Achievable Rates Of Asymmetrical Multiple Access

Achievable Rates of Asymmetrical Multiple Access in Block-Fading Channels Thomas Deckert and Gerhard Fettweis Vodafone Chair Mobile Communications Systems Technische Universit¨at Dresden, 01062 Dresden, Germany {deckert,fettweis}@ifn.et.tu-dresden.de

Abstract— Based on the specification of a high-throughput wireless local area network system we consider the achievable rate regions of time, frequency and code division multiple accessing schemes. We are especially interested in how the schemes compare in a two user scenario with one high-rate and one lowrate user. The channel is modeled as block-fading and frequency selective. We focus on the case in which the transmitters do not know the channel but the receiver does. The transmitters are subject to individual peak or average power constraints. In the context of code division multiple access we also look at successive decoding and interference cancellation (SIC) in addition to optimum joint decoding. We argue that CDMA-like superimposed transmission of a high rate user and low rate users with a SIC receiver is suboptimal yet appropriate to improve the throughput in scenarios similar to our application.

I. I NTRODUCTION In this paper we consider the multiple access scenario, i. e., multiple users want to transmit messages to one common receiver. We put particular emphasis on a practical application and want to answer what multiple-access scheme is best ratewise given our system constraints. We consider a wireless local area network (WLAN) system based on orthogonal frequency division multiplexing (OFDM) being developed within the context of WIGWAM [1]. It is designed to deliver high throughput by providing high spectral efficiency to transmit large amounts of user data. Unfortunately, a typical (W)LAN data profile consists not only of large packets but also many very short messages (e. g., [2], more than 60% of all packets are less than 50 bytes). The latter are due to control signaling (e. g., acknowledgments) and low-rate user traffic typical for, e. g., web browsing. They also may require low latency and a given user may have very few packets to send during a given time interval. Thus the data available to a transmitter may very well be limited. Currently, in OFDM-based WLANs packets are timemultiplexed though typically via random access methods. For the system laid out in [1] a user with favorable channel conditions may be able to transmit several hundred bytes in a single OFDM symbol. However, the user may not be able to exploit that if it has only little data to send (see also [3]). Then, time multiplexing under stringent delay constraints is disadvantageous for achieving high overall throughput. Thus here we are interested in how time division multiple access (TDMA) compares to (orthogonal) frequency division

(OFDMA) and code division multiple access (CDMA) in terms of achievable rate regions under the constraints given by a practical system. Here, “achievable rate region” refers to the set of all joint data rates of the users at which the transmitted messages can be decoded with arbitrarily low probability of error. We use “rate” rather than “capacity” to distinguish between the maximum rate of some transmission scheme and the maximum rate the channel supports. We consider simple yet fair schemes that share the resources based on the requested data rates and delay requirements rather than instantaneous channel conditions (as, e. g., “greedy” TDMA in [4], [5]). The important consequence is that a user is guaranteed a transmit opportunity even if its channel conditions are poor or it has not enough data to exploit all channel resources. In particular our schemes are periodic, i. e., they use the same resource allocation pattern repeatedly. Many researchers have considered the capacity region of the multiple access channel and achievable rates of specific multiaccess schemes (for an overview see [6], [7] and references therein). Which scheme achieves the largest achievable rate region depends strongly on what the transmitter knows about the channels of the users, and whether the decoding delay requirements are loose or stringent [7]. For the channels being known at the transmitters [4], [8] discuss the case of tolerable delay much longer than the time scale of variation of the fading processes and [9], [10] consider capacity if we can tolerate short delays only. Multiple access in the absence of channel knowledge at the transmitter and with stringent delay constraints is examined in [11], [12], where [12] shows that for users with equal rates and powers and time-varying frequency-selective channels CDMA is best. In our case the users can tolerate short delay with respect to the temporal variations of the channel only, i. e., the fading conditions stay roughly constant during one code word transmission. In this situation a meaningful measure is the instantaneous capacity that depends on the actual channel state. We assume that the receiver knows the channels perfectly but that the channel states are unknown to the transmitters. As a consequence the best the transmitters can do is to fix the data rate and transmission power at which to send. As the channel varies so does the instantaneous capacity. In particular, it may be lower than the transmitted rate in which case no reliable transmission is possible and the channel is in outage. Thus we

consider the maximum rates that can be supported given the maximum tolerable probability of such outage events. In many works the users are rather symmetrical in the sense that they have the same data rate requirements and transmission powers. Here, we are confronted with an asymmetrical setup where in a given time interval a few users need to transmit large amounts of data and several other users need to transmit small packets only. Thus, we look at channel access of a high-rate and a low-rate user as was done in [3]. We consider constraints on the power per sub-carrier as well as averaged over one OFDM symbol or one multi-access period. We describe our system and comparison setup in Section II and the outage rate regions of the considered schemes in Section III. In Section IV we apply the obtained expressions to the two user case with a special emphasis on asymmetrical users. We summarize our discussion in Section V and argue that the CDMA-like superimposed transmission scheme introduced in [3] beats TDMA and OFDMA in our application. II. S YSTEM M ODEL We now introduce the setup of our multiple access system, the channel model and the constraints used in the analysis of the following sections. For ease of exposition we will focus on the two-user case later on but the results may be extended to more users. Thus we consider several users here. All considered multiple access schemes are assumed to be designed for ideal operation, i. e., there are no guard intervals necessary in TDMA, no guard sub-carriers in OFDMA and we assume symbol and frequency synchronous reception of all users. In OFDMA each user gets a block of adjacent subcarriers. We do not consider “interleaved” sub-carrier assignments or frequency hopping as they require rather complex synchronization of the users in practice. In all schemes U users share an available total bandwidth B and a total transmission (frame) duration T . In TDMA the entire bandwidth is used exclusively by user
u ∈ U = {1, . . . , U } for a duration µu T , 0 ≤ µu ≤ 1,
u µu = 1. In FDMA a frequency band of µu B, 0 ≤ µu ≤ 1, u µu = 1 is used exclusively by user u during the entire time T . Finally, in CDMA each user is allocated all of the bandwidth during all of the time where we do not require explicitly the usual spreading operation. As we consider discrete time and frequency, T corresponds to N temporal samples and B to M sub-carriers. The transmitters send sequences of symbols from a zeromean complex Gaussian alphabet corresponding to code words. All sequences of a transmitter are equally likely. In line with [1] a sequence is mapped to all of the user’s subcarriers rather than to a single subcarrier. The transmitters do not know the instantaneous channel state. Hence they fix their rate and power for the entire transmission. The transmit power of a user u at OFDM symbol n and subcarrier m, Su (n, m), may be constrained either (a) per subcarrier, (b) per OFDM symbol, or (c) averaged over T and B. These constraints, in turn, may be due to, e. g., spectral mask requirements, power amplifier limitations, or limited battery life, respectively. In our case Su (n, m) = Su for all (n, m)

{ˆ x1 } {y}

Fig. 1.

Decoder User 1

{ˆ x2 } Decoder User 2

{h1 }

{h2 }

-

-

Successive decoding with interference cancellation for two users.

user u employs, and constraint (b) is equivalent to (a) in TDMA and equivalent to (c) in OFDMA, and all constraints are equivalent in CDMA. Thus we consider peak power ˆ constraints,
N
M Su (n, m) ≤ Su , and average power constraints, n=1 m=1 Su (n, m)/(N M ) ≤ Su . We assume that all users are subject to stringent delay requirements and thus encounter slow fading; a scenario typical of indoor applications like WLANs. Hence, we model the channel as being constant during the transmission of one code word (block fading). The channel may be frequency selective but we assume frequency-flat fading on each sub-carrier. The receiver performs optimum maximum likelihood sequence detection on the received signal y(n, m) =

U 

hu (m)xu (n, m) + v(n, m),

(1)

u=1

where xu (n, m) is the symbol sent by user u at time n and sub-carrier m, hu (m) is the corresponding channel coefficient with E[|hu (m)|2 ] = 1 and v(n, m) denotes the complex white Gaussian receiver noise with power spectral density N0 . In TDMA and OFDMA, for every (n, m) only one of the xu (n, m) is nonzero, and single-user detection suffices. In CDMA joint maximum likelihood decoding (JD) of the {xu } is optimal but often too complex. Thus we also look at successive decoding with interference cancellation (SIC). As in Fig. 1 the user signals are decoded one-by-one with subsequent decoding steps operating on the received signal with the interference by all preceding users canceled out. As known from, e. g., [8], [9] SIC achieves the points on the capacity region boundary if the channel states are known to the transmitters and average power constraints apply. However, as we will see, it is sub-optimal in our scenario. III. O UTAGE R ATE R EGIONS In this section we describe the rate regions for each scheme if the block-fading channel is unknown to the transmitters. Capacities and rates are considered in terms of spectral efficiency averaged over the total transmission duration T and bandwidth B with unit bits per complex dimension. In our block fading case the instantaneous capacity of a single link conditioned on a particular channel state is [13]   κ(m)S 1  log2 1 + (2) C(S, M, κ) = |M| BN0 m∈M

where M is an index set denoting the sub-carriers used by the link and κ is the M -dimensional channel power vector with κ(m) = |h(m)|2 . For rates r ≤ C(S, M, κ) the message can be decoded successfully, i. e., with arbitrarily low probability

of error. These rates are achievable over channel κ. For larger rates decoding fails, i. e., the probability of error is close to 1. In our context, to the transmitter, C(S, M, κ) is a random variable through the dependence on the unknown channel powers. As C(S, M, κ) is unknown and transmission occurs at some constant rate r it is possible that C(S, M, κ) < r. Let the probability of this channel outage event be p. We call a rate achievable if it is at most equal to the instantaneous capacity with probability 1 − p [11], [13]. In general, the rate region R of a multi-access scheme is a sub-set of the set RU + of U -dimensional element-wise nonnegative real rate vectors r = [r1 , . . . , rU ] such that the ru are achievable with that scheme. As in TDMA and OFDMA a user experiences single-link conditions in its time slot or sub-band, respectively, we have independent conditions on the individual user rates given µ. The rate vectors achievable for individual outage probabilities given by p = [p1 , . . . , pU ] are simply     ∀u ∈ U : C• (S, p) = r ∈ RU +
U µ∈[0,1] , u µu =1  Pr (µu C(Su (µu ), Mu , κu ) ≥ ru ) ≥ 1 − pu , (3) Element µu of the resource allocation vector µ denotes the proportion of T or B user u is allotted in TDMA or OFDMA, respectively. The sets Mu are all equal to M = {1, . . . , M } in TDMA. In OFDMA they are disjoint subsets – Mu ⊆ M, Mu ∩ Mj = {∅}, u = j – with |Mu |/M = µu . Finally S is a set of functions Su (µ) giving the power of user u based on µ. For peak power constraints we have Su (µ) = Sˆu while for average power constraints Su (µ) = Su /µ. In CDMA the conditions on the individual user rates cannot be easily decoupled and it is harder to derive an expression similar to (3). Given the channel states the instantaneous rate region of CDMA achieved by JD (CDMA/JD) is known to be (e. g.[7], [14])    U  ru ≤ RJD (S, {κu }) = r ∈ R+  u∈J

  κu (m)Su 1  , ∀J ⊆ U . (4) log2 1 + |M| BN0 m∈M

u∈J

Acknowledging the constraints on the partial sum rates of any user sub-set J in (4) we apply the notion of decoding success jointly to all users, i. e., all users or none are decoded successfully at the same time. Then the outage rate region of CDMA with joint decoding can be described by (cf. [10]) 

  CJD (S, p) = r ∈ RU + Pr (r ∈ RJD (S, {κu })) ≥ 1 − p . (5) We may use the SIC receiver of Fig. 1 in CDMA. Here users are decoded in order π = [π(1), . . . , π(U )] which is a permutation of U. User π(1) is decoded in the presence of all other users. Consequently, it should transmit at a rate lower than its single-user capacity to make up for the additional interference. User π(U ) is decoded lastly. If all previous decoding stages are successful it will experience single-user

conditions and may transmit at rates up to its single-user capacity. Given π and the channel the instantaneous rate region of CDMA with SIC receiver (CDMA/SIC) is    rπ(u) ≤ Rπ (S, {κu }) = r ∈ RU +  ∀u ∈ U : 

  1  log2  1 + |M|  m∈M



κπ(u) (m)Sπ(u) π(U
) j=π(u+1)

κj (m)Sj + BN0

   . (6)  

In (6) π is applied on all sub-carriers. Adapting π for every sub-carrier requires a code word be mapped to exactly one subcarrier; but we do not do that here. Also, the rate constraint of user π(u) is coupled to all previously decoded users. They must have been successfully decoded and cancelled. In the following we neglect that user π(u) may be decoded even if a previous stage failed. We further neglect that the receiver can adapt the decoding order based on its channel knowledge. We will see that these simplifications have little effect in the scenario we are interested in. Now the outage rate region of CDMA/SIC in order π, Cπ , is given by (5) with Rπ replacing RJD . As the firstly decoded user must not transmit at its singleuser rate, Cπ is a subset of CJD only. However by time sharing between different decoding orders π the convex hull of all Cπ can be achieved; we denote this rate region by CSIC . IV. A PPLICATION TO A SYMMETRIC U SERS In the following we investigate numerically the two user case and apply the results to a system where one user aims for a much higher rate than the other one. This setting is in line with our application of Section I. We look at the boundaries of the p-outage rate regions of our schemes by evaluating (3), (5) at the same outage probability p = 0.01 for each user. According to the system specification in [1] we model the channel as block fading and frequency selective with an IEEE 802.11n channel D power delay profile and 100 MHz bandwidth. The channel transfer function is sampled at M = 512 sub-carriers1 . As it is instructive to do so we compare the frequency-selective to the frequency-flat fading scenario where all sub-carriers are in the same complex normally distributed fading state and the average over the logarithms in (2), (6) collapses to one log function. We consider an equal power scenario with average received signal-to-noise ratios (SNR) of Su /(BN0 ) = 20 dB for each user and an unequal power scenario with user 1 at 20 dB and user 2 at 0 dB. All boundaries have been obtained by checking the constraints in (3), (5) for appropriate rate region samples where the probabilities are estimated over 1000 multi-access channel realizations. We begin with the frequency selective equal power case. Fig. 2 shows the rate region boundaries, i. e., the maximum rate of user 2 given a rate of user 1. TDMA, OFDMA correspond 1 While this is less than the 640 sub-carriers specified within 100 MHz in [1] the argument here should not be affected by that.

CDMA/JD

3 TDMA OFDMA CDMA/SIC peak average pwr. constr.

r2 / bits/complex dimension

r2 / bits/complex dimension

4

CDMA/JD 0.4 0.3

1 4 2 3 r1 / bits/complex dimension Fig. 2. Outage rate regions for frequency selective channels with success probability 1 − p = 0.99 and equal received SNR of 20 dB.

TDMA OFDMA CDMA/SIC peak 0.1 average [1, 2] 0 1 4 0 2 3 r1 / bits/complex dimension Fig. 3. Outage rate regions for frequency selective channels. Success probability 1 − p = 0.99. Received SNRs: 20 dB for user 1, 0 dB for user 2.

to single-user operation at any particular time or frequency, respectively, with division of time or frequency between the users determined by µ (s.t. µ1 = µ, µ2 = 1 − µ). For peak power constrained TDMA the instantaneous capacity C is independent of µ as the channel is time-invariant (see (2)). By varying µ we obtain as rate region boundary the straight line between the single-user outage capacities. In average power constrained TDMA and given µu user u’s power is larger than for constrained peak power by 1/µu . Thus its instantaneous capacity is also increased and we obtain a larger rate region. Note that TDMA is uniformly advantageous to OFDMA. An OFDMA user only gets a contiguous fraction of the bandwidth B that each TDMA user is allotted. So in (2) we average the instantaneous capacity Cu over more channel states in TDMA than in OFDMA. Thus, given µ, the variance of Cu is lower in TDMA than in OFDMA while the mean Cu is the same. Therefore in TDMA it is more likely that Cu is above a given rate ru , ru < Cu . As we aim for low outage probability we aim for rates below Cu and we see that the TDMA rate region must enclose that of OFDMA under either power constraint. As could be expected from the usual discussion in, e. g., [7], [14] jointly decoded CDMA achieves the largest rate region. However, the large degree of diversity we have here plays a significant role in ensuring this result. We will elaborate on that when we consider the frequency flat case where we will see that CDMA might not be best. For successful operation of CDMA/SIC the user for which decoding is most likely to be successful should be decoded first. As we must apply the same decoding order to all sub-carriers this likelihood depends on the average of the instantaneous capacity over all sub-carriers, Cu . Here, in high frequency diversity, Cu is near its mean and depends mainly on the SNRs. Thus, the decoding order can be fixed depending on the SNRs and rates of the users. Naturally, for equal power users the firstly decoded user should transmit at lower rate than the second. This is proved by the boundary of C[2,1] in Fig. 2. User 2 transmits at low rates as it is decoded first while user 1 can transmit at high rates. This boundary also shows that requiring all users be decoded successfully is not critical with high diversity. Even if all constraints in (6) were decoupled the best CDMA/SIC

could achieve would be that user 1 transmits at its single user rate and user 2 at the maximum rate given the interference by user 1. The rate region would be rectangular with the maximum rates as in C[2,1] . From the figure we observe that C[2,1] almost achieves as much. Note that the SIC scheme does not achieve the maximum sum rate, r1 + r2 , of CDMA/JD. This is in contrast to the case when the transmitters know the channels and the SIC scheme achieves the same rates as JD by, e. g., time-sharing between decoding orders [2, 1] and [1, 2] [8], [9]. CDMA/SIC lies in between average and peak power constrained TDMA. However, for the combination of a high-rate and a low-rate user it may be as good as CDMA/JD and better than TDMA/OFDMA. Note that this is the scenario SIC is designed for. Again diversity affects how CDMA/SIC and TDMA/OFDMA compare and we will comment on it when we look at flat fading. Now we turn to the frequency selective unequal power case in Fig. 3. Here the average received powers correspond directly to the desired rates of the users. Not surprisingly TDMA/OFDMA and CDMA/JD qualitatively display the same performance as in the equal SNR case. We observe that CDMA/SIC achieves nearly the same rate region as joint decoding and encloses almost the entire rate region of average power constrained TDMA. As evidenced by the boundary of C[1,2] decoding order [1, 2] is the right choice for a large set of rate vectors as user 1’s power is much larger than that of user 2. The only critical vectors are those for which user 1’s rate is near its single-user outage capacity. As we decode user 1 in the presence of user 2’s interference they are not supported by this scheme. If user 1 must transmit at that high a rate the decoding order must be [2, 1] again. But as the power of user 2 is much smaller than that of user 1, user 2 faces massive interference and must transmit at very low rates. Practically, in that case, the system supports the single user 1 only. Thus, CDMA/SIC may be used to transmit at full rate a low-rate user while still maintaining a high-rate user transmitting somewhat below its single-user capacity. Note that contrary to the equal power case the high-rate user should be decoded first. Hitherto we discussed a rich diversity scenario and saw that CDMA/SIC has advantages over TDMA and OFDMA

2 1

[2, 1] 0

0

0.2

TDMA/OFDMA average

0.8

CDMA/JD

0.6

[1, 2]

0.4

TDMA/OFDMA peak

0.2 0

0.2

depending on the power constraints. To gain some more insight into how CDMA and TDMA/OFDMA compare we consider frequency flat fading now, i. e., we have no diversity. The main consequence is that the instantaneous capacity Cu depends on a single channel state (applying to all sub-carriers at all times) rather than being an average over several states. Thus the variance of Cu is higher than in the frequency selective case. Apart from reducing the rate that can be guaranteed with probability 1 − p we will see that the widely varying Cu significantly affects CDMA behavior. Another effect of having a single fading state on all sub-carriers at all times is that the instantaneous capacity of a user in TDMA equals that in OFDMA given µ and a power constraint. Consequently the corresponding rate regions are identical. Consider the flat fading equal power case in Fig. 4. Firstly, CDMA/JD achieves a larger rate region than peak powerconstrained but a much smaller one than average powerconstrained TDMA/OFDMA. Note that the results from, e. g., [7] saying that CDMA/JD achieves capacity are based on transmitter-side channel knowledge. As we do not assume that here there is no guarantee that CDMA/JD achieves the largest rate region. The CDMA/JD region can be made to encompass that of TDMA/OFDMA again if the average SNR (per user) is increased. The SNR at which CDMA and average power constrained TDMA/OFDMA have the same maximum sum rate r1 + r2 given an outage probability p is2 Su /(BN0 ) = p 2 ) which amounts to 26 dB here. 4p/(1−p)/(2 ln(1−p)+ 1−p In the frequency selective equal power case this threshold is much lower. Comparing CDMA with TDMA we numerically determined it to be about −3 dB. This behavior is due to two counteracting effects. In TDMA/OFDMA the instantaneous single-user capacities need to be larger than the corresponding rates independently of each other (3). In CDMA/JD the same constraints exist but without scaling of the rates (and possibly powers) by 1/µ (4). Hence, based on individually satisfying the single-user constraints for the same SNRs and p, CDMA could support larger sum rates than TDMA/OFDMA can. This difference becomes more pronounced with higher SNR. However, there is an 2 Starting from the outage probabilities of TDMA and CDMA for same SNRs and rates, eliminate the dependency on rate while solving for SNR.

TDMA/OFDMA average 0.01 TDMA/OFDMA peak

CDMA/JD SIC

0.005

CDMA/SIC

[2, 1] 1 0.4 0.6 0.8 r1 / bits/complex dimension Fig. 4. Outage rate regions for flat block-fading with success probability 1 − p = 0.99 and equal received SNR of 20 dB. 0

r2 / bits/complex dimension

r2 / bits/complex dimension

0.015

1

0 0

[1, 2]

1.5 1 2 2.5 3 3.5 r1 / bits/complex dimension Fig. 5. Outage rate regions for flat fading with success probability 1 − p = 0.99 and received SNR of 30 dB for user 1 and 0 dB for user 2. 0.5

additional sum rate constraint in CDMA and, moreover, all three constraints need to be satisfied jointly. This is harder to achieve than independently satisfying the constraints and the more so the larger the variation of the individual capacities is. In flat fading without the possibility to average over several independent κ(m) in (2) these variations are largest. In Fig. 4 the latter effect dominates and CDMA achieves a smaller set of rates than average power constrained TDMA/OFDMA. The rate region of CDMA/SIC in Fig. 4 is equal to that of peak power constrained TDMA/OFDMA. Considering a specific decoding order this observation means the firstly decoded user needs to transmit at negligible rate. Then the (joint) outage probability is determined by the secondly decoded user only and it can transmit at rates up to its single-user outage capacity. Obviously, in flat fading the individual instantaneous capacities vary that much that we cannot ensure decoding both equal power users jointly in all but trivial cases. We would need to take the hint of Fig. 3 and increase the average power of the firstly decoded user to improve the performance of the first decoding stage. This is demonstrated in Fig. 5 where the SNR is 30 dB3 for user 1 and 0 dB for user 2. We observe only a small advantage of CDMA with SIC over peak power constrained TDMA/OFDMA for low rates of user 1. Considering our original aim of having one user transmit at high rate and another at low rate, CDMA is better suited than TDMA/OFDMA for high diversity channels while for low diversity this conclusion is reversed. V. C ONCLUSION In the previous sections we considered the rate regions achievable with TDMA, OFDMA and CDMA under peak and average constraints on the power use. To summarize our comparison we trace back these constraints to the physical limitations on transmission power listed in Section II: (a) per sub-carrier, (b) per OFDM symbol, (c) averaged over T and B. We regard (a) and (b) as more important in practice. We base this final comparison on the results of Section IV so all previously made assumptions on the system, the channels and average received powers apply. We consider TDMA, OFDMA and CDMA with SIC as jointly decoded CDMA is 3 We

determined that a 20 dB difference would not suffice.

TABLE I S CHEMES ACHIEVING THE LARGEST RATE REGION IN FREQUENCY SELECTIVE FADING . Transmit Power Constrained

Scheme

(a) per sub-carrier

CDMA

(b) per OFDM symbol

CDMA

(c) per T B

CDMA/TDMA

TABLE II S CHEMES ACHIEVING THE LARGEST RATE REGION IN FREQUENCY FLAT FADING . Transmit Power Constrained

Scheme

(a) per sub-carrier

All

(b) per OFDM symbol

OFDMA

(c) per T B

OFDMA/TDMA

a very complex technique. In interpreting the following recall that (b) corresponds to the relaxed average power constraint in OFDMA and to the stringent peak power constraint in TDMA. Hence, OFDMA may be advantageous to TDMA in that case depending on the fading conditions. Concerning the frequency-selective case Table I lists for each transmission power constraint the scheme achieving the largest rate region. “Largest” ideally refers to a rate region enclosing all competing rate regions. This notion is ill-defined if the transmission power is limited on average over T and B only. Then TDMA is average power-constrained and TDMA as well as CDMA with SIC achieve rate vectors the other of the two cannot. Looking at very asymmetrical rates in the case of equal average received powers CDMA with SIC has a slight advantage whereas TDMA is clearly better if the user rates are similar (Fig. 2). If the average received powers are different CDMA with SIC is best if the low rate user is to get all its capacity and the high rate user can dispense with some of its speed (Fig. 3). However, TDMA can still compete and may be more attractive for allocating near full rate to the high rate user and some marginal rate to the low rate user. Note, however, that (c) may not be as relevant as (a) and (b). In all other cases of Table I CDMA with SIC wins quite clearly. In the frequency flat scenario the picture changes with OFDMA universally achieving the largest rate regions in all cases. This is summarized in Table II. As observed in Figs. 4, 5 CDMA with SIC performs no better than peak power-constrained TDMA/OFDMA and should not be used in this scenario if data rates are what matters only. Note that in OFDMA frequency hopping, i. e., switching to another sub-band during transmission of one code word, or interleaved sub-carrier assignments help in exploiting frequency diversity. Then OFDMA could achieve nearly the same rate regions as TDMA does in the frequency selective scenario. It would compare to CDMA with SIC in scenario (b) as TDMA compares to CDMA with SIC in (c). OFDMA may seem the better alternative to CDMA then as it may be simpler to implement. However, OFDMA requires the users to be

precisely synchronized in time and frequency to avoid interference. Otherwise interference cancellation or some other form of multi-user detection may be necessary which may make it as complex as CDMA with SIC. Alternatively one could use guard bands to lessen the synchronization requirements by reducing interference. Thus, a simple OFDMA implementation implies a loss of spectral efficiency. In the simple analysis in Section IV the CDMA variants have been treated rather restrictively. By adapting the decoding order based on the channel knowledge and trying to decouple the constraints on the individual user rates one may be able to demonstrate enlarged rate regions especially for the SIC receiver in frequency flat fading. The results for CDMA should be regarded as lower bounds. Concluding this discussion, CDMA with successive decoding at the receiver is the best scheme rate-wise to accommodate high rate transmission and low rate transmission in the multiple access channel given our system constraints. A key factor contributing to this result proved to be the frequency diversity which in our case is rather large. ACKNOWLEDGMENT This research was supported by the German Ministry of Education and Research within the project Wireless Gigabit with Advanced Multimedia Support (WIGWAM) under grant 01 BU 370. R EFERENCES [1] G. Fettweis and R. Irmer, “WIGWAM: System concept development for 1 Gbit/s air interface,” in WWRF 14, San Diego, USA, July 2005. [2] K. Claffy, G. Miller, and K. Thompson, “The nature of the beast: Recent traffic measurements from an Internet backbone,” in INET’98, Geneva, 1998. [Online]. Available: http://www.caida.org/Papers [3] T. Deckert, W. Rave, and G. Fettweis, “Superposed signaling option for bandwidth efficient wireless LANs,” in Proc. WPMC 2004, Sept. 2004. [4] R. Knopp and P. Humblet, “Information capacity and power control in single-cell multiuser communications,” in Proc. Intern. Conf. Commun., ICC 1995, vol. 1. IEEE, June 1995, pp. 331–335. [5] A. S. Leong, J. S. Evans, and S. Dey, “Power control and multiuser diversity in multiple access channels with two time-scale fading,” in Proc. WIOPT 2005. IEEE, Apr. 2005, pp. 86–95. [6] E. Biglieri, J. Proakis, and S. Shamai (Shitz), “Fading channels: Information-theoretic and communications aspects,” IEEE Trans. Inform. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998. [7] A. Goldsmith, Wireless Communications. New York: Cambridge Univ. Press, 2005. [8] D. N. Tse and S. V. Hanly, “Multiaccess fading channels–Part I: Polymatroid structure, optimal resource allocation and throughput capacities,” IEEE Trans. Inform. Theory, vol. 44, no. 7, pp. 2796–2815, Nov. 1998. [9] S. V. Hanly and D. N. Tse, “Multiaccess fading channels–Part II: Delaylimited capacities,” IEEE Trans. Inform. Theory, vol. 44, no. 7, pp. 2816– 2831, Nov. 1998. [10] L. Lifang, N. Jindal, and A. Goldsmith, “Outage capacities and optimal power allocation for fading multiple-access channels,” IEEE Trans. Inform. Theory, vol. 51, no. 4, pp. 1326–1347, Apr. 2005. [11] L. H. Ozarow, S. Shamai, and A. D. Wyner, “Information theoretic considerations for cellular mobile radio,” IEEE Trans. Veh. Technol., vol. 43, no. 2, pp. 359–378, May 1994. [12] E. Erkip and B. Aazhang, “A comparitive study of multiple accessing schemes,” in Proc. 31st Asilomar Conf. Signals, Systems and Computers, ACSSC 1997, vol. 1. IEEE, 1997, pp. 614–619. [13] G. Caire, G. Taricco, and E. Biglieri, “Optimum power control over fading channels,” IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1468– 1489, July 1999. [14] T. M. Cover and J. A. Thomas, Elements of Information Theory, 1st ed. New York: John Wiley and sons, 1991.