rate constraint (12) for user
`is satisfied. Then, we
`switch to another user and go through the users one at a
`time. This process repeats for all users until the data rate
`constraint for all users are satisfied. This algorithm converges
`because for a given , as
`increases,
`for all
`decreases, and more
`in (19) become one while
`(18) increases for those where
`. Hence,
`may
`increases. During this process, some of the other
`change from one to zero and consequently decrease the total
`data rate for other users. However, as all the
`increase,
`increases accordingly. As long as the total data rate is
`less than
`bits/symbol, which is the total number of bits
`possibly transmitted within an OFDM symbol, the algorithm
`
`in
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`
`will converge to a solution that satisfies all the constraints.
`Since the optimization problem is a convex optimization
`problem over a convex set, the set of necessary conditions is
`also sufficient, and the solution that satisfies all the necessary
`conditions is the unique optimal solution.
`In the process of adjusting
`, the
`for
`situation where, for a fixed , more than one
`has
`the same values cannot be ignored. In that case,
`has to
`take values within the interval (0, 1). This solution suggests
`that the subcarrier should be shared by multiple users. In
`practice, this can be done by having these users with
`time share the th subcarrier, and the ratio of the symbols used
`. The detailed
`by different users are set proportionally to
`flow chart of the algorithm is given in the Appendix.
`Now, we have an algorithm to obtain the optimal values of
`and
`
`if
`otherwise.
`
`(23)
`
`This solution, when substituted in (7), gives a lower bound to
`the minimum overall transmit power. However, we cannot use
`may
`these results immediately in (2). One problem is that
`not be in
`, and the other is that some
`may be within
`(0, 1), indicating a time-sharing solution. Furthermore, simply
`quantizing
`and
`will not satisfy the individual rate
`constraints in (3).
`To solve this problem, we propose a multiuser adaptive
`OFDM (MAO) scheme where the subcarrier allocation follows
`essentially the solution to the lower bound in (7), and then
`the single-user bit allocation algorithm given in Section III is
`applied to each user on the allocated subcarriers. Specifically,
`we modify
`for the optimization problem in (7) by letting
`for each
`where
`, and
`for
`. Then, we apply the single-user bit allocation
`algorithm on each user using the assigned subcarriers. We
`denote the total transmit power (in energy/symbol) obtained
`using this MAO scheme by
`. It is easy to see that
`, where
`is the minimum power in the original
`problem, and
`is the minimum power for the modified
`problem with the relaxed constraints. More specifically, the
`difference between
`and the minimum
`gives an upper
`bound to how far away our MAO scheme is from the solution
`of our original optimization problem.
`
`V. PERFORMANCE COMPARISON
`In this section, we obtain and compare the performance
`of the MAO scheme with other static subcarrier allocation
`schemes. We consider a system that employs M-ary quadra-
`ture amplitude modulation (MQAM) with
`.
`Square signal constellations (4-QAM, 16-QAM, and 64-QAM)
`are used to carry two, four, or six bits/symbol. The bit-error
`probability is upper bounded by the symbol error probability,
`which is tightly approximated by
`[12, p.
`281], where
`is the minimum distance between the points
`in the signal constellation. Since the average energy of a M-
`QAM symbol is equal to
`, it follows that the
`required power for supporting
`bits/symbol at a given BER
`
`is
`
`where we recall that
`
`and
`
`is convex and increasing in
`
`It is easy to see that
`that
`.
`To evaluate the performance of our scheme, we have
`simulated 1000 sets of five-path frequency selective Rayleigh
`fading channels with an exponential power delay profile. Each
`set of channels consists of
`independent channels, one for
`each user. We use an OFDM system with 128 subcarriers over
`a 5 MHz band along with a total (over all users) transmission
`rate equal to 512 bits/symbol (or equivalently, an average
`of four bits/subcarrier). Recall that the single-sided power
`spectral density level
`is equal to unity, and we assume
`that the average subcarrier channel gain
`is equal to
`unity for all
`and
`.
`For comparison purposes, we have also considered three
`other static multiuser subcarrier allocation methods. Two of
`them are based on the multiple access methods described in
`[7]. The methods are presented as follows.
`• OFDM-TDMA: each user is assigned a predetermined
`TDMA time slot and can use all the subcarriers within
`that time slot exclusively.
`• OFDM-FDMA: each user is assigned a predetermined
`band of subcarriers and can only use those subcarriers
`exclusively in every OFDM symbol.
`In a frequency selective fading channel, there is a
`high correlation between the channel gains of adjacent
`subcarriers. In order to avoid the situation where all
`subcarriers of a user are in deep fade, we propose an
`enhanced version of OFDM-FDMA, which we shall refer
`to as OFDM Interleaved-FDMA.
`• OFDM Interleaved-FDMA: this is the same as OFDM-
`FDMA except that subcarriers assigned to a user are
`interlaced with other users’ subcarriers in the frequency
`domain.
`The time and subcarrier assignment of these three multiuser
`OFDM schemes are illustrated in Fig. 2. Note that these static
`schemes have predetermined subcarrier allocations which are
`independent of the channel gains of the users. The main
`difference between the proposed MAO scheme and these static
`schemes is that MAO assigns subcarriers adaptively based on
`the instantaneous channel gains. To ensure a fair comparison,
`we use the optimal single-user bit allocation (OBA) for each
`user on the assigned subcarriers. For comparison purposes,
`we also show the results when equal bit allocation (EBA) is
`employed on the assigned subcarriers for these three OFDM
`schemes. Notice that when using EBA, all three schemes will
`have the same performance in an uncoded system. This is
`because the average bit signal-to-noise ration (SNR) needed
`is a function of only the marginal probability density function
`of each subcarrier gain.
`
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`Fig. 2. Subcarrier and time-slot allocations of OFDM-TDMA, OFDM-FDMA, and OFDM interleaved-FDMA schemes.
`
`Fig. 3. Average bit signal-to-noise ration (SNR) required by different schemes in various root mean square (RMS) delay spreads in a five-user system
`with Pe = 104.
`
`Fig. 3 shows the average bit SNR needed to achieve a BER
`for a five-user system versus the root mean
`at
`square (RMS) delay spread (for definition, see for example
`[18, p. 160]) for different multiuser OFDM schemes. The
`average required transmit power (in energy per bit) is defined
`as the ratio of the overall transmit energy per OFDM symbol
`(including all subcarriers and all users) to the total number of
`bits transmitted per OFDM symbol. Moreover, we define the
`average bit SNR as the ratio of the average transmit power to
`. As we assume that the data rate is
`the noise PSD level
`is just a constant, the overall transmit power
`fixed and that
`is proportional to the average bit SNR. For ease of comparison,
`we have used the average bit SNR for comparison. We find
`in Fig. 3 that the MAO scheme is never more than 0.6 dB
`from the lower bound. Since the bit SNR of the optimal
`combined subcarrier, bit, and power allocation algorithm must
`lie between the bit SNR’s achieved by the lower bound and the
`MAO scheme, we find that the MAO scheme is never more
`than 0.6 dB away from the optimal solution. On the other
`hand, we observe that our proposed MAO scheme is 3–5 dB
`better than the static subcarrier allocation schemes with OBA,
`which are in turn 5–10 dB better than that with EBA. We also
`find that when OBA is used, the OFDM interleaved-FDMA
`
`scheme and the OFDM-TDMA scheme have very similar
`performance, and both of them outperform the OFDM-FDMA
`scheme.2 A closer observation of Fig. 3 also indicates that the
`gains achieved by optimal bit allocation and optimal multiuser
`subcarrier allocation increase with the RMS delay spread. This
`is mainly because the larger the RMS delay spread, the more
`the fading variation and hence higher gains can be obtained
`when the allocation is performed adaptively.
`Fig. 4 shows the average bit SNR (in dB) needed to achieve
`the same BER versus the number of users when the RMS delay
`spread is 100 ns. We find that the savings in the required bit
`SNR achieved by MAO when compared to other schemes are
`roughly the same, independent of the number of users in the
`system.
`in the
`While these two figures show the improvement
`required bit SNR, the results can perhaps be more easily
`understood using the more familiar BER versus bit SNR
`curves. For each BER requirement, we compute
`for all
`and then use our algorithm to calculate the subcarrier
`
`2 OFDM-FDMA refers only to the specific FDMA scheme that assigns to
`each user a contiguous band of subcarriers as shown in Fig. 2, but not the
`general FDMA schemes. In fact, both OFDM interleaved-FDMA and MAO
`can be considered as different forms of FDMA and they are not outperformed
`by the OFDM-TDMA scheme.
`
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`
`Fig. 4. Average bit SNR required by different schemes versus the number of users in a multiuser OFDM system with 100 ns RMS delay spread,
`and Pe = 104.
`
`Fig. 5. BER versus average bit SNR for various subcarrier allocation schemes.
`
`allocation for the MAO case. For all other static subcarrier
`allocation schemes, the allocations are independent of the
`BER. Once the subcarrier allocation is fixed, we apply the
`optimal bit and power allocation algorithm to every user.
`The final average power per bit divided by the noise power
`spectral density level gives the average bit SNR. We repeat
`this procedure for different BER values, and the results are
`
`plotted in Fig. 5 for a five-user system with an RMS delay
`spread equal to 100 ns. We find that our proposed MAO has
`at least 3–4 dB advantage over all other schemes.
`Another way to illustrate the impact of the bit and subcarrier
`allocation is