A precoded OFDMA system with user cooperation

We propose a two user OFDMA scheme with interblock precoding and user cooperation.We show that the proposed scheme can achieve maximum diversity of order twice the number of fading paths. A non-cooperative precoded OFDMA system that uses the same transmission energy per information block would require at least two transmit antennas to achieve the same diversity. In cases where there is a significant difference in SNR between each user/destination pair, the proposed scheme can improve the capacity of the weak user.


I. INTRODUCTION
Multiuser Cooperation is a promising technology for improving the performance of wireless communication systems. It was shown in [1][2] that cooperation has the potential to increase the data rate. Furthermore, the work by [3] shows that, in theory, the achievable diversity order is equal to the number of cooperating users. Two types of cooperation has been used in the past, decode-and-forward, and amplify-andforward. In [4] it is shown that the former performs better than other schemes in terms of Bit Error Rate (BER) but is more demanding on each user. Therefore, in certain situations where complexity needs to be kept low, it might be preferable to use an amplify-and-forward scheme.
OFDM systems have gained popularity due to their ability to handle fading. Although OFDM systems can turn a frequency-selective fading channel into several parallel flatfading subchannels, each information symbol is transmitted over one channel only. This can be a problem when some subchannels experience deep fading. Several ways have been proposed in the literature for introducing path diversity in OFDM systems. In [5], it is shown that a single user OFDM system with non-redundant block precoding can achieve diversity gain up to L, the number of channel fading paths. For maximum diversity, the performance gain is exploitable using a Maximum Likelihood (ML) decoder. Smaller than L diversity gains are also possible via subcarrier grouping [5], which results in reduced complexity decoding at the receiver. If one used redundant precoding or oversampling at the receiver, the maximum diversity gain is still possible via a linear receiver [6] and [7].
In this paper we propose a cooperative multiuser OFDM scheme with inter-block precoding. Inter-block precoding was used in [8] to exploit time diversity introduced by time varying channels. Here, even if the channel is completely static, inter-block precoding allows us to exploit spatial diversity introduced by cooperation. We study the advantages of the proposed scheme as compared to a multiuser OFDM scheme without cooperation, which uses the same transmission energy per information block as the proposed scheme. The main findings of this paper can be summarized as follows.
• When there is a difference in the SNR between the users and the receiver, cooperation improves the capacity of the weak user. When both users face high signal-tonoise ratio (SNR) between each other and the receiver, cooperation has no effect on capacity. • We show that a two-user precoded OFDM system with cooperation, with each user operating with a single transmit antenna, can achieve maximum diversity of order 2L. The non-cooperative OFDM system with intra-block precoding would need to require two transmit antennas to achieve the same diversity. Notation: The small and capital bold letters respectively denote vectors and matrices. We denote the N × N identity matrix as I N and all-zero matrix as 0 N . The superscripts (·) T and (·) H denote transposition and Hermitian, respectively. We use diag{x} and to denote a diagonal matrix with the vector x on the diagonal and element-wise multiplication, respectively.

A. Background
Let us consider a two-user OFDMA system where users are assigned disjoint carriers. User 1 transmits over subcarriers in set I 1 , and receives over subcarriers in set I 2 , where I 1 I 2 = {0, 1, ..., N − 1} and I 1 I 2 = {}. User 2 transmits over subcarriers in set I 2 , and receives over subcarriers in set I 1 . |I| denotes the cardinality of set I.
Let s i j denote the ith OFDM block of user j with the length |I j |, and y i p denote the corresponding signal received by user p over the carriers in set I j . The time-domain multipath channel between j and p is denoted by h pj (n) n ∈ [0, L − 1], with zero mean i.i.d. Gaussian and unit variance. The taps h pj (n) are uncorrelated for different p, j pairs, and also for different values of n. Let the frequency-domain channel be H pj (k) = where with I j (k) denoting the kth element of the set I j according to some predefined ordering. We assume the channel is slowly varying, i.e., the channel remains constant for a few OFDM blocks, and the receivers have perfect knowledge of the channel; n i pj denotes noise between users p and j during the transmission of the ith block. We assume that the noise is circularly complex Gaussian with zero mean, uncorrelated between different i's (slots), and different p, j's. For simplicity we assume that for the noise variance it holds: (σ i kl ) 2 = (σ i+1 kl ) 2 = σ 2 kl . We will next discuss a scenario where users both transmit and receive simultaneously using the same antenna, i.e., in full duplex mode. Since there could be practical hardware difficulties related to such scenario, we will later discuss an approach of using time division multiplexing to achieve full duplex operation. The only change that this will introduce is to change the effective channel between a transmitter and a receiver. We should stress that this will not change the following analysis or the conclusions drawn in this paper. Thus we continue to present our method assuming full duplex operation for simplicity.

B. Proposed scheme
We perform inter-block precoding on the two successive data blocks before they enter the OFDM system. Precoding [5] [8] can help exploit the multipath diversity and spatial diversity that will be introduced by the cooperative retransmissions. The inter-block precoding is implemented as follows.
If the uncoded blocks of user j are d i j , d i+1 j , the precoded blocks are The ith block of user j consists of the user's own transmission, and also data from the other user, m. Since each user is allocated a different set and number of carriers, the received block sent by user m, (j = m) will be of size |I m | × 1 and will have to be mapped to a block of size |I j | × 1 before it is transmitted by user j. Let us represent this mapping by premultiplying the received block by a matrix P j . If |I 1 | = |I 2 | then the mapping matrix could be a permutation matrix, or simply the identity matrix. In order to maximize cooperation, we let |I 1 | = |I 2 | = N/2 in this paper. Then, P 1 and P 2 become square permutation matrices.
The proposed cooperation scheme is implemented in cycles of 3 blocks (or slots): Slot i: Both users transmit their own data s i 1 and s i 2 , respectively. These will be received as H 21 s i 1 + n 21 and H 12 s i 2 + n 12 , respectively, by the other user. Slot i + 1: Users transmit their own data s i+1 1 and s i+1 2 plus the received signal after it has been scaled and mapped from the incoming carriers to outgoing carriers. The scaling depends on the amount of power being allocated for cooperation. Under a fixed energy constraint, allocating too much power to the forwarding signal would weaken the user's own signal. We use α and β as the scalars to denote this scaling of signals. Hence, the amount of power allocated for cooperation at users 1 and 2 are proportional to α 2 and β 2 , respectively.
Slot i + 2: Both users again transmit R 1 s i 1 and R 2 s i 2 as their own data, plus the received signal. Note that there is a component of s i 1 (s i 2 ) in the received signals at users 1 (2). In order to eliminate this, the precoding is modified as for this block. The received signal during this block is ignored. The echo component gain αβP 1 H 12 P 2 H 21 or (αβP 2 H 21 P 1 H 12 ) can be tracked at each user by correlating the received signal at the (i + 2)th block by the transmitted signal at the ith time block.
In the (i + 3)th slot, this cycle is repeated with two new data blocks. Hence the rate of transmission is 2/3.
Let us consider the signal at the base station (BS) during three slot intervals. For simplicity, P 1 = P 2 = I, then the received signal on carriers in I 1 equals: where on the (i+1)th block, (4) and (5) have been substituted for R 1 and R 2 .
Similarly, the received signal over carriers in I 2 is: In vector form, we have: where and:  (12) and (15), and keeping in mind that each of s i m and s i+1 m are functions of both d i m and d i+1 m , one can clearly see the motivation behind using inter-block precoding. By cooperation, we have effectively created two transmission paths for our information, given by (12) and (15). This is analogous to effectively employing two transmitters. Hence we can exploit this by transmitting our information on both channels given by (12) and (15).
We should note that inter-block precoding was used in [8] to exploit time diversity introduced by time varying channels. Here, even if the channel is completely static, inter-block precoding allows us to exploit spatial diversity introduced by cooperation.
At the receiver we combine decoding of (12) and (15) as: where

C. Maintaining constant transmission energy between cooperative and non-cooperative schemes
Letσ 2 i , σ 2 i be the signal power of user i without and with cooperation, respectively. For simplicity let us takeσ 1 =σ 2 = σ and σ 1 = σ 2 = σ. In the cooperative OFDM scheme, transmission of s i j and s i+1 j requires 3 slots, as opposed to 2 slots in the non-cooperative scheme. To maintain the energy used by the two schemes on these two block at the same level, we need to adjust the transmission power. If we treat the channel taps as zero-mean Gaussian random variables, the magnitudes |H ij (k)| 2 will be i.i.d. Rayleigh distributed with E{|H 21 (k)| 2 } = 1. In addition, we assume the inter-user channels are very good. Then σ 2 = E{σ 2 } ≈ 2 3+(α 2 +β 2 )σ 2 .

D. Time division duplexing
The cooperation scheme described above is strongly dependent on the users being able to both receive and transmit simultaneously. However, in most situations this might be difficult. Nevertheless it is possible to effectively achieve full duplex operation by time division duplexing. In the original scheme both users transmit during the entire duration of time slot i (N symbols plus the cyclic prefix). However, we can allocate time for each user proportional to the length of the data vectors s i 1 and s i 2 or |I 1 | and |I 2 |. During time slot i, user 1 will first transmit |I 1 | data symbols plus the cyclic prefix. Next, user 2 will transmit his own |I 2 | data symbols plus the cyclic prefix. During each transmission, all the other users will be in the receiving mode and there will not be any hardware limitations. Instead of having a common OFDM block length of N , each user has its own OFDM block length or DFT length which is in general different from other users. Hence (2) will be reduced to (22) where H ij (k) are the |I j | point DFT coefficients of channel h ij (n), because user j used an OFDM block length of |I j |. Moreover, Apart from this change in the effective channel, there will be no difference in the proposed scheme and hence the analysis and the conclusions will hold even for this case.

III. CHANNEL CAPACITY
In this section, we assume the noise between the users is negligible compared to the noise between the users and the receiver, i.e., E{|n 12 | 2 } E{|n 01 | 2 } and E{|n 21 | 2 } E{|n 02 | 2 }. Since we assume equal transmission power for both users, the SNR between the users is much higher compared to the SNR between the receiver and the users.
Conditioned on the channel, the channel capacities on the carriers of I 1 and I 2 and the combined channel capacity, are respectively [9]: where R n1 , R n2 and R n are respectively the correlation matrices of n 1 , n 2 and n, expressed as and Since we use unitary precoding matrices, the inter-block precoding will have no effect on channel capacity. The user cooperation is advantageous because, if either C 1 or C 2 is very small compared to the other, we can increase the capacity to (C 1 + C 2 )/2 for the poor user. Typically, the lower capacity acts as a bottleneck in improving system performance but we can eliminate this bottleneck via cooperation.
The non-cooperative channel capacities are [9]: In the cooperative scheme, C co = C 1 + C 2 is the capacity available to transmit 4 blocks using 3 time slots. On the other hand, in the non-cooperative scheme, C non−co =C 1 +C 2 is the capacity available to transmit 4 blocks using 2 time slots.

IV. DIVERSITY
It is shown in [5] that for a single user OFDM system, the maximum diversity gain achievable with one transmit antenna is equal to the number of independent fading paths of the channel. In this section, we follow a similar procedure and study the diversity gain achieved by (18). Unlike the multiplexing gain which is related to the data rate, diversity is related to the bit error rate performance [10]. Diversity is usually increased by adding more transmitters and receivers. We show that (18) achieves the full spatial diversity available, i.e. 2L without addition of transmitters.
Following [11], the probability ofd being detected when d is transmitted is where d 2 (y,ŷ) = y −ŷ 2 , y = R and F 1 and F 2 are submatrices of N -point DFT matrix and dependent on I 1 and I 2 .
Because R n is generally invertible, it is reasonable to assume G to have full rank, conditioned on the inter-user channels H 12 and H 21 . Then the pairwise error probability is [11] P (d →d|H 12 We see that for high SNR the decay of the error probability has order 2L. In order to achieve this diversity it is necessary to do inter-block precoding. Intuitively, using inter-block precoding the data within a block and between blocks can share all the available channels equally, and thus the receiver can obtain the maximum number of copies of those data. Mathematically, if the inter-user channels, i.e., H 12 and H 12 encounter deep fading on some subcarriers and e contains zeros on the same subcarriers, without inter-block precoding G will become ill-conditioned and lose diversity on those subcarriers. Furthermore, we need α, β = 0 to achieve the full diversity. If we choose α = 0, β = 0 as an example, we loose the diversity for e 1 = 0. However, if the channel of one user is very bad, this user should terminate cooperation, i.e., α or β equals zero. Therefore, although the capacity of the cooperative scheme is larger than the non-cooperative scheme when the SNR difference between two users is significant, the diversity gain may be partially lost. Unlike pure transmit diversity, where we always have a good (wired) channel between the transmitters, cooperation can exhibit the same performance only when the inter-user channel is good. In OFDM, where we have multiple carriers, some carriers will enjoy the full diversity gain by cooperation while some carriers will not.
In general, ML decoding has prohibitively high complexity especially when the number of subcarriers is large. Thus, we implement ML by subcarrier grouping [5]. In noncooperative OFDM system, only symbols within a block for each user is grouped for detection. However, the proposed cooperation scheme combines symbols on different subcarriers within a block and between blocks of two users to decode at the receiver. The number of grouping subcarriers K for each block of each user must be not less than L in order to exploit full diversity. For example, for L = 2 the precoding matrix can be expressed as where U n is a n × n unitary Vandermonde matrix defined as in [5]. However, two users can exchange their subcarriers to transmit data in the next slot, i.e., in slot i+1, user 1 transmits over subcarriers in set I 2 , and receives over subcarriers in set I 1 ; user 2 transmits over subcarriers in set I 1 , and receives over subcarriers in set I 2 . By this way the minimum K is reduced to L/2 to achieve the diversity on the order of 2L and the precoding matrix becomes W = U 2 I N/2 .

V. SIMULATION RESULTS
In this section, we compare the performance of cooperative and non-cooperative systems. We consider the OFDM system with N = 16 subcarriers and 4QAM signals. We use Zheng and Xiao's model in [12] to generate channels consisting of two equal power taps with the normalized Doppler shift equal to 0.001. The channel is virtually static in order to eliminate temporal diversity due to by channel variation and thus highlight multipath and cooperation diversity. We allocate equal number of subcarriers to each user keep P 1 = I and P 2 = I. The SNR of inter-user channel is fixed at 30 dB. For the inter-block precoding, we use 2 × 2 unitary matrices and group carriers into blocks of 2. Two users exchange their subcarriers in slot i+1. This method can greatly reduce the complexity of the maximum likelihood decoder and exploit the full diversity.
1) Increase in capacity by cooperation: In this section, we compare the capacity of the cooperative OFDMA system and non-cooperative OFDMA system. scheme. We consider four cases where the SNR of user 2 (SNR02) is below the SNR of user 1 (SNR01) by 30, 20, 10 and 0 dB. α and β in these four cases are listed Tab.1. Fig.1 shows significant improvement in capacity for the weak user (user 2) due to cooperation. Without cooperation the capacity of user two isC 2 given by (30), while with cooperation it is (C 1 + C 2 )/2.
2) Increase in BER performances by cooperation: Fig. 2 shows the BER performance of the cooperation scheme with inter-block precoding and the non-cooperation scheme with the same inter-block precoding. In order to achieve full diversity, two users exchange subcarriers to transmit data in slot i+1 for both cooperative and noncooperative scheme. We see that there is significant improvement in performance due to cooperation diversity. In this paper we propose an OFDMA system that uses cooperation and inter-block precoding to improve performance. In terms of diversity, we have shown that cooperation can double the diversity available given a fixed inter-user channel. In terms of channel capacity, there is an increase of effective channel capacity for the weak user when the SNR of the users towards the receiver are significantly different from each other.