Alternative relaying for cooperative multipleaccess channels in wireless vehicular networks
 Peng Xu^{1}Email author,
 Zhiguo Ding^{2},
 Xuchu Dai^{1},
 Ioannis Krikidis^{3} and
 Athanasios V Vasilakos^{4}
https://doi.org/10.1186/168714992014165
© Xu et al.; licensee Springer. 2014
Received: 28 December 2013
Accepted: 1 October 2014
Published: 11 October 2014
Abstract
In this paper, a novel spectrally efficient halfduplex cooperative transmission protocol is proposed for cooperative multipleaccess channels in wireless vehicular networks, where multiple sources (vehicles) deliver messages to a common destination (roadside base station or roadside infrastructure) with the help of multiple decodeandforward relays (roadside stations). The basic idea is to apply superposition coding at each transmitter in order to achieve the full diversity gain, where the linear zeroforcing detection is used at each relay to combat interrelay interference. Compared to existing uplink cooperative protocols, the proposed scheme can exploit the cooperation involving both the relays and the sources. An achievable diversitymultiplexing tradeoff is developed for the proposed transmission protocol. Even with strong interrelay interference which has been ignored by many existing works, the proposed scheme can still approach the optimal multipleinput singleoutput upper bound. Numerical results have also been provided to demonstrate the performance of the proposed protocol.
Keywords
Cooperative multiple access Wireless vehicular network Alternative relaying Diversitymultiplexing tradeoff1 Introduction
1.1 Background
Recent researches have paid much attention to wireless vehicular ad hoc networks (VANETs) since they are important to intelligent transportation, environment monitoring, etc [1, 2]. Unlike the cellular networks, there does not exist a powerful base station in a vehicular ad hoc network, so peertopeer transmission is relied on. Correspondingly, a key problem is how to design the routing protocols to efficiently forward the source messages to the destination. One potential method is to design crosslayer protocols for mobile vehicular ad hoc networks. For example, the work in [3] has shown that cooperative transmission, a lowcost and effective alternative to multipleinput multipleoutput (MIMO) techniques to enhance the physical layer performance, can bring its physical layer benefits to the upper layer, such as the design of routing protocols. Motivated by these benefits of cooperative transmission, this paper aims to design a novel cooperative transmission scheme for the multiple access channel (MAC) in the wireless vehicular networks.
Exploring cooperation among the source nodes has been studied in [4, 5] for the MACs. The work in [4] proposed a cooperative multipleaccess (CMA) transmission protocol using amplifyandforward (AF) strategy, and its decodeandforward (DF) version has been proposed in [5] by using superposition (SP) coding at each source, where the idea of superposition coding was first proposed in [6] to increase the system capacity. Specifically, the key idea of such superimposed cooperative schemes is that each user will share parts of its power with other users and transmit a mixture of its own and others’ information. The diversitymultiplexing tradeoff (DMT) [7] achieved by these schemes can approach the optimal multipleinput singleoutput (MISO) upper bound. However, such schemes only exploit the cooperation among the sources and are therefore constrained by the number of sources.
1.2 Related work
To further improve the reception robustness, exploiting relays as an extra dimension has been recognized as a costeffective way, since the number of idle users is always larger than the active ones in a typical wireless network. However, extra time slots may be consumed to repeat source messages in relay transmission. For example, due to the halfduplex constraint, there exists a loss of spectral efficiency for the classical orthogonal transmission schemes [8, 9]. The standard nonorthogonal transmission strategies in [4, 10] can make some improvement and achieve the full multiplexing gain but still suffer a significant diversity loss for large multiplexing gains, if compared with the optimal MISO upper bound. To overcome limitation of the available diversity, a successive relaying concept which aims to physically reuse the relaydestination channels was studied in [11–14]. The basic idea is to arrange two or more relays to alternatively retransmit the source messages. To deal with the interrelay interference, the interrelay link was usually assumed to be either sufficiently strong or weak in these DFrelaying works. A cooperative transmission protocol named as shifted successive DF relaying (SSDFR) was proposed in [14], where maximum likelihood (ML) decoding is utilized at two multipleantenna relays. In [15], another cooperative transmission protocol, called relayreuse DF (RRDF), has been proposed to the scenario with a single source node, which yields furthermore improvement in comparison with [14] due to the linear zeroforcing (ZF) detection which can reduce the computational complexity.
For a general uplink scenario with multiple users and relays, it is still a challenging problem to integrate the cooperation among the sources and the cooperation from the relays. In [16, 17], each relay utilized superposition coding to assist each source. However, the developed upper bound of DMT in [16, 17] can only be achieved with some assumptions on the conditions of interrelay channels. The work in [18] considered a more general scenario and proposed a cooperative protocol, called relayassisted CMA (RCMA), which carefully schedules the multiple sources and relays’ transmission, but the optimal MISO bound can only be approached for a large number of relays. More works that considered such multiuser multirelay networks can be found in [19–21], where the assistance of relays has been well exploited. Generally, these works only use the assistance of relays without the cooperation between sources, which cannot fully exploit the capacity of the uplink system.
1.3 Contribution
This paper aims to design a novel cooperative transmission protocol for a CMA channel to integrate the cooperation from dedicated relays and the mutual cooperation between users.
Specially, we consider a cooperative multipleaccess communication scenario in which multiple source nodes communicate with the common destination with multiple halfduplex relays. Such a communication pattern is an important building block in wireless VANETs. For example, some proposals of intelligent vehicle highway systems (IVHS) have introduced a variety of vehicletovehicle and vehicletoroadsideinfrastructure communications. In this case, cooperation can take place between vehicles or between vehicles and roadside stations. When some vehicles wish to send messages to a common roadside basestation, they can cooperate with each other and also obtain external assistances from more powerful roadside stations with multiple antennas.
The proposed cooperative transmission protocol is termed as alternativerelaying decodeandforward (ARDF) throughout this paper. The main novelty is that a new cooperative multipleaccess transmission protocol has been designed by carefully applying these lowcost tools (zeroforcing detection, antenna selection, superposition coding, repetitioncoded decodeandforward, etc.) and fully exploiting the potentials of relay and source cooperation. To the best of the authors’ knowledge, no existing works can effectively achieve this full cooperation. The basic idea of the proposed protocol is to combine the superposition coding [6] with RRDF scheme [15]. By carefully arranging the transmission process and applying superposition coding at transmitters, each source can be assisted by all the other nodes, including relays and other sources; by utilizing the linear ZF decoding in terms of RRDF transmission scheme, the reuse of the relaydestination links can be achievable in the scenario with a general interrelay interference. Hence, the potential of such a system can be fully exploited, and the full diversity gain can be achievable. Antenna selection is also applied at each relay to select the best antenna for transmitting, which can preserve the full diversity gain and save transmit power.
DMT is used to evaluate the spectral efficiency of the proposed CMA protocol, since it has been recognized as an effective informationtheoretic criterion. The achievable DMT for the proposed ARDF protocol for the CMA channel is developed by characterizing the associated outage probabilities of outage events. From the derived result, we can show that the DMT of the proposed scheme can approach the optimal MISO upper bound when the length of one cooperative data frame is large. Compared to the previous CMA schemes in [4, 5] and RCMA in [18], the proposed cooperative scheme achieves a better DMT curve at most multiplexing gain ranges, especially when the frame length is large enough. Monte Carlo simulation results have also been presented to demonstrate the outage probability performance of the proposed transmission protocol.
for some product distribution p_{1}(x_{1})⋯p_{ M }(x_{ M }), where R_{ i } the data rate of source i (i=1,⋯,M) and H is the corresponding sourcedestination channel matrix. As can be observed from the above equation, the capacity region of MAC is a set of many inequalities with different conditions, which makes the calculation of DMT much more difficult than the singleuser scenario in [15]. These challenges also make the corresponding analysis more valuable.
This paper is organized as follows. Section 2 describes the proposed CMAARDF transmission protocol. Section 3 presents the details of the ARDF transmission process of the proposed scheme. Section 4 outlines the achievable DMT and the Monte Carlo simulation results for the proposed protocol. Section 5 offers concluding remarks. The proofs of the final DMT result are provided in the ‘Appendices’ section.
2 Protocol description
Every antenna at each relay is marked with a distinct index number from 1 to M_{ r }. As shown in Figure 1, the column vector h_{i,R j} denotes the channel vector between the ith source and R_{ j }, where i, j=1,2, and h_{ Si } represent the sourcedestination channel. The channel vector between R_{ j } and D is denoted as a column vector ${\mathbf{g}}_{j}=\{{g}_{j,1},\cdots \phantom{\rule{0.3em}{0ex}},{g}_{j,{M}_{r}}\}$. Moreover, the column vector h_{j,m} denotes the interrelay channel vector between the mth antenna of R_{ j } and the other relay. Each relay uses all the M_{ r } antennas to receive messages but only chose the best antenna to transmit signals for simplicity. The index of such a transmit antenna is defined as r_{ j } at R_{ j }. Note that this antenna selection method can preserve the diversity gain achieved by all the antennas at each relay in the high SNR regions.
The proposed protocol consists of two stages, initialization and data transmission.
2.1 Initialization
During the initialization stage, each relay channel is assumed to have the knowledge of its incoming and outgoing channel state information (CSI). Based on the sourcerelay CSI, using the relay selection method in [8], we known that R_{ j } can decode the messages from S_{ i } when the sourcerelay channel satisfies $log(1+\rho {\mathbf{h}}_{i,\mathit{\text{Rj}}}{}^{2})>\stackrel{~}{R}$. Here, log(·) is taken to base 2, ρ denotes the transmit signaltonoise ratio (SNR), where we have assumed that all transmitters are under the same and fixed power constraint, · denotes the Euclidean norm of a vector, and $\stackrel{~}{R}$ denotes the number of bits in each codeword transmitted by sources^{a}. If R_{ j } can correctly decode the messages from both sources, i.e., ${\mathbf{h}}_{i,\mathit{\text{Rj}}}{}^{2}\ge \frac{{2}^{\stackrel{~}{R}}1}{\rho}$ for ∀i∈{1,2}, it will broadcast a onebit indicator, which is denoted as ACK1. Otherwise, R_{ j } broadcasts a onebit indicator NACK1, assumed that all the other nodes can correctly receive each feedback signal. Denote A_{ k } as the situation in which k relays broadcast ACK1 signals. On the other hand, based on the relaydestination CSI, each relay can determine the index of the best antenna as ${r}_{j}\triangleq arg\underset{r=1\dots {M}_{r}}{max}{g}_{j,r}{}^{2}$.
Otherwise, the system lies in event $\overline{E}$ if any relay sends back a NACK2 signal.
2.2 Cooperative transmission
For situation A_{0} in which both relays send back NACK1 signals, we can only utilize the mutual cooperation between the two sources. In one cooperative data frame, each source intends to send L codewords, and the system works in mode 1. In time slot n (2≤n≤2L), S_{ i } will transmit a mixed message^{b}α_{2}x(n−1)+α_{1}x(n) and the other source listens, where i=2 if n is even and i=1 if n is odd. The data rate per channel use here is $\frac{2L}{2L+1}\stackrel{~}{R}$.
For situation A_{1} in which only one relay sends back ACK1 signal, we assume that R_{1} is this relay without loss of generality. Each source intends to transmit only one codeword, and the system works in mode 2. For the first two time slots, the transmission process is the same as that in mode 1, and R_{1} can correctly decode x(1) and x(2) since it sends back ACK1. In the third time slot, S_{1} and R_{1} will retransmit x(2) and x(1), respectively. In the fourth time slot, R_{1} will retransmit x(2). The transmission data rate in this situation is $\frac{2}{4}\stackrel{~}{R}$.
For situation A_{2}, if the event $\overline{E}$ occurs, the system works in mode 3. The transmission scheduling during the first three time slots is the same as that in mode 2. Then, during the fourth time slot, R_{1} and R_{2} will retransmit x(2) and x(1), respectively. During the fifth time slot, R_{2} retransmits x(2). The data rate for the event is $\frac{2}{5}\stackrel{~}{R}$. For the event E, both relays broadcast ACK2 signals. As shown in Equation (2), each relay can correctly decode two unknown codewords transmitted by a source and the other relay, so the ARDF transmission process can be performed. The two sources will transmit 2L codewords to the destination during 2L+3 time slots, so the data rate is $\frac{2L}{2L+3}\stackrel{~}{R}$. The details of such an ARDF process are put into in to the next section, i.e., Section 3, and the system works in mode 4.
The transmission modes for each situation, where ${y}_{i},{n}_{i}\in {\mathcal{C}}^{i},i=4,5,2L+1,2L+3;{x}_{j}\in {\mathcal{C}}^{j},j=2,2L$
Situation  Transmission mode  Equivalent channel matrix 

Situation A_{0}  $\underset{\mathit{\text{Mode}}\phantom{\rule{1em}{0ex}}1}{\underset{\u23df}{{\mathbf{y}}_{2L+1}={\mathbf{H}}_{{A}_{0}}{\mathbf{x}}_{2L}+{\mathbf{n}}_{2L+1}}}$  ${\mathbf{H}}_{{A}_{0}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\underset{(2L+1)\times 2L\phantom{\rule{1em}{0ex}}\text{Matrix}}{\underset{\u23df}{\left[\begin{array}{lllll}{h}_{S1}& 0& 0& \cdots & 0\\ {\alpha}_{2}{h}_{S2}& {\alpha}_{1}{h}_{S2}& 0& \cdots & 0\\ 0& {\alpha}_{2}{h}_{S1}& {\alpha}_{1}{h}_{S1}& \cdots & 0\\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0& \cdots & 0& {\alpha}_{2}{h}_{S2}& {\alpha}_{1}{h}_{S2}\\ 0& \cdots & 0& 0& {h}_{S1}\end{array}\right]}}$ 
Situation A_{1}  $\underset{\mathit{\text{Mode}}\phantom{\rule{1em}{0ex}}2}{\underset{\u23df}{{\mathbf{y}}_{4}={\mathbf{H}}_{{A}_{1}}{\mathbf{x}}_{2}+{\mathbf{n}}_{4}}}$  ${\mathbf{H}}_{{A}_{1}}=\left[\begin{array}{ll}{h}_{S1}& 0\\ {\alpha}_{2}{h}_{S2}& {\alpha}_{1}{h}_{S2}\\ {g}_{1,{r}_{1}}& {h}_{S1}\\ 0& {g}_{1,{r}_{1}}\end{array}\right]$ 
Situation A_{2} event $\overline{E}$  $\underset{\mathit{\text{Mode}}\phantom{\rule{1em}{0ex}}3}{\underset{\u23df}{{\mathbf{y}}_{5}={\mathbf{H}}_{{A}_{2},\overline{E}}{\mathbf{x}}_{2}+{\mathbf{n}}_{5}}}$  ${\mathbf{H}}_{{A}_{2}},\overline{E}=\left[\begin{array}{ll}{h}_{S1}& 0\\ {\alpha}_{2}{h}_{S2}& {\alpha}_{1}{h}_{S2}\\ {g}_{1,{r}_{1}}& {h}_{S1}\\ {g}_{2,{r}_{2}}& {g}_{1,{r}_{1}}\\ 0& {g}_{2,{r}_{2}}\end{array}\right]$ 
Situation A_{2} event E  $\underset{\mathit{\text{Mode}}\phantom{\rule{1em}{0ex}}4}{\underset{\u23df}{{\mathbf{y}}_{2L+3}={\mathbf{H}}_{{A}_{2},E}{\mathbf{x}}_{2L}+{\mathbf{n}}_{2L+3}}}$  ${\mathbf{H}}_{{A}_{2},E}=\mathbf{H}$, refer to Eqution (3). 
3 Alternativerelaying DF process
3.1 Transmission process
The proposed ARDF transmission process follows the concept of successive relaying [13], where the two halfduplex relays alternatively retransmit the source messages to mimic a fullduplex node. As shown in Figure 1, the sending source sends a superposition codeword α_{2}x(n−3)+α_{1}x(n) in any one time slot n (n>3), where x(n) is its own codeword and x(n−3) is the codeword it received from the other source in time slot n−3. At the same time, the sending relay will transmit a superposition codeword β_{2}x(n−1)+β_{1}x(n), where x(n−1) and x(n−2) are codewords it received from the source during the previous two time slots. The other source (i.e., the listening source) and the other relay (i.e., the listening relay) listen to the transmissions in this time slot. In the next time slot, i.e., time slot n+1, both the two sources and relays will exchange their working roles, where the listening ones become the sending ones and the sending ones become the listening ones. At the sources, different to [5, 6], the sending source retransmit the codeword x(n−3) instead of x(n−1) in time slot n. This is because we have arranged the sending relay to transmit x(n−1), so such a codeword should not be transmitted by the sending source again in this time slot to avoid the potential mutual cancellation. In the proposed ARDF process, each codeword can be transmitted by both the sources and relays during four successive time slots, which ensures that the ‘full’ cooperation can be achieved. In the following paragraphs, we will describe the transmission process during the four successive time slots in details.
From the above ARDF transmission process, one can observe that the listening relay may need to decode at most two unknown codewords in a time slot. For instance, during time slot n (2≤n≤2L), in addition to the new source codeword x(n), the listening relay also does not know x(n−1) transmitted by the sending relay. This is because it is transmitting during the previous time slot and cannot simultaneously receive the source codeword x(n−1) due to the halfduplex constraint. After canceling the known codewords, this relay will decode two independent unknown codewords, i.e., x(n) and x(n−1), from one source and the other relay, respectively. Such a transmission mode mimics a twouser MAC with two singleantenna transmitters and one M_{ r }antenna receiver. At each relay, simple linear zeroforcing detection approaches can be used to separate the source message and the interference from the other relay, where each relay is equipped with multiple antennas and therefore has the capability to separate the messages under the ZF condition in Equation (2).
3.2 Data model
Note that we have used a fixed power allocation strategy in each time slot, and the use of more advanced adaptive power allocation shall yield better outage performance at finite SNRs. However, for the infinite SNR region, the precise power allocation does not affect the final DMT result [27].
4 Diversitymultiplexing tradeoff and numerical results
In this section, the analytical performance evaluation of the proposed protocol will be first shown by using DMT. Then, the Monte Carlo simulations will be carried out to demonstrate the outage performance of the proposed cooperative ARDF protocol.
4.1 Achievable diversitymultiplexing tradeoff
where P_{ e } is the ML detection error probability and R is the target data rate per channel use. Following similar steps in [22], it can be proved that the ML error probability in the MAC scenario can be tightly bounded by the outage probability at high SNR. So the outage probability will be focused in this paper. When $\underset{\rho \to \infty}{lim}\frac{logf\left(\rho \right)}{log\rho}=d$, f(ρ) is said to be exponentially equal to ρ^{ d }, denoted as $f\left(\rho \right)\doteq {\rho}^{d}$ ($\stackrel{\u0307}{\le}$ and $\stackrel{\u0307}{\ge}$ are similarly defined). The following theorem gives the final DMT performance of the proposed cooperative protocol.
Theorem 1
where (x)^{+} denotes max{x,0}.
Proof 1
We have briefly provided the proof of this theorem in [28] without formal derivations. In this journal paper, the details of the proof for this theorem will be blue presented in Appendix 2.

Remark 1: The optimal MISO DMT for the addressed CMA scenario can be expressed as${d}_{\mathit{\text{MISO}}}\left(r\right)=(2{M}_{r}+2)(1r),\phantom{\rule{1em}{0ex}}0\le r\le 1.$

According to Equation (5), when the frame length L is sufficiently large, the achievable DMT of the proposed scheme yields $\frac{2L+3}{2L}\approx 1$, which demonstrates that the optimal MISO tradeoff d_{ MISO }(r) can be asymptotically approached by the proposed scheme.
where M denotes the number of the sources. When there only exist two active users in a network, d_{ CMA }(r)=2(1−r). Hence, the schemes in [4, 5] cannot perform well for a small number of sources, whereas the proposed ARDFCMA scheme can further enhance the reception robustness by exploiting the relay nodes.

Remark 3: On the other hand, the RCMA protocol in [18] can be straightforwardly extended to the twouser tworelay scenario considered in this paper. It means that each relay apply a simple antenna selection, and the DMT result presented in Equation (7) in [18] can be accordingly updated as${d}_{\mathit{\text{R\_CMA}}}\left(r\right)=(1r)+{M}_{r}{\left(24r\right)}^{+},\phantom{\rule{1em}{0ex}}0\le r\le 1.$

Compared to the RCMA scheme, the proposed scheme achieves one more diversity gain, because it exploits the help of two M_{ r }antenna relays while preserves the cooperation between two sources. Although the proposed protocol can only achieve a maximal multiplexing gain $\frac{2L}{2L+3}$ while the schemes in [18] can achieve 1, the difference is negligible for a large L.
4.2 Numerical results
In this section, some numerical results based on the Monte Carlo simulations are provided for the addressed CMA scenario. The targeted data rate is set as 2 or 4 bits per channel use (BPCU). Each relay is equipped with two antennas, i.e., M_{ r }=2. For the proposed CMAARDF protocol, set L=25,α_{1}=β_{1}=0.87 for simplicity. Note that all the channels are assumed to be Rayleigh fading with unit variance and all the nodes have the same transmit power.
5 Conclusions
In this paper, a new cooperative transmission protocol has been proposed for a CMA scenario in the wireless vehicular networks. Without the use of assumptions of strong interrelay channel in [13], the relay reuse has still been realized using the linear ZF detection at the halfduplex relays to combat a general interrelay interference. In addition, to fully exploit the cooperation between all the transmitters, the superposition coding strategy has also been carefully applied at both sources and relays. To evaluate the spectral efficiency, an achievable DMT of the proposed scheme was developed, which demonstrated that the proposed ARDF protocol can outperform the existing related schemes and approximately achieve the optimal MISO upper bound. The derived analytical result and the numerical results have demonstrated that the proposed scheme has a better performance in comparison with the existing related schemes in most conditions. The impact of adaptive power allocation has not been analyzed due to the high SNR considered in this paper, where a promising future direction is to carry out the study for the optimal design of power allocation to improve the overall system throughput at intermediate SNR.
Endnotes
^{a} For a particular situation, the data rate should be $\frac{\stackrel{~}{R}\times {N}_{x}}{{N}_{t}}$, where N_{ x } and N_{ t } are the number of transmitted codewords and the number of required time slots, respectively. So the targeted data rate R (i.e., the average data rate) can be calculated from $\stackrel{~}{R}$. We will provide more discussions about the relationship between the two rates later in this paper.
^{b} In this paper, x(n) denotes the new source codeword sent by S_{ i } (i=1,2) in time slot n (1≤n≤2L). Obviously, the mapping criterion is $x\left(n\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\left\{\begin{array}{ll}{s}_{1}\left(\frac{n+1}{2}\right),& \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}n\phantom{\rule{1em}{0ex}}\text{is odd}\\ {s}_{2}\left(\frac{n}{2}\right),& \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}n\phantom{\rule{1em}{0ex}}\text{is even}\end{array}\right.$, where s_{ i }(l) denotes the lth (1≤l≤L) codeword of S_{ i } in one cooperative frame.
Appendices
Appendix 1
Preliminary results
We first present four lemmas and their proof steps in this section. These results will be useful for the proofs of Theorem 1 in the next section.
Lemma 2
Lemma 3
where $\gamma =\frac{{2}^{\stackrel{~}{R}}1}{\rho {\alpha}_{1}^{2}{\delta}^{2}}$, δ^{2} is the variance of each channel coefficient. Furthermore, the probability of the event $\overline{E}$ can be obviously obtained as $P\left(\overline{E}\right)\doteq {\rho}^{({M}_{r}1)(1\stackrel{~}{r})}$.
Lemma 4
When X_{1} and X_{2} are subjected to exponential distribution, the probability can be approximated as P(X_{1}X_{2}<a)≈−a lna, where a>0 and is sufficiently small.
Lemma 5
Assuming that $z=max\{{z}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{z}_{{M}_{r}}\}$, where z_{ i } is exponentially distributed with unit variance, the expectation of e^{−c z}(c>0) can be revealed as $\epsilon \left[{e}^{\mathit{\text{cz}}}\right]=\frac{{M}_{r}!}{{\mathrm{\Pi}}_{i=1}^{{M}_{r}}(c+i)},\epsilon [\xb7]$ denotes the expectation of a random variable.
The proofs steps of the above lemmas are provided in the following section.
Proof of Lemma 2
Proof of Lemma 3
According to Equation (A.3), Lemma 3 can be proved.
Proof of Lemma 4
where ${B}_{i}\left(u\right)={\int}_{u}^{+\infty}\frac{{e}^{x}}{{x}^{i}}\mathrm{d}x$. Then, the improper integral Φ_{ a } can be calculated as ${\mathrm{\Phi}}_{a}={lim}_{u\to {0}^{+}}{\mathrm{\Phi}}_{a}\left(u\right)$.
For a sufficiently large ρ, Φ_{ a } can be approximately calculated as Φ_{ a }≈−a lna.
Proof of Lemma 5
Appendix 2
Proof of theorem 1
where $\left\mathcal{S}\right$ denotes number of users in , Q denotes the number of codewords transmitted by each source in one cooperative frame, and h_{ l } is a channel vector, both the structure of h_{ l } and the set are a function of and the details of their relationship to will be discussed in the next few subsections. The outage events occur when any constraint in Equation (B.2) is not met, and the highest outage probability achieved by each constraint is the dominant factor [22]. In the following subsections, different values of each parameter in Equation (B.2) will be considered for different situations.
Situation A_{0}
Thus, $P\left({\mathcal{O}}_{{A}_{0}}\right)\stackrel{\u0307}{\le}{\rho}^{2(1\stackrel{~}{r})}$ can be easily obtained by combining ${P}_{1}\left({\mathcal{O}}_{{A}_{0}}\right)$ and ${P}_{2}\left({\mathcal{O}}_{{A}_{0}}\right)$.
Situation A_{1}
where the last relationship is based on Lemma 4. Hence, $P\left({\mathcal{O}}_{{A}_{1}}\right)\stackrel{\u0307}{\le}{\rho}^{({M}_{r}+2)(1\stackrel{~}{r})}$ can be obtained.
Situation A_{2}event $\overline{E}$
Therefore, it can be shown that $P\left({\mathcal{O}}_{{A}_{2},\overline{E}}\right)\stackrel{\u0307}{\le}{\rho}^{(2{M}_{r}+2)(1\stackrel{~}{r})}$.
Situation A_{2}event E
The outage probability in situation A_{2} event E is difficult to be obtained using the above method, but the upper bound can be calculated by following the similar analysis in [17]. Firstly, model 4 in Table 1 is assumed to be a symmetric 2Luser multipleaccess system where the codeword x(n) in x_{2L} is transmitted by S_{ n }, and such an assumption will make the analysis tractable. For the twouser case considered in this paper, the performance would not be worse than the performance of the former one. So that $\mathcal{\mathcal{L}}=\mathcal{S}\subseteq \{1,2,\cdots \phantom{\rule{0.3em}{0ex}},2L\}$, Q=1, h_{ l } denotes the lth column vector of H in Equation (3), and there are (2^{2L}−1) source data constrains in Equation (B.2). For each constraint, there exists a probability that the channel condition cannot satisfy it, and the highest outage probability is the dominant factor and achieve the system’s DMT.
In order to calculate the outage probability of each constraint, a (m+3)×m MIMO channel is first considered as y_{ m }=F_{ m }s_{ m }+w_{ m }, where ${\mathbf{s}}_{m}\in {\mathcal{C}}^{m},{\mathbf{y}}_{m},{\mathbf{w}}_{m}\in {\mathcal{C}}^{m+3}$ and F_{ m } is the (m+3)×m topleft submatrix from H in Equation (3). According to [17], the outage probability achieved by this system for every 1≤m≤2L is the same as the highest outage probability for each constraint in Equation (B.2). When m=1, ${P}_{1}\left({\mathcal{O}}_{{A}_{2},E}\right)\doteq {\rho}^{(2{M}_{r}+2)(1\stackrel{~}{r})}$ can be easily obtained, following the similar steps in Equation (B.7).
where ${P}_{1}={\rho}^{{M}_{r}(1\stackrel{~}{r})},{P}_{2}={\rho}^{({M}_{r}1)(1\stackrel{~}{r})}$. Following the analysis in Claim 3 of [8] and the similar steps in [14], $d\left(r\right)=(2{M}_{r}+2){\left(1\frac{2L+3}{2L}r\right)}^{+}$ can be proved by substituting $\stackrel{~}{r}=\frac{2L+3}{2L}r$ into Equation (B.18).
Declarations
Acknowledgements
The work of Peng Xu and Xuchu Dai was supported by the National Natural Science Foundation of China (NSFC) under grant number 61471334, the National High Technology Research and Development Program of China (863 Program) under grant number 2012AA01A502, and the National Basic Research Program of China (973 Program: 2013CB329004). The work of Zhiguo Ding was supported by a Marie Curie International Fellowship within the 7th European Community Framework Programme and the UK EPSRC under grant number EP/I037423/1. This paper has been presented in part at IEEE International Conference on Communications (ICC), June, Canada, 2012 [28].
Authors’ Affiliations
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