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Erschienen in: EURASIP Journal on Wireless Communications and Networking 1/2010

Open Access 01.12.2010 | Research Article

A Precoded OFDMA System with User Cooperation

verfasst von: Yao Yu, Sarod Yatawatta, AthinaP Petropulu

Erschienen in: EURASIP Journal on Wireless Communications and Networking | Ausgabe 1/2010

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Abstract

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.

1. Introduction

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 [3]. Three types of cooperation have been used in the past, decode-and-forward (DF) [1, 4], amplify-and-forward (AF) [5], and coded cooperation [6]. In [4], 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 [4], 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 [5], 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 [7], 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 [8] (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 [8] that the performance of the cooperative protocol depends on the number of relays and relay selection. In [9], 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 [10] 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq1_HTML.gif taps. Maximum diversity gain, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq2_HTML.gif , was achieved in [11, 12] via a linear receiver using redundant precoding, or oversampling at the receiver. In [13] it was shown that a single user OFDM system with nonredundant block precoding can achieve diversity gain up to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq3_HTML.gif . The performance gain is exploitable using a Maximum Likelihood (ML) decoder. Reduced complexity decoding at the receiver is possible via subcarrier grouping [13], which may result in smaller than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq4_HTML.gif diversity gains. Other nonredundant precoding techniques were also considered in [1416]. A multirelay cooperative OFDM system with nonredundant precoding and AF relaying was investigated in [17]. 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, 710, 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 [16] 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 [17], intrablock precoding [13] 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq5_HTML.gif slots on the average; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq6_HTML.gif 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.

1.3. Notation

The small and capital letters in bold denote vectors and matrices. We denote the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq7_HTML.gif identity matrix as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq8_HTML.gif and all-zero matrix as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq9_HTML.gif . The statistical expectation of a random variable is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq10_HTML.gif . The superscripts https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq11_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq12_HTML.gif denote the conjugation and Hermitian respectively. We use https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq13_HTML.gif to denote element-wise multiplication.

2. Signal Model and Assumptions

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq14_HTML.gif and receives over subcarriers in set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq15_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq17_HTML.gif . User 2 transmits over subcarriers in set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq18_HTML.gif and receives over subcarriers in set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq19_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq20_HTML.gif denotes the cardinality of set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq21_HTML.gif . We assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq22_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq23_HTML.gif denote the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq24_HTML.gif th OFDM block of user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq25_HTML.gif with the length https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq26_HTML.gif , that is transmitted over the subcarriers in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq27_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq28_HTML.gif denote the corresponding signal received by user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq29_HTML.gif in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq30_HTML.gif th time slot over the carriers in set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq31_HTML.gif .
The time-domain multipath channel between user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq32_HTML.gif and user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq33_HTML.gif is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq35_HTML.gif ; each channel tap is assumed to be zero-mean i.i.d. Gaussian with unit variance. The taps https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq36_HTML.gif are assumed to be uncorrelated for different https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq37_HTML.gif pairs, and also for different discrete times https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq38_HTML.gif . We assume that the channel is slowly varying, that is, the channel remains constant over several OFDM blocks. The BS has perfect knowledge of the interuser and user-to-BS channel. Let the frequency-domain channel be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq39_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq40_HTML.gif . Then the received signal by user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq41_HTML.gif from user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq42_HTML.gif in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq43_HTML.gif th slot is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ1_HTML.gif
(1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ2_HTML.gif
(2)
with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq44_HTML.gif denoting the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq45_HTML.gif th element of the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq46_HTML.gif according to some predefined ordering; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq47_HTML.gif denotes noise at user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq48_HTML.gif during the transmission of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq49_HTML.gif th block from user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq50_HTML.gif with the variance of its entries being https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq51_HTML.gif . We assume that the noise is circularly complex Gaussian with zero mean, temporally and spatially white, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ3_HTML.gif
(3)
For simplicity we assume that for the noise variance it holds: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq52_HTML.gif .
The signal-to-noise ratio (SNR) throughout this paper is defined as the ratio of the power of transmitted signal to the power of additive noise as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ4_HTML.gif
(4)
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.

3. The Precoded Cooperation Scheme

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.
Let us express https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq53_HTML.gif be the unitary precoding matrix for user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq54_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ5_HTML.gif
(5)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq55_HTML.gif contains the first half of the rows of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq56_HTML.gif while https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq57_HTML.gif contains the other half. On denoting the uncoded blocks of user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq58_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq59_HTML.gif , the precoded blocks are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ6_HTML.gif
(6)
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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq61_HTML.gif are transmitted and recovered in three time slots as follows.
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq62_HTML.gif
Both users transmit their own data, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq64_HTML.gif , respectively. These are received as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq65_HTML.gif + https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq67_HTML.gif + https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq68_HTML.gif , respectively, by the other user.
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq69_HTML.gif
The users transmit a weighted combination of their own data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq70_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq71_HTML.gif ) and the signal that they received during the previous slot after it has been scaled by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq72_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq73_HTML.gif ) and mapped from the incoming carriers to outgoing carriers. The amount of power allocated for cooperation by users https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq75_HTML.gif is proportional to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq77_HTML.gif , respectively. The selection of those weights is formulated as an optimization problem in Section 3.5. The transmitted signals of both users, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq79_HTML.gif are given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ7_HTML.gif
(7)
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq80_HTML.gif
Both users again transmit https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq82_HTML.gif as their own data, plus the signal that they received during slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq83_HTML.gif . Note that there is a component of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq84_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq85_HTML.gif ) in the received signals by users https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq86_HTML.gif (2). In order to eliminate that component, the precoding for that block is modified as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ8_HTML.gif
(8)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq87_HTML.gif can be obtained at each user by correlating the signal that was received in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq88_HTML.gif th slot with the signal that was transmitted in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq89_HTML.gif th time slot. Therefore, the transmitted signals https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq91_HTML.gif can be expressed as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ9_HTML.gif
(9)
In the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq92_HTML.gif th slot, the cycle is repeated with two new data blocks. Table 2 shows the transmit signals of each user during three slots.
The signals received at the BS during slots https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq93_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq94_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq95_HTML.gif over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq96_HTML.gif are:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ10_HTML.gif
(10)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ11_HTML.gif
(11)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ12_HTML.gif
(12)
Similarly, the received signals over carriers in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq97_HTML.gif during slots https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq98_HTML.gif are:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ13_HTML.gif
(13)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ14_HTML.gif
(14)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ15_HTML.gif
(15)
Based on (10), (12), and (14), let us form the matrix equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ16_HTML.gif
(16)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ17_HTML.gif
(17)
Similarly, based on (13), (15), and (11), let us form the matrix equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ18_HTML.gif
(18)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ19_HTML.gif
(19)
By observing (16) and (18), and keeping in mind that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq99_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq100_HTML.gif are functions of both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq103_HTML.gif , 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 [16] 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.
Combining (16) and (18), the following MIMO problem can be formulated at the receiver:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ20_HTML.gif
(20)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ21_HTML.gif
(21)
Assuming knowledge of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq104_HTML.gif , recovery of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq105_HTML.gif based on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq106_HTML.gif is discussed in Section 3.4.

3.1.1. Transmission Energy Adjustment

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq107_HTML.gif be the power of one data block transmitted by user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq108_HTML.gif without and with cooperation, respectively. For simplicity let us take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq110_HTML.gif . In the cooperative OFDM scheme, the transmission of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq111_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq112_HTML.gif requires three slots, as opposed to the two slots required in the no-cooperation scheme. To maintain the energy used by the two schemes for the transmission of a block pair at the same level, we need to adjust the transmission power. In the noncooperative case, the transmission of 2 blocks requires energy equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq113_HTML.gif . Under cooperation, the energy spent by user 1 and user 2 to transmit 3 blocks is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ22_HTML.gif
(22)
To ensure that the energy spent is the same in cooperative and noncooperative cases it should hold: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq114_HTML.gif .
Since the channel taps are assumed to be zero-mean unit-variance Gaussian random variables, the magnitudes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq115_HTML.gif are i.i.d. Rayleigh distributed, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq116_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq117_HTML.gif be the average https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq118_HTML.gif over the interuser channel coefficients. It holds
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ23_HTML.gif
(23)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq119_HTML.gif are sufficiently small (23) can be approximated as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq120_HTML.gif .

3.1.2. Diversity Analysis

It is shown in [13] 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 [18] and is usually increased by adding more transmitters and receivers. In this section, we follow a similar procedure as in [13] to study the diversity gain achieved by (20) and show that (20) achieves the full spatial diversity available, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq121_HTML.gif without adding more transmitters.
The probability of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq122_HTML.gif being detected when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq123_HTML.gif is transmitted is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ24_HTML.gif
(24)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq124_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq125_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq126_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq127_HTML.gif is the noise variance. Then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ25_HTML.gif
(25)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq128_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ26_HTML.gif
(26)
Let us define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ27_HTML.gif
(27)
and partition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq129_HTML.gif into two https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq130_HTML.gif vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq131_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq132_HTML.gif . Then, (25) can be further rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ28_HTML.gif
(28)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ29_HTML.gif
(29)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq134_HTML.gif are submatrices of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq135_HTML.gif -point DFT matrix corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq136_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq137_HTML.gif .
Because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq138_HTML.gif is generally invertible, it is reasonable to assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq139_HTML.gif has full rank. Conditioned on the interuser channels https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq140_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq141_HTML.gif , the pairwise error probability is [19]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ30_HTML.gif
(30)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq142_HTML.gif denotes the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq143_HTML.gif th eigenvalue of a matrix in the decreasing order.
It can be seen that for high SNR the decay of the error probability is of the order of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq144_HTML.gif . We should emphasize that interblock precoding is essential in achieving this diversity. Intuitively, using interblock 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. This can also been seen analytically as follows. Without interblock precoding, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq145_HTML.gif , the signal model in (20) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ31_HTML.gif
(31)
Since both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq146_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq147_HTML.gif can be partitioned into six https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq148_HTML.gif diagonal matrices as seen in (17), (19), the ML decoding algorithm is performed on a pair of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq149_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq150_HTML.gif per subcarrier:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ32_HTML.gif
(32)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ33_HTML.gif
(33)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq151_HTML.gif denotes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq152_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq153_HTML.gif denotes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq154_HTML.gif . Repeating the diversity analysis as above, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ34_HTML.gif
(34)
Similar to (28), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ35_HTML.gif
(35)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq155_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ36_HTML.gif
(36)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq156_HTML.gif contains the first https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq157_HTML.gif entries of the column in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq158_HTML.gif -point DFT matrix corresponding to the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq159_HTML.gif th subcarrier in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq160_HTML.gif . Since the rank of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq161_HTML.gif is two, the maximum rank of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq162_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq163_HTML.gif . If we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq164_HTML.gif as an example, user 1 cannot achieve the maximum diversity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq165_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq166_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq167_HTML.gif 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.
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq168_HTML.gif
Both users transmit their own data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq169_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq170_HTML.gif , respectively. User 1 receives https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq171_HTML.gif from user 2.
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq172_HTML.gif
Both users transmit their own data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq173_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq174_HTML.gif , respectively. At the time of transmission, user 1 receives https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq175_HTML.gif from user 2.
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq176_HTML.gif
User 2 terminates transmission. User 1 transmits the signal that he received in the previous two slots over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq178_HTML.gif .
In the ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq179_HTML.gif )th slot, the cycle is repeated with two new data blocks. Table 3 shows the transmit signals of each user during the three slots. The received signals at the BS containing user 1's data and user 2's data are, respectively, equal
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ37_HTML.gif
(37)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ38_HTML.gif
(38)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq180_HTML.gif represents the addictive Gaussian noise on the user-to-BS channel for user 1 in the ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq181_HTML.gif )th slot over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq182_HTML.gif .

3.2.1. Transmission Energy Adjustment

Under cooperation, the energy spent by user 1 and user 2 to transmit https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq183_HTML.gif blocks is:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ39_HTML.gif
(39)
Similar to the full-cooperation scheme, the average signal power over the interuser channel coefficients https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq184_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ40_HTML.gif
(40)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq185_HTML.gif , (40) can be approximated as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq186_HTML.gif . In the full-cooperation scheme, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq187_HTML.gif . 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 [13], the maximum diversity gain of user 1 in (37) is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq188_HTML.gif . From the analysis of Section 3.1.2, the maximum diversity gain of user 2 in (38) is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq189_HTML.gif . 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.
Table 1 summarizes the maximum diversity gain of the full-cooperation scheme (FC), the full-cooperation scheme without precoding (FC-no precoding), the half-cooperation scheme (HC), the no-cooperation scheme with precoding (NC) and the no-cooperation scheme without precoding (NC-no precoding).
Table 1
Maximum diversity gain for the various schemes.
 
FC
FC-no precoding
HC
NC
NC-no precoding
Maximum diversity gain of user 1
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq190_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq191_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq192_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq193_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq194_HTML.gif
Maximum diversity gain of user 2
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq195_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq196_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq197_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq198_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq199_HTML.gif
Table 2
Transmitted signal for the full-cooperation scheme.
 
User 1
User 2
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq200_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq201_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq202_HTML.gif
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq203_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq204_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq205_HTML.gif
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq206_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq207_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq208_HTML.gif
Table 3
Transmit Signal for the half-cooperation scheme.
 
User 1
User 2
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq209_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq210_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq211_HTML.gif
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq212_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq213_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq214_HTML.gif
 
User 1
User 1
Slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq215_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq216_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq217_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq218_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq219_HTML.gif symbols plus the cyclic prefix). However, we can allocate half a time slot for each user to the data vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq220_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq221_HTML.gif . During time slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq222_HTML.gif , user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq223_HTML.gif will first transmit https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq224_HTML.gif data symbols plus the cyclic prefix. Next, user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq225_HTML.gif will transmit his own https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq226_HTML.gif 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

The maximum diversity can be best exploited using ML decoding. In general, ML decoding has prohibitively high complexity especially when the number of subcarriers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq227_HTML.gif is large. Here, following the main idea of [13], we implement ML by optimal subcarrier grouping. The set of all subcarriers is divided into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq228_HTML.gif equally spaced groups. Each group contains https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq229_HTML.gif subcarriers. In order to achieve the maximum multipath diversity, it should hold that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq230_HTML.gif (see Section 4 for diversity analysis). To reduce the complexity further, we let two users exchange their subcarriers to transmit data in the second slot, that is, in slot https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq231_HTML.gif , user 1 transmits over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq232_HTML.gif and receives over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq233_HTML.gif , while user 2 transmits over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq234_HTML.gif and receives over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq235_HTML.gif . By this way, the minimum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq236_HTML.gif can be reduced to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq237_HTML.gif to achieve diversity of the order of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq238_HTML.gif . For simplicity, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq239_HTML.gif is an integer. The sets of subcarriers for two users https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq240_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq241_HTML.gif are divided into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq242_HTML.gif groups, each group containing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq243_HTML.gif equally spaced subcarriers. Let us define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ41_HTML.gif
(41)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq244_HTML.gif denotes the subcarrier pattern for the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq245_HTML.gif th group; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq246_HTML.gif represents the transmitted signal of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq247_HTML.gif th user in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq248_HTML.gif th slot over the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq249_HTML.gif th group of subcarriers; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq250_HTML.gif denotes the received signal at BS in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq251_HTML.gif th slot over the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq252_HTML.gif th group of subcarriers in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq253_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq254_HTML.gif denotes the noise at user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq255_HTML.gif over the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq256_HTML.gif th group of subcarriers during the transmission of the data block from user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq257_HTML.gif in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq258_HTML.gif th slot (The https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq259_HTML.gif th user represents the BS); https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq260_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq261_HTML.gif are the fading coefficients of the channel from the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq262_HTML.gif th user to the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq263_HTML.gif th user over the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq264_HTML.gif th group of subcarriers in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq265_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq266_HTML.gif , respectively. If we take the full-cooperation scheme as an example, the model for the received signal by grouping subcarriers can be reduced to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ42_HTML.gif
(42)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ43_HTML.gif
(43)
By optimal subcarrier grouping, we only perform the precoding on a group of subcarriers.
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq267_HTML.gif be an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq268_HTML.gif unitary Vandermonde matrix defined as in [13]. The precoding matrixes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq269_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq270_HTML.gif in (6) can be simplified as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq271_HTML.gif .

3.5. Optimal Allocation of Power during Allocation

In this section, we discuss the optimization of power allocation parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq272_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq273_HTML.gif 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.
Let us define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq274_HTML.gif to be the vector of the parameters to be determined. The objective function is defined in terms of the SINR of user 1 and user 2. In order to derive the SINR for the users, we need to separate the data of each user in the model of (42) as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ44_HTML.gif
(44)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq275_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq276_HTML.gif contain the first half and the second half of the columns of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq277_HTML.gif , respectively. On letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq278_HTML.gif denote the SINR of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq279_HTML.gif th user at the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq280_HTML.gif th subcarrier, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ45_HTML.gif
(45)
where the numerators of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq281_HTML.gif are linear functions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq282_HTML.gif and the denominators are the polynomials of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq283_HTML.gif .
The optimization problem is formulated as follows. We wish to maximize the SINR on the worst subcarriers of the weak user, subject to the constraint that the SINR on all subcarriers of the strong user is above some threshold https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq284_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ46_HTML.gif
(46)
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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq285_HTML.gif , the situation on those subcarriers will improve.
A more standard form of the above problem is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ47_HTML.gif
(47)
or equivalently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ48_HTML.gif
(48)
Since the denominators of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq286_HTML.gif are polynomials of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq287_HTML.gif , finding the solution of the problem of (48) is not easy. We will proceed by making some simplifying assumptions. Let us assume that the interuser channels are quite good and that the noise at the user end is very small so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq288_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq289_HTML.gif are negligible as compared to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq290_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq291_HTML.gif . Since the coefficients of the high orders of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq292_HTML.gif are linear combinations of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq293_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq294_HTML.gif , the high orders of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq295_HTML.gif can be ignored. Therefore, the denominators of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq296_HTML.gif can be approximated as a linear function in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq297_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq298_HTML.gif be represented by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq299_HTML.gif . Finding the solution of (48) is based on the following observations.
(1)
The constraints (a)–(c) are linear so they give rise to the feasible set shown by a polyhedron https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq301_HTML.gif in Figure 1. The irregular pentagon https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq302_HTML.gif and a triangle https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq303_HTML.gif are formed by constraints (a) and (b)-(c), respectively.
 
(2)
With a fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq305_HTML.gif , the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq306_HTML.gif th inequality in the constraint (d) yields a halfspace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq307_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ49_HTML.gif
(49)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq308_HTML.gif .
 
(3)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq310_HTML.gif takes the minimum value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq311_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq312_HTML.gif and thus the feasible set is empty. As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq313_HTML.gif increases, the dimension of the feasible set increases. There are several different scenarios.
 
(a)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq314_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ50_HTML.gif
(50)
(i)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq315_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq316_HTML.gif , the halfspace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq317_HTML.gif approaches https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq318_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq319_HTML.gif is increasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq320_HTML.gif , and finally https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq321_HTML.gif intersects with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq322_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq323_HTML.gif denote the minimum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq324_HTML.gif that the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq325_HTML.gif th inequality of the constraint (d) yields. Then this https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq326_HTML.gif is achieved when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq327_HTML.gif touches a vertex of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq328_HTML.gif .
 
(ii)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq329_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq330_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq331_HTML.gif are decreasing functions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq332_HTML.gif . Therefore, the halfspace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq333_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq334_HTML.gif do not intersect within https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq335_HTML.gif .
 
(iii)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq336_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq337_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq338_HTML.gif is achieved on a vertex of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq339_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq340_HTML.gif .
 
 
(b)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq341_HTML.gif does not exist when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq342_HTML.gif , we have to consider the scenario in which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq343_HTML.gif . First, we consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq344_HTML.gif and so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq345_HTML.gif .
(i)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq346_HTML.gif , the intersection of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq347_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq348_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq349_HTML.gif itself. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq350_HTML.gif .
 
(ii)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq351_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq352_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ51_HTML.gif
(51)
 
(iii)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq353_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq354_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq355_HTML.gif then refer to (c).
 
 
(c)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq356_HTML.gif does not exist when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq357_HTML.gif , we have to consider the scenarios in which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq358_HTML.gif and so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq359_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq360_HTML.gif , we can always find a feasible https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq361_HTML.gif .
 
In conclusion, we know
(i)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq362_HTML.gif , any elements in the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq363_HTML.gif can give rise to the minimum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq364_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq365_HTML.gif . However, this case happens with small probability.
 
(ii)
Otherwise, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq366_HTML.gif always falls on a vertex of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq367_HTML.gif .
 
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq368_HTML.gif inequalities of constraint (d) need to be satisfied simultaneously, the optimal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq369_HTML.gif is a vertex of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq370_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_Equ52_HTML.gif
(52)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq371_HTML.gif is the maximum value of the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq372_HTML.gif obtained from the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq373_HTML.gif inequalities. Therefore, we can determine the optimal power allocation parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq374_HTML.gif with the aid of the geometric interpretation of (48). The minimum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq375_HTML.gif is the maximum value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq376_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq377_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq378_HTML.gif .

4. Simulation Results

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq379_HTML.gif subcarriers and 4QAM signals. We use the model of [20] to generate channels consisting of two equal power taps with normalized Doppler shift equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq380_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq381_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq382_HTML.gif . We force user 1 and user 2 to have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq383_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq384_HTML.gif deep-fading subcarriers, respectively. The variance of nondeep-fading subcarriers is set to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq385_HTML.gif while the variance of subcarriers in deep fade is set to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq386_HTML.gif . We consider three cases where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq387_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq388_HTML.gif .
Figures 2, 3, and 4 compare the BER performances of each user for FC and NC in three cases described above for SNR01 = SNR02. Since our proposed approach of optimizing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq389_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq390_HTML.gif holds only when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq391_HTML.gif , we consider the scenarios of relatively small SNR01 and SNR02. Let the threshold in (48) for the SINR of user 2 over all the subcarriers be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq392_HTML.gif corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq393_HTML.gif , respectively. In each channel realization, we update the optimal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq394_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq395_HTML.gif with knowledge of channel coefficients and noise variance. The procedure to determine the optimal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq396_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq397_HTML.gif based on the analysis in Section 3.5 is sketched as follows:
(1)
We first find the vertices of the feasible sets of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq399_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq400_HTML.gif satisfying the constraints (a)–(c) of (48);
 
(2)
We determine the vertex that gives rise to the the minimum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq402_HTML.gif for the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq403_HTML.gif th constraint of (d) in (48), and record the value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq404_HTML.gif ;
 
(3)
The optimal solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq406_HTML.gif of (48) is the maximum element of the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq407_HTML.gif . Based on that maximum value for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq408_HTML.gif the optimal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq409_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq410_HTML.gif are found via (52).
 
Figures 24 show that FC can significantly improve the performances of both users at higher SNR.
Figures 5, 6, and 7 show the BER performance of each user for the FC, HC and NC for SNR01 = SNR02 = 5 dB~35 dB. Both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq417_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq418_HTML.gif are fixed to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq419_HTML.gif for HC, while for FC it is taken https://static-content.springer.com/image/art%3A10.1155%2F2010%2F843745/MediaObjects/13638_2009_Article_2042_IEq420_HTML.gif . One can see that HC can significantly improve the performances of user 2 with a negligible penalty on the other user's performance as compared to NC. At low SNR, HC performs slightly better than FC with regards to user 2's performances. When the two users encounter relatively high SNR, the FC scheme can improve the performance of both users. When the antennas are not able to switch from one scheme to another, the FC scheme is always a wise choice regardless of the environment.

5. Conclusion

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.

Acknowledgments

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 [21].
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Metadaten
Titel
A Precoded OFDMA System with User Cooperation
verfasst von
Yao Yu
Sarod Yatawatta
AthinaP Petropulu
Publikationsdatum
01.12.2010
Verlag
Springer International Publishing
DOI
https://doi.org/10.1155/2010/843745

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