Performance analysis of relay selection in cooperative networks over Rayleigh flat fading channels
- Hanan Al-Tous^{1} and
- Imad Barhumi^{1}Email author
https://doi.org/10.1186/1687-1499-2012-224
© Al-Tous and Barhumi; licensee Springer. 2012
Received: 1 November 2011
Accepted: 18 June 2012
Published: 19 July 2012
Abstract
Performance analysis of an up-link cooperative diversity system is investigated; sharing of the two ordered best relays over Rayleigh flat fading channel is introduced to establish full diversity order for both users. The two users are competing for the same best relay, so assigning the best relay for one user, and the next-best relay for the other shows different diversity orders. The relays are ordered based on the end-to-end signal to noise ratio (SNR) of the source-relay-destination links. In this sense, relay selection is examined under different criteria. Mainly, the ordered best relay, the ordered next-best relay, and equally sharing the two best ordered relays. To this end, analytical expression, for the moment generating function (MGF) is derived, and used to find the probability density function (PDF), and the cumulative density function of the end-to-end SNR for decode and forward (DF) sharing scenario. Furthermore, the MGF of the upper bound of the end-to-end SNR for amplify and forward (AF) sharing scenario is also derived. Sharing the two ordered best relays shows better performance in the bit error probability (BEP) than using the next-best relay alone in DF relay systems, while exploiting the full diversity of the system. Sharing of the two ordered best relays in AF relay systems shows better BEP performance than using the best relay alone. Distributed space time block coding and distributed beamforming (BF) scenarios at the relays that utilize the bandwidth more efficiently are also explored. It is found that the BEP performance of the two ordered best relays distributed BF with equal power allocation for AF (at high SNR) and DF (for all SNRs) schemes outperforms the BEP performance of the best ordered relay alone. The BEP performance for the DF distributed BF scheme with equal power allocation approaches the BEP performance of the optimum power assignment under global power sum constraint. Numerical simulations are used to validate the analysis.
Keywords
Introduction
Cooperative wireless networks have been adopted in order to address the requested increase in capacity and to improve the wireless link performance. In cooperative communication, multiple relays are utilized to provide reliable, high data rate, and efficient communication. Recent activities on cooperative communication mostly stem from the potential of wireless applications and are motivated by many recent articles [1–4]. In most practical cooperative diversity protocols, transmission is accomplished in two phases: a broadcast phase and a multi-access phase. In the broadcast phase, the source node broadcasts its message to the assisting relays and to the destination node. Whereas in the second phase, the relays collaboratively transmit the received information to the destination node. Two relaying schemes are generally used for cooperative diversity networks: amplify-and-forward (AF) and decode-and-forward (DF). In AF, the received noisy message is amplified and forwarded to the destination node. The destination node combines the information sent by the user (source) and the partner relay(s), and makes a final decision on the transmitted symbol. Although noise is amplified by cooperation, the base station receives two or more independently faded versions of the signal and can make better decision on the detection of the transmitted information. In DF, the received noisy message at the relay is decoded first, and then the relay re-encodes the decoded message, and forwards it to the destination node. In the case of accurate symbol estimation, DF outperforms AF relaying, where the relay can reliably decode the source node message as the noise will not be amplified [5–8]. Diversity order two is achieved at high signal to noise ratio (SNR) using one relay for AF and adaptive DF schemes, using maximum ratio combining (MRC) technique to demodulate the received signals. In adaptive DF, the relay transmits if it decodes the message correctly (e.g., by using cyclic redundancy check), or otherwise keeps silent [9, 10]. In common cooperative diversity networks with _{12}N relaying nodes, N + 1 orthogonal channels or time slots are used to provide N + 1 diversity order, which encounters bandwidth penalty [11–14]. In [15], opportunistic best relay selection is used to utilize the resources efficiently; only two channels or time slots are required despite the number of relays, while maintaining a full diversity order N + 1, when the best relay is only used. Laneman et al. [3] proved that best relay selection achieves full diversity order for AF and DF cooperation schemes under flat fading Rayleigh channels. In there, the performance was measured in terms of the outage events and the associated outage probabilities. Different criteria for relay selection are investigated with their achievable diversity order in [16]. The average symbol error probability (SEP) of the cooperative system is used to analyze the performance of various systems and channel models as in [8].
In [17], Ikki and Ahmed investigated the k th ordered best relay and proved that the diversity order increases with the number of relays and decreases linearly with the order of the relay, i.e., the best ordered relay is denoted as k=0 that achieves full diversity order N + 1, whereas the next-best ordered relay is denoted as k=1 which achieves a diversity order N. Closed form expressions for the error and outage probabilities over identical and non-identical Rayleigh fading channels were derived. Single and multiple relay selection schemes were investigated. Several SNR sub-optimum multiple relay selection schemes with linear complexity in the number of relays were used [16]. Multiple potential relays and multiple simultaneous transmissions were introduced; where each source pairs with a single best relay, the outage probability of the proposed scheme is derived in [18]. In [19], joint selection scheme in multi-source multi-relay networks was considered, by selecting the best source node and the best relay to access the channel. Diversity of multiuser two-hop cooperative relay network was developed for different relaying protocols and tight closed-form expressions for the outage probability and the SEP were derived [20].
In this article, we consider the following scenario. A selected best relay of one user is the same best relay for another user in a power limited relay system. The probability of two users competing for the same relay is significant and comparable to the probability of having different best relays for a multi-user system with similar average channel conditions as proved in [18]. This case becomes visible in multi-user scenarios, or multi-carrier systems like OFDMA. In this sense, two solutions were proposed in literature; the first solution is to use the best relay for user one, and the next-best relay for the other user. So, it is transformed to the k th best ordered relay selection problem, which was investigated in [17, 21–23], by this solution the users achieve different diversity orders. The second solution is to use the best relay for the two users with half power for each user if this choice gives better performance than using the next-best relay alone with full power, which was proposed in [18]. However, no analytical results for the SEP, or outage probability have been derived because of the mathematical complexity. In summary, the two solutions stand on using only one relay for each user.
In here, we propose a different scheme, in which the best and the next-best relays are equally shared between the two users/carriers. The relays are ordered based on the instantaneous end-to-end SNR of the source-relay-destination links. Each user will have the chance to use the best and the next-best ordered relays in a predetermined manner. Equal opportunity of using the best relay is the fairground to build up this scheme. This solution is motivated by the observation that the bit error probability (BEP) performance of the two best order relays AF systems with half power for each relay in a three-time slots scenario outperforms the best relay BEP performance. The three-time-slots are used as follows: the first-time-slot is used to transmit the sources’ data to the relays and the destination, the second-time-slot is used to relay the processed data from the best relay to the destination, and the third-time-slot is used to relay the processed data from the next-best relay to the destination. In addition, sharing the two ordered best relays for both AF and DF cooperative schemes for independent identically flat fading Rayleigh channels, utilizing two-time slots using distributed space time block coding (STBC) or distributed beamforming (BF) are also investigated to exploit the channel efficiently. The novelty of this work, is that relays are selected to attain full diversity order for both users. Our scheme places full diversity selection at the core of the design scheme, and takes into consideration the limited available power at the relay. The available power at the relay is used to support the two users, which is equally split between the two users.
The moment generating function (MGF) formula of the received SNR of the two ordered best relays (the best, and the next-best) after using MRC is derived assuming equal power sharing in which each relay transmits with half power. Optimal power allocation for the two users could also be used, however, the problem is not analytically tractable for AF. Equal power allocation for orthogonal sharing DF is a fair allocation that maximizes the minimum end-to-end SNR of the two users over the source-relay-destination links. Exact expressions for the BEP, and the outage probability are derived based on the MGF, which can also be used for performance evaluation or comparison purposes for the following cases: orthogonal DF, STBC-DF, and BF-DF. Upper and lower bounds of the end-to-end SNR for orthogonal AF, and BF-AF are also derived. The BEP performance of sharing the two ordered best relays is compared with the BEP performance of the best, and the next-best ordered relays using simulations.
The remaining of the article is organized as follows. Section “System and channel model” presents the system model. Analytical expressions of the bit error and outage probabilities for the three-time slots scenarios are derived in Section “Scenario one (orthogonal three-time slots scenario)”. Analytical expressions of the BEP for the two-time slots scenarios are derived in Section “Scenario two (two-time slots scenario)”. Numerical results and conclusions are presented in Section “Numerical results and discussion” and Section “Conclusions”, respectively.
System and channel model
For relay selection schemes, the channel characterizing the link between the source S_{ j } and the i th ordered best relay R_{ bi }is denoted as ${h}_{{S}_{j}R}^{\left(i\right)}$, and the channel characterizing the link between the i th ordered best relay and the destination is denoted as ${h}_{\mathrm{RD}}^{\left(i\right)}$, where i∈{0,1}. In all scenarios, sources and relays have the same power capability. In relay selection, we assume that the best relay is the same for the two users, but the next-best relay may not be the same. However, this will not affect the analysis. For distributed STBC scenario, the best and the next-best relays are assumed to be the same for the two users in order to implement this scenario.
In this article, time division multiple access (TDMA) is considered. For a two-users case, each time slot is divided into two time sub-slots. Frequency division multiple access (FDMA) can also be considered in a similar fashion, where each frequency sub-band in FDMA corresponds to a time sub-slot in TDMA. The sharing scenarios are classified into two categories: three-time slots scenario (orthogonal), and two-time slots scenario (non-orthogonal). In both categories, the first-time slot is used for the sources’ transmission to the destination node and the relay nodes (broadcast phase). Orthogonal here refers to relay transmission, where the best relay transmits in the second-time slot, and the next-best relay transmits in the third-time slot (no interference). The second and third time slots are subdivided into T_{k 1}, and T_{k 2} for k∈{2,3} to transmit user’s 1 and 2 data, respectively. For two-time slots scenario, distributed STBC or distributed BF transmission schemes are used to relay the data from the two best ordered relays to the destination node for the two users simultaneously. Non-orthogonal here refers to the transmission in the second-time slot; the best, and the next-best relays transmit at the same time. Sub-slot T_{2j} is used by the best and next-best relays for transmitting user’s j data simultaneously for BF scenarios. The instantaneous value of the phase of the channel state information (CSI) of the source-relay and the relay-destination links are required to be available at the best, and the next-best relays to perform distributed BF. For distributed STBC the best and next-best relays transmit the re-encoded signal or the complex conjugate of the the re-encoded signal in a predetermined way as will be explained in Section “DF with distributed STBC (STBC-DF)”. The destination node for all scenarios, combines the directed and the relayed signals using MRC, where the received signals from all independent paths are co-phased, weighted, and combined, assuming the destination knows the instantaneous CSI from the sources, and the relays.
The instantaneous CSI is kept invariant over multiple transmission intervals. So, the selection of the best, and the next-best relays is performed once for multiple transmissions. Relay selection is performed before data transmission. The best relay and next-best relay can be determined for both AF and DF in a centralized or distributed fashion, depending on where the decision is carried out. In centralized relay selection, the destination node based on the end-to-end SNR of the source-relay-destination links, determines the best, and the next-best relays, and informs the selected relays through feedback channels. In distributed relay selection, each relay acquires the instantaneous CSI of the two links (relay-destination, and source-relay), the CSI of the relay-destination link can be acquired by allowing the destination to transmit a pilot signal. The relay then can determine the CSI of the relay-destination link assuming that the relay-destination link is symmetric. Besides, the CSI of the source-relay link can be determined at the relay from the source request to transmit. Based on CSI of the two links (relay-destination, and source-relay), the relay sets a timer and remains silent inversely proportional to the end-to-end SNR. The relay whose timer expires first or second will broadcast a signal to other relays, indicating that they can go to a sleep mode for the rest of the current transmission period. If the relay receives two signals from other relays before its timer goes to zero, it can go to a sleep mode, otherwise, it will be the best relay or the next-best relay [15, 24].
Scenario one (orthogonal three-time slots scenario)
Three-time slots scenario
T _{11} | T _{12} | T _{21} | T _{22} | T _{31} | T _{32} | |
---|---|---|---|---|---|---|
S_{1} | ${T}_{{x}_{1}}$ | – | – | – | – | – |
S_{2} | – | ${T}_{{x}_{2}}$ | – | – | – | – |
R_{b0} | ${R}_{{x}_{1}}$ | ${R}_{{x}_{2}}$ | ${T}_{{x}_{1}}$ | ${T}_{{x}_{2}}$ | – | – |
R_{b1} | ${R}_{{x}_{1}}$ | ${R}_{{x}_{2}}$ | – | – | ${T}_{{x}_{1}}$ | ${T}_{{x}_{2}}$ |
D | ${R}_{{x}_{1}}$ | ${R}_{{x}_{2}}$ | ${R}_{{x}_{1}}$ | ${R}_{{x}_{2}}$ | ${R}_{{x}_{1}}$ | ${R}_{{x}_{2}}$ |
where ${n}_{1j}^{\left(D\right)}$, ${n}_{1j}^{\left({R}_{l}\right)}$, and ${n}_{1j}^{\left({R}_{\mathrm{bi}}\right)}$ are the additive noise at the destination D, at the the relays R_{l}, and at the best ordered relays R_{bi}, respectively. The relays are ordered based on the end-to-end SNR of the source-relay-destination links as explained for DF as well as for AF scenarios in Sections “DF orthogonal three-time slots scenario” and “AF orthogonal three-time slots scenario”, respectively. The transmitted symbol x_{j} is drawn from a constellation with unit energy, P_{S}is the source transmitted power. The instantaneous received SNR at the destination node from the source S_{j}through the direct link over Rayleigh flat fading channel is defined as ${\gamma}_{{S}_{j}D}$. It is computed using (1) as ${\gamma}_{{S}_{j}D}={\gamma}_{0}|{h}_{{S}_{j}D}{|}^{2}$, where ${\gamma}_{0}=\frac{{P}_{S}}{{N}_{0}}$. ${\gamma}_{{S}_{j}D}$ is a random variable exponentially distributed with parameter ${\lambda}_{{S}_{j}D}$. To simplify the forthcoming analysis, the source-destination links of users 1 and 2 are assumed iid, i.e., ${\lambda}_{{S}_{1}D}={\lambda}_{{S}_{2}D}={\lambda}_{\mathrm{SD}}=\frac{1}{{\gamma}_{0}E\left\{\right|{h}_{{S}_{j}D}{|}^{2}\}}$, where E{·} stands for statistical expectation. Similarly, the instantaneous received SNR at the relay R_{l} from the jth source is computed using (2) as ${\gamma}_{{S}_{j}{R}_{l}}={\gamma}_{0}|{h}_{{S}_{j}{R}_{l}}{|}^{2}$, where ${\gamma}_{{S}_{j}{R}_{l}}$ is also exponentially distributed random variable with parameter ${\lambda}_{{S}_{j}{R}_{l}}$. The source relay links are assumed to have the same average ${\lambda}_{{S}_{j}{R}_{l}}={\lambda}_{\mathrm{SR}}=\frac{1}{{\gamma}_{0}E\left\{\right|{h}_{{S}_{j}{R}_{l}}{|}^{2}\}}$, $\forall l\in \{1,\dots ,N\}$, and $\forall j\in \{1,2\}$. For an exponentially distributed random variable X with parameter $\lambda $, the mean is given as ${\mu}_{X}=E\left\{X\right\}=\frac{1}{\lambda}$.
The transmission in the second, and the third-time slots depend on the cooperation scheme, DF or AF. The three-time slots DF scenario is investigated in Section “DF orthogonal three-time slots scenario”, and the three-time slots AF scenario is investigated in Section “AF orthogonal three-time slots scenario”.
DF orthogonal three-time slots scenario
where ${\gamma}_{\mathrm{sum}}^{\left(\mathrm{DF}\right)}$, is defined as ${\gamma}_{\mathrm{sum}}^{\left(\mathrm{DF}\right)}={\gamma}_{b0}^{\left(\mathrm{DF}\right)}+{\gamma}_{b1}^{\left(\mathrm{DF}\right)}$, and ${\gamma}_{b0}^{\left(\mathrm{DF}\right)}$, ${\gamma}_{b1}^{\left(\mathrm{DF}\right)}$ are the instantaneous end-to-end SNR of the best and the next-best relays, respectively. The factor $\frac{1}{2}$ in (5) is due to the fact that the best and next-best relays are shared between the two users with equal power, where ${P}_{R}^{\left(i\right)}={P}_{S}/2$ for i∈{0,1}.
The best relay R_{b0}is the relay with the maximum instantaneous end-to-end SNR at the destination node, i.e., ${\gamma}_{b0}^{\left(\mathrm{DF}\right)}=\underset{l}{\text{max}}\left({\gamma}_{0}\right|{h}_{{R}_{l}D}{|}^{2})$. The next-best relay R_{b1}is the relay with the next-maximum instantaneous end-to-end SNR at the destination node, i.e., ${\gamma}_{b1}^{\left(\mathrm{DF}\right)}=\underset{l,\phantom{\rule{2.77695pt}{0ex}}l\ne b0}{\text{max}}\left({\gamma}_{0}\right|{h}_{{R}_{l}D}{|}^{2})$, where l=1,…,N (l is used as an index for the relay without ordering). The selection of the best relay R_{b0} and next-best relay R_{b1}from the N available relays is determined by ordering the instantaneous end-to-end SNRs from the N relays as follows ${\gamma}_{b0}^{\left(\mathrm{DF}\right)}>{\gamma}_{b1}^{\left(\mathrm{DF}\right)}>{\gamma}_{b2}^{\left(\mathrm{DF}\right)}>\cdots >{\gamma}_{\mathrm{bN}-1}^{\left(\mathrm{DF}\right)}$.^{a} In the following, the probability density function (PDF) and the MGF of the end-to-end SNR γ^{(0.5D)} are derived in order to evaluate the BEP and outage probability performances of the proposed scenario.
where μ(x) is the unit step function, δ(x) is Dirac’s delta function, and λ_{RD}is the parameter of the exponential random variable characterizing the received SNR at the destination node from the relay. All relay-destination links are assumed to be iid random variables with ${\lambda}_{\mathrm{RD}}=\frac{1}{{\gamma}_{0}E\left\{\right|{h}_{{R}_{l}D}{|}^{2}\}}$.
Coefficients of the PDF, the MGF, the BEP, and the P _{out} for DF sharing scheme
Coeff. | Value | Coeff. | Value |
---|---|---|---|
A^{(DF)} | $\sum _{k=3}^{N}{\lambda}_{\text{RD}}{c}_{k}{(1-\beta )}^{k-1}\left(\beta +\frac{1-\beta}{k-2}\right)$ | a^{(DF)} | $\frac{1}{2{\lambda}_{\text{RD}}}\left(\frac{2{A}^{\text{(DF)}}{\lambda}_{\text{SD}}}{-2{\lambda}_{\text{RD}}+{\lambda}_{\text{SD}}}-\frac{4{B}^{\text{(DF)}}{\lambda}_{\text{SD}}}{{(-2{\lambda}_{\text{RD}}+{\lambda}_{\text{SD}})}^{2}}\right)$ |
+ c_{2}(1−β)βλ_{RD} | |||
B^{(DF)} | $\frac{{c}_{2}}{2}{(1-\beta )}^{2}{\lambda}_{\text{RD}}^{2}$ | b^{(DF)} | $\frac{1}{4{\lambda}_{\text{RD}}^{2}}\left(\frac{4{B}^{\text{(DF)}}{\lambda}_{\text{SD}}}{-2{\lambda}_{\text{RD}}+{\lambda}_{\text{SD}}}\right)$ |
D^{(DF)} | $\sum _{k=2}^{N}{c}_{k}{\beta}^{2}{(1-\beta )}^{k-2}$ | f^{(DF)} | $\frac{2{A}^{\text{(DF)}}}{2{\lambda}_{\text{RD}}-{\lambda}_{\text{SD}}}+\frac{4{B}^{\text{(DF)}}}{{(2{\lambda}_{\text{RD}}-{\lambda}_{\text{SD}})}^{2}}+$ |
$\sum _{k=3}^{N}\frac{2{E}_{k}^{\text{(DF)}}}{{\lambda}_{\text{RD}}k-{\lambda}_{\text{SD}}}+{D}^{\text{(DF)}}$ | |||
${E}_{k}^{\text{(DF)}}$ | $\frac{{c}_{k}}{2}{(1-\beta )}^{k}\left(\frac{-2}{k-2}\right){\lambda}_{\text{RD}}$ | ${E}_{k}^{\text{(DF)}}$ | $\frac{1}{{\lambda}_{\text{RD}}k}\left(\frac{2{E}_{k}^{\text{(DF)}}{\lambda}_{\text{SD}}}{-{\lambda}_{\text{RD}}k+{\lambda}_{\text{SD}}}\right)$ |
where a^{(DF)}, b^{(DF)}, f^{(DF)}, and ${e}_{k}^{\left(\mathrm{DF}\right)}$ (assuming that λ_{SD} ≠ λ_{RD} or multiple of it for simplicity, but the analysis can be easily extended), are given as in column 4 of Table 2.
The diversity order of sharing the two ordered best relays can be investigated using asymptotic analysis of the BEP or the outage probability P_{out}at high SNR values [3, 13]. Another approach, is to use the asymptotic analysis of the PDF or the MGF of the end-to-end SNR [31–33]. We follow the latter approach using the MGF of the end-to-end SNR at the output of the MRC. Using the results of [31], the MGF can be approximated as s→∞ by b|s|^{−d} + O(|s|^{−d}), ^{b} where d is the diversity order, and b is related to the coding gain. Writing ${\Psi}_{{\gamma}^{\left(0.5\mathrm{DF}\right)}}\left(s\right)$ as a division of two polynomials ${\Psi}_{{\gamma}^{\left(0.5\mathrm{DF}\right)}}\left(s\right)=\frac{B\left(s\right)}{A\left(s\right)}$, where B(s) and A(s) are the numerator and denominator polynomials, respectively. A(s) can be written as, $A\left(s\right)={\left(1+\frac{s}{2{\lambda}_{\mathrm{RD}}}\right)}^{2}\left(1+\frac{s}{{\lambda}_{\mathrm{SD}}}\right)\prod _{k=3}^{N}\left(1+\frac{s}{{\lambda}_{\mathrm{RD}}k}\right)$, which can be approximated for s→∞ as $A\left(s\right)\approx {\left(\frac{s}{2{\lambda}_{\mathrm{RD}}}\right)}^{2}\left(\frac{s}{{\lambda}_{\mathrm{SD}}}\right)\prod _{k=3}^{N}\left(\frac{s}{{\lambda}_{\mathrm{RD}}k}\right)={\left(2{\lambda}_{\mathrm{RD}}{\lambda}_{\mathrm{SD}}\prod _{k=3}^{N}{\lambda}_{\mathrm{RD}}k\right)}^{-1}{s}^{(N+1)}$. The numerator polynomial can be found by collecting and combining the corresponding terms, which is clearly of degree less than the denominator polynomial. Taking only the constant term of the numerator polynomial, and divide this term by the approximation of the denominator polynomial results in the term b|s|^{−(N + 1)}. This means that the diversity order is N + 1. Other terms which result from the division of the numerator polynomial with the approximation of the denominator polynomial contribute to O(|s|^{−(N + 1)}).
AF orthogonal three-time slots scenario
where ${n}_{\mathrm{kj}}^{\left(\mathrm{AF}\right)}$ is the additive noise at the destination, and ${G}_{{S}_{j}R}^{\left(i\right)}$ is the normalizing factor at the relay, which depends on the instantaneous CSI between the jth source and the ith ordered best relay. Assuming that each relay knows its instantaneous channel information ${h}_{{S}_{j}R}^{\left(i\right)}$, the normalizing factor using (3) is ${G}_{{S}_{j}R}^{\left(i\right)}=\frac{1}{\sqrt{{P}_{S}{\left|{h}_{{S}_{j}R}^{\left(i\right)}\right|}^{2}+{N}_{0}}}$.
where ${\gamma}_{\mathrm{sum}}^{\left(0.5\mathrm{AF}\right)}={\gamma}_{b0}^{\left(0.5\mathrm{AF}\right)}+{\gamma}_{b1}^{\left(0.5\mathrm{AF}\right)}$ with ${\gamma}_{b0}^{\left(0.5\mathrm{AF}\right)}$ and ${\gamma}_{b1}^{\left(0.5\mathrm{AF}\right)}$ are the upper bound of the end-to-end SNR from the best, and the next-best relays respectively. The best relay R_{b0}is selected as the relay with the maximum upper bound of the end-to-end SNR ${\gamma}_{b0}^{\left(0.5\mathrm{AF}\right)}$ at the the destination node, i.e., ${\gamma}_{b0}^{\left(0.5\mathrm{AF}\right)}=\underset{l}{\text{max}}\left(\text{min}({\gamma}_{{S}_{j}{R}_{l}},\frac{{\gamma}_{{R}_{l}D}}{2})\right)$, where l=1,…,N. Similarly, the next-best relay R_{b1}is selected as the relay with the next-maximum upper bound of the end-to-end SNR ${\gamma}_{b1}^{\left(0.5\mathrm{AF}\right)}$ at the the destination node, i.e., ${\gamma}_{b1}^{\left(0.5\mathrm{AF}\right)}=\underset{l,\phantom{\rule{2.77695pt}{0ex}}l\ne b0}{\text{max}}\left(\text{min}({\gamma}_{{S}_{j}{R}_{l}},\frac{{\gamma}_{{R}_{l}D}}{2})\right)$. The selection of the best and next-best relays R_{b0}and R_{b1}, respectively from the N available relays is done using the ordering of the upper bound of end-to-end SNRs from the N available relays as: ${\gamma}_{b0}^{\left(0.5\mathrm{AF}\right)}>{\gamma}_{b1}^{\left(0.5\mathrm{AF}\right)}>{\gamma}_{b2}^{\left(0.5\mathrm{AF}\right)}>\cdots >{\gamma}_{\mathrm{bN}-1}^{\left(0.5\mathrm{AF}\right)}$.
Coefficients of the PDF, the MGF, the BEP, and the P _{out} for AF sharing scheme
Coeff. | Value | Coeff. | Value |
---|---|---|---|
A^{(AF)} | $\sum _{k=3}^{N}{c}_{k}{\lambda}_{\text{eq}}\frac{1}{k-2}$ | a^{(AF)} | $\frac{1}{{\lambda}_{\text{eq}}}\left(\frac{{A}^{\text{(AF)}}{\lambda}_{\text{SD}}}{-{\lambda}_{\text{eq}}+{\lambda}_{\text{SD}}}-\frac{{B}^{\text{(AF)}}{\lambda}_{\text{SD}}}{{(-{\lambda}_{\text{eq}}+{\lambda}_{\text{SD}})}^{2}}\right)$ |
B^{(AF)} | $\frac{{c}_{2}{\lambda}_{\text{eq}}^{2}}{2}$ | b^{(AF)} | $\frac{1}{{\lambda}_{\text{eq}}^{2}}\left(\frac{{B}^{\text{(AF)}}{\lambda}_{\text{SD}}}{-{\lambda}_{\text{eq}}+{\lambda}_{\text{SD}}}\right)$ |
${E}_{k}^{\text{(AF)}}$ | $-\frac{{c}_{k}{\lambda}_{\text{eq}}}{k-2}$ | ${E}_{k}^{\text{(AF)}}$ | $\frac{2}{{\lambda}_{\text{eq}}k}\left(\frac{{E}_{k}^{\text{(AF)}}{\lambda}_{\text{SD}}}{-\frac{{\lambda}_{\text{eq}}k}{2}+{\lambda}_{\text{SD}}}\right)$ |
f^{(AF)} | $\frac{{A}^{\text{(AF)}}}{{\lambda}_{\text{eq}}-{\lambda}_{\text{SD}}}+\frac{{B}^{\text{(AF)}}}{{({\lambda}_{\text{eq}}-{\lambda}_{\text{SD}})}^{2}}+\sum _{k=3}^{N}\frac{{E}_{k}^{\text{(AF)}}}{\left(\frac{{\lambda}_{\text{eq}}k}{2}-{\lambda}_{\text{SD}}\right)}$ |
The diversity order of sharing the two ordered best relays for AF scheme is also N + 1, which can be found from the similarity between BEP_{γ(0.5AF)} for AF sharing (30) and the BEP_{γ(0.5DF)}for DF sharing (22).
Best relay selection criterion for AF scheme
The case | The best relay | The best relay | Comments |
---|---|---|---|
with sharing | without sharing | ||
${\gamma}_{{S}_{j}{R}_{l}}<\frac{{\gamma}_{{R}_{l}D}}{2}<{\gamma}_{{R}_{l}D}$ | $\underset{l}{\text{max}}\left({\gamma}_{{S}_{j}{R}_{l}}\right)$ | $\underset{l}{\text{max}}\left({\gamma}_{{S}_{j}{R}_{l}}\right)$ | Same |
$\frac{{\gamma}_{{R}_{l}D}}{2}<{\gamma}_{{S}_{j}{R}_{l}}<{\gamma}_{{R}_{l}D}$ | $\underset{l}{\text{max}}\left(\frac{{\gamma}_{{R}_{l}D}}{2}\right)$ | $\underset{l}{\text{max}}\left({\gamma}_{{S}_{j}{R}_{l}}\right)$ | Different |
$\frac{{\gamma}_{{R}_{l}D}}{2}<{\gamma}_{{R}_{l}D}<{\gamma}_{{S}_{j}{R}_{l}}$ | $\underset{l}{\text{max}}\left(\frac{{\gamma}_{{R}_{l}D}}{2}\right)$ | $\underset{l}{\text{max}}\left({\gamma}_{{R}_{l}D}\right)$ | Different |
Scenario two (two-time slots scenario)
So far, the transmission schemes discussed do not utilize the resources efficiently. The sources and the relays need to wait for three-time slots to start new transmission. In this section we discuss more efficient transmission schemes where only two time slots are required. Three types of such transmission schemes are discussed next. These schemes are DF with distributed STBC, DF, and AF with distributed BF.
DF with distributed STBC (STBC-DF)
Distributed STBC-DF scenario
T _{11} | T _{12} | T _{21} | T _{22} | |
---|---|---|---|---|
S_{1} | ${T}_{{x}_{1}}$ | – | – | – |
S_{2} | – | ${T}_{{x}_{2}}$ | – | – |
R_{b0} | ${R}_{{x}_{1}}$ | ${R}_{{x}_{2}}$ | ${T}_{{\stackrel{~}{x}}_{1}}$ | ${T}_{-{\stackrel{~}{x}}_{2}^{\ast}}$ |
R_{b1} | ${R}_{{x}_{1}}$ | ${R}_{{x}_{2}}$ | ${T}_{{\stackrel{~}{x}}_{2}}$ | ${T}_{{\stackrel{~}{x}}_{1}^{\ast}}$ |
D | ${R}_{{x}_{1}}$ | ${R}_{{x}_{2}}$ | ${R}_{{T}_{22}}^{\text{(DS)}}$ | ${R}_{{T}_{23}}^{\text{(DS)}}$ |
Defining ${\gamma}_{\mathrm{sum}}^{\left(\mathrm{DS}\right)}={\gamma}_{b0}^{\left(\mathrm{DS}\right)}+{\gamma}_{b1}^{\left(\mathrm{DS}\right)}$, the end-to-end SNR at the destination node after using MRC is obtained as ${\gamma}^{\left(\mathrm{DS}\right)}={\gamma}_{{S}_{j}D}+{\gamma}_{\mathrm{sum}}^{\left(\mathrm{DS}\right)}$, which is similar to the SNR expression obtained in (5). Hence, the same analysis can be carried out; the BEP performance is the same as (22). The goal of this analysis is not to investigate distributed STBC, but to examine the BEP performance of sharing the two ordered best relays. Detailed analysis of distributed STBC for multi-relay systems using pairwise error probability can be found in [37].
The two best order relays STBC-AF requires more investigation for the selection criterion, and the amplification gain at the relays using the instantaneous CSI. This however, is outside the scope of this article and will be a subject for further investigation.
Distributed BF for DF scheme
Distributed BF scenarios
T _{11} | T _{12} | T _{21} | T _{22} | |
---|---|---|---|---|
S_{1} | ${T}_{{x}_{1}}$ | – | – | – |
S_{2} | – | ${T}_{{x}_{2}}$ | – | – |
R_{b0} | ${R}_{{x}_{1}}$ | ${R}_{{x}_{2}}$ | ${T}_{{\stackrel{~}{x}}_{1}}$ / ${T}_{{x}_{1}}$ | ${T}_{{\stackrel{~}{x}}_{2}}$ / ${T}_{{x}_{2}}$ |
R_{b1} | ${R}_{{x}_{1}}$ | ${R}_{{x}_{2}}$ | ${T}_{{\stackrel{~}{x}}_{1}}$ /${T}_{{x}_{1}}$ | ${T}_{{\stackrel{~}{x}}_{2}}$ / ${T}_{{x}_{2}}$ |
D | ${R}_{{x}_{1}}$ | ${R}_{{x}_{2}}$ | ${R}_{{T}_{21}}^{\text{(BD)}}$ / ${R}_{{T}_{21}}^{\text{(BA)}}$ | ${R}_{{T}_{22}}^{\text{(BD)}}$ / ${R}_{{T}_{22}}^{\text{(BA)}}$ |
Coefficients of the PDF, the MGF and the BEP for the upper and the lower bounds of BD
Coeff. | S = U | S = L |
---|---|---|
A ^{(S)} | $\frac{{\lambda}_{\text{RD}}}{3}\sum _{k=2}^{N}{c}_{k}{(1-\beta )}^{k-1}\left(\frac{1-\beta}{k-\frac{4}{3}}+\beta \right)$ | ${c}_{4}{(1-\beta )}^{3}\beta {\lambda}_{\text{RD}}+{\lambda}_{\text{RD}}\sum _{\underset{k\ne 4}{k=2}}^{N}{c}_{k}{(1-\beta )}^{k-1}\left(\frac{1-\beta}{k-4}+\beta \right)$ |
B^{(S)} | - | $\frac{{c}_{4}{(1-\beta )}^{4}{\lambda}_{\text{RD}}^{2}}{4}$ |
${E}_{k}^{\left(S\right)}$ | $\frac{{c}_{k}{(1-\beta )}^{k}{\lambda}_{\text{RD}}}{4}\left(1-\frac{k}{k-\frac{4}{3}}\right)$ | $\frac{{c}_{k}{(1-\beta )}^{k}{\lambda}_{\text{RD}}}{4}\left(1-\frac{k}{k-\frac{4}{3}}\right),\phantom{\rule{2.36043pt}{0ex}}k\ne 4$ |
a^{(S)} | $\frac{3}{2{\lambda}_{\text{RD}}}\left(\frac{2{A}^{\left(S\right)}{\lambda}_{\text{SD}}}{\frac{-2{\lambda}_{\text{RD}}}{3}+{\lambda}_{\text{SD}}}\right)$ | $\frac{1}{2{\lambda}_{\text{RD}}}\left(\frac{2{A}^{\left(S\right)}{\lambda}_{\text{SD}}}{-2{\lambda}_{\text{RD}}+{\lambda}_{\text{SD}}}-\frac{4{B}^{\left(S\right)}{\lambda}_{\text{SD}}}{{(-2{\lambda}_{\text{RD}}+{\lambda}_{\text{SD}})}^{2}}\right)$ |
b^{(S)} | - | $\frac{1}{4{\lambda}_{\text{RD}}^{2}}\left(\frac{4{B}^{\left(S\right)}{\lambda}_{\text{SD}}}{-2{\lambda}_{\text{RD}}+{\lambda}_{\text{SD}}}\right)$ |
${e}_{k}^{\left(S\right)}$ | $\frac{2}{{\lambda}_{\text{RD}}k}\left(\frac{2{E}_{k}^{\left(S\right)}{\lambda}_{\text{SD}}}{-\frac{{\lambda}_{\text{RD}}k}{2}+{\lambda}_{\text{SD}}}\right)$ | $\frac{2}{{\lambda}_{\text{RD}}k}\left(\frac{2{E}_{k}^{\left(S\right)}{\lambda}_{\text{SD}}}{\frac{-{\lambda}_{\text{RD}}k}{2}+{\lambda}_{\text{SD}}}\right)$ |
f^{(S)} | $\frac{2{A}^{\left(S\right)}}{\frac{2{\lambda}_{\text{RD}}}{3}-{\lambda}_{\text{SD}}}+\sum _{k=2}^{N}\frac{2{E}_{k}^{\left(S\right)}}{(\frac{{\lambda}_{\text{RD}}k}{2}-{\lambda}_{\text{SD}})}+{D}^{\text{(DF)}}$ | $\frac{2{A}^{\left(S\right)}}{2{\lambda}_{\text{RD}}-{\lambda}_{\text{SD}}}+\frac{4{B}^{\left(S\right)}}{{(2{\lambda}_{\text{RD}}-{\lambda}_{\text{SD}})}^{2}}+\underset{k\ne 4}{\underset{23DF}{\sum _{k=2}^{N}}}\frac{2{E}_{k}^{\left(S\right)}}{\frac{{\lambda}_{\text{RD}}k}{2}-{\lambda}_{\text{SD}}}+{D}^{\text{(DF)}}$ |
where A^{(L)}, B^{(L)}, ${E}_{k}^{\left(L\right)}$, a^{(L)}, f^{(L)}, and ${E}_{k}^{\left(L\right)}$ are as defined in column 3 in Table 7.
where ${\gamma}_{\mathrm{Op}}^{\left(\mathrm{BD}\right)}$ is defined as the maximum received SNR using optimum power assignment for distributed BF scenario using the best and the next-best relays under the constraint ${P}_{R}^{\left(0\right)}+{P}_{R}^{\left(1\right)}={P}_{S}$. This upper bound can also be used to compute the ${\mathrm{BEP}}_{\mathrm{Op}}^{\left(\mathrm{BD}\right)}$ using (22) by replacing 2λ_{RD}with λ_{RD} in all terms. Hence, the BEP^{(BD)} can be lower bounded by ${\mathrm{BEP}}_{\mathrm{Op}}^{\left(\mathrm{BD}\right)}$, i.e., ${\mathrm{BEP}}_{\mathrm{Op}}^{\left(\mathrm{BD}\right)}\le {\mathrm{BEP}}^{\left(\mathrm{BD}\right)}$. However, in this article we are only concerned with equal power sharing to simplify the analysis.
Distributed BF for AF scheme
where n_{0j}, n_{1j} and ${n}_{2j}^{\left(\mathrm{BA}\right)}$ are the additive noise at the best and next-best relays in time slot T_{1}and at the destination in time slot T_{2}, respectively.
where ${\gamma}_{\mathrm{Op}}^{\left(\mathrm{BA}\right)}$ is defined as the maximum received SNR using optimum power assignment for the best, and the next-best ordered relays under the constraint ${P}_{R}^{\left(0\right)}+{P}_{R}^{\left(1\right)}={P}_{S}$. ${\gamma}_{{b}_{0}}^{\left(\mathrm{BA}\right)}=\underset{l}{\text{max}}\left(\text{min}({\gamma}_{{S}_{j}{R}_{l}},{\gamma}_{{R}_{l}D})\right)$, and ${\gamma}_{{b}_{1}}^{\left(\mathrm{BA}\right)}=\underset{l,l\ne {b}_{0}}{\text{max}}\left(\text{min}({\gamma}_{{S}_{j}{R}_{l}},{\gamma}_{{R}_{l}D})\right)$. Hence, the lower bound of the ^{(BA)} can be found using (30) by replacing λ_{eq} = 2λ_{SR} + λ_{RD}with λ_{eq} = λ_{SR} + λ_{RD}.
It is worth noting that the weights for BF AF and DF are chosen as w_{1}=1 and w_{2}=1 for the following reasons. First, the weights w_{1}and w_{2} can be considered as power adjustment factors, and in this study optimal power adjustment is not considered, only equal power assignment is investigated for all the scenarios. Second, the weights are not similar to the weights given by [16, 38] which were derived for one user scenario. The weights for this problem need to be selected to maximize the minimum received SNR of the two users under individual relay power constraint and user sum power constraint. Finally, if power adjustment is to be used, there is no benefit from relay ordering, relay ordering in this article is based on relays’ equal power transmission.
Numerical results and discussion
Figure 2 shows results of the analytical BEP performance for three channel conditions for DF with sharing the best and next best ordered relays. In setup 1,λ λ_{RD}λ_{SR}λ_{SD}λ_{RD}λ_{SR}λ_{SD}λ_{RD}λ_{SR}The BEP performance of channel setup 3 outperforms the BEP performance of the other setups. An SNR gain of 1.3022dB and 1.9301dB at BEP = 10^{−4}is observed compared to setup 2 and setup 1 respectively. It is well observed that, a better BEP performance is achieved when the source-relay and the relay-destination links are in better channel conditions than source-destination link.
Diversity calculations
Scheme | BEP_{1} | BEP_{2} | Diversity order |
---|---|---|---|
Best AF | 2.4777×10^{−8} | 8.7832×10^{−10} | 4.8181 |
Next-best AF | 1.0507×10^{−6} | 7.4121×10^{−8} | 3.8253 |
Sharing AF | 1.8862×10^{−8} | 6.2973×10^{−10} | 4.9046 |
Best DF | 3.3840×10^{−9} | 1.0909×10^{−10} | 4.9551 |
Next-best DF | 2.5017×10^{−7} | 1.6573×10^{−8} | 3.9160 |
Sharing DF | 6.6892×10^{−9} | 2.0673×10^{−10} | 5.0160 |
Conclusions
In this article, relay selection for cooperative networks is investigated. Assuming two users scenario, and N relays are available, BEP and outage probability are computed for the case of sharing the two ordered best relays for AF and DF relaying schemes. Analytical expressions for the BEP and the outage probability are derived for the different scenarios. Simulation results validate the analytical expressions of the BEP performance. The BEP performance of the proposed schemes were also compared with the BEP performances of the best and next-best relays. In DF scheme, the BEP performance of the best relay outperforms the BEP performance of the sharing scenario. But, the BEP performance of BF-DF scenario outperforms the BEP performance of the ordered best relay. Furthermore, the BEP performance of equally sharing the two best ordered relays for AF outperforms the BEP performance of the ordered best relay. Efficient channel utilization is achieved by using STBC and BF.
Endnotes
^{a}The decoding set C is a subset of the N available relays. If a relay is not in the decoding set, the end-to-end SNR value is set to zero.^{b}A function a(x) is written as O(x) if $\underset{x\to 0}{lim}\frac{a\left(x\right)}{x}=0$.
Declarations
Authors’ Affiliations
References
- Sendonaris A, Erkip E, Aazhang B: User cooperation diversity: part I system description. IEEE Trans. Commun 2003, 51(11):1939-1948. 10.1109/TCOMM.2003.819238View ArticleGoogle Scholar
- Sendonaris A, Erkip E, Aazhang B: User cooperation diversity: part II implementation aspects and performance analysis. IEEE Trans. Commun 2003, 51(11):1939-1948. 10.1109/TCOMM.2003.819238View ArticleGoogle Scholar
- Laneman J, Tse D, Wornell G: Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Trans. Inf. Theory 2004, 50(12):3062-3080. 10.1109/TIT.2004.838089MathSciNetView ArticleMATHGoogle Scholar
- Laneman J: Cooperative diversity in wireless networks: algorithms and architectures. Ph.d. thesis, Massachusetts Institute of Technology, Cambridge, MA, Aug 2002Google Scholar
- Hu J, Chen X: Performance of decode-and-forward cooperative communications with channel estimation errors over Rayleigh fading channels. 2nd International Conference on Future Computer and Communication (ICFCC), vol. 1 (2010), vol. 1 pp. 164–167Google Scholar
- Laneman J: Network coding gain of cooperative diversity. Military Communications Conference, MILCOM 2004, vol. 1, pp. 106–112Google Scholar
- Anghel P, Kaveh M: Exact symbol error probability of a cooperative network in a Rayleigh-fading environment. IEEE Trans. Wirel. Commun 2004, 3(5):1416-1421. 10.1109/TWC.2004.833431View ArticleGoogle Scholar
- Ribeiro A, Cai X, Giannakis G: Symbol error probabilities for general cooperative links. IEEE Trans. Wirel. Commun 2005, 4(3):1264-1273.View ArticleGoogle Scholar
- Vien N, Nguyen H, Le-Ngoc T: Diversity analysis of smart relaying. IEEE Trans. Veh. Technol 2009, 58(6):2849-2862.View ArticleMATHGoogle Scholar
- Wang T, Cano A, Giannakis G, Laneman JN: High-performance cooperative demodulation with decode-and-forward relays. IEEE Trans. Commun 2007, 55(7):1427-1438.View ArticleGoogle Scholar
- Ribeiro A, Cai X, Giannakis G: Opportunistic multipath for bandwidth-efficient cooperative multiple access. IEEE Trans. Wirel. Commun 2006, 5(9):2321-2327.View ArticleGoogle Scholar
- Torabi1 M, Ajib W, Haccoun D: Performance analysis of amplify-and-forward cooperative networks with relay selection over Rayleigh fading channels. IEEE 69th Vehicular Technology Conference (2009), pp. 1–5Google Scholar
- Zhao Y, Adve R, Lim T: Symbol error rate of selection amplify-and-forward relay systems. IEEE Commun. Lett 2006, 10(11):757-759.View ArticleGoogle Scholar
- Ikki S, Ahmed M: Performance of multiple-relay cooperative diversity systems with best relay selection over Rayleigh fading channels. EURASIP J. Adv. Signal Process. vol.1 2008, 145: 1-7.Google Scholar
- Bletsas A, Shin H, Win M: Cooperative communications with outage-optimal opportunistic relaying. IEEE Trans. Wirel. Commun 2007, 6(9):3450-3460.View ArticleGoogle Scholar
- Jing Y, Jafarkhani H: Single and multiple relay selection schemes and their achievable diversity orders. IEEE Trans. Wirel. Commun 2009, 8(3):1414-1423.View ArticleGoogle Scholar
- Ikki S, Ahmed M: On the performance of cooperative-diversity networks with the Nth best-relay selection scheme. IEEE Trans. Commun 2010, 58(11):3062-3069.View ArticleGoogle Scholar
- Beres E, Adve R: Selection cooperation in multi-source cooperative networks. IEEE Trans. Wirel. Commun 2008, 7(1):118-127.View ArticleGoogle Scholar
- Guo W, Liu J, Zheng L, Liu Y, Zhang G: Performance analysis of a selection cooperation scheme in multi-source multi-relay networks. International Conference on Wireless Communications and Signal Processing (WCSP) (2010), pp. 1–6Google Scholar
- Zhang X, Wang W, Ji X: Multiuser diversity in multiuser two-hop cooperative relay wireless networks: system model and performance analysis. IEEE Trans. Veh. Technol 2009, 58(2):1031-1036.View ArticleGoogle Scholar
- Ikki S, Ahmed M: On the performance of adaptive decode-and-forward cooperative diversity with the Nth best-relay selection scheme. IEEE Global Telecommunications Conference, GLOBECOM 2009 (2009), pp. 1–6Google Scholar
- Ikki S, Ahmed M: On the performance of amplify-and-forward cooperative diversity with the Nth best-relay selection scheme. IEEE International Conference on Communications, ICC ’09 (2009), pp. 1–6Google Scholar
- Yang C, Zhao S, Wang W, Peng M: Performance of decode-and-forward opportunistic cooperation with the Nth best relay selected. Proceedings of the 6th International Wireless Communications and Mobile Computing Conference IWCMC 2010. pp. 1253-1257Google Scholar
- Woradit K, Quek T, Suwansantisuk W, Wymeersch H, Wuttisittikulkij L, Win M: Outage behavior of selective relaying schemes. IEEE Trans. Wirel. Commun 2009, 8(8):3890-3895.View ArticleGoogle Scholar
- Ikki S, Ahmed M: Performance analysis of adaptive decode-and-forward cooperative diversity networks with best-relay selection. IEEE Trans. Commun 2010, 58(1):68-72.View ArticleGoogle Scholar
- Beaulieu N, Hu J: A closed-form expression for the outage probability of decode-and-forward relaying in dissimilar Rayleigh fading channels. IEEE Commun. Lett 2006, 10(12):813-815.View ArticleGoogle Scholar
- David H, Nagaraja H: Order Statistics,. (Wiley, New York, 2003)View ArticleMATHGoogle Scholar
- Ibe O: Fundamentals of Applied Probability and Random Processes. (Elsevier Academic Press, Amsterdam, 2005)MATHGoogle Scholar
- Papoulis A: Probability, Random Variables, and Stochastic Processes,. (McGraw-Hill, New York, 1991)MATHGoogle Scholar
- Simon M, M Alouini: Digital Communication Over Fading Channels. (Wiley, New York, 2000)View ArticleGoogle Scholar
- Wang Z, G Giannakis: A simple and general parameterization quantifying performance in fading channels. IEEE Trans. Commun 2003, 51(8):1389-1398. 10.1109/TCOMM.2003.815053View ArticleGoogle Scholar
- Annamalai A, Deora G, Tellambura C: Analysis of generalized selection diversity systems in wireless channels. IEEE Trans. Veh. Technol 2006, 55(6):1765-1775.View ArticleGoogle Scholar
- Liu Q, Ma X, Zhou G: A general diversity gain function and its application in amplify-and-forward cooperative networks. IEEE Trans. Signal Process 2011, 59(2):859-863.MathSciNetView ArticleGoogle Scholar
- Hasna M, Alouini M: End-to-end performance of transmission systems with relays over Rayleigh-fading channels. IEEE Trans. Wirel. Commun 2003, 2(6):1126-1131. 10.1109/TWC.2003.819030View ArticleGoogle Scholar
- Alamouti S: A simple transmit diversity technique for wireless communications. IEEE J Sel. Areas Commun 1998, 16(8):1451-1458. 10.1109/49.730453View ArticleGoogle Scholar
- Bai Z, Yuan D, Kwak K: Performance evaluation of STBC based cooperative systems over slow Rayleigh fading channel. Comput. Commun 2008, 31(17):4206-4211. http://www.sciencedirect.com/science/article/B6TYP-4TF7CB9-3/2/360a5d1dfe9a667ca91e1a2295786f17 10.1016/j.comcom.2008.09.010View ArticleGoogle Scholar
- Jing Y, Hassibi B: Distributed space-time coding in wireless relay networks. IEEE Trans. Wirel. Commun 2006, 5(12):3524-3536.View ArticleGoogle Scholar
- Larsson P, Rong H: Large-scale cooperative relay network with optimal coherent combining under aggregate relay power constraints. The Future Telecommunications Conference, Beijing FTC2003 2003, 166170-166170.Google Scholar
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