MIMO-STBC Based Multiple Relay Cooperative Communication Over

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MIMO-STBC Based Multiple Relay Cooperative Communication Over Time-Selective Rayleigh Fading Links With Imperfect Channel Estimates Neeraj Varshney, Student Member, IEEE, and Aditya K. Jagannatham, Member, IEEE

Abstract—This paper analyzes the effect of time-selective fading arising due to node mobility and imperfect channel estimates on the end-to-end performance of multiple-input multiple-output space-time block coded multiple relay cooperative communication systems. Both dual-phase and multi-phase selective decode-andforward relaying protocols are considered for the end-to-end communication in a multiple relay cooperative system, followed by presentation of complete analyses for the same. For each protocol, closed-form expressions are derived for the per-frame average pair-wise error probability and asymptotic error floor over independent and nonidentical time-selective Rayleigh fading links. A framework is also developed for obtaining the optimal source relay power factors for each of the above protocols, which significantly improve the end-to-end reliability of the system for a given power budget. It is shown that both the multirelay systems achieve the full diversity order (DO) when all the nodes are static and perfect channel estimates are available at each receiving node. The DO of the system reduces to that of the direct source–destination link for a scenario when only the relays are mobile. Interestingly, however, in other mobile scenarios, both the systems are limited by an asymptotic error floor with increasing signal to noise ratio. Further, the impact of DO of the source–relay and relay–destination links on the optimal source–relay power allocation is also explicitly demonstrated. Simulation results yield several important insights into the end-to-end error performance for different mobility conditions and also validate the derived analytical results. Index Terms—Cooperative systems, multiple-input multipleoutput (MIMO), mobility, multiple relays, optimal power allocation, space-time block coded (STBC), time-selective fading.

I. INTRODUCTION ELAY-BASED cooperation, a recent advancement in wireless communication technologies [1]–[5], has been adopted in IEEE 802.11s and Third-Generation Partnership Project Long Term Evolution (3GPP LTE)—Release 10, i.e., LTE-Advanced [6] due to its ability to potentially enable higher data rates of over 300 Mb/s in the uplink and 75 Mb/s in the downlink together with enhanced reliability, improved bandwidth utilization, and reduced battery consumption. Moreover,

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Manuscript received June 8, 2016; revised October 5, 2016; accepted November 28, 2016. Date of publication December 1, 2016; date of current version July 14, 2017. This research was supported in part by funding from IIMA IDEA Telecom Centre of Excellence (IITCOE). The review of this paper was coordinated by Dr. H-F. Lu. The authors are with the Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India (e-mail: neerajv@iitk. ac.in; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2016.2634924

relay-based cooperation can also enhance the coverage area of conventional cellular networks by leveraging the previously unexploited user equipments (UEs) of other mobile users as relay nodes [7]. This eliminates the need for deploying additional relay nodes in pre-existing cellular infrastructure toward improving coverage, thus reducing the costs. Since typically there are several UEs in the network at any given instant, the communication signal broadcast by the source (eNodeB/UE) can be forwarded by multiple UEs to the desired destination node through several independent channels, combining which results in a significant decrease in the end-to-end bit error rate (BER) at the destination. However, in general, these cooperative UEs as well as the source and/or destination nodes are mobile during signal transmission and reception, since the users are mobile. This mobile nature of the nodes results in time-selective fading and leads to a significant increase in the error rate at the receiver. Further, due to channel estimation errors, only imperfect channel state information (CSI) is available at the various nodes frequently in practice. However, related works in the existing literature on cooperative communication [8]–[12] consider only ideal scenarios with static nodes and perfect CSI availability. This limits the applicability of the results presented and theory developed in the above works in practical scenarios with timevarying channels due to mobile nodes and imperfect CSI due to limited number of training symbols and pilot power. Hence, performance analysis of cooperative communication protocols considering time-selective links arising due to node mobility together with imperfect CSI is critically important from a practical perspective and is, therefore, one of the central aims of the work presented in this paper. In a cooperative system, each relay retransmits the symbols received from the source after some simplistic signal processing which characterizes the various cooperative relaying protocols and fundamentally affects their overall performance. Amongst them, the selective decode-and-forward (DF) protocol [9], in which the relay forwards the symbol only if the received signal to noise ratio (SNR) is greater than a threshold, stands out as a better choice for high-rate next generation wireless networks since it does not result in either noise amplification or error propagation at the relay, which can be mitigating factors for high data rate transmission. Moreover, the end-to-end reliability as well as data rate of selective DF cooperative systems can be further enhanced by exploiting spatial diversity and spatial multiplexing, respectively, through multiple-input multiple-output (MIMO)

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wireless links. Cooperative MIMO wireless communication is currently an active research area and several recent works have focused on the analysis of various aspects of such systems. The work in [13] considers beamforming and combining for MIMO relay networks, while recent advancements on low complexity detection for MIMO systems with DF relays has been reported in [14]. Other recent works in this area include [15]–[17], which consider the impact of imperfect channel gains, channel and noise correlations, spatial, i.e., antenna correlation respectively on the performance of orthogonal space-time block coded (OSTBC) based amplify-and-forward (AF) MIMO relay systems. It is also worth noting that in contrast to beamforming, i.e., maximum ratio transmission [18] or other transmit precoding schemes [19]–[21], STBC-based transmission does not require CSI at the transmitting nodes and still achieves the full diversity order (DO) of the system. Therefore, STBC is ideally suited for high data rate transmission since it eliminates the need for CSI feedback which can be very high due to the low coherence time arising from node mobility. Also, massive MIMO has recently gained significant attention toward providing higher data rates to users in 5G wireless systems. However, implementing massive MIMO is very challenging in a highly mobile environment such as vehicle to vehicle (V2V) communication [22]. This arises due to the fact that the number of orthogonal pilot sequences required consumes an enormous fraction of time frequency resources. Further, feeding back the estimated responses to the base station requires a significant amount of resources on the uplink. In addition, the quality of the channel estimate in massive MIMO systems can be significantly compromised due to pilot contamination. Interestingly, the recent work in [23] demonstrates that continuous ultradense networks are better suited than massive MIMO-based macro cells to provide higher throughput in 5G wireless networks to mobile users. This arises because the channel varies very quickly in mobile wireless scenarios due to time selectivity, thus leading to a poor performance of massive MIMO in mobile macro cell scenarios. Hence, at present, it is significantly challenging to implement massive MIMO in mobile scenarios and further research is necessary before it can be successfully implemented in mobile cooperative communication systems. Therefore, this paper presents a comprehensive analysis to characterize and enhance the performance of multiple relay selective DF MIMO-STBC cooperative systems over time-selective Rayleigh faded links with imperfect channel estimates, which has so far, to the best of our knowledge, not been considered in any of the works in the existing literature. Further, it has been shown in [9] that allocating equal power to all the transmitting nodes in a cooperative scenario is optimal only when the relay–destination link is weaker than the source–relay link. However, for other scenarios, i.e., either when the source–relay link is weaker than the relay– destination link or when both the links have identical channel conditions, equal power allocation is not the optimal solution in order to minimize the end-to-end error rate. For these scenarios, it is therefore necessary to employ optimal power factors at the transmitting nodes, which can significantly improve the end-toend performance of the cooperative system. In this context, the presented end-to-end pair-wise error probability (PEP) analysis

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is also used to derive the optimal power factors which minimize the PEP, thus leading to optimal performance. Section I-A presents a brief summary of related works and their contributions in current literature. A. Related Work Vehicular communication is a key area where node mobility has a significant impact on the performance of the wireless system, as shown in [24]. Vehicular communication primarily occurs in two modes: V2V, which involves multiple hops between several mobile vehicle nodes or vehicle to infrastructure (V2I) for telematics in intelligent transportation systems (ITS). The 802.11p standard proposes techniques to enhance the performance of Wi-Fi in vehicular environments for ITS toward achieving wireless access in vehicular environments (WAVE). This includes data communication in V2I and V2V systems in the ITS band, i.e., 5.9 GHz. However, several works such as [24] and [25] have shown that node mobility has a significant impact on the wireless channel, thereby affecting the performance of the wireless communication system. Thus, it is key to develop optimal communication strategies to enhance the quality of wireless communication in vehicular scenarios. Cooperative communication, which significantly improves the end-to-end error rate through cooperative diversity is particularly suited for application in such systems. Therefore, cooperative wireless communication systems with mobile nodes have been the focus of several recent research works. Works such as [26]–[33], consider cooperative systems with time-selective Rayleigh faded links together with imperfect channel estimates. In [26] and [27], Khattabi and Matalgah consider a multiple relay AF cooperative system and obtain the expressions for the BER, Shannon capacity, and outage performance. In [28] and [29], a conventional relaying scheme is presented, in which all the relays participate in a single cooperative phase, followed by the pertinent analysis of the end-to-end performance in terms of the resulting outage probability and Shannon capacity. The BER performance of the same has been analyzed in [30] and [29]. However, existing works [26]–[30] in the literature suffer from several significant drawbacks. First, all the systems considered are based on the AF protocol which is limited by the noise amplification at the relay. Also, the results derived in these works are not exact owing to the fact that they employ only upper bounds for the end-to-end SNR. A more fundamental drawback of these systems and the analyses presented therein is that they are limited to single-input single-output systems, which are inadequate for high data rate next generation wireless scenarios. In the context of DF relaying, in [31]–[33] a conventional fixed DF cooperative relaying scheme, which results in a high endto-end error rate due to the error propagation at the relay(s) is considered. Hence, selective DF relaying is considered in this paper in order to overcome the limitations of AF/fixed DF cooperation in [26]–[33]. Further, another significant contrast with respect to the works mentioned earlier is that the analysis presented in this paper considers an optimal power allocation framework which can be employed to enhance the end-to-end performance of the selective DF cooperative system. Fikadu

VARSHNEY AND JAGANNATHAM: MIMO-STBC-BASED MULTI RELAY COOPERATIVE COMMUNICATION

et al. [8] recently demonstrated the SER performance for multirelay regenerative networks over independent and identical η − μ fading channels. The analysis therein involves the Lauricella function and is limited to single antenna nodes. Moreover, the work in [8] does not consider practical distortions such as time-selective fading due to node mobility or the availability of imperfect CSI. The performance of a multirelay system with imperfect CSI and distributed STBC employed at the relay nodes has been analyzed in [34]. However, the system model in [34] is fundamentally different from the proposed work and considers only imperfect CSI ignoring node mobility. Analytical results for an OSTBC-based AF MIMO relay system considering perfect channel estimates at the relay and destination nodes were derived in [35]–[37]. Further, the analysis has recently been extended to the imperfect CSI case in [15] where tools from finite-dimensional random matrix theory are used to characterize the statistical properties of the SNR. However, the works in [15], [35]–[37] suffer from several drawbacks. First, the analyses therein are limited to orthogonal OSTBCs which cannot be readily extended to the class of general STBCs. Second, the presence of a single relay with AF relaying which results in noise amplification at the relay in contrast to the selective DF protocol considered in our work is considered in [15], [35]–[37]. The works in [10], [38] have analyzed the outage performance of selective DF MIMO-OSTBC-based cooperative systems considering the presence of single and multiple relays, respectively. However, these works are also limited to OSTBC-based transmission and do not consider practical distortions such as time-selective fading due to node mobility or the availability of imperfect CSI, while analyzing the end-to-end outage performance. Further, another related work in the area of MIMO STBC cooperative communication is [39]. However, the work therein neither considers imperfect CSI nor considers node mobility. Moreover, [39] employs suboptimal near maximum likelihood (ML) and minimum distance (MD) decoders at each receiving node, which lead to a degradation of the end-to-end PEP performance in comparison to selective DF MIMO STBC cooperative communication with ML decoding considered in our paper. In this paper, the time variation of the wireless fading channel is captured by a first-order autoregressive (AR1) model. This model best captures the wireless channel for mobile nodes, as described in the work on vehicular communication [24] for WAVE related studies. Also, this has been well researched for mobile-to-mobile communication scenarios in [40] and empirical models based on Rayleigh fading channel taps have been proposed in [41]. Several related existing works in cooperative communication with mobile nodes such as [26]–[30] are based on the AR1 process for time-varying channels, with the complex channel coefficients modeled as having a Rayleigh envelope and uniform phase. Further, the work in [42] has verified the applicability of the first-order Markovian assumption for a Rayleigh fading channel model, establishing its theoretical accuracy. Thus, the Rayleigh fading channel along with the first-order autoregressive (AR) process is a standard model and is best suited to study the behavior and analyze the performance of wireless communication systems in time-varying wireless scenarios with node mobility. Moreover, the Rayleigh fading

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channel model is also valid for practical scenarios where nodes are assumed to be far away from each other. B. Contributions The various contributions of the paper are itemized below. 1) The work begins by deriving novel closed-form expressions for the per-frame average PEP in Section III-A, (16) and the asymptotic error floor arising due to mobility with imperfect channel estimates in Section III-B, (25) for the dual-phase multirelay selective DF MIMO-STBC cooperative system over independent nonidentical time-selective Rayleigh faded links. 2) The derived asymptotic floor is subsequently used to analytically demonstrate that except for the scenarios when either all the nodes are static or only the relay nodes are mobile, the considered multirelay system with perfect CSI saturates at the asymptotic floor derived in Section III-B, (25) and achieves zero DO due to the time-varying nature of the links. 3) Next, analysis is presented for the ensuing DO when the system does not experience the asymptotic floor and it is shown in Section III-B, (26) and Section III-C, (34) that the system with perfect CSI achieves the DO of Ns Nd when only relays are mobile and the full DO, i.e., Ns Nd + Nr K min{Ns , Nd } when all nodes are static, where Ns , Nr , Nd are the number of antennas at the source, relay, destination, respectively, and K denotes the number of relays. 4) Subsequently, a framework is developed for optimal source–relay power allocation in Section III-C, (31) which significantly improves the end-to-end performance of the system with node mobility and imperfect CSI in the low as well as moderate SNR regimes. The impact of DOs of the source–relay and relay–destination links on the optimal power allocation is also explicitly demonstrated in Section III-C. 5) In the second part of the paper, the results above are extended to a multirelay scenario with a multi-phase protocol in which each relay combines the signals received from the source and at most m previous relays. In contrast to the dual-phase protocol, this multi-phase protocol termed as Pm , 1 ≤ m < K, requires K + 1 phases for end-to-end communication, where K denotes the number of relays. 6) Similar results as above for the dual-phase protocol, i.e., end-to-end PEP analysis with node mobility and imperfect channel estimates, asymptotic floor for per-frame average PEP, DO analysis and optimal source–relay power allocation are derived also for the multi-phase multirelay scenario in Section IV-A, (37); Section IV-B, (43); Section IV-C, (52) and Section IV-C, (51), respectively. The organization of the rest of the paper is as follows. The system model for the multiple relay MIMO-STBC cooperative setup with mobile nodes and imperfect CSI is presented in Section II. Section III presents a comprehensive analysis of the per-frame average end-to-end PEP, DO, and optimal power allocation in the dual-phase multiple relay cooperative system. This is followed by the same for the multi-phase multiple relay

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protocol in Section IV. Simulation results for all the scenarios above are presented in Section V, followed by the conclusion in Section VI. II. MIMO-STBC SELECTIVE DF SYSTEM MODEL WITH MOBILE NODES AND IMPERFECT CHANNEL ESTIMATES Consider a selective DF cooperative communication system with a source and K relay nodes having Ns = Nr = N antennas, since the source and relays employ the same STBC code, and a destination node with Nd antennas. Let C ∈ {Xi  [k]}, 1 ≤ i ≤ |C| denote the MIMO STBC codeword set, where |C| denotes the total number of possible codewords. In this system, the source transmits an STBC codeword Xi  [k] ∈ C N ×T during the first phase, where T is the block length and k represents the kth codeword matrix in a frame of Nb codewords. The received (r ) codeword matrices Ysd [k] ∈ C N d ×T and Ysr [k] ∈ C N ×T at the destination and rth-relay, respectively, where 1 ≤ r ≤ K, can be expressed as,  P0 Hsd [k]Xi  [k] + Wsd [k] (1) Ysd [k] = N Rc  P0 (r ) Ysr [k] = H(r ) [k]Xi  [k] + Wsr(r ) [k] (2) N Rc sr where Rc denotes the rate of the STBC and P0 represents the power transmitted by the source. The source–destination and source–rth relay MIMO channel matrices Hsd [k] ∈ C N d ×N and (r ) (sd) (sr ) Hsr [k] ∈ C N ×N comprise of entries hn ,m [k], hl,m [k] which 2 are independent and Rayleigh distributed with average gains δsd (r ) 2 and (δsr ) , respectively. In the successive phase(s), a set of relays forward the codeword only if the decoding SNR exceeds a threshold similar to the one considered in works such as [9] (r ) and [11]. The received codeword Yrd [k] ∈ C N d ×T at the destination corresponding to the transmission by the rth relay can be expressed as,  Pr (r ) (r ) (r ) H [k]Xi  [k] + Wrd [k]. (3) Yrd [k] = N Rc rd Similar to the source–destination and source–rth relay MIMO (r ) channel matrices, the MIMO channel matrix Hrd [k] ∈ C N d ×N (rd) comprises of entries hn ,l [k] which are independent and modeled (r )

as complex Gaussian with variance (δrd )2 . The quantity Pr denotes the transmit power of the rth relay. The entries of the addi(r ) tive white noise matrices Wsd [k] ∈ C N d ×T , Wsr [k] ∈ C N ×T , (r ) and Wrd [k] ∈ C N d ×T in (1), (2), and (3), respectively, are symmetric complex Gaussian with variance η0 /2 per dimension. This paper considers a general scenario when all the nodes including the source and destination are mobile. Hence, all the links between the source, relay, and destination nodes are time selective in nature, which can be modeled using a first-order AR process [42]. Further, it is assumed that the channel is approximately constant over the duration of a single codeword while varying in a time-selective manner from one codeword to another. The variation of the codeword matrix within a frame can

therefore be modeled as, Hsd [k] = ρsd Hsd [k − 1] +



1 − ρ2sd Esd [k]   (r ) 2 ) (r ) (r ) ) H(r 1 − ρsr E(r sr [k] = ρsr Hsr [k − 1] + sr [k]   (r ) 2 (r ) (r ) (r ) (r ) Hrd [k] = ρrd Hrd [k − 1] + 1 − ρrd Erd [k] (r )

(4) (5) (6)

(r )

where the quantities ρsd , ρsr , and ρrd are the correlation parameters of the source–destination, source–rth relay, and rth relay–destination links, respectively. These parameters can be evaluated using the standard Jakes’ model as, ρ = J0 ( 2πRfs ccν ), where J0 (·) is the zeroth-order Bessel function of the first kind, fc is the carrier frequency, ν is the relative speed between two mobile terminals, c is the speed of light, and Rs is (r ) (r ) the symbol rate. The matrices Esd [k], Esr [k], and Erd [k] are the time-varying components of the source–destination, source– rth relay, and rth relay–destination links whose entries can be modeled as zero mean circularly symmetric complex Gaussian (r ) (r ) random variables with variances σe2sd , (σe sr )2 , and (σe rd )2 , respectively. On the other hand, it can also be noted that, while STBC-based transmission enables optimal ML symbol detection at the relay and destination nodes, it is often difficult, if not impossible, to estimate the instantaneous CSI corresponding to the transmission of each coded block Xi  [k] due to the time-selective nature of the link. Hence, similar to the works [26]–[31], it is assumed that the rth-relay and destination nodes can only avail imperfect estimates of the channel matrices ) (r ) (r )   (r H sr [1] = Hsr [1] + H,sr [1] and Hsd [1] = Hsd [1] + H,sd [1], (r ) (r ) (r )  [1] = H [1] + H [1], respectively, estimated once in H rd rd ,rd the beginning of each frame and subsequently used to detect each coded block Xi  [k], 1 ≤ k ≤ Nb in the corresponding frame. This is a standard assumption in typical wireless systems with a preamble of pilots for channel estimation. The entries of (r ) (r ) the error matrices H,sr [1], H,sd [1], and H,rd [1] are complex circularly symmetric Gaussian with zero mean and variances (r ) (r ) (σ sr )2 , σ2sd , and (σ rd )2 , respectively. Using (4), one can now obtain the following expression for Hsd [k] as, Hsd [k] = ρksd−1 Hsd [1] +

k −1   1 − ρ2sd ρksd−i−1 Esd [i], i=1

 sd [1] − ρk −1 H,sd [1] = ρksd−1 H sd +

k −1   1 − ρ2sd ρksd−i−1 Esd [i].

(7)

i=1

Substituting the above expression in (1), the received codeword matrix Ysd [k] can be expressed as,  P0 k −1   sd [k]. Ysd [k] = ρ Hsd [1]Xi  [k] + W (8) N Rc sd  sd [k] = Wsd [k] The effective noise matrix is defined as, W  M I I + Wsd [k] − Wsd [k] where the terms Wsd [k] = NPR0 c ρksd−1  P 0 (1−ρ 2sd ) k −1 k −i−1 M H,sd [1]Xi  [k] and Wsd [k] = Esd [i] i=1 ρsd N Rc

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Xi  [k] are the noise components arising due to imperfect channel estimation and node mobility, respectively. The effective noise power ηsd can be evaluated as, ηsd = η0 +

P0 2(k −1) P0 2(k −1) ρ Na σ2sd + (1−ρsd )Na σe2sd N Rc sd N Rc

(9) where Na is the number of nonzero symbol transmissions per codeword instant. Similarly, one can obtain the expression for (r ) (r ) the received codeword matrices Ysr [k] and Yrd [k] at the rthrelay and destination corresponding to the transmission by the source and the rth-relay, respectively, as  P0  (r ) k −1  (r ) (r )  (r ) [k] (10) ρ Ysr [k] = Hsr [1]Xi  [k] + W sr N Rc sr  Pr  (r ) k −1  (r ) (r )  (r ) [k] (11) Yrd [k] = ρ Hrd [1]Xi  [k] + W rd N Rc rd  sr(r ) [k] and W  (r ) [k] are where the effective noise matrices W rd defined as,  k −1 P0  (r ) 2    (r ) k −i−1 (r ) (r )  1− ρsr ρsr Wsr [k] = Wsr [k] + N Rc i=1  P0  (r ) k −1 (r ) )  ρ × E(r [i]X [k]− H,sr [1]Xi  [k] i sr N Rc sr (12)  k −1 Pr  (r ) 2    (r ) k −i−1  (r ) [k] = W(r ) [k] + 1− ρ ρrd W rd rd rd N Rc i=1  Pr  (r ) k −1 (r ) (r ) ρ × Erd [i]Xi  [k]− H,rd [1]Xi  [k] N Rc rd (13) (r )

(r )

and the respective effective noise powers ηsr and ηrd can be obtained as, P0  (r ) 2(k −1)  (r ) 2 P0 ρsr Na σ sr + ηsr(r ) = η0 + N Rc N Rc   (r ) 2(k −1)   (r ) 2 Na σe sr , × 1− ρsr (14) (r ) ηrd

Pr  (r ) 2(k −1)  (r ) 2 Pr ρ = η0 + Na σ rd + N Rc rd N Rc   (r ) 2(k −1)   (r ) 2 Na σe rd . × 1 − ρrd

Fig. 1. Dual-phase selective DF-based multiple-relay cooperative system with each of the relays participating in the second phase.

initially broadcasts an STBC codeword Xi  [k] to the destination and K relays. In the second phase, the set of relays which decode the received codeword successfully, retransmit the codeword to the destination over orthogonal channels. Let ξjk (r) denote the state of relay r where ξjk (r) = 1 if relay r decodes the transmitted codeword matrix Xi  [k] correctly and 0 otherwise. Let ξjk = [ξjk (1), ξjk (2), . . . , ξjk (K)]T , 0 ≤ j ≤ 2K −1 denote the set of all the possible 2K binary vector states. The binary vector ξ0k = [0, 0, . . . , 0, 0]T represents the state when all the relays decode codeword Xi  [k] in error whereas ξ2kK −1 = [1, 1, . . . , 1, 1]T represents the state when all the relays decode the kth codeword correctly. Further, let the set Ψkj defined as Ψkj = {r|ξjk (r) = 1, r = 1, 2, . . . , K} include all the relays which decode the codeword Xi  [k] correctly. Employing the above framework, the per-frame average PEP for end-toend decoding at the destination for the dual-phase multirelay cooperative wireless system is given by the result below. Theorem 1: The per-frame average PEP P e for end-toend decoding in the selective DF relaying based multiple relay cooperative system with the dual-phase protocol is given by, |C| |C| Nb   1   Pe ≤ Pr(Xi [k]) Pr(Xi  [k] → Xi [k]) (16) Nb   k =1 i =1

i=1,i= i

(15)

In the following section, expressions are obtained for the perframe average PEP in a multirelay MIMO-STBC cooperative wireless system employing the dual-phase selective DF protocol as shown in Fig. 1 under conditions of node mobility and imperfect CSI at each receiving node. III. DUAL-PHASE MULTIPLE-RELAY-BASED SELECTIVE DF COOPERATIVE SYSTEM A. Per-Frame Average PEP Analysis With Node Mobility and Imperfect Channel Estimates Consider a multirelay wireless system employing the dualphase cooperative protocol in which a multi-antenna source

where Pr(Xi  [k]) is the probability for the transmission of the codeword Xi  [k] at the source, which is equal to 1/|C|, when all the codewords are equally likely to be transmitted. The quantity Pr(Xi  [k] → Xi [k]), which denotes the endto-end average PEP of confusing Xi  [k] as Xi [k] at the destination, is given in Eq. (17) as shown at the top of (r ) (r ) (r ) 2 2 = δsd + σ2sd , (δ˜sr )2 = (δsr )2 + (σ sr )2 , next page, where δ˜sd (r ) (r ) (r ) (δ˜rd )2 = (δrd )2 + (σ rd )2 and the function G(ν(θ)) is defined π 1 as G(ν(θ)) = π1 0 2 ν (θ ) dθ [9]. Proof: The end-to-end conditional PEP of confusing Xi  [k] ∈ C N ×T with Xi [k] ∈ C N ×T at the destination for a (1) (1) (K ) (K ) given channel H = {Hsd , Hsr , Hrd , . . . , Hsr , Hrd } can be

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⎧ ⎞ ⎛ ⎛

2(k −1)

2 ⎞ N ⎞ ⎫ ⎪ ⎪ (r ) (r ) ⎪ ⎪ 2 |C| ˜ N ⎬ ⎨ λln δsr P0 ρsr ⎟ ⎢⎜   ⎜  ⎜ ⎟ ⎟ ⎢⎜ ⎟ ⎟ G⎜ ⎝1 + ⎠ ⎠ ⎣⎝ ⎠ ⎝ (r ) ⎪ ⎪ 4N Rc ηsr sin2 θ ⎪ ¯k ⎪ n =1 ⎭ ⎩ l=1 r ∈Ψ j ⎡⎛

Pr(Xi  [k] → Xi [k]) =

K 2 −1

j =0

⎫ ⎧ ⎛ ⎛

2(k −1)

2 ⎞N ⎞⎪⎞ ⎪ (r ) (r ) ⎪ ⎪ 2 |C| N ⎬⎟ λln δ˜sr P0 ρsr  ⎜ ⎨ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ×⎝ 1− G⎝ ⎝1 + ⎠ ⎠⎪⎠ (r ) 2 ⎪ 4N R η sin θ ⎪ ⎪ c sr n =1 l=1 ⎭ r ∈Ψ kj ⎩ ⎛

⎛

⎡ ⎛

2(k −1)

2 ⎞N d ⎤⎞⎤   (r ) (r ) N 2 d ˜ N 2(k −1) 2 ˜2 λin δrd Pr ρrd  ⎜ ⎜ ⎢ ⎟⎥ λin δsd ⎟ ⎥ ⎢ 1 + P0 ρsd ⎟⎥ ×G ⎜ 1 + ⎝ ⎠ ⎥ 2 ⎝ ⎣ ⎦ ⎠⎦ (r ) 2 4N R η sin θ c sd 4N R η sin θ k c n =1 rd

(17)

r ∈Ψ j

the average PEP of confusing Xi  [k] by Xl [k] as,

written as [11], [12],

Pr(Xi  [k] → Xi [k]|H) =

K 2 −1

Pr(Xi  [k] → Xi [k]|ξjk , H)

j =0

× Pr(ξjk |H), where Pr(Xi  [k] → Xi [k]|ξjk , H) denotes the end-to-end conditional PEP for the scenario when the system is in state ξjk and the quantity Pr(ξjk |H) represents the conditional probability that the system is in state ξjk . Averaging over the channel H along with the assumption that each of the links experiences independent (but not-necessarily identical) fading, the end-to-end PEP of confusing Xi  [k] by Xi [k] at the destination can be written as, Pr(Xi  [k] → Xi [k]) =

K 2 −1

K 2 −1

EH {Pr(Xi  [k] → Xi [k]|ξjk , H)}EH {Pr(ξjk |H)}

j =0

(18) where EH {·} denotes the expectation operation over the PDF of H. Since the links between the source and relays fade independently, the above expression can be further simplified as,

Pr(X [k] → Xi [k]) =

K 2 −1

i

EH {Pr(Xi  [k] → Xi [k]|ξjk , H)}

j =0

×

K 

EH {Pr(ξjk (r)|H)}

if ξjk (r) = 0, if ξjk (r) = 1. (20)

The average PEP of confusing Xi  [k] by Xl [k] at relay r when codeword matrix Xi  [k] is transmitted by the source, can be evaluated as follows. The conditional PEP of confusing Xi  [k] by Xl [k] at relay r where l = i can be written using (10) as,  (r ) (1)) PeS →R r (Xi  [k] → Xl [k]|H sr ⎛# $  & &2 && $ && (r ) 2(k −1) &&  (r )  [k] − Xl [k])&& H ρ P (1)(X $ & & sr sr 0 i ⎜% F ⎜ = Q⎝ (r ) 2N Rc ηsr

⎞ ⎟ ⎟ ⎠

where Q(·) denotes the Gaussian Q function [9] and || · ||F is the Frobenius norm of a matrix. Using the singular value decomposition of the codeword difference matrix Xi  [k] − Xl [k], the above expression can be simplified similar to [43] as,

EH {Pr(Xi  [k] → Xi [k]|ξjk , H)Pr(ξjk |H)}

j =0

=

EH {Pr(ξjk (r)|H)} ⎧ ⎨ |C| P S →R r (Xi  [k] → Xl [k]), l=1 e = ⎩ 1 − |C| P S →R r (X  [k] → X [k]), i l l=1 e

(19)

r =1

where EH {Pr(ξjk (r)|H)} denotes the average probability of relay r being in the state ξjk (r) and can be expressed in terms of

) ˜ (r PeS →R r (Xi  [k] → Xl [k]|H sr (1)) # ⎛$ ⎞ N &&˜ (sr) &&2 $  (r ) 2(k −1) N 2 P ρ ⎜$ n =1 λln n ˜ =1 &hn ˜ ,n (1)& ⎟ % 0 sr ⎟ = Q⎜ ⎝ ⎠ (r ) 2N Rc ηsr

where λl1 , λl2 , . . . λlN denote the non-negative singular values of the codeword difference matrix Xi  [k] − Xl [k], the quan) ˜ (sr) (1) is the (˜ ˜ (r tity h n, n) coefficient of the matrix H sr (1) = n ˜ ,n )  (r ˜ , n ≤ N and Ul ∈ C N ×N is a unitary maH sr (1)Ul for 1 ≤ n H trix, i.e., Ul Ul = UH l Ul = IN . It follows from the result in ˜ (sr) (1) of the effec[43] that the envelope of each coefficient h n ˜ ,n ) ˜ (r (1) is Rayleigh distributed with avertive channel matrix H sr (r ) 2 (r ) 2 (r ) 2 (r ) ˜ ˜ age power gain (δsr ) , where (δsr ) = (δsr ) + (σ sr )2 . Fur(sr) ˜ (sr) (1)|2 ther, it can be readily seen that the gain g˜ (1) = |h n ˜ ,n

n ˜ ,n

(r ) is exponential distributed with mean (δ˜sr )2 . Using the identity

VARSHNEY AND JAGANNATHAM: MIMO-STBC-BASED MULTI RELAY COOPERATIVE COMMUNICATION

EH {Pr(ξjk |H)} =

K  r =1

6015

⎧ ⎛   (r ) 2(k −1) 2  (r ) 2N ⎞⎫ |C| N ⎬   ⎨ ρsr P λln δ˜sr 0 ⎠ EH {Pr(ξjk (r)|H)} = G⎝ 1+ (r ) ⎭ ⎩ 4N Rc ηsr sin2 θ ¯k r ∈Ψ j

n =1

l=1

⎧ ⎛   (r ) 2(k −1) 2  (r ) 2N ⎞⎫ |C| N ⎬  ⎨   P0 ρsr λln δ˜sr ⎠ × G⎝ 1+ 1− (r ) ⎭ ⎩ 4N Rc ηsr sin2 θ k r ∈Ψ j

⎞ ⎛# $ &2 &2 Nd & N Nd & N $   & & 2(k −1) (sd) &˜ (rd) & ⎟ (r ) 2(k −1) ˜ (1)& ⎜$ P0 ρsd λ2in &h λ2in &hl,n (1)& ⎟ ˜l,n ⎜$  Pr ρrd n =1 ˜ ⎟ ⎜ n =1 l=1 l=1 $ k  Pr(Xi [k] → Xi [k]|ξj , H) = Q ⎜% + ⎟ (r ) ⎟ ⎜ 2N Rc ηsd 2N Rc ηrd ⎠ ⎝ r ∈Ψ kj ⎛

N 

⎡

2 λ2in δ˜sd ⎣ 1 + P0 ρsd EH {Pr(Xi  [k] → Xi [k]|ξjk , H)} = G ⎝ 4N Rc ηsd sin2 θ n =1

PeF

2(k −1)

(22)

n =1

l=1

N d

 r ∈Ψ kj



(23)

 (r ) 2(k −1) 2  (r ) 2 N d ⎤⎞ Pr ρrd λin δ˜rd ⎦⎠ (24) 1+ (r ) 4N Rc ηrd sin2 θ

⎫ ⎧ ⎛ '⎛   (r ) 2(k −1) 2  (r ) 2 N ⎞⎬⎞ |C| |C| 2K −1 Nb  N  ⎨   ρsr λln δ˜sr 1  ⎠ ⎠ ⎝ = G⎝ 1+ (r ) 2 ⎭ ⎩ Nb 4N η ˜ sin θ k sr a ¯ n =1 k =1 i=1 j =0 l=1 r ∈Ψ j ( )* + I 1k (j )

⎧ ⎞ ⎛   (r ) 2(k −1) 2  (r ) 2 N ⎞⎫ |C| N ⎬ ˜  ⎨   ρsr λln δsr ⎠ ⎠ ×⎝ G⎝ 1+ 1− (r ) ⎭ ⎩ 4Na η˜sr sin2 θ k ⎛

(

r ∈Ψ j

⎛

N 

l=1

n =1

⎡

)*

+

I 2k (j )

2(k −1) 2 ˜2 λin δsd ⎣ 1 + ρsd × G⎝ 4Na η˜sd sin2 θ n =1 (

N d





r ∈Ψ kj

)*

1+



(r ) 2(k −1) 2  ˜(r ) 2 λin δrd (r ) 4Na η˜rd sin2 θ

ρrd

N d ⎤⎞ , ⎦⎠

(25)

+

I 3k (j )

π x2 Q(x) = π1 0 2 exp(− 2 sin 2 )dθ for the Gaussian Q(·) function θ [9, pp. 158] in the above instantaneous PEP expression and inte(sr) grating it over the exponential distribution of the gains g˜n˜ ,n (1), the average PEP for erroneous decoding at relay r can be simplified as, PeS →R r (Xi  [k] → Xl [k]) ⎛   (r ) 2(k −1) 2  (r ) 2N ⎞ N  P0 ρsr λln δ˜sr ⎠. = G⎝ 1+ (r ) 4N Rc ηsr sin2 θ n =1

˜ (rd) (1) are the complex Gaussian channel coeffi˜ (sd) (1), h h ˜ln ln cients of the resulting source–destination and relay r-destination  sd (1)Ui and H  (r ) (1)Ui , re˜ (r ) (1) = H ˜ sd (1) = H matrices H rd rd spectively. Averaging now over the exponential distributed (sd) ˜ (sd) (1)|2 , gains of the fading channel coefficients g˜˜l,n (1) = |h ˜l,n (rd) (rd) 2 ˜ g˜ (1) = |h (1)| , the average PEP of confusing Xi  [k] by l,n

(21)

Using the above expression along with (20), the final expression for the average probability of the system being in state ξjk given in (19) can be written as Eq. (22) as shown at the top of the ¯ k contains all the relays which decode the page, where the set Ψ j codeword matrix Xi  [k] erroneously. Considering the transmission of all the relays from the set Ψkj along with the source, the conditional PEP of confusing Xi  [k] with Xi [k] at the destination, when the system is in state ξjk , can be derived as Eq. (23) as shown at the top of the page, where

l,n

Xi [k] at the destination when the system is in state ξjk can be simplified as Eq. (24) as shown at the top of the page, where (r ) (r ) (r ) (δ˜rd )2 = (δrd )2 + (σ rd )2 . Finally, after substituting this expression for EH {Pr(Xi  [k] → Xi [k]|ξjk , H)} along with the expression for EH {Pr(ξjk |H)} from (22) in (19), one can readily obtain the end-to-end PEP of confusing Xi  [k] by Xi [k] at the destination as (17).  B. Asymptotic Floor for Per-Frame Average PEP In order to demonstrate the impact of node mobility and channel estimation errors on the end-to-end system performance, one can derive the asymptotic floor given in Eq. (25) as shown at the top of the page for a multirelay dual-phase cooperative system

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 66, NO. 7, JULY 2017

by neglecting η0 in (9), (14), and (15) at high SNR and substituting the resulting expressions in (17), where the quantities (r ) (r ) η˜sd , η˜sr , and η˜rd are defined as,  2(k −1) 2 2(k −1)  2 σe sd , σ sd + 1 − ρsd η˜sd = ρsd  ) 2(k −1)  (r ) 2   ) 2(k −1)  (r ) 2 σ sr + 1 − ρ(r σe sr , η˜sr(r ) = ρ(r sr sr  (r ) 2(k −1)  (r ) 2   (r ) 2(k −1)  (r ) 2 (r ) σ rd + 1 − ρrd σe rd . η˜rd = ρrd Consider now the specific scenario when each of the receiving nodes have perfect CSI for ML detection and all the nodes except (r ) (r ) the source are static which implies ρsd , ρsr < 1 and ρrd = 1∀r. Under this condition, it can be readily seen that the terms η˜sd , (r ) (r ) (r ) η˜sr above are nonzero, whereas η˜rd is zero since ρrd = 1∀r. Therefore, all the PEP terms corresponding to the states ξ1k = [0, 0, . . . , 0, 1]T , ξ2k = [0, 0, . . . , 1, 0]T , . . . , ξ2kK −1 = [1, 1, . . . , 1, 1]T in (25) reduce to zero since I3k (j) = 0 for 1 ≤ j ≤ 2K −1. However, only the PEP term corresponding to the state ξ0k = [0, 0, . . . , 0, 0]T contributes in (25) which demonstrates that the system experiences an asymptotic error floor due to the mobility of the source node. For the scenario when each of the receiving nodes have perfect CSI and all the nodes except the destination are (r ) (r ) static, i.e., ρsd , ρrd < 1, and ρsr = 1∀r, the terms η˜sd , (r ) (r ) (r ) η˜rd above are nonzero, whereas η˜sr = 0 since ρsr = 1∀r. Therefore, it can be readily seen in (25) that all the PEP terms corresponding to the states ξ0k = [0, 0, . . . , 0, 0]T , ξ1k = [0, 0, . . . , 0, 1]T , . . . , ξ2kK −2 = [1, 1, . . . , 1, 0]T in (25) reduce to zero since I1k (j) = 0 for 0 ≤ j ≤ 2K − 2 and only one PEP term corresponding to the state ξ2kK −1 = [1, 1, . . . , 1, 1]T in (25) contributes to the asymptotic error floor due to the destination node mobility. Further, consider the scenario when all the nodes except (r ) (r ) the K relays are static, i.e., ρsr , ρrd < 1 and ρsd = 1 with the availability of perfect CSI for detection. For this sce(r ) (r ) nario, the terms η˜sr , η˜rd above are nonzero and η˜sd = 0 for ρsd = 1. Therefore, it can be readily observed that the asymptotic error floor PeF in (25) reduces to zero since all the PEP terms corresponding to the states ξ0k = [0, 0, . . . , 0, 0]T , ξ1k = [0, 0, . . . , 0, 1]T , . . . , ξ2kK −1 = [1, 1, . . . , 1, 1]T are zero. This is due to the fact that the terms I3k (j), 0 ≤ j ≤ 2K − 1 are zero for η˜sd = 0. Moreover, the asymptotic PEP and DO achieved for end-to-end decoding in this scenario are shown in the following result. Lemma 1: The asymptotic per-frame average PEP for the scenario with all the nodes except the K relays static, i.e., (r ) (r ) ρsr , ρrd < 1, ρsd = 1 and perfect CSI available in the dualphase cooperative system is given by, |C| 2 −1 . Nb    K

/N N d

CN N d +N N d |Ψ kj | 0N k ( n =1 λ2in )N d +N d |Ψ j | k =1 i=1 j =0  N N d ' (r )   4Na η˜rd × CN 2  (r ) 2(k −1)  (r ) 2 ρ δ k k ¯ rd rd r ∈Ψ r ∈Ψ

1 Pe ≤ Nb

j

4N Rc 2 β0 δsd

j

⎫

 ×

N 2 |C|  N −N , (r ) ⎬ η N N d   4Na η˜sr 0 2 λ  (r ) 2(k −1)  (r ) 2 ln ⎭ P ρsr δsr n =1 l (26)

and the net achieved diversity for this scenario is N Nd . |C| 0N 0 Proof: At high SNR, r ∈Ψ k {1 − l=1 G( n =1 (1 + j

(r ) (r ) P 0 (ρ sr ) 2( k −1) λ2l n ( δ˜sr ) 2 N ) )} (r ) 2 4N R c η sr sin θ

≈ 1 which implies that the decoding at the relays is error free. Using this approximation and ignoring the additive unity factor in each of the terms in (17) along with Pr(Xi  [k]) = 1/|C|, the expression for P e in (16) can be simplified at high SNR as (26), where β0 = P0 /P is the power factor at the source. The constant Cz is defined as π −1) Cz = π1 0 2 (sin2 θ)z dθ = 1×3×···×(2z 2×2×4×···×(2z ) [44, Eq. 3.621.3]. Further, using (26), it can be readily seen that the DO achieved for  this scenario is N Nd . Using the observations above, one can conclude that except for the scenarios when either all the nodes are static or only the relay nodes are mobile, the multirelay system with perfect CSI is limited by the asymptotic error floor derived in (25). Therefore, the resulting DO for the system with the error floor is zero since the SER is constant at high SNR. The DO for a scenario with all the nodes static and perfect CSI at each receive node is derived next. Toward this end, a high SNR asymptotic end-to-end PEP expression is derived initially. This is also used to determine the optimal source–relay power factors which can be subsequently employed at the source and relays to minimize the end-to-end PEP at the destination. It is further demonstrated through simulations in Section V that the derived optimal factors not only improve the performance of the system with static nodes and perfect CSI but also significantly improve the error performance of the system with node mobility and imperfect CSI. C. Optimal Source–Relay Power Allocation and DO Analysis The asymptotic end-to-end per-frame average PEP for the dual-phase cooperative system at high SNR is given in the result below. Lemma 2: At high SNR, the end-to-end per-frame average PEP for the dual-phase multiple-relay-based system considering (r ) (r ) the scenario with all the nodes static, i.e., ρsd = ρsr = ρrd = 1 (r ) and perfect CSI at each receiving node, i.e., σ2sd = (σ sr )2 = (r )

(σ rd )2 = 0 for detection is given by, Pe ≤

|C| |C| Nb   1  Pr(Xi  [k]) Nb  k =1 i =1

|C| 2 −1   K

=

i=1 j =0

GC (σi2 , j)

K 2 −1

i=1,i= i  j =0

η D(j ) 0

P

GC (σi2 , j)

η D(j ) 0

P

(27)

where the exponent D(j) for the state ξj is defined as D(j) = ¯ j | and the quantity GC (σ 2 , j) is given N Nd + N Nd |Ψj | + N 2 |Ψ i in Eq. (28) as shown at the top of next page, where σi2 and κ are 0 |C| 2 2 −N defined as σi2 = N , respectively. n =1 λin , κ = l=1 (σl )

VARSHNEY AND JAGANNATHAM: MIMO-STBC-BASED MULTI RELAY COOPERATIVE COMMUNICATION

(4N Rc )N N d +N N d |Ψ j |+N

6017

¯j| |Ψ

¯

CN N d +N N d |Ψ j | (ζ(N 2 )κ)|Ψ j |

N N d 0

N 2 . 0 2 ¯ (r ) 2 (r ) 2 2 )N N d β N N d +N |Ψ j | β (δ (σi2 )N d +N d |Ψ j | (δsd (δ ) ) ¯j sr r rd r ∈Ψ j r ∈Ψ 0 ⎧   K N Nd /N N d C 2 −1 ⎨. N N d +N N d |Ψ kj |  4N Rc 4N Rc Pr(Xi  [k] → Xi [k]) ≈ k 2 (r ) ⎩ β0 δsd (σi2 )N d +N d |Ψ j | βr (δ )2 k j =0 GC (σi2 , j) =

2

r ∈Ψ j

rd

 N 2 ⎤⎫ ⎬ η N N d +N N d |Ψ kj |+N 2 |Ψ¯ kj |  4N R c 0 ⎣CN 2 κ ⎦ × (r ) 2 ⎭ P β (δ ) 0 sr ¯k r ∈Ψ ⎡

(28)

(29)

j

Proof: At high SNR,

0

(r ) (r ) P 0 (ρ sr ) 2( k −1) λ2l n ( δ˜sr ) 2 N ) )} (r ) 4N R c η sr sin 2 θ

r ∈Ψ kj

{1 −

|C| l=1

0 G( N n =1 (1 +

≈ 1 and ignoring the additive unity factor from each of the terms in (17), the expression for Pr(Xi  [k] → Xi [k]) can be simplified at high SNR as Eq. (29) as shown at the top of page, where β0 = P0 /P, βr = Pr /P, 1 ≤ r ≤ K are the power factors at the source and relay r, respectively. Further, it can be noted that the state of the network Ψkj for each of the codewords Xi  [k], k = 1, 2, . . . , Nb is the same for this condition with each of the links time-invariant (quasi-static) or all nodes static. Considering now Ψkj = Ψj , ¯k = Ψ ¯ j ∀k in (29) and substituting the resulting expression Ψ j along with Pr(Xi  [k]) = 1/|C| in (16) leads to the result shown in (27) above.  The expression in (27) can be further upper bounded using 2 GC (σi2 , j) ≤ GC (σm in , j) as, P e ≤ |C|

K 2 −1

GC (j)

j =0

η D(j ) 0

(30)

P

where GC (j) is defined as GC (j) = GC (σi2 , j)|σ i2 =σ m2 i n and 0N 2 2 σm in = mini { n =1 λin }. Using the above asymptotic PEP expression, the optimal power allocation for end-to-end per-frame PEP minimization in the dual-phase multirelay system is now given by the result below. Theorem 2: The optimal power factors β0 , βr , r = 1, 2, . . . , K for the dual-phase P0 protocol-based multirelay selective DF MIMO-STBC cooperative system can be obtained as the solution of the optimization problem below, min

β 0 ,β 1 ,...,β r

K 2 −1

j =0

s.t. β0 +

(β0 K 

ψj 0

2 )N N d +N |Ψ¯ j |

r ∈Ψ j

(βr )N N d

,

βr = 1, β0 ≥ 0, β0 ≥ 0, 1 ≤ r ≤ K, (31)

r =1

where ψj = GC (j)|β 0 =1,β r =1∀r (η0 /P )D(j ) = 2 GC (σi , j)|σ i2 =σ m2 i n ,β 0 =1,β r =1∀r (η0 /P )D(j ) . The above optimization problem is a Geometric program which can be solved using any standard convex solver such as CVX [45] for K ≥ 2 relays. However, one can explicitly

obtain the optimal power factors for a single relay, i.e., K = 1 as shown in [46, Appendix A]. It can also be seen that the term corresponding to the minimum value of the exponent D(j) dominates in (27) at high SNR. Therefore, the DO d for the dual-phase multirelay cooperative system can be obtained as, d=

min

0≤j ≤2K −1

= N Nd +

D(j) min

0≤j ≤2K

1 −1

= N Nd + N K + 2

2 ¯j| N Nd |Ψj | + N 2 |Ψ min

0≤j ≤2K −1

1

2 (N Nd − N 2 )|Ψj | . (32)

Now, the following results demonstrate the impact of the source–relay and relay–destination DOs on the end-to-end performance as well as the optimal power allocation. Lemma 3: The asymptotic per-frame average end-to-end PEP for the scenario with the DOs of the source–relay links lower than the relay–destination links, i.e., N 2 < N Nd in the dual-phase cooperative system is given by, ⎡ ⎤ |C| N N d +N 2 K K  (4N Rc ) CN N d (CN 2 κ) ⎢ ⎥ Pe ≈ ⎣

N 2 ⎦ 0 2 (r ) K 2 i=1 (σ 2 )N d (δ 2 )N N d β N N d +N K i r =1 (δsr ) sd 0 ×

η N N d +N 2 K 0

P

(33)

and the net achieved DO for this scenario is N Nd + N 2 K. Proof: For the scenario with N 2 < N Nd , it can be readily seen from (32) that min0≤j ≤2K −1 {(N Nd − N 2 )|Ψj |} = 0 occurs when the system is in state ξ0 = [0, 0, . . . , 0, 0]T . Therefore, using (32), the net achieved DO for this scenario can be seen to equal N Nd + N 2 K and the per-frame average end-toend PEP at high SNR can be obtained as (33) by substituting ¯ j | = K in (27).  |Ψj | = 0 and |Ψ Further, one can also observe from the above expression that the end-to-end performance of the system in the scenario with N 2 < N Nd can be enhanced by allocating almost all the power to the source, i.e., β0 ≈ 1. However, the end-to-end performance for the scenario with N 2 > N Nd can be optimized by assigning equal power to each of the transmitting nodes, i.e., β0 = βr = 1/(K + 1), ∀r which is shown in the result below.

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Lemma 4: The asymptotic per-frame average end-to-end PEP for the scenario with the DO of the source–relay links higher than the relay–destination links, i.e., N 2 > N Nd in the dual-phase cooperative system is given as, ⎡ ⎤ |C| N N d +N N d K  (4N Rc ) CN N d +N N d K ⎢ ⎥ Pe ≈ ⎣

  N N d ⎦ 0 2 (r ) K N N 2 )N N d β d i=1 (σi2 )N d +N d K (δsd r =1 βr δrd 0 ×

η N N d +N N d K 0

(34)

P

and the net achieved DO for this scenario is N Nd + N Nd K. Proof: Proof is similar to that of Lemma 3 above and can be derived by noting that min0≤j ≤2K −1 {(N Nd − N 2 )|Ψj |} = (N Nd − N 2 )K for N 2 > N Nd occurs when the system is in state ξ2K −1 = [1, 1, . . . , 1, 1]T . Therefore, the per-frame average end-to-end PEP at high SNR can be obtained as (34) by ¯ j | = 0 in (27). Further, it can be substituting |Ψj | = K and |Ψ seen from (34) that the achieved diversity for this scenario is  N Nd + N Nd K. Next, for the scenario with N 2 = N Nd , the expression for P e can be obtained as (27) in which all the terms corresponding to the states ξj , 0 ≤ j ≤ 2K − 1 dominate. Moreover, using (32), the DO for this scenario can be seen to equal N Nd + N 2 K since N 2 − N Nd = 0. Finally, the net DO for the dual-phase cooperative system can be obtained as N Nd + N Kmin{N, Nd } by considering the minimum of the DOs obtained above since the DO corresponds to the minimum exponent of the inverse SNR term η0 /P . The above multiple relay system is a dual-hop cooperative system. The analytical framework developed above can be similarly used to also analyze a multihop (> 2) multirelay selective DF cooperative wireless system with K + 1 phases for end-toend communication as demonstrated next. IV. MULTIPLE-PHASE MULTIPLE-RELAY Pm SELECTIVE DF COOPERATIVE SYSTEM

Fig. 2. tem.

Multi-Phase Pm selective DF-based multiple relay cooperative sys-

The schematic diagram for the Pm -based multi-phase cooperative system is shown in Fig. 2 assuming that each of the relays decode the codeword correctly and retransmit in the subsequent phases. A. PEP Analysis With Node Mobility and Imperfect Channel Estimates Consider now terminology similar to the one used in the previous section with additionally the set Ψkj (r) defined as, Ψkj (r) = {˜ r |ξjk (˜ r) = 0; max{1, r − m} ≤ r˜ < r}, corresponding to the relay r, which contains all the previous relays that decode the codeword without any error. Further, let the average gain, correlation parameter, variances of the estimation error, and timevarying error component for the link between relay r˜ and relay r ( r˜,r ) ( r˜,r ) ( r˜,r ) ( r˜,r ) be denoted by (δrr )2 , ρrr , (σ r r )2 , and (σe r r )2 , respectively. The per-frame average end-to-end PEP for the multiphase multirelay cooperative wireless system is given by the result below. Theorem 3: The union bound for the per-frame average PEP P e at the destination in the selective DF relaying cooperative system with the multiple-phase protocol Pm is given by,

In contrast to the dual-phase multirelay cooperative system, each relay r in the Pm system receives the STBC codeword from the previous min{m, r − 1} relays along with the source and selectively retransmits the codeword in phase r + 1. Af|C| |C| Nb   1  ter K + 1 phases, the destination finally decodes the STBC Pe ≤ Pr(Xi  [k]) Pr(Xi  [k] → Xi [k]) Nb codeword using the signals received from the source and a set k =1 i  =1 i=1,i= i  (35) of relays which selectively retransmitted in the previous phases. ⎫⎞⎛ ⎫⎞ ⎧ ⎧ ⎛ K |C| |C| 2 −1  ⎨ ⎬ ⎬  ⎨  ⎝ Pr(Xi  [k] → Xi [k]) = EH {PeS →R r (Xi  [k] → Xl [k])} ⎠⎝ EH {PeS →R r (Xi  [k] → Xl [k])} ⎠ 1− ⎭ ⎭ ⎩ ⎩ k ¯k j =0

r ∈Ψ j

⎛

N 

r ∈Ψ j

l=1

⎡

2(k −1) 2 ˜2 λin δsd ⎣ 1 + P0 (ρsd ) × G⎝ 4N Rc ηsd sin2 θ n =1

N d

 r ∈Ψ kj

l=1

 (r ) 2(k −1) 2  (r ) 2 N d ⎤⎞ Pr ρrd λin δ˜rd ⎦⎠ 1+ (r ) 4N Rc ηrd sin2 θ



(36)

⎛ ⎡   (r ) 2(k −1) 2  (r ) 2 N  ( r˜,r ) 2(k −1) 2  ( r˜,r ) 2 N ⎤⎞ N ˜sr   P ρ δ ρrr P λ λln δ˜rr sr 0 r ˜ ln ⎦⎠ 1+ EH {PeS →R r (Xi  [k] → Xl [k])} = G⎝ ⎣ 1+ (r ) ( r˜,r ) 2 2 4N R η sin θ 4N R η sin θ rr k sr c c n =1 r˜∈Ψ (r ) j

(37)

VARSHNEY AND JAGANNATHAM: MIMO-STBC-BASED MULTI RELAY COOPERATIVE COMMUNICATION

where the end-to-end average PEP of confusing Xi  [k] by Xi [k] is given in Eq. (36) as shown at the bottom of the previous page. The quantity EH {PeS →R r (Xi  [k] → Xl [k])} in (36) is defined ( r˜,r ) ( r˜,r ) ( r˜,r ) in (37), where (δ˜rr )2 = (δrr )2 + (σ r r )2 and the noise ( r˜,r ) power ηrr is given by,  ( r˜,r ) 2(k −1)  ( r˜,r ) 2 ρrr Na σ  r r

=

Proof: Similar to the dual-phase system, the end-to-end PEP of confusing Xi  [k] by Xi [k] in a multi-phase cooperative system can be obtained as,

=

EH {Pr(Xi  [k] →

Xi [k]|ξjk , H)}EH {Pr(ξjk |H)}

j =0

=

K 2 −1

⎩

l=1

(38)

r =1

where EH {Pr(Xi  [k] → Xi [k]|ξjk , H)}, the average end-to-end PEP of confusing Xi  [k] by Xi [k], is given in (24) and is same for both the dual-phase and multiple phase systems. The quantity EH {Pr(ξjk (r)|ξ, H)} denotes the average probability of relay r being in state ξjk (r) conditioned on the states of the previous m relays denoted by ξ = [ξjk (r−1), ξjk (r−2), . . . , ξjk (r−m)] (r )

(r )

( r˜,r )

and the channel H = {Hsd , Hsr , Hrd , Hrr , 1 ≤ r ≤ ( r˜,r ) K, max{1, r − m} ≤ r˜ DO of RD links N Nd + K N Nd Each node static N N d + N K m in {N , N d }

Lemma 6: The end-to-end SERs for N Nd < N 2 and N Nd > N 2 , respectively, at high SNR are given as, P e |N N d < N 2 ≈

|C|  (4N Rc )N N d +N N d K CN N d +N N d K  N d +N d K 2 N N d 0K  2 NN d (r ) β i=1 σi2 δsd β0 r r =1 δ

× P e |N N d > N 2 ≈

η N N d +K N N d 0

P

|C|  i=1

×

,

(50)

(4N Rc )N N d +N K CN N d (CN 2 κ ˜ )K  2 N d  2 N N d 0K  (r ) 2 N 2 δsr β0 σi δsd β0 r =1 2

η N N d +K N 2 0

P

Fig. 3. Per-block average PEP performance of Alamouti coded (a) dual-phase system and (b) multi-phase Pm , m = 1 system with K = 2 relays, N b = 5, (r ) ( r˜ , r ) (r ) channel gains δs2d = (δsr )2 = (δrr )2 = 10, (δr d )2 = 1, and error vari2 2 ances σ e = 0.1, σ  = 0.01.

(51)

and the respective DOs are, d|N N d < N 2 = N Nd + KN Nd

(52)

d|N N d > N 2 = N Nd + KN 2 .

(53)

Proof: Given in Appendix C in the technical report [46].  Similar to the dual-phase protocol for N Nd < N 2 , the optimal power allocation is given as β0 = βr = K 1+1 , 1 ≤ r ≤ K, while for N Nd > N 2 , β0 = 1 is optimal. For the scenario with N Nd = N 2 , the diversity follows as N Nd + N Nd K. Finally, the DO for this system can be seen to be dPm = N Nd + N Kmin{N, Nd } which can be arrived at by considering the minimum of the DOs obtained for the scenarios with N Nd = N 2 , N Nd > N 2 , and N Nd < N 2 . Finally, the achieved DOs in multiple relay dual/multi phase cooperative systems under various mobile scenarios with perfect channel estimates are summarized in Table 1. V. SIMULATION RESULTS This section presents simulation results to demonstrate the end-to-end performance and also to validate the analytical results derived for the multiple relay cooperative scenarios described above. For simulation purposes, consider the transmission of either Alamouti STBC or Golden space time code [47] with QPSK modulated symbols. The other parameters are set as, fc = 5.9 GHz, Rc = 9.6 kb/s, noise power η0 = 1, number of antennas N = Nd = 2, and correlation parameter ρ = 0.9915, 0.9724 corresponding to relative speeds of 32, 58 m/h, respectively. Further, the variances of the error terms due to estimation

Fig. 4. Per-block average PEP performance of 2 × 2 Golden code based dualphase and multi-phase Pm , m = 1 systems with K = 1, N b = 20, channel 2 = δ 2 = 10, δ 2 = 1, and error variances σ 2 = 0.01, σ 2 = 0.01. gains δsd sr e  rd (r )

errors and time-selective fading are set to be σ2sd = (σ sr )2 = (r )

( r˜,r )

(r ) (σe rd )2

( r˜,r ) (σe r r )2

(r )

(σ rd )2 = (σ r r )2 = σ2 ∈ {0, 0.01, 0.001} and σe2sd = (σe sr )2 = = = σe2 ∈ {0.1, 0.01}, respectively, where 2 σ = 0 represents the availability of perfect channel estimates at each receiving node. Figs. 3–9 demonstrate the end-to-end performance of the dual-phase and multiple-phase Pm cooperative systems with K = 2 relays and it can be seen therein that the per-frame average PEP values derived using ((16)) and (35) for the dual-phase and multiple-phase systems, respectively, match with those obtained from the simulation. Further, it can also be noted that the systems with the dual-phase and multi-phase Pm protocol are equivalent for the single-relay scenario, i.e., K = 1 and achieve the same end-to-end performance. Simulation results in Figs. 3–6 considering K = 2 relays yield various interesting insights which are as follows.

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 66, NO. 7, JULY 2017

Fig. 5. Per-block average PEP performance of Alamouti coded dual-phase system with K = 2 relays, N b = 5, error variances σ e2 = 0.1, σ 2 = 0.01 and channel 2 = 10, (a) (δ (r ) ) 2 = (δ (r ) ) 2 = 1, ∀r (b) (δ (r ) ) 2 = 1, (δ (r ) ) 2 = 10, ∀r (c) (δ (r ) ) 2 = 10, (δ (r ) ) 2 = 1, ∀r. gains δsd sr sr sr rd rd rd

Fig. 6. Per-block average PEP performance of Alamouti coded: (a) Dualphase system. (b) Multi-Phase Pm , m = 1 system with K = 2 relays, N b = 2 = (δ (r ) ) 2 = (δ (1, 2) ) 2 = 10, (δ (r ) ) 2 = 1, ∀r, and error 5, channel gains δsd rr sr rd variances σ e2 = 0.1, σ 2 = 0.

Fig. 3 demonstrates that the time-selective nature of the links due to mobility significantly degrades the end-to-end performance compared to the scenario when all the nodes are static, i.e., ρsd = ρsr = ρrd = 1. Moreover, the severity of degradation increases with increasing relative speed, which leads to a decrease in the channel correlation. These results clearly validate the fact that both the systems are limited by the asymptotic error floor derived in (25) and (41) for mobile nodes. However, the dual-phase and multi-phase systems achieve a DO of N Nd + N Kmin{N, Nd } = 12 when all the nodes are static and have perfect channel estimates for detection as shown in

Sections III-C and IV-C. Similar observations can also be made for the 2 × 2 Golden code [47] based dual-phase and multiphase Pm , m = 1 cooperative systems as shown in Fig. 4. These plots also validate the fact that the analyses presented for both the systems are general and not limited to orthogonal STBCs such as the Alamouti code. From Fig. 5(a), it can be seen that for symmetric coopera2 , the system performance with only the tive links, i.e., δsr2 = δrd source mobile is worse than that of the scenario with only the relay mobile. Moreover, since both the links are symmetric, both source and destination mobility, for the same speeds, have an identical effect on the system performance. However, for the asymmetric cooperative scenario when the source–relay links are weak in comparison to the relay–destination links as shown in Fig. 5(b), source mobility drastically degrades the performance when compared to the scenario with only the destination node mobile. Moreover, for the scenario with the source– relay links stronger than the relay–destination links as shown in Fig. 5(c), destination mobility drastically degrades the performance when compared to the scenario with only the source node mobile. Also, it can be seen in Fig. 5(a) and (b) that there is a slight gap between the theoretical and simulated PEP bound in the low SNR regime. A similar trend can be observed in related works on the performance of STBC-coded systems such as [36], [48]–[51]. This arises since the union bound for PEP is only tight at high SNR. Fig. 5(c) shows the average PEP with the average channel gains of the source–relay and source–destination link increased. It can be seen that the gap decreases significantly with increasing strengths of the source–relay and source–destination links. Further, it can be seen from Fig. 6 that when only the relay node is in motion with perfect CSI available at each node, the performance of the system is not limited by an asymptotic error floor and it achieves a DO of N Nd = 4 as shown in Section III-B. However, the performances of the systems employing the dual-phase and multi-phase Pm protocols are limited by the asymptotic error floors derived in (25) and (41), respectively, which arise due to the time-selective nature of the links when either the source or the destination is mobile. Figs. 7–9 plot the per-frame average PEP with the optimal power factors β0 , βk , 1 ≤ k ≤ 2 obtained by solving the optimization problems in (31) and (48) using the CVX solver for the dual-phase and multiple-phase Pm protocols with K = 2

VARSHNEY AND JAGANNATHAM: MIMO-STBC-BASED MULTI RELAY COOPERATIVE COMMUNICATION

Fig. 7. Per-block average PEP performance of Alamouti coded cooperative systems employing: (a) Dual-phase protocol with optimal power factors β 0 = 0.7951, β1 = β 2 = 0.1024, N b = 5 and error variances σ e2 = 0.01, σ 2 = 0.001. (b) Multi-Phase Pm , m = 1 protocol with optimal power factors β 0 = 0.8018, β1 = 0.0761, β2 = 0.1221, N b = 5 and error variances (1, 2) (1, 2) σ e2 = σ 2 = (σ  r r )2 = (σ e r r )2 = 0.01.

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relays, respectively. It can be observed that the optimal power factors significantly enhance the end-to-end performance in comparison to equal power allocation, i.e., β0 = β1 = β2 = 13 for various channel conditions. Moreover, the performance gain with the optimal power factors is more pronounced when the relay–destination link is stronger than the source–relay link, as shown in Fig. 7. This arises due to the fact that the relay and destination nodes can decode the transmitted codewords accurately with very high probability, since higher power is employed at the source. On the other hand, when the source–relay links are in good channel conditions compared to the relay– destination links as considered in Fig. 8, equal power allocation, i.e., β0 = β1 = β2 = 1/3 is optimal. Further, one can also observe from Figs. 7 and 9 that the optimal power factors derived using the optimization problems in (31) and (48) can significantly reduce the error rate in the low as well as moderate SNR regimes. However, at high SNR, the performance of the system using both equal and optimal power allocation approaches the asymptotic floor derived in (25). This is because of the fact that the system performance is independent of the transmit power at high SNR due to node mobility and imperfect CSI. In Fig. 9, the performances of the suboptimal fixed DF relaying based near ML decoder and MD decoder in [39] are also demonstrated in the presence of node mobility and imperfect CSI, which can be seen to result in a significantly higher PEP in comparison to the proposed schemes. VI. CONCLUSION

Fig. 8. Per-block average PEP performance of Alamouti coded multirelay system employing: (a) Dual-phase protocol with K = 2 relays, N b = 5, and error variances σ e2 = 0.01, σ 2 = 0.001. (b) Multi-Phase Pm , m = 1 protocol (1, 2) with K = 2 relays, N b = 5, and error variances σ e2 = (σ e r r )2 = 0.01, σ 2 = (1, 2)

(σ  r r )2 = 0.01.

This paper comprehensively analyzes the performance of dual-phase and multiple-phase Pm multirelay MIMO-STBC cooperative wireless systems over time-selective and possibly nonidentical Rayleigh faded links. Closed-form expressions are derived for the per-frame average end-to-end PEP, DO and optimal power factors for both the systems. Further, this paper explicitly demonstrates the impact of the DOs of the source–relay and relay–destination links on the end-to-end PEP performance as well as optimal power allocation. It is analytically demonstrated that except for the scenarios when either all the nodes are static or only the relay nodes are mobile, both the multirelay systems with perfect CSI are limited by the asymptotic error floor derived. Finally, simulation results have been presented to demonstrate the performance of both the systems under various node mobility conditions and also to validate the analysis. REFERENCES

Fig. 9. Per-block average PEP performance of Alamouti coded cooperative systems employing (a) dual-phase protocol with optimal power factors β 0 = 0.4250, β1 = β 2 = 0.2875 and error variances σ e2 = 0.01, σ 2 = 0.001 (b) multi-phase Pm , m = 1 protocol with optimal power factors β 0 = 0.4055, β1 = 0.2754, β2 = 0.3191 and error variances σ e2 = σ 2 = (1, 2) (1, 2) (σ  r r )2 = (σ e r r )2 = 0.01.

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VARSHNEY AND JAGANNATHAM: MIMO-STBC-BASED MULTI RELAY COOPERATIVE COMMUNICATION

Neeraj Varshney (S’10) received the B.Tech. degree in electronics and communication engineering from the Uttar Pradesh Technical University, Lucknow, India, in 2008 and the M.Tech. degree in electronics and communication engineering from the Jaypee Institute of Information Technology, Noida, India, in 2011. He is currently working toward the Ph.D. degree in electrical engineering at the Indian Institute of Technology, Kanpur, India. His research interests include signal processing, communications, and networks which include digital communication, MIMO technology, cooperative communication. From October 2011 to August 2012, he was a Project Research Fellow at Jaypee Institute of Information Technology, Noida, India.

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Aditya K. Jagannatham (S’04–M’05) received the Bachelor’s degree in electrical engineering from the Indian Institute of Technology, Bombay, Mumbai, India, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of California, San Diego, CA, USA. From April 2007 to May 2009, he was employed as a Senior Wireless Systems Engineer at Qualcomm Inc., San Diego, CA, USA, where he worked on developing 3G UMTS/WCDMA/HSDPA mobile chipsets as part of the Qualcomm CDMA technologies division. His research interests include the area of next-generation wireless communications and networking, sensor and ad-hoc networks, digital video processing for wireless systems, wireless 3G/4G cellular standards, and CDMA/OFDM/MIMO wireless technologies. He has contributed to the 802.11n high throughput wireless LAN standard. Dr. Jagannatham received the CAL(IT)2 Fellowship for pursuing graduate studies at the University of California San Diego and, in 2009, he received the Upendra Patel Achievement Award for his efforts toward developing HSDPA/HSUPA/HSPA+ WCDMA technologies at Qualcomm. Since 2009, he has been a Faculty Member in the Electrical Engineering Department, IIT Kanpur, Kanpur, India, where he is currently an Associate Professor, and is also associated with the BSNL-IITK Telecom Center of Excellence. At IIT Kanpur, he received the P. K. Kelkar Young Faculty Research Fellowship (June 2012 to May 2015) for excellence in research and the Gopal Das Bhandari Memorial Distinguished Teacher Award for the year 2012–2013 for excellence in teaching. He has also delivered a set of video lectures on Advanced 3G and 4G Wireless Mobile Communications for the Ministry of Human Resource Development funded initiative National Programme on Technology Enhanced Learning.