Using Unclaimed Sub-carriers in Opportunistic OFDMA Systems Patrick Svedman∗ , Leonard J. Cimini, Jr.† and Bj¨orn Ottersten∗ ∗ KTH School of Electrical Engineering
† Electrical and Computer Engineering Dept.
SE-100 44 Stockholm, SWEDEN.
[email protected]
University of Delaware Newark, DE 19716, USA.
Abstract— In this paper, we consider the unclaimed subcarriers that no user feeds back in an opportunistic OFDMA system with a per sub-carrier power constraint. Unclaimed subcarriers appear in OFDMA systems with reduced feedback, where the users concentrate the feedback on the sub-carriers with good channel quality. Here we propose and evaluate two ways to use these sub-carriers to improve the system performance. One approach is to transmit pilots symbols on the unclaimed subcarriers. A second approach is to schedule users based on their feedback about adjacent sub-carriers. Simulation results show that for low to moderate channel rms delay spread, it is more advantageous to transmit data on unclaimed sub-carriers, even at a low rate, than to use them for additional pilot symbols. For high delay spreads however, transmitting pilots on the unclaimed sub-carriers gives higher system throughput.
I. I NTRODUCTION Opportunistic communication promises increased throughput in multiuser systems [1]. It relies on the principle of multiuser diversity, which exploits the fact that users fade independently. By giving channel access to users with favorable channel conditions and using adaptive modulation, the system throughput can be increased compared with a system that uses fixed user scheduling. Opportunistic scheduling has already entered existing single-carrier standards, like 1xEV-DO [2] and HSDPA [3]. A strong candidate for future wireless cellular systems is orthogonal frequency division multiple access (OFDMA), in particular for the downlink [4]. In OFDMA, users can be simultaneously scheduled on different frequency sub-carriers. This opens the possibility to exploit the fact that users typically experience frequency fading. By scheduling users on time instants and frequencies where their channel is strong, transmit power is not wasted on channels with poor conditions. A problem with channel-aware scheduling for the OFDMA downlink in frequency division duplex (FDD) systems is that the total feedback load can be very high. To schedule users on different sub-carriers, the base-station requires knowledge of the channel quality on the sub-carriers of all active users. If the number of sub-carriers is high and the users fade rapidly, the total feedback rate will be overwhelming, even if the channel quality is highly quantized. Different approaches to deal with this problem were treated in [5]–[7]. In [5], each user fed back the channel quality of only its best sub-carriers. More sub-carriers per user were fed back when many users were
active, whereas fewer sub-carriers were fed back when there were many active users [6]. A positive effect of this adaptive feedback rate is that, for different numbers of active users, the total feedback rate remains fairly constant as well as the probability that a sub-carrier is not fed back by any user. In [7], the approach was instead to let each user feed back only one bit per sub-carrier that signifies if the sub-carrier channel quality is above a threshold. The threshold was adapted as a function of the number of users in order to allow for higher order modulation when many users were active. In addition to these approaches, the feedback load can be further reduced by grouping adjacent sub-carriers into sub-bands or clusters, which can be used as feedback units instead of the separate sub-carriers. There is a risk that all users experience good channels on the same sub-carriers. If a reduced feedback scheme is used, as described above, the consequence could be that the scheduler, for many sub-carriers, receives no channel quality information. What to do with these unclaimed sub-carriers that no user claims is the topic of this paper. One option is to move transmit power from unclaimed sub-carriers to subcarriers where users can be scheduled. In this paper however, we assume a power constraint per sub-carrier, since power constraints typically are in the form of spectral masks. We propose to use the unclaimed sub-carriers for pilot symbols in addition to the grid of fixed pilot sub-carriers. We also propose a simple scheme that assigns each unclaimed sub-carrier to a user that has fed back sub-carrier channel quality information close to the unclaimed sub-carrier. The system model is presented is Section II. The proposed methods to use the unclaimed sub-carriers is presented in Section III. They are evaluated by means of simulations in Section IV and the paper is concluded in Section V. II. S YSTEM M ODEL The OFDMA downlink of an FDD cellular wireless communication system is considered. Perfect time and frequency synchronization is assumed, as well as a wide-sense quasistationary channel (constant during each OFDM symbol). It is also assumed that the channel impulse response is shorter than the cyclic prefix. Then, the received signal on subcarrier n ∈ N = {1, . . . , N } at OFDM symbol t for user
k ∈ {1, . . . , K} can be written as yk (n, t) = c(n, t)Gk (n, t) + wk (n, t)
B. Random (1)
where c(n, t) is the transmitted symbol, Gk (n, t) is the subcarrier frequency response and wk (n, t) is AWGN with vari2 . Inter-cell interference is not considered in this paper. ance σw In order to compensate for the effect of the channel, each user has to estimate the channel frequency response for each sub-carrier that they will use for data reception. The channel estimation is based on a-priori known pilot symbols that are transmitted on some sub-carriers. Linear MMSE channel estimation is used in this paper, which is described in more detail in the Appendix. On each sub-carrier, a different user can be scheduled. The scheduling is updated regularly with an interval of several OFDM symbols, here called a block. As in [5], the scheduling is based on limited feedback from each active user in the form of the supportable number of bits per symbol for a number of the strongest sub-carriers. The sub-carriers that no user fed back are here called unclaimed sub-carriers. For each block, the set of sub-carriers can be divided into three disjoint sets, N p ∪ N c ∪ N u = N = {1, . . . , N }, where N p is the predefined set of fixed pilot sub-carriers, N c is the set of claimed sub-carriers, and N u is the set of unclaimed sub-carriers. For the claimed sub-carriers, at least one user fed back the instantaneous channel quality, and for the unclaimed sub-carriers, the scheduler has received no instantaneous channel information. Note that the users do not consider the fixed pilot sub-carriers in N p for feedback. Also note that the sets N c and N u are random, since they depend on the instantaneous channels of the users. It is here assumed that transmit power cannot be transferred from N u to N c . How to use the bandwidth in N u is the topic of this paper. III. U SING U NCLAIMED S UB - CARRIERS In this section, we propose three alternatives which are evaluated in Section IV. In the Pilot method, pilots are transmitted on the unclaimed sub-carriers, in addition to the fixed sub-carriers. In the Random method, no fixed pilots are transmitted. Instead, the users have to rely only on the pilots on the unclaimed sub-carriers. In the Data method, users are scheduled on the unclaimed sub-carriers. A. Pilot In the Pilot method, the fixed pilots in N p are accompanied by pilot symbols in the set of unclaimed sub-carriers, N u . The users can detect which sub-carriers are in the set N u if the base-station transmits a user index (e.g. 0) that is reserved for the dynamically assigned pilot sub-carriers. For the subcarriers in N c , the base-station transmits the user index of the scheduled user, either on a control channel or directly on the sub-carrier. The Pilot approach could increase the accuracy of the channel estimation for the sub-carriers in N c , leading to lower error rates.
It is possible that the pilot sub-carriers in N p would be suitable for high-order modulation for some users. If the opportunistic feedback and scheduling principle is extended to all sub-carriers, the set of fixed pilot sub-carriers, N p , is empty. Instead, pilots are only transmitted in the random set N u . The disadvantage of this method is that the channel estimation quality is more random. The advantage is that the users can choose sub-carriers to feed back from N instead of only N \ N p , thereby increasing the multiuser diversity effect. C. Data It is difficult for the scheduler to assign a user and modulation level to an unclaimed sub-carrier without jeopardizing the transmitted packet. No specific channel quality information is available to the scheduler, other than that the unclaimed subcarrier is weaker than all the claimed sub-carriers. However, if the unclaimed sub-carrier is almost adjacent to a claimed sub-carrier, it not likely that the quality of the unclaimed subcarrier deviates much from the quality of the claimed subcarrier (see [5] for a derivation of this probability, which is a function of the channel delay-spread and the distance to the adjacent sub-carrier). Then, it seems to be a good choice to schedule the user from the adjacent claimed sub-carrier, but with a one-step lower modulation level. We call this the Data method. It requires no extra forward signaling of modulation levels for the sub-carriers in N u , since the user knows that it did not feed back that sub-carrier, and will therefore try to detect symbols from a constellation-size one level below the constellation-size of the adjacent claimed sub-carrier. Only the scheduling decision has to be signaled. IV. S IMULATIONS AND D ISCUSSION The methods described above, as well as two reference methods, have been evaluated by means of simulation. They are shortly summarized here. • Fixed : Only N p is used for pilots. N u is unused. • Perfect : Similar to the Fixed method, but the channel is perfectly estimated at the receivers. • Pilot : Both N p and N u are used for pilots. • Random : Only N u is used for pilots, whereas N p is empty. • Data : Only N p is used for pilots and N u is used for data symbols, assigned to the scheduled user closest in frequency, but at a lower rate. A cell with K users has been simulated. Time-dispersive sample-spaced i.i.d. Rayleigh fading channels are assumed, gk (m, t) =
L−1
βk,l (t)δ(m − l)
l=0
with an exponentially decaying power delay profile E |βk,l (t)|2 = Ae−lT /2ν where ν is the root-mean-square (rms) delay-spread, L = 60 is the number of taps and T = 260 ns is the tap spacing,
Throughput [Mbps]
4
3
2
1
0 0
Perfect Fixed Pilot Random Data 500
1000 1500 RMS Delay-spread [ns]
2000
Fig. 1. The throughput for the proposed methods to use the unclaimed subcarriers. The frequency selectivity increases along the x-axis. The number of users in the cell is 10.
−1
10
−2
10
BER
which is equal to the sampling time. The constant A is used to normalize the average channel power to one. The number of sub-carriers is 256, the OFDM symbol time is 82.16 µs, including the 15.6 µs cyclic prefix. The carrier-frequency is 2 GHz. Jakes’ model is used for temporal block-fading [8]. We assume that the receivers use a separate power prediction algorithm for the sub-carrier SNRs. In this numerical evaluation, the fed back supportable rates available at the scheduler are assumed to be without errors or delay. On each sub-carrier, the user with the highest reported supportable rate is scheduled and assigned a modulation level from BPSK, QPSK, 16-QAM, 64-QAM or 256-QAM. The adaptive modulation thresholds are based on a target bit-error rate of 10−4 . The throughput is computed as the number of bits in the successfully received packets, which are 128 bits long. A packet is considered erroneous if at least one bit is erroneous. The users estimate the channel on each sub-carrier using the LMMSE estimator described in the Appendix using P = 6 pilot symbols. The pilots are taken from the same OFDM symbol as the sub-carrier to be estimated and chosen so that the sub-carrier to be estimated is located in the middle of the 6 pilots. The fixed pilots in N p are spaced 8 sub-carriers apart in each OFDM symbol. The average SNR for all users is set to 10 dB. Each of the K users feeds back the supportable number of bits per symbol for their S < N strongest sub-carriers in each block. The allowed number of bits per symbol is 1, 2, 4, 6 and 8. For each sub-carrier, the base-station schedules the user with the highest supportable number of bits per symbol. The scenarios K = 10 and K = 50 users in the cell have been studied. For 10 users, each user feeds back S = 28 sub-carriers, which gives E [| N u |] ≈ 59 sub-carriers [6]. For 50 users, S = 6, which gives E [| N u |] ≈ 58. An important difference in the structure of N u between few and many users is that for few users, the unclaimed sub-carriers tend to be adjacent. The reason is that the fed back sub-carriers of a user often are adjacent, especially for lower delay-spreads. For many users, the sub-carriers in N u tend to be more uniformly spread in the OFDM symbol, since more users feedback fewer sub-carriers. The pilot-spacing of 8 gives | N p | = 32 subcarriers. Figures 1-4 show the system throughput and bit-error rate (BER) for the different methods as a function of the channel rms delay spread ν. The performance loss due to channel estimation errors can be seen as the difference between the Perfect and Fixed methods, both of which do not utilize the unclaimed sub-carriers. In Figure 1, where there are 10 users in the cell, the throughput of Perfect increases with the increased channel frequency variability due to the opportunistic scheduling. For Fixed, the throughput initially increases, but as the delay spread reaches a certain point, the effect of the reduced channel estimation quality dominates. By using the unclaimed subcarriers for additional pilots, Pilot manages to reduce some of the loss due to channel estimation errors at high delay spreads.
−3
10
Perfect Fixed Pilot Random Data
−4
10
−5
10
0
500
1000 1500 RMS Delay-spread [ns]
2000
Fig. 2. The bit-error rate for the proposed methods to use the unclaimed sub-carriers. The frequency selectivity increases along the x-axis. The number of users in the cell is 10.
For the Data method, which schedules users on unclaimed sub-carriers, the throughput is even higher than for Perfect for low delay-spreads. This is due to the fact that Perfect does not use the unclaimed sub-carriers. However, as the frequency variability increases with the delay spread, the reduced channel estimation quality as well as the loss in adaptive modulation accuracy on the unclaimed sub-carriers makes the throughput of Data reach that of Fixed. The overall performance for the Random scheme is poor, except when the channels are almost flat, i.e. for low delay spread, where the penalty of the random pilot placement is very small. The underlying bit-error rates for the 10-user scenario is shown in Figure 2. The BER increases with increased channel variability, except for Perfect, which decreases sligthly. This is due to the slightly higher channel gains for the scheduled
6
Throughput [Mbps]
5
Perfect Fixed Pilot Random Data
4 3 2 1 0 0
500
1000 1500 RMS Delay-spread [ns]
2000
Fig. 3. The throughput for the proposed methods to use the unclaimed subcarriers. The frequency selectivity increases along the x-axis. The number of users in the cell is 50.
−1
10
−2
BER
10
Perfect Fixed Pilot Random Data
−3
10
−4
10
0
500
1000 1500 RMS Delay-spread [ns]
2000
Fig. 4. The bit-error rate for the proposed methods to use the unclaimed sub-carriers. The frequency selectivity increases along the x-axis. The number of users in the cell is 50.
users, which for this particular average SNR (10 dB) and 10 users implies more margin in the adaptive modulation. The higher modulation level thresholds are separated further apart (in dB) than the lower. Of the methods that do not enjoy perfect channel estimation, Pilot has the lowest error rates, due to the additional transmitted pilots on the sub-carriers that no user fed back. Even though the BER of Data is higher than for Fixed, the throughput is higher, since more bits are transmitted. In Figures 3 and 4, the similar results are given, but for 50 users. The throughput for Perfect is higher than for 10 users, due to the increased multiuser diversity. Note that the BER for Perfect is slightly higher than for 10 users. The reason is that, for the particular average SNR is these simulations and 50 users, the highest modulation level is selected more often, but with a smaller margin. An important difference to the 10-user
case is that Random performs relatively much better. This is due to the fact that for 50 users, the sub-carriers in | N u | are typically more uniformly spaced in the OFDM symbol. For 10 users, the unclaimed sub-carriers are often located on adjacent sub-carriers, which leads to worse channel estimation on the other sub-carriers for Random. In the simulations in this paper, the fixed pilot spacing has not been varied. The effect of varying the fixed pilot spacing is equivalent to varying the channel rms delay spread, which has been done here. Furthermore, the selected scheduling method has an impact on performance. In this paper, the maxthroughput scheduling has been used. Since the additional pilots in the Pilot scheme are benificial for all users, it is reasonable to assume that this method would perform as well with other schedulers. The Data scheme should also perform similarly with other scheduling methods. However, the user-scheduling on the unclaimed sub-carriers should not be changed. For the Random scheme, the situation would not change much with another scheduling method, due to the poor channel estimation performance. In the simulations in this work, intercell interference was not considered. The Fixed scheme could be expected to perform relatively slightly better if intercell interference would have been included, since not using the unclaimed sub-carriers at all would give a lower average intercell interference. V. C ONCLUSIONS In this paper, we investigated a situation that can occur in OFDMA systems with limited feedback from each user, i.e. that the scheduler has no channel state information about some of the available sub-carriers. We proposed to schedule users also on the unclaimed sub-carriers, but at a lower rate, by using the available channel state information about adjacent subcarriers. As an alternative, we suggested to use the unclaimed sub-carriers for transmission of additional pilot symbols, in order to improve the channel estimation quality at all receivers. Finally, a more extreme option was considered in which only the unclaimed sub-carriers were used for pilot symbols. This could enable users to claim sub-carriers that were previously reserved for pilot symbols, at the cost of a more random channel estimation quality. Simulation results with max-throughput scheduling and LMMSE channel estimation were shown. The approach to transmit low-rate data on the sub-carriers that no user claimed in the feedback showed superior performance for low to moderate channel rms delay spread. For the higher delay spreads, additional pilots on the unclaimed sub-carriers gave the highest system throughput due to the improved channel estimation. The scheme with completely random pilot placement proved to be the worst in terms of throughput, especially for channels with high frequency variability. In order to improve the accuracy of the scheduling on the unclaimed sub-carriers, channel estimation methods could also be used at the transmitter. This was investigated in [9] in a different context. Also, the incorporation of previously fed back channel quality information could be used for a more successful resource allocation.
A PPENDIX Here, we describe the linear minimum mean square error (LMMSE) channel estimator that the receivers use in order to be able to coherently detect the transmitted symbols. The correlation between two sub-carriers can be computed from the channel model in Section IV as rf (∆n) = E Gk (n + ∆n, t)Gk (n, t)H 1 − e−T /2ν 1 − e−LT /2ν−j2πL∆n/N 1 − e−LT /2ν 1 − e−T /2ν−j2π∆n/N 1 − e−T /2ν ≈ (2) 1 − e−T /2ν−j2π∆n/N where ∆n is the spacing in sub-carriers. The approximation is valid when L is large. The temporal correlation is given by rt (∆t) = J0 (2πfd ∆t), where J0 (x) is the zero-order Bessel function of the first kind, ∆t is the time-difference and fd is the maximum Doppler frequency. If the channel impulse response taps gk (m, t) are zero-mean complex Gaussian distributed and independent of the noise, so will the sub-carrier frequency response Gk (n, t). Then, the LMMSE estimate of Gk (n, t) is [10] =
ˆ k (n, t) = rGp (n, t)R−1 (n, t)pk (n, t) G pp
(3)
where pk (n, t) = [pk (n1 , t1 ) · · · pk (nP , tP )]T ∈ CP ×1 is the vector of least-squares estimated frequency responses at the P pilot positions (n1 , t1 ) · · · (nP , tP ). These P pilots are used to estimate the channel on sub-carrier n and OFDM symbol t. The estimate of the channel at pilot sub-carrier ni and OFDM symbol ti is given by pk (ni , ti ) =
yk (ni , ti ) wk (ni , ti ) = Gk (ni , ti ) + c(ni , ti ) c(ni , ti )
(4)
where c(ni , ti ) is a known unit-energy pilot symbol. The cross covariance vector between the non-pilot sub-carrier that is to be estimated and the pilot sub-carriers, rGp (n, t) ∈ C1×P is given by rGp (n, t)
=
E[Gk (n, t)pk (n, t)H ]
=
[rG (n−n1 ,t−t1 )···rG (n−nP ,t−tP )]
(5)
and the covariance matrix of the pilot sub-carriers, Rpp (n, t) ∈ CP ×P is given by Rpp (n, t) = E[pk (n, t)pk (n, t)H ] = rG (0,0) · · · rG (n1 −nP ,t1 −tP ) .. .. 2 .. + σw IP (6) . . .
rG (nP −n1 ,tP −t1 )
···
rG (0,0)
H
where denotes Hermitian transpose. The frequency response correlation function (7) rG (∆n, ∆t) = E Gk (n + ∆n, t + ∆t)Gk (n, t)H can be factored into the frequency and temporal correlation functions of the frequency response [11] 2 rG (∆n, ∆t) = σG rf (∆n)rt (∆t)
(8)
2 is the total average power of the channel impulse where σG response. The frequency correlation depends on the impulse response rms delay spread and the temporal correlation depends on the maximum Doppler frequency of the user. The P pilot sub-carriers used for estimation of the channel on (n, t) should be chosen wisely, for instance to maximize the l1 norm of the cross covariance (5). The filter order P is a trade-off between estimation accuracy and complexity. The complexity of the LMMSE estimator is high, since it requires a matrix inversion of a P ×P matrix, but several lowcomplexity alternatives exist, for instance [12], [13]. In this paper we consider the optimal LMMSE as described above, even though low-complexity approximations could be applied. Here, we assume that the users employ a low-complexity algorithm to predict the channel qualities on all sub-carriers (for the feedback) and the more complex LMMSE algorithm only on the sub-carriers where the users are scheduled (for data detection). A further problem with the LMMSE estimator is that the channel correlation and the noise power are required. In this paper, we assume them to be known by the receivers.
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