Distributed Amplify-and-Forward cooperation while maintaining transmission freedom

This paper proposes a novel Distributed Amplify-and-Forward (DAF) cooperative scheme, achieving higher diversity order and yet maintaining the same transmission freedom as the conventional Amplify-and-Forward (AF) scheme. In the DAF scheme, a user's transmitted symbols are partitioned into several sequences in order to be relayed by different users. In the cooperative network, each user still uses half of their transmission time for relaying others' signals. But instead of relaying one user's entire transmitted sequence, it helps different users for the relaying. Theoretical analysis of the DAF scheme is carried out in order to justify its advantages over the existing schemes. The outage behaviour and diversity-multiplexing tradeoff (DMT) analyses of the DAF scheme are presented. Through outage behaviour analysis, it is shown the DAF scheme achieves substantial diversity gains over the AF scheme. Furthermore, the DMT analysis justifies both the scheme's achievable diversity gains and its ability to maintain the same transmission freedom as the AF scheme. The theoretical analyses are then extended to a general cooperative network consisting of N(N ≥ 2) relays, showing the diversity order can be increased with respect to the number of relays but not at the expense of each user's transmission freedom.


I. INTRODUCTION
S PATIAL diversity is a crucial technique to improve com munication quality. The multiple transmit/receive antennas (MIMO) system [1] and the cooperative system [2] are intro duced due to their nature of creating spatial diversity. Among them, the cooperative system can be applied more widely since it does not impose any size constraint and extra cost to the mobile unit. In the cooperative communication network, each user equipped with a single antenna not only transmits their own information, but also relays other's information, creating a virtual mUltiple transmit antennas array. So far, there are three types of cooperative schemes: Amplify-and-Forward (AF) [3], Decode-and-Forward (DF) [3], [2] and Coded Cooperation (CC) [4], [5]. The authors have carried out a comparative investigation of the existing schemes in [6].
For a cooperative network, each user achieves diversity gains at the expense of their transmission freedom, since some of their transmission time is sacrificed relaying other's signals. In a practical communication system, this freedom loss is translated into spectral efficiency loss. To compensate, Chatzi georgiou et. al [7] proposed a high-order modulation scheme to L. Chen  978-1-86135-369-6/101$25.00 ©2010 IEEE be employed in the cooperative system. Later, the authors pro posed Trellis Coded Modulation (TCM) scheme [8], achieving better coding gains for high spectral efficiency systems [9]. Laneman et. al [10] showed that better diversity gains can be achieved if cooperation is performed in a distributed manner, meaning more relays are involved for signal retransmission. In [10], two types of distributed cooperation schemes were proposed: repetition-based cooperation and space-time-coded cooperation. For the repetition-based cooperation, extra spatial diversity is created at the cost of extra transmission freedom (time) loss. As a result, the achievable diversity gain can not be increased according to the number of users. For space time-coded cooperation, the transmission freedom loss does not apply and hence diversity gains could be further achieved accordingly. Space-time-coded cooperation is operated in DF mode requiring decoding and re-encoding at the relays. The diversity gain is achieved with substantial system complexity increase. Therefore, a distributed cooperative scheme with out sacrificing transmission freedom or system complexity, inspires the design of the proposed DAF scheme.
This paper introduces the DAF cooperative scheme, in which each user uses half of their transmission time to relay more than one other user. To achieve this, the transmitted sequence of a user is partitioned into several parts, each of which is relayed by a different user in AF mode. Therefore, diverse transmission paths are created for the relayed symbols while each user still maintains their transmission freedom, half of their transmission time. Theoretical analysis of the DAF scheme is presented in order to verify its performance advantage. Our analysis is drawn from a network with two relaying users and then extended to a larger network with N(N :::: 2) users. The outage behaviour analysis of the DAF scheme shows that substantial diversity gains could be achieved over the conventional AF scheme. The achievable diversity gains can be increased with respect to the number of relays. The DMT analysis was first introduced by Zheng and Tse in [11], analysing the balance between the performance gain and the transmission freedom loss in MIMO systems. It was then applied to cooperative systems in [2] [10] [12]. Our DMT analysis shows that the DAF scheme has the same max imal multiplexing gain as the AF scheme, but achieves further diversity gains. The diversity gain is increased according to the number of relaying users. Prior to the writing of this paper, the authors' earlier work [13] of integrating the DAF scheme with channel coding showed significant coding gains can be achieved over the AF scheme.
The paper is organised as follows: Section II will present the preliminaries of the paper. Section III will present the DAF system model. Section IV will present the outage behaviour analysis; Section V will present the DMT analysis. Both the outage behaviour analysis and DMT analysis are extended in a larger cooperative network in Section VI. Section VII concludes the paper and presents our future work.

II. PRELIMINARY
This section presents the preliminaries of the paper. It includes definitions of commonly used parameters and an introduction to the conventional AF scheme which will be used to compare with the proposed DAF scheme.

A. Parameterisations
The anlaysed cooperative network is assumed to operate in half-duplex mode, requiring orthogonal time division channel allocation for the receiving and transmitting of each user. The channel quality is measured by the Signal-to-Noise Ratio (SNR) which can be defined as: where E denotes the average transmitted symbol energy and 0' 2 denotes the variance of noise at the receiver. For sim plicity of the analysis, it is assumed that the network has symmetric channels meaning all of them have similar SNR values. The channel between transmitting user a and receiving user b is assumed to be Quasi-static Rayleigh fading with fading coefficient D:ab. All channels within the cooperative network are assumed to be statistically independent. D:ab is a Gaussian random variable with zero mean and unit variance.
The exponential order of 1/ID:abI 2 is defined as: (2) where the base of the logarithm is 2.
If the cooperative system operates with a transmission rate of R(p) bitslslHz, which is a function of the SNR, the multiplexing gain of the system can be defined as: where r is a normalised value representing the ratio of effective transmission. At a SNR of p, if the system can achieve a maximum-likelihood (ML) error probability of Pe(p), its diversity gain is defined as: The derived result of the relationship between d and r is called the Diversity-Multiplexing Tradeoff (DMT), denoted as d(r).
We could further claim protocol A is superior to protocol B if for any multiplexing gain r, dA(r) � dB(r). � N and e N denote the set of real and complex N -tuples. � N + denotes the set of nonnegative N -tuples. 0 is used to denote the set of outage events in the cooperative system, 0 � � N and 0+ = On� N +. According to [11] and [12], the DMT (d( r) of a cooperative system with N users is upper bounded by:

B. AmplifY-and-Forward
The AF scheme consists of three users: Source (8) and its signal Destination (D), Relay (R) helps 8 for the transmission.
A classical cooperative process contains two Time Slots (TS) with equal duration. The first TS is for initial transmission when 8 transmits its information to D and R. The second TS is for relaying transmission when R amplifies its received signal from 8 and transmits it to D. D would combine the received signals using ML detection [3]. If D:SD, D:SR and D:RD denote the fading coefficients of the channels between 8 -D, 8 -R and R -D respectively, and R denotes the transmission rate of the cooperative system, the outage behaviour of the AF scheme can be modelled as [2]: f..l and /J are random variables. The DMT characteristics of the AF scheme can be described by [2]: It can be seen that diversity order of 2 can be obtained from the AF scheme.

III. SYSTEM MODEL
This section presents the system model for the DAF scheme, detailing this novel transmission protocol. In general, if a DAF cooperative network has N relaying users, the transmitted signal of 8 will be equally partitioned into N sections, each of which will be relayed by a different user. It is not difficult to realise that when N = 1, it becomes the conventional AF scheme. For simplicity, the description of the DAF system model is given with N = 2, and it could be easily extended into a larger cooperative network.
A complete cooperative process of the DAF scheme also consists of two TSs, which is shown in Fig.l. In the first TS, 8 transmits its signal to D as well as to two different relays (Rl and R2)' It is assumed that Rl and R2 are perfectly synchronised with 8, and Rl received the first half of 8's signal while R2 received the second half. Y l,k = aSDxI,k + Vl,k , k = 1, 2, ... , l/2, where l denotes the length of the signal transmitted during the two TSs and l12N. The second TS is also partitioned into two equal halves for relaying transmission. In the first half of the second TS, R I amplifies its received signal with gain f3l and re-transmits to D as: Y 2 ,k = aR1Df3l ( asR1xI,k -l / 2 + WLk -l / 2 ) + V 2 ,k, where k = l/2 + 1, l/2 + 2, ... , 3l/4, lasRl 1 2 c + 0' ;1 .
Similaly, in the second half of the second TS, R 2 re-transmit S's signal to D as: where C f3 2 � 1 1 2 2 ' (13) aSR2 C + O'W2 After the two TSs, D combines YI,k(k = 1, 2, ... , l/4) with Y 2 ,k of (10) and Y l,k (k = l/4 + 1, l/4 + 2, ... , l/2) with Y 2 ,k of (12) for further signal processing in order to retrieve the transmitted information. Fig.2  half of a user's total transmission time is used for their own transmission, while the other half is partitiioned into smaller divisions in order to help different users. It maintains the same transmission freedom as the conventional AF scheme [2].

IV. OUTAGE BEHAVIOUR
This section presents the outage behaviour analysis for the DAF scheme with two relaying users. Its extension to larger networks will be mentioned in Section VI. The conclusion is drawn by first formalising the transmission signal model, then determining the scheme's mutual information and finally modelling its outage behaviour. The following theorem models the scheme's outage behaviour. Proof To prove Theorem 1, it is necessary to formalise the system model of the DAF scheme into matrix form. Equations (9) to (13) can be alternatively expressed as: where x E C l / 2 X l denotes vector of transmitted signals, fj E C l x I denotes the vector of received signals at D, v E C l x I denotes the vector of noise samples at D, and W l and W 2 E C l / 4X l denote the vector of noise samples at R I and R 2 . Let X i and Yi denote the enties ofx and fj, the relationship between X i and Yi can be categorised into the following two sets. For  Now, it is straight forward to see that the mutual information between x and 1} can be determined by: The mutual information between X i and fJf is determined by: I(x, y) = 42: log(l + lasDI 2 p + f ( lasRt 1 2 p, laRtDI 2 p)) .
(28) By substituting equation (27) into (28), it is not difficult to derive equation (14) and the proof is complete. Fig.3 shows the Monte-Carlo simulation results comparing the outage behaviour between the AF and DAF schemes. They are obtained by using equation (6) and (14)  can achieve a 5dB diversity gain over the AF scheme. It is also important to emphasise that unlike most distributed cooperative schemes, this diversity gain is not achieved at the cost of each user's transmission freedom. Its ability to maintain the same transmission freedom as the AF scheme will be justified in the next section.

v. DIVERSITy-MULTIPLEXING TRADEOFF
This section presents the DMT analysis for the proposed scheme. Through the DMT analysis, we are able to determine the scheme's maximal multiplexing gain and diversity gain. The tradeoff between them can also be reflected from the analysis. First of all, the following theorem is proposed to describe the DMT of the scheme. Proof Referring to equations (3) and (4), both d and r describe the system's asymptotic behaviour when p -+ 00. To analyse d DA F ( 2 ) (r), we shall also analyse I(x, y) 's asymptotic behaviour since it is also a function of p. Based on equation (26), it can be derived that: 1 .
When p -+ 00, we could claim the following approximations: 1 (./ 1 2 � 1 a 2 / a 2 � 1 la 1 2 � p -O SD fJ t -' Wt V -, SD - Based on equations (32) and knowing asymptotically R r log p, we can further define: t=l From equation (34), it can be calculated that 1-2r < OSD < 1 and 2 -4r < "L,; = l (OS Rt + ORtD) < 2. According to d(r) definition given by equation (5), we can easily conclude its upper bound expression which is given by equation (29). The proof is complete. Fig.4 shows the DMT analysis result of the DAF scheme. Its performance is compared with the conventional AF scheme with one and two relays respectively. Notice that the AF scheme with two relays is identical to the distributed coop erative scheme using repetition re-transmission proposed in [10]. Observing this, we conclude the following comments: First, cooperation achieves further diversity gain compared to direct transmission, but loses mUltiplexing gain, i.e. loss of transmission freedom; Second, the DAF scheme achieves the same maximal multiplexing gain as the AF scheme with one relay, but achieves higher diversity gain. It verifies the achieved diversity gains shown in Fig.3. Third, the DAF scheme achieves the same maximal diversity gain as the AF scheme with two relays, but achieves higher maximal multiplexing gain -maintaining higher transmission freedom compared to the existing distributed cooperative scheme.

VI. EXTENSION TO MULTIPLE USERS NETWORK
This section extends the analysis proposed in the above two sections to a larger cooperative network with N (N � 2) relaying users. Two theorems describing its outage behaviour and diversity-multiplexing tradeoff will be presented. Since the same methodology to prove Theorem 1 and Theorem 2 is used, the proof given in this section will only state the important generalised equations. In order to substantiate the theorems, the corresponding simulation and analysis results will also be shown.
In a DAF cooperative network with N relaying users, S will partition its sequence into N equal parts of length l/2N, each of which will be relayed by a different user. Its system model can be easily extended from Section III. The following theorem describes its outage behaviour. Proof Similar to the proof of Theorem 1, by formalising its system model into matrix form, we can have the following N signal tuples (Xi, fit), (Xi, fir),···, (Xi, fir) with length l/2N. The mutual information between the transmitted signal x and received signal y can be determined by: Applying the derived results of equations (24) and (25), we have: I(x, y) = _ l _ t 10g(1 + lasvl 2 c + laRtDI 2 1 ,B t I 2 IaSRtI 2 c ).
The Monte-Carlo simulation results of the DAF scheme with different numbers of relaying users is shown in Fig.5. It can be seen that further diversity gains can be achieved by increasing the number of relaying users.
By defining its outage event set of 0 and 0+, it is not difficult to calculate that 1 -2r < bSD < 1 and N -2Nr < 2:!1 (bS Rt + bRtD) < N . Therefore, the d D AF( N) (r) upper bound given by equation (38) can obtained and the proof is complete. Fig.6 shows the DMT analysis of the DAF scheme with different numbers of relaying users. It verifies the achievable diversity gains shown by Fig.5. More importantly, it shows the maximal multiplexing gain remains the same as the AF scheme regardless of the number of relaying users. The same transmission freedom is maintained.

VII. CONCLUSIONS AND FUTURE WORK
A novel distributed amplify-and-forward cooperation scheme was proposed. For each user in the DAF cooperative