- Research Article
- Open Access
A Precoded OFDMA System with User Cooperation
© Yao Yu et al. 2010
- Received: 1 November 2009
- Accepted: 30 March 2010
- Published: 10 May 2010
A new cooperative scheme for a two-user orthogonal frequency division multiple access (OFDMA) uplink communication scenario is proposed. Each user is equipped with one transmit/receive antenna. Before transmission, inter-block linear precoding is introduced to pairs of blocks. The cooperative transmission is implemented in cycles of three time slots. During each slot, a user transmits either his data, or a weighted mixture of his data and the data that he received in previous slots of the same cycle. The weights are obtained in an optimum fashion, so that a user that faces deep fading on certain subcarriers can benefit from the other user's channel, without taxing significantly the resources of that user. It is shown that the proposed scheme achieves the maximum available diversity for both users (full cooperation), or for the weak user (half cooperation) without increasing the number of antennas needed as compared to an energy-equivalent noncooperative OFDMA system that also uses inter-block precoding. Further, the proposed use of inter-block precoding allows one to exploit the cooperation induced diversity in 1.5 slots on the average; 2 slots would be needed if intra-block precoding was used instead.
- Data Block
- Relay Selection
- Orthogonal Frequency Division Multiple Access
- OFDM System
- Cooperative Transmission
Multiuser Cooperation is a promising technology for improving the performance of wireless communication systems, as it has the potential to increase the data rate [1, 2], and achieve diversity order equal to the number of cooperating users . Three types of cooperation have been used in the past, decode-and-forward (DF) [1, 4], amplify-and-forward (AF) , and coded cooperation . In , a two-user cooperative system was considered and in that context it was shown that the AF approach performs better than the DF, with the performance gap closing as the SNR increases. Also in , it was shown that coded cooperation based on channel coding can in general outperform both AF and DF schemes at all SNR levels, while it is comparable to the noncooperative system at low SNR.
OFDM systems have gained popularity due to their ability to handle frequency selective fading. Various forms of cooperation in the context of OFDM systems have been considered. In , a hybrid forwarding scheme was proposed for cooperative relaying in OFDM-based networks that adaptively decides between AF, DF, or no relaying at all, based on the instantaneous SNR on each subcarrier. An OFDM cooperative scheme for multihop networks was proposed in , where in order to achieve full spatial diversity, relay selection is performed on a per-subcarrier basis instead of the entire block. Each subcarrier can determine the best relay independently at each hop, so that different subcarriers experience different paths. In  (Chapter 17), a general two-phase cooperative protocol for OFDM networks was studied, where in phase 1 each user transmits its own data and in phase 2 the relay decodes the source symbols that are not decoded successfully by the central node, according to feedback information sent by the central node. In order to resolve multiple users at the central code, the users can send their information in different time slots or utilize different sets of subcarriers in phase 1. It was shown in  that the performance of the cooperative protocol depends on the number of relays and relay selection. In , a multiuser OFDM network was considered where some users serve as AF relays by offering some of their subcarriers to other users. Optimal schemes of power control, subcarrier allocation, and relay selection were considered in the same paper. A DF cooperation strategy and resource-allocation algorithm for two-user OFDMA systems was proposed in  and was shown to achieve the capacity region upper bound of two-user OFDMA systems.
It is well known that OFDM systems loose multipath diversity as each symbol is transmitted on one subcarrier only. Several ways have been proposed in the literature for introducing path diversity in OFDM systems. Suppose that the multipath channel is finite impulse response (FIR) with taps. Maximum diversity gain, , was achieved in [11, 12] via a linear receiver using redundant precoding, or oversampling at the receiver. In  it was shown that a single user OFDM system with nonredundant block precoding can achieve diversity gain up to . The performance gain is exploitable using a Maximum Likelihood (ML) decoder. Reduced complexity decoding at the receiver is possible via subcarrier grouping , which may result in smaller than diversity gains. Other nonredundant precoding techniques were also considered in [14–16]. A multirelay cooperative OFDM system with nonredundant precoding and AF relaying was investigated in . Based on the expression of pairwise error probability (PEP), it was demonstrated that the maximum diversity order is the sum of the source-to-destination channel length and the length of the shortest channel among the relay links.
In this paper we propose a cooperative approach for a two-user OFDMA system that combines linear interblock precoding and user cooperation. The transmission occurs in cycles of three time slots each; two new precoded data blocks for each user are transmitted in each cycle. In the first slot, both users transmit their own data. In the two subsequent slots, each user transmits a weighted combination of the user's own precoded data and also data from the other user that were received in the previous slot. The weights are obtained as the solution of a constrained optimization problem that allows the user that faces a bad channel on certain subcarriers to benefit from the user that has a better channel, without taxing significantly the resources of that user. Two methods are proposed to implement this scheme: the full cooperation and the half cooperation. In the full-cooperation scheme, both users are involved in the cooperation. The base station (BS) recovers the transmitted symbols after it has collected data from both users in the three slots. In the half-cooperation scheme, only the strong user transmits cooperative information. We show that the proposed cooperative schemes combined with interblock precoding can achieve the maximum available diversity, that is, twice the length of the multipath channel. To achieve the same diversity order, a noncooperative OFDMA system that uses the same transmission energy per block pair and the same interblock precoding scheme would require at least two transmit antennas. Further, the proposed use of interblock precoding allows one to exploit the diversity induced by cooperation in 1.5 slots on the average; 2 slots would be needed if intrablock precoding was used instead.
1.1. Relation of Contribution to the Literature
For most existing cooperative OFDM techniques [5, 7–10, 17], the users serving as relays transmit only the data of other users during the cooperation phase. The main difference between the proposed approach and these techniques lies in the fact that each cooperating user transmits a linear combination of the user's own data and also data from the other user. Superposing user's own data and data from the other user can double the maximum diversity gain of each user.
In this paper, we propose to use interblock precoding for our proposed cooperation scheme. Inter-block precoding was previously applied to channel estimation for OFDM systems in  to exploit time diversity introduced by time varying channels. However, here, even if the channel is completely static, interblock precoding allows one to exploit the spatial diversity that is introduced by cooperation. In , intrablock precoding  was employed to achieve multipath diversity for multirelay cooperative OFDM system. The proposed use of interblock precoding allows one to exploit the diversity induced by cooperation in slots on the average; slots would be needed if intrablock precoding was used instead.
1.2. Paper Organization
The paper is organized as follows. In Section 2 we describe the signal model of a multiuser OFDM system. In Section 3, we propose a full-cooperation scheme and a half-cooperation scheme for a two-user OFDMA system and provide diversity analysis. Further, we describe a modified ML decoder based on subcarrier grouping. We provide simulation results of two cooperative schemes in Section 4, and finally make some concluding remarks in Section 5.
The small and capital letters in bold denote vectors and matrices. We denote the identity matrix as and all-zero matrix as . The statistical expectation of a random variable is denoted by . The superscripts and denote the conjugation and Hermitian respectively. We use to denote element-wise multiplication.
Let us consider a two-user OFDMA system where users communicate with a BS. The users are assigned disjoint carriers. User 1 transmits over subcarriers in set and receives over subcarriers in set , where and . User 2 transmits over subcarriers in set and receives over subcarriers in set . denotes the cardinality of set . We assume that . Let denote the th OFDM block of user with the length , that is transmitted over the subcarriers in , and denote the corresponding signal received by user in the th time slot over the carriers in set .
For simplicity we assume that for the noise variance it holds: .
It is well known that a good interuser channel is a pre-requisite for cooperation. In a multiuser system, the partners are selected to have a good channel between them. Therefore, throughout this paper we assume that the interuser channels are sufficiently good.
We will next discuss a scenario where both users transmit and receive simultaneously using the same antenna, that is, in full duplex mode. Since there could be practical difficulties in such scenario, we will later discuss an approach where time division multiplexing is used to achieve full duplex operation. As that approach does not change the following analysis nor the conclusions drawn in this paper, for simplicity, we continue to present our methods assuming full duplex operation.
First, the users perform interblock precoding on pairs of successive data blocks before they enter the OFDM system. As it will be shown in a subsequent subsection, the purpose of the precoding is to exploit the multipath diversity and spatial diversity that is introduced by the cooperative retransmissions.
Second, each user transmits two precoded data blocks in a cycle of 3 slots. Two cooperative transmission schemes are considered for a three-slot cycle, namely, the full-cooperation scheme and the half-cooperation scheme.
3.1. The Full-Cooperation Scheme
In this scheme, the two users superimpose their own data to the data received from the other user. Two blocks from each user, that is, and are transmitted and recovered in three time slots as follows.
Both users transmit their own data, and , respectively. These are received as + and + , respectively, by the other user.
In the th slot, the cycle is repeated with two new data blocks. Table 2 shows the transmit signals of each user during three slots.
By observing (16) and (18), and keeping in mind that and are functions of both and , , one can see that cooperation has effectively created two transmission paths for the information blocks. This effect is analogous to employing two transmitters. We should note that interblock precoding was used in  to exploit time diversity introduced by time varying channels. Here, even if the channel is completely static, interblock precoding allows us to exploit spatial diversity introduced by cooperation. The proposed scheme with interblock precoding requires on the average 1.5 slots for each data block to achieve the double diversity induced by cooperation. Without interblock precoding, two slots would be required.
Assuming knowledge of , recovery of based on is discussed in Section 3.4.
3.1.1. Transmission Energy Adjustment
To ensure that the energy spent is the same in cooperative and noncooperative cases it should hold: .
When are sufficiently small (23) can be approximated as .
3.1.2. Diversity Analysis
It is shown in  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. Diversity is related to the bit error rate performance  and is usually increased by adding more transmitters and receivers. In this section, we follow a similar procedure as in  to study the diversity gain achieved by (20) and show that (20) achieves the full spatial diversity available, that is, without adding more transmitters.
and and are submatrices of the -point DFT matrix corresponding to and .
where denotes the th eigenvalue of a matrix in the decreasing order.
and contains the first entries of the column in the -point DFT matrix corresponding to the th subcarrier in . Since the rank of is two, the maximum rank of is two. Thus, without interblock precoding, the maximum diversity gain that the cooperation scheme could achieve would be two.
To achieve the full diversity for both users we need . If we choose as an example, user 1 cannot achieve the maximum diversity . However, if the channel of one user is very bad, this user should terminate cooperation to maintain its own signal power at a certain level, that is, set or to zero. Unlike pure transmit diversity, where we always have a good (wired) channel between the transmitters, cooperation can exhibit the same performance only when the interuser 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.
3.2. The Half-Cooperation Scheme
In the full cooperation scheme, both users are involved in the cooperation. In order to keep the total energy consumed by the full-cooperation scheme equal to that of the no-cooperation scheme, we have to reduce the power assigned to each data block. Therefore, the maximum diversity gain is doubled at the price of SNR. It is expected that, at low SNR, the full-cooperation scheme will yield worse performance in terms of BER than the no-cooperation scheme. One might wonder whether the performance at low SNR can be improved by sacrificing diversity to some extent. Next, we investigate another scheme in which only the strongest of the two users cooperates. In particular, user 1 serves as a relay for user 2, while user 2 does not help user 1. Unlike the full-cooperation scheme, users send their information separately to the BS. Again, three slots are required for two users to transmit two blocks of data as follows.
Both users transmit their own data and , respectively. User 1 receives from user 2.
Both users transmit their own data and , respectively. At the time of transmission, user 1 receives from user 2.
User 2 terminates transmission. User 1 transmits the signal that he received in the previous two slots over and .
where represents the addictive Gaussian noise on the user-to-BS channel for user 1 in the ( )th slot over .
3.2.1. Transmission Energy Adjustment
When , (40) can be approximated as . In the full-cooperation scheme, . Therefore, the half-cooperation scheme can save more transmission power for each data block. It is expected that when SNR is relatively low, the half-cooperation scheme can yield better performance than the full-cooperation scheme.
3.2.2. Diversity Analysis
Similar to the scenarios discussed in , the maximum diversity gain of user 1 in (37) is . From the analysis of Section 3.1.2, the maximum diversity gain of user 2 in (38) is . This indicates that user 1 has to sacrifice its performances for the sake of user 2. On the other hand, the full-cooperation scheme is a win-win situation for both users when the SNR is relatively high.
Maximum diversity gain for the various schemes.
Maximum diversity gain of user 1
Maximum diversity gain of user 2
Transmitted signal for the full-cooperation scheme.
Transmit Signal for the half-cooperation scheme.
3.3. Time Division Duplexing
The cooperation scheme described above is strongly dependent on the users being able to both receive and transmit simultaneously. However, in a practical situation 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 ( symbols plus the cyclic prefix). However, we can allocate half a time slot for each user to the data vectors and . During time slot , user will first transmit data symbols plus the cyclic prefix. Next, user will transmit his own data symbols plus the cyclic prefix. During each transmission, all the other users will be in receive mode. Therefore, there is no difference between this time division approach (half duplex) and the full duplex one, and the analysis and conclusions hold in this case too.
3.4. Symbol Recovery
By optimal subcarrier grouping, we only perform the precoding on a group of subcarriers.
Let be an unitary Vandermonde matrix defined as in . The precoding matrixes and in (6) can be simplified as .
3.5. Optimal Allocation of Power during Allocation
In this section, we discuss the optimization of power allocation parameter and based on the model of (42). We assume that user 1 is the strong user, that is, user 1 has less subcarriers in deep fade as compared to user 2 (weak user). During the cooperation, user 1 will assist user 2, while at the same time, will also receive some help.
where the numerators of are linear functions of and the denominators are the polynomials of .
where the last constraint means that user 2 never spends more energy than user 1 when helping user 1. The threshold depends on the applications that user 1 needs to transmit, and is here assumed given. The lower the threshold, the more help user 1 will provide. The advantage for user 1 is that if the user has subcarriers on which the SINR is less than , the situation on those subcarriers will improve.
The constraints (a)–(c) are linear so they give rise to the feasible set shown by a polyhedron in Figure 1. The irregular pentagon and a triangle are formed by constraints (a) and (b)-(c), respectively.
When takes the minimum value , and thus the feasible set is empty. As increases, the dimension of the feasible set increases. There are several different scenarios.
- (a)If ,
If or , the halfspace approaches as is increasing in , and finally intersects with . Let denote the minimum that the th inequality of the constraint (d) yields. Then this is achieved when touches a vertex of .
If , and are decreasing functions of . Therefore, the halfspace and do not intersect within .
If , or , is achieved on a vertex of if .
If does not exist when , we have to consider the scenarios in which and so . When , we can always find a feasible .
If , any elements in the set can give rise to the minimum , . However, this case happens with small probability.
Otherwise, always falls on a vertex of .
where is the maximum value of the set obtained from the inequalities. Therefore, we can determine the optimal power allocation parameters with the aid of the geometric interpretation of (48). The minimum is the maximum value of = , .
In this section, we provide simulation result to illustrate the performance of the proposed full-cooperation (FC) and half-cooperation (HC) schemes. To illustrate the advantages of cooperation in addition to precoding, we compare the two proposed approaches to a noncooperative scheme (NC) that uses the same interblock precoding strategy and is equivalent in terms of power consumption.
We consider an OFDM system with subcarriers and 4QAM signals. We use the model of  to generate channels consisting of two equal power taps with normalized Doppler shift equal to . The channel is virtually static in order to eliminate temporal diversity due to by channel variation and thus highlight diversity due to cooperation and multipath. The SNR of the interuser channel is fixed at 30 dB. For the interblock precoding, we use unitary matrices and group carriers into blocks of two. Two users exchange their subcarriers as described in Section 3.4.
In our simulations, we assign unit power to each OFDM symbol for the noncooperative scheme. The power of each OFDM block in the proposed cooperative schemes is determined by (23) and (40). This guarantees that cooperative and noncooperative schemes consume the same energy during a cycle of three slots. In the following figures and discussion the term SNR refers to the SNR for the noncooperative scheme, that is, the reciprocal of the noise power. We assume . We force user 1 and user 2 to have and deep-fading subcarriers, respectively. The variance of nondeep-fading subcarriers is set to while the variance of subcarriers in deep fade is set to . We consider three cases where , .
We first find the vertices of the feasible sets of and satisfying the constraints (a)–(c) of (48);
We determine the vertex that gives rise to the the minimum for the th constraint of (d) in (48), and record the value of ;
The optimal solution of (48) is the maximum element of the set . Based on that maximum value for the optimal and are found via (52).
In this paper, we have proposed and compared two precoded schemes with user cooperation for two-user OFDMA systems. By analyzing the pairwise error probability of the proposed system, we have shown that the full-cooperation scheme can double the diversity available to both users without requiring additional transmitters. Therefore, the full-cooperation scheme can improve the BER performance of both users when the SNR of the users towards the receiver is relatively high so that the fading dominates the performance. On the other hand, when the SNR of two users is low, the half-cooperation scheme can achieve slightly better performance than the full-cooperation scheme. Furthermore, the use of interblock precoding, as compared to intrablock precoding, reduces the number of time slots required by the cooperative OFDM system to achieve the maximum diversity induced by cooperation. The extension of the proposed scheme to the multiuser case is not trivial; it involves selecting the users to cooperate with each other, or modifying the proposed scheme to render the cooperation of more than two users feasible. Such extension will be part of future work.
This work was supported by the Office of Naval Research under Grant ONR-N-00014-07-1-0500 and the National Science Foundation under Grant CNS-0905425. Preliminary results of this work were presented at the 2004 Asilomar Conference on Signals, Systems, and Computers .
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