Rate Regions Of Asymmetrical Multiple Access

Rate Regions of Asymmetrical Multiple Access with Receive Diversity Thomas Deckert, Sebastian Kaiser and Gerhard Fettweis Vodafone Chair Mobile Communications Systems Technische Universit¨at Dresden, 01062 Dresden, Germany {deckert,kaiser,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 if the receiver has multiple antennas. Our results show that given a fixed total number of transmitting antennas putting more antennas at the receiver helps tremendously in separating simultaneously transmitting users. Then the individual rates at which the users can transmit simultaneously are increased and code division (CDMA) becomes increasingly attractive over the other two. We focus on the case in which the transmitters do not know the channel but the receiver does. Then the preferred successive decoding and interference cancellation CDMA receiver is suboptimal. However, code division still performs best.

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. A typical (W)LAN data profile consists 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 or low-rate user traffic and may require low latency. Also, a given user may have few packets to send during a given time interval. Thus the data available to a transmitter can be quite 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 our 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. Another ingredient of our discussion is the use of multiple antennas at the transmitters and the receiver. The system of [1] specifies up to 4 antennas at either side of the link. Inspired by [4] we focus on the case where the number of receiving antennas and the total number of transmitting antennas match. As the number of observations equals the number of impinging signals the receiver should be able to separate the signals easily. So in CDMA where the users are superimposed we can increase the simultaneously achievable rates. As TDMA and OFDMA keep the users separate by design they will not benefit from this aspect of multi antenna designs. This is in line with the sum capacity results in [4]. Somewhat similarly, [5] finds that if the channels are known at both sides of the link that in the single-antenna case only one user should be active at any time while with multiple antennas more users can be allowed to access the medium simultaneously. 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 [6]). 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 [7], [8] 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 [8]. For the channels known at the transmitters [6], [9] discuss the case of tolerable delay much longer than the time scale of variation of the fading and [10], [11] consider short tolerable delays. Multiple access in the absence of channel knowledge at the transmitter is examined in [12], [13] for stringent delay constraints and in [14] under an ergodic scenario. For

users with equal rates and powers and frequency-selective channels [13], [14] show that CDMA is best. 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. In our case the users can tolerate short delay with respect to the temporal variations of the channel only. In this situation a meaningful measure is the instantaneous capacity that depends on the actual channel state. 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 asymmetrical two user case with multiple antennas. There we also look at the performance of a practical system to supplement the theoretic discussion. 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 in the multi-antenna case. 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. The 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. 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) 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 ˆ peak power
constraints,
M Su (n, m) ≤ Su , and average power N constraints, n=1 m=1 Su (n, m)/(N M ) ≤ Su . The transmitters as well as the receiver may have multiple antennas. If a transmitter employs NT antennas its power is equally divided among them. 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 NT -dimensional vector of symbols sent by user u at time n and sub-carrier m, Hu (m) is the corresponding NR xNT -matrix of unit average power channel coefficients and NR -dimensional vector v(n, m) denotes the independent complex white Gaussian receiver noise samples per receiving antenna 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 of the {xu } is optimal but often too complex. Thus we 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., [15] SIC achieves the points on the capacity region boundary if the channel states are known to the transmitters and average power constraints apply. Here, the transmitters do not know the channel but, e. g., [3] showed that it performs quite well for users with significantly different rate requirements. 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

{ˆ x1 } {y}

Fig. 1.

Decoder User 1

{ˆ x2 } Decoder User 2

{H1 } -

{H2 } -

Successive decoding with interference cancellation for two users.

averaged over the total transmission duration T and bandwidth B with unit bits per sub-carrier use. In our block fading case the instantaneous capacity of a single link conditioned on a particular channel state is [16]   1  1 log2 det INR + K(m) (2) C(M, K) = |M| BN0 m∈M

where index set M denotes the sub-carriers used by the link and the block-diagonal matrix K = diag(K(1), . . . , K(M )) with K(m) = S/NT · H(m)HH (m) represents the channel cross correlation scaled with the power per transmit antenna. For achievable rates r ≤ C(M, K) the message can be decoded successfully, i. e., with arbitrarily low error probability. For larger rates decoding fails, i. e., the error probability is close to 1. In our context, to the transmitter, C(M, K) is random through the dependence on the unknown channel powers. As C(M, K) is unknown and transmission occurs at some constant rate r it is possible that C(M, K) < r with some probability p. We call a rate achievable if it is at most equal to the instantaneous capacity with probability 1 − p [12], [16]. 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 : r ∈ RU C• (S, p) = +
µ∈[0,1]U , u µu =1  Pr (µu C(Mu , Ku ) ≥ 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. Ku depends on Su as before. In CDMA the conditions on the individual user rates cannot be easily decoupled and it is harder to derive an expression similar to (3). The SIC receiver of Fig. 1 decodes the users one by one in some 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

sufficiently 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 a decoding order and the channel states the instantaneous rate region of CDMA with SIC receiver is    rπ(u) ≤ Rπ (S, {κu }) = r ∈ RU +  ∀u ∈ U :     π(U ) π(U )   C M, Kj  − C M, Kj  . (4) j=π(u)

j=π(u+1)

In (4) the individual rate constraint of user π(u) is coupled to all previously decoded users as they must have been successfully decoded and cancelled. In the following we neglect that user π(u) may be decoded successfully even if some previous decoding stage failed. We further neglect that the receiver can adapt the decoding order based on its channel knowledge. With these simplifications we can describe the outage rate region of CDMA with SIC decoding in order π, Cπ ,     Cπ (S, p) = r ∈ RU + Pr (r ∈ Rπ (S, {Ku })) ≥ 1 − p . (5) IV. E FFECT OF R ECEIVE D IVERSITY We now investigate numerically a two user system that employs as much antennas at the receiver as there are transmitting antennas in the system. 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. There are M = 512 sub-carriers1 . Multiple antennas are assumed to be uncorrelated. The rate values in the following correspond to the boundaries of the rate regions for an outage probability p = 0.01. They have been obtained by checking the constraints in (3), (5) for appropriate rate vector samples where the probabilities are estimated over 1000 multi-access channel realizations. Firstly, consider the case that each user has an average received signal-to-noise ratio (SNR) of Su /(BN0 ) = 20 dB. This represents the ideal power control case. Fig. 2 shows the boundaries of the rate regions, i. e., the maximum rate of user 1 given a rate of user 2, if the transmitters and the receiver have a single antenna only. The general behavior of TDMA and OFDMA for flat channels is known from, e. g., [8] and largely extends to this frequency-selective scenario. For peak power constraints TDMA achieves the straight line between the maximum achievable single-user rates. For average power constrained TDMA, if the share of resources µu of user u drops it can send at inverse proportionally higher power and its instantaneous capacity increases. Thus we obtain a larger rate region. 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.

TDMA OFDMA

3 2 [2, 1] 1

CDMA

r1 / bits/sub-carrier use

r1 / bits/complex dimension

peak average pwr. constr.

4

16 12

2x4 8 4

[1, 2] 0

TDMA peak TDMA average CDMA [2, 1]

3/1x4

0 0

1x1

1 4 2 3 r2 / bits/complex dimension Fig. 2. Outage rate regions for TDMA, OFDMA and CDMA with SIC. Target success probability 1 − p = 0.99 and equal received SNR of 20 dB.

0.6 0.8 1 r2 /r1 Fig. 3. Rate of user 1 versus rate ratio of users 2 and 1 for various schemes, success probability 1 − p = 0.99 and equal received SNRs of 20 dB.

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. Hence in (2) we average the instantaneous capacity Cu of a user 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 in both schemes. 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 TDMA performance upper bounds that of OFDMA we concentrate on TDMA. In CDMA SIC reception is successful if the user which is most likely to be decoded successful is decoded first. Thus, for equal SNRs, the firstly decoded user should transmit at a rate lower than the second one. This can be observed in Fig. 2. This observation is in line with our application of [1] and we assume user 1 aims for a much higher rate than user 2. Hence the correct decoding order is [2, 1] here. Note that CDMA with SIC is advantageous to TDMA or OFDMA for asymmetrical users but only marginally so. In the following we compare the characteristics of the single-antenna reference, denoted “1x1”, to the case where the receiver has four antennas and the transmitters either have two antennas each (“2x4”) or user 1 has three antennas and user 2 one antenna (“3/1x4”). For the equal-SNR scenario Fig. 3 shows the rate r1 user 1 achieves with each scheme given the rate r2 of user 2 normalized on r1 . We compare the schemes for the same normalized user 2 rate. As expected r1 grows with the number of transmitting antennas of user 1. Even more interesting is how CDMA and TDMA compare for r2 growing from 0 to larger values. In the 1x1 reference CDMA has a slight advantage over TDMA for r2 /r1
0.15. For larger user 2 rates decoding user 2’s message in the presence of interference by user 1 gets more difficult. To maintain a given ratio r2 /r1 the users have to “back off” from their capacity limit causing the rapid decline of r1 for r2 /r1 > 0.15. In the other antenna scenarios the CDMA-“cut-off” r2 /r1 where r1 begins to drop is significantly larger than in 1x1. Now CDMA with SIC offers a clear advantage over TDMA

for asymmetric users. Since the numbers of channel inputs and outputs are equal the receiver may separate the signals by simple channel inversion. This is considerably easier than solely relying on exploiting a priori knowledge of the difference in the users’ coding schemes and powers – as has to be done in the 1x1 reference. Consequently user 1 can transmit at its maximum rate up to relatively high r2 . On the other hand, in TDMA the users are orthogonal by design. So multiple antennas at either side of the link do not help the receiver distinguish between them and the general behavior of r1 versus r2 /r1 equals that of the reference case. Note that for CDMA r1 cuts off at smaller r2 /r1 in 3/1x4 than in 2x4. This is because in 3/1x4 the ratio of the maximum achievable rates of users 2 and 1 is 0.44 (not shown) rather than 1 as in 2x4 due to the different numbers of antennas the users employ. Consequently r2 /r1 > 0.44 necessitates that r1 be chosen less than its maximum in 3/1x4 while for 2x4 this mechanism applies to r2 /r1 > 1. The 3/1x4 case a priori gives a rate advantage to user 1 and is designed for use at low r2 /r1
0.3 while the 2x4 case sacrifices some rate of user 1 in exchange for higher user 2 rates. We now turn to an unequal power scenario with user 1 at 20 dB and user 2 at 0 dB due to, e. g., bad propagation conditions for user 2 or insufficient power control. The maximum achievable single-user rate rˆ2 of user 2 is much smaller than that of user 1, rˆ1 . We want to have user 2 transmit its data rather than maximizing the system sum rate. The goal is to have r2 as large as possible while not disturbing user 1 too much. This and the fact that user 1 is the stronger one mean that the CDMA decoding order should be [1, 2]. Fig. 4 zooms into the corresponding part of the rate regions (see Fig. 2). Here we use normalized rates ρu = ru /rˆu . Clearly, CDMA outperforms TDMA for high ρ2 . For a wide range of ρ2 user 1 can be offered 0.8 of its maximum rate in 1x1 and over 0.9 in the multi-antenna cases. Average power constrained TDMA seems to suffer if multiple antennas are used. Note that the absolute rates r1 are still much higher for 3/1x4 and 2x4 than for 1x1 (at ρ2 = 0.7: 10.6, 8.0 and 3.7) but multi-antenna users loose proportionally more of their capacity than single-antenna users. In this form of TDMA user 1 can scale its power proportionally to the transmission opportunity

0

0.2

0.4

1 r1 / bits/complex dimension

5 CDMA [1, 2]

0.8 0.6 ρ1

TDMA

average

0.4 0.2 TDMA peak 0 0.7

0.8

ρ2

0.9

1

Fig. 4. Normalized rate regions. Success probability 1 − p = 0.99. User 1 at 20 dB, user 2 at 0 dB.

it lost, i. e., ∼ 1/(1−ρ2 ). However, capacity scales logarithmic with power only. Thus, while the power scaling depends solely on ρ2 , ρ1 depends on the absolute value of r1 and decreases with r1 increasing and ρ2 fixed. The information-theoretic treatment has been verified by simulation of a two user multi-access system in which each user has one transmitting antenna and the receiver has one or two antennas. The signals superimpose but are not spread. Thus the scheme conforms to the notion of CDMA as we defined it in Section II. The transmitted data rate is varied by adapting the code rate of the convolutional encoder (1/2, 2/3, 3/4) and the modulation scheme (BPSK, 16QAM, 64QAM). There are 512 sub-carriers and the channel is modeled as before. One code word corresponds to 3 OFDM symbols. In the 1x1 case SIC reception is used as in [3] whereas in the 1x2 case a zero-forcer inverts the channel and the signals are decoded independently. For a given transmitted data rate R the code word error probability Pe is estimated over 10000 code words. From these the effective data rate r that represents successful transmission is obtained by r = R(1 − Pe ). Fig. 5 shows the effective rate regions for equal average SNRs of 25 dB. We note that the SIC receiver in 1x1 supports high rate transmission of a user only for rather low rates of the other one. In particular, transmission close to the maximum achievable rate requires the other user not to transmit. Thus for 1x1 TDMA is the better choice practically. If the number of receiving antennas is increased to match the number of all transmit antennas the picture changes significantly and it is possible to have large rates for both users simultaneously. These results are completely in line with Section IV. V. C ONCLUSION Considering our original aim of having one user transmit at high rate and another at low rate, CDMA is better suited than TDMA (and thus OFDMA) if the number of receiving antennas is on the order of the total number of transmitting antennas in the system or if low rate users are expected to have small power. Then with CDMA it is possible to have a highrate user transmit at rates close to its achievable maximum while at the same time supporting low-rate transmission. For power controlled single antenna systems TDMA and OFDMA are viable alternatives.

1x2

4 3 2 1 0 0

[2, 1]

1x1

[1, 2]

1 4 2 3 r2 / bits/complex dimension

5

Fig. 5. Rate region of a practical two user system. Equal received SNRs of 20 dB. ru = transmitted data rate · (1 − code word error rate).

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] A. Mantravadi, V. V. Veeravalli, and H. Viswanathan, “Spectral efficiency of MIMO multiaccess systems with single-user decoding,” IEEE J. Select. Areas Commun., vol. 21, no. 3, pp. 382–394, Apr. 2003. [5] H. Boche and E. A. Jorswieck, “Multiple antenna multiple user channels: Optimisation in low SNR,” in Proc. WCNC 2004, vol. 1. IEEE, 2004, pp. 513–518. [6] 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. [7] 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. [8] A. Goldsmith, Wireless Communications. New York: Cambridge Univ. Press, 2005, to be published. [Online]. Available: http: //wsl.stanford.edu/∼andrea/Wireless/Book.ps [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] S. Visuri and H. B¨olcskei, “MIMO-OFDM multiple access with variable amount of collision,” in Proc. ICC 2004, vol. 1. IEEE, 2004, pp. 286– 291. [15] S. Vishwanath, S. A. Jafar, and A. Goldsmith, “Optimum power and rate allocation strategies for multiple access fading channels,” in Proc. VTC 2001-Spring, vol. 4. IEEE, May 2001, pp. 2888–2892. [16] 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.