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Alternative relaying for cooperative multipleaccess channels in wireless vehicular networks
EURASIP Journal on Wireless Communications and Networking volume 2014, Article number: 165 (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.
1 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.
Compared to the pointtopoint communication in [15], we consider a multiuser case for the addressed MAC scenario in this paper. As a consequence, DMT for the MAC channel in [22] should be adopted, the informationtheoretic criterion, which is developed based on the capacity region of MAC [23]. Specifically, define {X}_{\mathcal{S}}=\{{X}_{i}:i\in \mathcal{S}\}, and X_{ i }(Y) is the channel input (output) variable. For a instantaneous channel state information (CSI) which is known at the receiver, the capacity region of the Muser MAC can be written as following [23]:
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
A cooperative multipleaccess scenario is considered in this paper, where multiple sources wish to communicate a common destination under the help of relays. In particular, the addressed scenario consists of two singleantenna sources (i.e., vehicles, denoted as S_{1} and S_{2}), two M_{ r }antenna (M_{ r }≥2) relays (i.e., fixed roadside stations, denoted as R_{1} and R_{2}), and one common singleantenna destination (i.e., base station, denoted as D). The abstract system model and its application to the wireless vehicular networks are shown in Figures 1 and 2, respectively. All channels are assumed to be frequencyflat quasistatic Rayleigh fading. Such a Rayleigh fading assumption is valid in a practical vehicular network since it is very likely that there does not exist a lineofsight (LOS) component between a pair of VANET transceivers due to their relatively low antenna height [3, 24, 25]. All nodes are constrained by the halfduplex assumption and use the same transmission power. Time division multiplexing access (TDMA) is utilized in this scenario for its simplicity.
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}.
For situation A_{2} in which both relays broadcast ACK1 signals, two subevents will be further considered. According to the sourcerelay/interrelay CSI, R_{ j } will determine whether it can use ZF detection to separate the mixture of two unknown streams which are from S_{ j } and the other relay, respectively. For the ZF receiver, the qualification criterion [26] is {\left[{\left({\mathbf{H}}_{\mathit{\text{Rj}}}^{H}{\mathbf{H}}_{\mathit{\text{Rj}}}\right)}^{1}\right]}_{k,k}<\rho /{2}^{\stackrel{~}{R}1}, where [X]_{k,k} represents the kth diagonal element of matrix X, {\mathbf{H}}_{\mathit{\text{Rj}}}=\left[\begin{array}{cc}{\beta}_{1}{\mathbf{h}}_{2,{r}_{2}}& {\alpha}_{1}{\mathbf{h}}_{1,R1}\end{array}\right]for j=1, and \left[\begin{array}{cc}{\beta}_{1}{\mathbf{h}}_{1,{r}_{1}}& {\alpha}_{1}{\mathbf{h}}_{2,R2}\end{array}\right]for j=2. Note that α_{ i } and β_{ i } are the power weighting factors at the sources and relays, respectively, which are constrained by 0<α_{ i },β_{ i }<1 and \sum _{i=1}^{2}{\alpha}_{i}^{2}=\sum _{i=1}^{2}{\beta}_{i}^{2}=1. If R_{ j } satisfies this ZF condition, it will send back another onebit indicator ACK2. Otherwise, it sends back NACK2. Let E denote the event that both relays send back ACK2 signals, and such an event can be represented as [26]
Otherwise, the system lies in event \overline{E} if any relay sends back a NACK2 signal.
2.2 Cooperative transmission
Based on the initialization stage, the transmission is divided into four different modes according to the indicators sent back from the two relays. The flow chart of the proposed protocol is shown in Figure 3, where each of the sources is assumed to correctly decode the messages of the other source, which is valid for high SNR regions as shown in [5]. In the next paragraphs, we will describe the cooperative transmission mode for each situation in details.
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.
From the protocol description, the transmission mode and equivalent channel matrix for each situation can be easily shown in Table 1.
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.
Assume n to be odd without loss of generality. In time slot n (1≤n≤2L), the sending source S_{1} first transmits x(n), while R_{1} and S_{2} decode x(n) from S_{1}. In time slot n+1, R_{1} is the sending relay and uses its best antenna to retransmit x(n); R_{2} receives x(n) from R_{1}. In time slot n+2, R_{2} becomes the sending relay and uses its best antenna to repeat x(n). In time slot n+3, the sending source S_{2} retransmits x(n) again. Therefore, during such four successive time slots, x(n) can be transmitted four times, by both two sources and relays. Similar to the transmission of x(n), the codeword of S_{2}, x(n+1), will be transmitted by S_{2}, R_{2}, R_{1}, and S_{1} in time slot n+1, n+2, n+3, and n+4, respectively. When a transmitting node wishes to transmit two codewords simultaneously, it will transmit the superposition of these two codewords and use an amplitude factor to constrain the power of each codeword. Figure 4 illustrates the whole ARDF transmission process, where 2L+3 time slots are required for a cooperative data frame.
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].
For the ARDF process with respect to situation A_{2} event E, the data model can be expressed as y=H x+n, where x=[x(1),⋯,x(2L)]^{T}, which is the codeword vector of the two sources transmitted during a data frame; y=[y(1),⋯,y(2L+3)]^{T}, which is the observation vector at the destination during a data frame that lasts 2L+3 time slots; n= [n(1),⋯,n(2L+3)]^{T}, which is the Gaussian additive noise vector; and the (2L+3)×(2L) equivalent channel matrix is as follows:
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
Firstly, recall that the diversity and multiplexing gain are defined as [7]
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
When each relay is equipped with at least two antennas, i.e., M_{ r }≥2, the achievable DMT of the proposed ARDF protocol for the twouser and tworelay CMA scenario can be expressed as
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.

Remark 2: The achievable DMT of the CMA schemes in [4, 5] can be written as
{d}_{\mathit{\text{CMA}}}\left(r\right)=M(1r),\phantom{\rule{1em}{0ex}}0\le r\le 1,
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.
Figure 5 illustrates the achievable DMTs of various cooperative schemes. As can be seen from this figure, the proposed scheme can approach the optimal MISO upper bound for a sufficiently large L, and hence, outperforms the comparable ones in most multiplexing range.
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.
In addition to the superpositioncodingbased CMA in [5], the RCMA scheme in [18] is also used to compare with the proposed ARDF protocol. As shown in Figure 6, ARDF has limited performance at low SNR. This is because the performance of the proposed scheme is much more sensitive to the sourcerelay/interrelay channel condition, and the outage event is prone to occur at each relay at low SNR, which results in some degradation to the outage performance. However, the proposed scheme achieves the smallest outage probability among the comparative ones when we increase SNR. This performance gain is due to the fact that the proposed ARDF scheme can exploit the available relays to provide an extra dimension to improve the outage performance, whereas the scheme in [5] only considers the cooperation between the two sources and the RCMA scheme in [18] does not consider the mutual source cooperation. Therefore, larger diversity gains can be achieved by the proposed transmission protocol, which guarantee the superior performance particularly at high SNR. In Figure 7, different antenna numbers, i.e., M_{ r }, is consider at each relay to demonstrate the outage performance of the proposed scheme. One can observe that increasing the number of relay antennas can further improve the robustness of the transmission scheme, since more diversity gains can be provided.
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
By assuming quasistatic and flat Rayleighfading channels, the probability of each situation A_{ k } with kqualified relays broadcasting ACK1 signals can be expressed as P\left({A}_{k}\right)\doteq {\rho}^{(2k){M}_{r}(1\stackrel{~}{r})}, where
Lemma 3
By assuming quasistatic and flat Rayleighfading channels and {\alpha}_{1}={\beta}_{1}=\sqrt{\frac{1}{2}} for simplicity, the probability of the event E that both relays can successfully perform ZF detection and broadcast ACK2 signals, as expressed in Equation (2), can be bounded as
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^{−cz}(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
Denote B as the event that a relay R_{ j } is not qualified, i.e., it can be expressed as the event: B\triangleq \bigcup _{i=1}^{2}\left\{log(1+\rho \underset{1}{\overset{2}{\alpha}}{\mathbf{h}}_{i,\mathit{\text{Rj}}}{}^{2})<\stackrel{~}{R}\right\}. The probability of the event B can be easily calculated as P\left(B\right)\doteq {\rho}^{{M}_{r}(1\stackrel{~}{r})}. Thus, the probability of each situation A_{ k } can be expressed as follows:
Proof of Lemma 3
Consider a signal model in which the r_{2}th antenna of R_{2} transmits a message {\stackrel{~}{s}}_{1} with a power lever {\beta}_{1}^{2} while the first source S_{1} broadcasts a message {\stackrel{~}{s}}_{2} with a power lever {\alpha}_{1}^{2}. We set {\alpha}_{1}={\beta}_{1}=\sqrt{\frac{1}{2}} for ease of explanation. Such a signal model can be found in most time slots with odd numbers in situation A_{2} event E. At this time, the signal vector received by R_{1} can be written as {\mathbf{r}}_{R1}={\mathbf{H}}_{R1}\stackrel{~}{\mathbf{s}}+{\mathbf{w}}_{1}, where {\mathbf{r}}_{R1},{\mathbf{w}}_{1}\in {\mathcal{C}}^{{M}_{r}}, {\mathbf{H}}_{R1}=\left[\begin{array}{ll}{\beta}_{1}{\mathbf{h}}_{2,{r}_{2}}& {\alpha}_{1}{\mathbf{h}}_{R1}\end{array}\right], \stackrel{~}{\mathbf{s}}={[{\stackrel{~}{s}}_{1},{\stackrel{~}{s}}_{2}]}^{T}, and w_{1} is the noise vector. This transmission model can be recognized as a special MIMO system with two transmit, and M_{ r } receive antennas when ZF detection is applied. Denote the event as {E}_{R1}\triangleq \left\{log\left(1\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\frac{\rho}{{\left[{\left(\underset{R1}{\overset{H}{\mathbf{H}}}{\mathbf{H}}_{R1}\right)}^{1}\right]}_{k,k}}\right)>\stackrel{~}{R},\phantom{\rule{0.3em}{0ex}}\forall \phantom{\rule{0.3em}{0ex}}k\in \{1,2\}\right\}. Applying ZF detection at the receiver and according to [26], the probability of E_{R 1} can be expressed as
when ρ is sufficiently large, where \gamma =\frac{{2}^{\stackrel{~}{R}}1}{\rho {\alpha}_{1}^{2}{\delta}^{2}}. Note that the similar ZF decoding can be applied at R_{2} when R_{1} and S_{2} are transmitting messages at the same time, and E_{R 2} is similarly defined as {E}_{R2}\triangleq \left\{log\left(1+\frac{\rho}{{\left[{\left(\underset{R2}{\overset{H}{\mathbf{H}}}{\mathbf{H}}_{R2}\right)}^{1}\right]}_{k,k}}\right)>\stackrel{~}{R},\forall k\in \{1,2\}\right\}, whose probability is the same as P(E_{R 1}), i.e.,
Then, the probability of the event E defined in Equation (2) can be revealed as P\left(E\right)=P\left({E}_{R1}\bigcap {E}_{R2}\right), where the events E_{R 1} and E_{R 2} are not strictly independent. Since reciprocal channel is assumed, the channel vectors {\mathbf{h}}_{1,{r}_{1}} and {\mathbf{h}}_{2,{r}_{2}} have one common element which is the channel coefficient between the r_{1}th antenna of R_{1} and the r_{2}th antenna of R_{2}, so that it is difficult to obtain the accurate value of P(E). However, when we notice that P(E) can be bounded as P(E_{R 1})+P(E_{R 2})−1≤P(E)≤P(E_{R 1}), we have
According to Equation (A.3), Lemma 3 can be proved.
Proof of Lemma 4
Assume X_{1} and X_{2} to be independently exponentially distributed with unit variance without loss of generality. Let D=\left\{\right({x}_{1},{x}_{2})\in {\mathbb{R}}^{2+}{x}_{1}{x}_{2}<a\} and {f}_{{x}_{1}{x}_{2}}({x}_{1},{x}_{2})={f}_{{x}_{1}}\left({x}_{1}\right){f}_{{x}_{2}}\left({x}_{2}\right) which is the joint density of {x_{1},x_{2}}, where {f}_{{x}_{i}}\left({x}_{i}\right) denotes X_{ i }’s probability density function (PDF) {f}_{{x}_{i}}={e}^{{x}_{i}}, i=1,2. Then, the probability P(X_{1}X_{2}<a) can be expressed as
Now, the function Φ_{ a }(u) is first defined as
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).
Moreover, from [29] (Equation 3.351.4), B_{ i }(u) can be calculated by the exponential integral function as
where i≥2, and exponential integral function can be shown as \text{Ei}(u)=ln\left(u\right)+{\sum}_{k=1}^{\infty}\frac{{(u)}^{k}}{k\xb7k!}. Now that, B_{ i }(u) can be obtained as
where o(1)→0 when u→0^{+}. By substituting B_{ i }(u) into Equation (A.7) and rearranging the infinite series, Φ_{ a }(u) can be rewritten as
where
Let v=a u^{−1}, ϕ_{ n }(v) can be expressed as
where
Following similar proof steps of Corollary 1 in [15], the limit {lim}_{v\to +\infty}{F}_{n}\left(v\right) can be proved to exist. Hence, by recalling Equation (A.10), Φ_{ a } can be expressed as
For a sufficiently large ρ, Φ_{ a } can be approximately calculated as Φ_{ a }≈−a lna.
Proof of Lemma 5
It is not difficult to obtain the PDF of z as {f}_{z}\left(z\right)={M}_{r}{e}^{z}{(1{e}^{z})}^{{M}_{r}1}, z>0, since z=max\{{z}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{z}_{{M}_{r}}\} and {f}_{{z}_{r}}\left({z}_{r}\right)={e}^{{z}_{r}}, r∈{1,⋯,M_{ r }}, so that
Using binomial theorem, ε[e^{−cz}] can be written as
where {C}_{p}^{q}=\frac{p!}{(pq)!q!}, p and q are positive integers, and p≥q. Splitting each term on the right side, Equation (A.16) can be calculated as
Furthermore, by iteratively repeating the similar process in Equation (A.17), the expectation of e^{−cz} can be finally obtained as
Appendix 2
Proof of theorem 1
As shown in [7, 22], the ML error probability can be tightly bounded by the outage probability at high SNR, so the outage probability will be analyzed in this section. According to [22] and the protocol description in Section 2, we can define the outage event of the proposed ARDF protocol as \mathcal{O}\triangleq \bigcup _{k=0}^{2}{\mathcal{O}}_{{A}_{k}}, where {\mathcal{O}}_{{A}_{2}}\triangleq {\mathcal{O}}_{{A}_{2},\overline{E}}\bigcup {\mathcal{O}}_{{A}_{2},E}. Here, {\mathcal{O}}_{{A}_{k}} denotes the outage event in the situation A_{ k } at the destination, {\mathcal{O}}_{{A}_{2},\overline{E}} and {\mathcal{O}}_{{A}_{2},E} are similarly defined. Thus, the overall outage probability of the proposed protocol can be expressed as
The probabilities of each situation A_{ k } and the event E have been presented in Lemma 2 and Lemma 3, respectively. Moveover, the outage probability in each situation will be analyzed in the terms of the MAC capacity region. Based on the definition in Equation (1), the source data rate constrains for the proposedARDF protocol can be further calculated as follows:
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}
\mathcal{S}\subseteq \{1,2\}, Q=L, and h_{ l } denotes the lth column vector of {\mathbf{H}}_{{A}_{0}} in Table 1. When \left\mathcal{S}\right=1, is assumed to be {1} without loss of generality, so \mathcal{\mathcal{L}}=\{1,3,\cdots \phantom{\rule{0.3em}{0ex}},2L1\}, and the outage probability at the destination in such a case can be calculated as
where X_{ i }=h_{ Si }^{2}, i=1,2; (a) holds since 0<{\alpha}_{1}^{2}<1, and (b) holds since X_{1} is independent of X_{2}. Otherwise, \mathcal{S}=\{1,2\} and \mathcal{\mathcal{L}}=\{1,2,\cdots \phantom{\rule{0.3em}{0ex}},2L\},
In order to make the analysis more tractable, a (2L+1)×(2L+1) square matrix {\stackrel{~}{\mathbf{H}}}_{{A}_{0}} is first defined as {\stackrel{~}{\mathbf{H}}}_{{A}_{0}}\triangleq [{\mathbf{H}}_{{A}_{0}},\mathbf{0}], where the zero column vector is (2L+1)dimensional. It is easy to show that {\mathbf{H}}_{{A}_{0}}{\mathbf{H}}_{{A}_{0}}^{H}={\stackrel{~}{\mathbf{H}}}_{{A}_{0}}{\stackrel{~}{\mathbf{H}}}_{{A}_{0}}^{H} and [{\mathbf{I}}_{2L+1}+\rho {\stackrel{~}{\mathbf{H}}}_{{A}_{0}}{\stackrel{~}{\mathbf{H}}}_{{A}_{0}}^{H}] is a tridiagonal matrix. According to [30], the determinant of the tridiagonal matrix can be shown iteratively as
where {D}_{n}=det[{\mathbf{I}}_{2L+1}+\rho {\stackrel{~}{\mathbf{H}}}_{n}{\stackrel{~}{\mathbf{H}}}_{n}^{H}], and {\stackrel{~}{\mathbf{H}}}_{n} denotes the n×n topleft submatrix from {\stackrel{~}{\mathbf{H}}}_{{A}_{0}}, x_{ n } and y_{ n } are the nth element on the principle diagonal and subdiagonal of {\stackrel{~}{\mathbf{H}}}_{{A}_{0}}, respectively. By using such a property and note that x_{2L+1}=0, the following inequality can be obtained
From Equation (B.5) and Lemma 4, {P}_{2}\left({\mathcal{O}}_{{A}_{0}}\right) can be upper bounded as
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}
\mathcal{\mathcal{L}}=\mathcal{S}\subseteq \{1,2\}, Q=1, h_{ l } denotes the lth column vector of {\mathbf{H}}_{{A}_{1}} in Table 1. When \left\mathcal{S}\right=1, assume that \mathcal{S}=\left\{1\right\}, {P}_{1}\left({\mathcal{O}}_{{A}_{1}}\right) can be easily obtained as
where {G}_{j}={g}_{j,{r}_{j}}{}^{2}\triangleq max\{{g}_{j,1}{}^{2},\cdots \phantom{\rule{0.3em}{0ex}},{g}_{j,{M}_{r}}{}^{2}\},j=1,2. Otherwise, \mathcal{S}=\{1,2\}, {P}_{2}\left({\mathcal{O}}_{{A}_{1}}\right) is
The determinant of [{\mathbf{I}}_{2}+\rho {\mathbf{H}}_{{A}_{1}}^{H}{\mathbf{H}}_{{A}_{1}}] can be first calculated as
where
So that det\left({\mathbf{I}}_{2}+\rho {\mathbf{H}}_{{A}_{1}}^{H}{\mathbf{H}}_{{A}_{1}}\right)\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}{\rho}^{2}\left[{G}_{1}^{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}(1\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\alpha}_{1}{\alpha}_{2}){X}_{1}{X}_{2}\right] can be obtained, where 0<{\alpha}_{1}{\alpha}_{2}\le \frac{1}{2}, and {P}_{2}\left({\mathcal{O}}_{{A}_{1}}\right) can be upper bounded as
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}
\mathcal{\mathcal{L}}=\mathcal{S}\subseteq \{1,2\}, Q=1, h_{ l } denotes the lth column vector of {\mathbf{H}}_{{A}_{2},\overline{E}} in Table 1. When \left\mathcal{S}\right=1, {P}_{1}\left({\mathcal{O}}_{{A}_{2},\overline{E}}\right)\doteq {\rho}^{(2{M}_{r}+2)(1\stackrel{~}{r})} can be easily obtained by following the similar steps in Equation (B.7). Otherwise, \mathcal{S}=\{1,2\}, {P}_{2}\left({\mathcal{O}}_{{A}_{2},\overline{E}}\right) can be written as
In the above equation, det\left({\mathbf{I}}_{2}+\rho {\mathbf{H}}_{{A}_{2},\overline{E}}^{H}{\mathbf{H}}_{{A}_{2},\overline{E}}\right)={K}_{1}{K}_{2}, where
so that det\left({\mathbf{I}}_{2}+\rho {\mathbf{H}}_{{A}_{2},\overline{E}}^{H}{\mathbf{H}}_{{A}_{2},\overline{E}}\right)>{\rho}^{2}\left[{G}_{1}^{2}+{G}_{2}^{2}+(1{\alpha}_{1}\right.\left(\right]separators="">\n \n \n \n \alpha \n \n \n 2\n \n \n )\n \n \n X\n \n \n 1\n \n \n \n \n X\n \n \n 2\n \n \n \n can be obtained. By using Lemma 4 and following the similar steps in Equation (B.11), {P}_{2}\left({\mathcal{O}}_{{A}_{2},\overline{E}}\right) can be upper bounded as
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).
When m>1, following the similar DMT analysis for the intersymbol interference (ISI) channel in [31] and the proof steps in [17], the average error probability can be upper bounded by
where c_{1} is a constant, \mathbf{f}={[{h}_{S1},{h}_{S2},{g}_{1,{r}_{1}},{g}_{2,{r}_{2}}]}^{T}, ε[·] denotes the expectation of a random variable, and \overline{\lambda}=\underset{\mathbf{f}\in {\mathcal{C}}^{4}}{inf}{\lambda}_{min}\left(\frac{{\mathbf{F}}_{m}}{\left\mathbf{f}\right}\right)>0, λ_{min}(·) denotes the minimum singular value of a matrix. By using Lemma 5, P_{ e } can be upper bounded as
By observing the fact that P\left({\mathcal{O}}_{{A}_{2},E}\right)\stackrel{\u0307}{\le}{P}_{e}[7], the outage probability of model 4 in situation A_{2} event E can be upper bounded as
Now that recalling Equation (B.1), Lemma 2, and Lemma 3 and considering all the situations, the overall outage probability can be written as
Depending on the variablerate strategy in Equation (34) of [8] and integrating the four transmission modes, the target transmission data rate R BPCU can be expressed as
One can also refer to [8] to get the mapping criterion from R to \stackrel{~}{R}. It is not difficult to prove the inequality P\left(\overline{E}\right)+P\left({A}_{2}\right)1\le P({A}_{2},\overline{E})\le P\left(\overline{E}\right), so P({A}_{2},\overline{E})\doteq P\left(\overline{E}\right) can be obtained in the largeSNR region. From Equation (A.2) and Lemma 3 and substituting R=r logρ, \stackrel{~}{R}=\stackrel{~}{r}log\rho, P({A}_{2},E)=P\left({A}_{2}\right)P({A}_{2},\overline{E}) into Equation (B.19), r can be revealed as
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).
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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].
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Xu, P., Ding, Z., Dai, X. et al. Alternative relaying for cooperative multipleaccess channels in wireless vehicular networks. J Wireless Com Network 2014, 165 (2014). https://doi.org/10.1186/168714992014165
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DOI: https://doi.org/10.1186/168714992014165