This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access Faing Correlations for Wireless Cooperative Communications: Diversity Coing Gains Shen Qian, Jiguang He, Markku Juntti Ta Matsumoto School of Information Science, Japan Avance Institute of Science Technology JAIST - Asahiai, Nomi, Ishikawa, 923-292 Japan Email: {shen.qian, matumoto}@jaist.ac.jp Centre for Wireless Communications, FI-9004 University of Oulu, Finl Email: {jiguang.he, markku.juntti}@oulu.fi Abstract We theoretically analyze age probabilities of the lossy-forwar LF, ecoe--forwar DF aaptive ecoe--forwar ADF relaying techniques, with the aim of characterizing the impact of the spatial temporal correlations of the faing variations. First, the exact age probability ressions are analytically obtaine for LF, DF, ADF relaying assuming each link suffers from statistically inepenent Rayleigh faing. Then, approximate, yet accurate close-form ressions of the age probabilities for the relaying schemes are erive. Base on the ressions, iversity coing gains are foun. The mathematical methoology for the erivation of the licit age probability is further extene to the case where the account is taken of the spatial temporal correlations of the faing variations. The iversity gains with LF, DF, ADF are then erive in the presence of the correlations. It is foun that the three techniques can achieve full iversity, as long as the faing variations are not fully correlate. We then investigate the optimal relay location the optimal power allocation for the three relaying techniques. It is shown that the optimal solutions can be obtaine uner the framework of convex optimization. It is reveal that in correlate faing, the relay shoul move close to the estination or allocate more transmit power to the relay for achieving lower age probabilities, compare to the case in inepenent faing. Keywors Cooperation techniques, relay channels, faing variation, age probability, iversity coing gains, spatial temporal correlations, source-channel separation theorem. I. INTRODUCTION A promising cooperative technique calle lossy-forwar LF relaying [] has gaine a lot of attention recently, since it improves throughput efficiency reuces the age probability, compare to the conventional ecoe--forwar DF relaying [2]. Unlike the conventional DF strategy [3], [4], the ecoe information sequence at the relay R is interleave, re-encoe, transmitte to estination D, even though errors may be etecte in the information sequence after ecoing at R. Iterative processing between This research was supporte in part by the European Unions FP7 project, ICT-69555 Links-on-the-fly Technology for Robust, Efficient Smart Communication in Unpreictable Environments RESCUE, in part by Acaemy of Finl project, No. 268209 the network compression base wireless cooperative communication systems NETCOBRA, in part by Nokia Founation HPY Founation, also in part by the JAIST Doctoral Research Fellow program. Fig.. S h R hsd hsd hrd s t The schematic scenario of one relay aie communication system. two ecoers at the estination, one for ecoing the signal receive via the source S-D link the other for that via the R-D link, with log-likelihoo ratio LLR exchange via a LLR moification function improves ecoing performance [5]. The LF technique can be viewe as a istribute joint source-channel coing system with sie information [6] [9]. It has been foun that LF can achieve turbo-cliff-like bit error rate BER performance over aitive white Gaussian noise AWGN channels [0]. In [], the initial iea for LF is provie by utilizing Slepian-Wolf type cooperation for wireless communications. The coing algorithms are propose for faing relay channel in [2], [3] with the purpose of achieving the turbo processing gain. The key concept of the coing technique for LF is introuce in [], where it assumes that the relay oes not nee to necessarily recover the information sent from S perfectly. In [4], a three-noe LF relaying over Rayleigh faing channels is stuie by ientifying the relationship between the DF protocol Slepian-Wolf coing [5]. However, a rawback of [], [4] is that the amissible rate region is etermine by the Slepian-Wolf theorem which oes not perfectly match the problem setup, since only the information of S nees to be recovere at the estination. Zhou et al. [2] eliminate the aforementione rawback by utilizing the theorem of the source coing with sie information in the network information theory. Base on [2], the technique is further extene to multiple access relay channel MARC [6], where a pair of correlate sources are transmitte to a estination with the ai D 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 2 of one relay. Furthermore, He et al. [7] Qian et al. [8] apply the LF technique to multi-source multi-relay systems. Quite recently, a two-relay LF TLF transmission system is propose a power allocation scheme for minimizing the age probability of the TLF system is presente in [20]. The technique is applie to wireless sensor networks WSNs, where a simple, yet efficient, power allocation scheme for an arbitrary number of sensors is erive [2]. Brulat et al. present a meium access protocol in [22] specifically for the LF-base network esign framework. Kosek-Szott et al. [23] propose a centralize meium access protocol specifically esigne to operate with LF relaying for coorinate wireless local area network WLAN. It is shown that the centralize approach provies consierable gains compare to the istribute approach. The major contributions uner the LF relaying framework are summarize in Table I. Even though the superiority in LF over Rayleigh faing channels in terms of age probability is shown in [2], [6], [8], all the evaluations relate to the LF technique are base on the area integral over the amissible rate region [2], [6], [20], [2], which nees to be evaluate numerically with respect to the probability ensity functions pfs of instantaneous signal-to-noise power ratio SNR. Therefore, the iversity coing gains [24] cannot be ientifie separately from the numerically obtaine age probability thereby it is impossible to erive insightful ressions inicating the performance gains ue to iversity coing in faing channels. Furthermore, in practice, it is often the case that the faing conitions erience by ifferent links are correlate ue to the insufficient separations in the space or time omains between the noes or transmissions, respectively [25] [27]. In fact, the channel capacity epens heavily on the faing correlation [28], [29]. Therefore, examining the performance of iversity techniques in correlate faing conitions is a long lasting problem with great importance. Cheng et al. [4] has analyze the performance of LF relaying in correlate faing channels. However, in [4], the S-R link is moele by a binary symmetric channel BSC moel with fixe crossover probability, which is not realistic in practical applications. This paper aims at filling the gap between numerically obtaine results the licit ressions inicating the iversity coing gains, investigating the impact of faing correlations on the system performance. We use aaptive ecoe--forwar ADF relaying [30] as a counterpart of LF relaying since they require the same transmit phases. The LF, DF, ADF relaying strategies specify the secon phase operation performe by the relay noe, upon the information part obtaine after ecoing at R. For faing correlations, we consier the following two cases: faing variations in S-D R-D links are spatially correlate for LF, DF, ADF; 2 faing variations on the two slots in S-D link are temporally correlate for ADF. The age probability is erive by assuming the S-R link is also suffering from faing variation hence the bit error probability of the information part after ecoing at R is also a rom variable. Block faing assumption is use through the paper,, hence the relationship between the S-R link error probability the S-R link instantaneous SNR can be resse by using the theorem of lossy source-channel separation. The iscussions are conucte from four aspects: age probability erivations for LF, DF, ADF relaying in inepenent correlate faing; 2 iversity coing gains analyses base on approximate, yet accurate close-form age ressions; 3 optimal relay location ientification by taking into account the geometric gains; 4 optimal power allocation between S R, uner the total transmit power constraint. The main contributions of this paper are summarize as follows: * The exact age probabilities for LF, DF, ADF relaying over inepenent correlate Rayleigh faing channels are erive. Approximate, yet accurate closeform age ressions for each relaying techniques are then obtaine. * Explicit iversity orer coing gains are erive for LF, DF, ADF over the links suffering from statistically inepenent faing. It is proven that all the three relaying techniques can achieve full iversity, the coing gain with LF is larger than that of DF, smaller than that of ADF. * Diversity orers of LF, DF, ADF relaying over the links suffering from correlate faing are obtaine. It is foun that, full iversity orer can be achieve unless faing is fully correlate. * Optimal relay locations for achieving the lowest age probability are investigate in inepenent correlate faing. It is shown that with high faing correlation values, the relay shoul move closer to the estination for obtaining better age performance. However, for specific relaying technique, the optimal relay location keeps the same e.g., at mipoint for LF, regarless the value of the S-D link average SNRs. * Optimal power allocations between S R for achieving the lowest age probability are also analyze for LF, DF, ADF relaying in inepenent correlate faing. It is foun that the higher the channel correlation, the more power shoul be allocate to R commonly to the relaying techniques consiere. The line of the rest of the paper is as follows. Section II presents the system channel moels, the assumptions use through this paper. Section III erives the age probabilities of DF, LF, ADF relaying in statistically inepenent faing. The iversity coing gains are also iscusse in Section III. Section IV erives the age probabilities in correlate faing. Optimal relay locations are investigate in Section V. Optimal power allocation analysis is provie in SectionVI. Section VII conclues the paper. II. SYSTEM AND CHANNELS MODELS The wireless relay network moel use in this paper is shown in Fig.. A source S communicates with a estination D with help of a relay R. Each terminal is equippe with single antenna. Transmission phases are orthogonal no 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 3 TABLE I. Year Authors Contributions SUMMARY OF MAJOR CONTRIBUTIONS ON LF RELAYING. 2005 Hu Li [] Propose Slepian-Wolf cooperation, which loits istribute source coing technologies in wireless cooperative communications. 2007 Wolegebreal Karl [9] Consiere a network-coing-base MARC in the presence of non-ieal source-relay links, analyze the age performance coverage. 2007 Sneessens et al. [3] Derive a ecoing algorithm which enables the use of turbo-coe DF relaying by taking into account the probability of error between the source the relay. 202 Anwar Matsumoto [] Propose a iterative ecoing technique, accumulator-assiste istribute turbo coe, where the correlation knowlege between the source the relay is estimate loite. 203 Cheng et al. [4] Propose a scheme for loiting the source-relay correlation in joint ecoing process at the estination, base on the Slepian-Wolf theorem. 204 Zhou et al. [2] Derive the exact age probabilities by utilizing the theorems of lossy source-channel separation source coing with sie information. 205 Wolf et al. [20] Propose an optimal power allocation strategy for a two-relay system base on convex optimization to minimize age probability. 205 Lu et al. [6] Derive the age probability for orthogonal MARC for correlate source transmission where erroneous source information estimates at the relay are forware. 206 Wolf et al. [2] Propose asymptotically optimal power allocation scheme for WSNs with correlate ata from an arbitrary amount of sensors. 206 Brulat et al. [22] Propose a meium access control MAC layer protocol for LF relaying introuce testbe implementation base on UP GNU raio frameworks. multiple access channel is involve. The cooperation protocols consiere in this paper are LF, DF, ADF relaying. A. Relaying In LF relaying, S broacasts the coe information sequence to D R at the first time slot. The information sequence, obtaine as the result of ecoing at R, is interleave, reencoe transmitte to D at the secon time slot, even if ecoer etects errors. The bit level correlation between the information sequences transmitte from S R is estimate utilize in an iterative joint ecoing process at D to retrieve the original message of S. In this paper, we assumes that the correlation is known to the receiver. For more etails ab the correlation estimation at D, reaers may refer to [], [5]. In the conventional DF relaying, S broacasts the coe information sequence to D R at the first time slot. R tries to fully recover the receive information sequence. If it is successfully recovere, the information sequence is forware to D at the secon time slot. R keeps silent if error is etecte after ecoing at R. In ADF relaying, S broacasts the coe information sequence to D R at the first time slot. If the transmitte information is successfully recovere at R, the recovere information sequence is interleave, re-encoe transmitte to D at the secon time slot. Then, D performs joint ecoing by exchanging LLRs between the two ecoers, one correspons to the S-D link, the other to the R-D link. If R etects error in the information part after ecoing, R notifies S of the information recovery failure via a feeback link, S interleaves the information sequence, re-encoes, retransmits the information sequence to D again. D performs joint ecoing by exchanging LLRs between two ecoers, one for the sequence transmitte at the first time slot the other for that at the secon time slot. B. Channel Moel The S-D R-D links are consiere to be spatially correlate with ρ s = h SD h RD 0 ρ s representing the spatial correlation between h SD h RD. h ij i {S, R}, j {R, D}, i j enotes the complex channel gain of i-j link. The faing gains with the two transmissions over the S-D links are also correlate with ρ t = h SD h S D 0 ρ t representing the temporal correlation between h SD h S D, where h S D enotes the complex faing gain of the S-D link at the secon time slot. The variation moel can be represente by the first-orer Gauss-Markov rom process [3]. Let k {, 2,, K} represent symbol inex. The receive symbol ysd k via the S-D link yk RD via the R-D link both at D, y k via the S-R link at R are resse as y k SD = G SD h SD x k S n k SD, y k RD = G RD h RD x k R n k RD, 2 y k = G h x k S n k, 3 respectively, where n k ij i {S, R}, j {R, D}, i j enotes the zero-mean white aitive Gaussian noise AWGN with variance of N 0 /2 per imension. x k S xk R are the coe moulate symbols transmitte from S R with E{ x k S 2 } = E{ x k R 2 } = E s, with E s representing the transmit symbol energy. It is assume that the complex Gaussian link gain has zero mean unit variance E[ h ij 2 ] =. h ij is kept constant over one block uration ue to the block faing assumption. Let G ij represents the geometric gains of the i-j link. The average instantaneous SNRs can be resse as γ ij = G ij E s /N 0 γ ij = h ij 2 γ ij, respectively. We assume that the channel state information CSI is only available at the receiver sies. Each link is assume to suffer from frequency non-selective Rayleigh faing. The pf of instantaneous SNR γ ij is given 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 4 RR HbR Amissible Region RR HbR Amissible Region If the rate pair R S, R R falls into the inamissible region, the age event occurs, D cannot reconstruct b S with an arbitrarily small error probability. Since p f = 0 0 < p f 0.5 are istinctive, the age probability of LF relaying can be resse as HbS RS a Rate region for S R when p f = 0. Fig. 2. by Rate region for LF relaying. Hpf Hpf HbS RS b Rate region for S R when p f 0. pγ ij = γ ij, i {S, R}, j {R, D}, i j, γ ij γ ij 4 For the sake of simplicity, the effect ue to shaowing is not taken into account. III. OUTAGE ANALYSIS IN INDEPENDENT FADING A. Outage Behavior of LF in Inepenent Faing Let b S b R be the binary information sequence of S ecoing result at R, respectively. Accoring to the theorem of source coing with sie information [32], b S can be reconstructe error-freely at D if the source coing rates of b S b R, R S R R, satisfy the following inequalities: { R S Hb S ˆb R, R R Ib R ; ˆb 5 R, where ˆb R is the estimate of b R at D, H I ; enote the conitional entropy the mutual information between the arguments, respectively. Let p f be a crossover probability of a BSC virtually moeling the S-R link. p f stays the same within one block but varies block by block. Base on the Shannon s lossy source channel separation theorem [33], [34], the relationship between p f the instantaneous channel SNR is given as [2] p f = H 2 log 2, 6 with H2 enoting the inverse function of the binary entropy. The minimum istortion is equivalent to p f [35] uner the Hamming istortion assumption. With p f = 0 inicating perfect ecoing at R, we have Hb S b R = Hb R b S = 0. Hence, the inamissible rate region becomes the triangle area A as shown in Fig. 2a. When 0 < p f 0.5, the inamissible region is shown in Fig. 2b which can be ivie into two areas, B C. The rate region efine in 5 inicates that, although b R contains error, with 0 R R Hb R, it can serve as the sie information for losslessly recovering b S. In the case R R > Hb R, the conition becomes to R S Hp f. = P A P B P C, 7 where P A, P B, P C enote the probabilities that the rate pair R S, R R falls into the inamissible areas A, B, C, respectively. Taking into account the impact of p f, P A, P B, P C can be resse as P A = Pr[p f = 0, 0 R S <, 0 R R < Hp f p f ], 8 P B = Pr[0 < p f 0.5, 0 R S < Hp f, R R 0], 9 P C = Pr[0 < p f 0.5, Hp f R S <, 0 R R < Hp f p f ], 0 where a per-block BSC moel is again use to represent the R-D link with crossover probability p f. p f p f = p f p f p f p f with representing convolution operation. Base on the Shannon s lossless source channel separation theorem, the relationship between the instantaneous channel SNR γ ij its corresponing rate R i is given by R i Θγ ij = log 2 γ ij, i {S, R}, j = D with its inverse function γ ij Θ R i = 2 Ri. 2 Solving 8, 9 0 base on the pfs of the instantaneous SNRs of the corresponing channels, the age probabilities of LF relaying over inepenent Rayleigh channels can be resse as P = Θ Θ Θ =Θ 0 Θ Θ =Θ 0 [ ] Θ Θ =Θ Θ ξ, Θ =Θ 0, 3 where ξ, = H 2 {H 2 [ Θ] H 2 [ Θ]}. Gaussian coebook is assume. With loss of generality, the signaling imensionality is equal to two the spectrum efficiency of the transmission chain, incluing the channel coing scheme moulation multiplicity in each link is set to the unity. For more generic representation, reaers may refer to [2]. 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 5 RR HbR Amissible Region HbS RS a Rate region for S R when p f = 0. Fig. 3. Rate region for ADF relaying. RS HbS E Amissible Region HbS RS b Rate region for twice transmit S when p f 0. B. Outage Behavior of DF in Inepenent Faing In DF relaying, relay keeps silent if error is etecte after ecoing. When p f = 0, the age analysis for DF is the same as that for LF. When p f 0, it is equivalent to that of pointto-point S-D transmission. Therefore the age probability of DF relaying can be resse as P =Pr[p f = 0, 0 R S <, 0 R R < Hp f ] Pr[0 < p f 0.5, R S < ] = Θ =Θ 0 Θ Θ γ RD Θ C. Outage Behavior of ADF in Inepenent Faing Θ 4 With ADF, S retransmits an interleave re-encoe version of the information, if it is notifie of the ecoing failure via a feeback link from R. The rate regions of ADF with p f = 0 p f 0 are shown in Fig. 3a Fig. 3b, respectively. Similarly to the case of LF, the age probability is efine as the probability that the source rate pairs of S R, R S, R R R S, R S, fall into the inamissible area D or E shown in Fig. 3a Fig. 3b, respectively. R S is the rate of retransmitte information sequence from S. Let P D P E enote the probabilities that R S, R R R S, R S fall into the inamissible areas D E, respectively. The age. probability of ADF relaying is then given by P =P D P E =Pr[p f = 0, 0 R S <, 0 R R < Hp f ] Pr[0 < p f 0.5, 0 R S <, 0 R S < Hp f ] Θ = γ S D =Θ 0 Θ Θ Θ Θ Θ =Θ 0. γ SD γ S D 5 The erivations for the licit ressions of 3, 4, 5 may not be possible. We use a numerical metho [36] to evaluate P, P, P. D. Approximations By invoking the property of onential function e x x for small x, corresponing to high SNR regime, the age probabilities of LF, DF, ADF relaying, respectively, can be approximate by the close-form ressions, as, P P P ln4 4 ln2 3 SD ln4 4 ln2 3, 6 2 ln4 4 ln2 3 2, 7 γ 2 SD ln4 4 ln2 3 SD ln4 4 ln2 3. 8 2 It is foun from Fig. 4 that in inepenent faing i.e., ρ s = 0 for LF DF, ρ s = ρ t = 0 for ADF the approximate age curves obtaine from 6, 7, 8 well match the numerically calculate age curves from 3, 4, 5. E. Diversity Orer Coing Gains By setting the geometric gain of each link to ientical replacing γ ij with a generic γ, corresponing to equilateral triangle noe locations i.e., S, R, D are locate to the vertex of an equilateral triangle, 6, 7, 8 reuce to 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 6 Outage Probablity 0 0 2 0 3 0 4 0 5, Approx, Approx, Approx ρ s =0 ρ s = ρ s =0.99 that G ADF c > G LF c > G DF c, which verifies the relationship between the age probabilities shown in Fig. 4, i.e., with the inepenent channels ρ s = 0, ρ t = 0, P > P > P. Theorem : The age probability of LF is smaller than that of DF, i.e., P < P. Proof: The ifference between P in 6 P in 7, is P P = ln4 2 4 ln2 3. 2 25 Let = ln4 2 4 ln2 3 2. Since 0 6 5 0 5 20 25 30 Average SNR of the S D link B Fig. 4. Comparison between the exact approximate age probabilities for LF, DF, ADF relaying over inepenent spatial correlate channels, where the S-R, S-D, R-D links have equal istances ρ t = 0. lim =.3069 < 0 26 = 3 4 ln2 γ 2 > 0, 27 P = 2 ln4 2 8 ln2 2 ln4 4 γ 2 2 γ 3 4 ln2 3 2 γ 4, P = ln4 4 ln2 2 ln4 2 2 γ 2 γ 3 4 ln2 3 2 γ 4, 9 20 P = ln4 γ 2 4 ln2 3 2 γ 3, 2 respectively. Then, in high average SNR regime, 9, 20, 2 can further be approximate as P = G LF c γ G LF, 22 P = G DF c γ G DF, 23 P = G ADF c γ G ADF, 24 where G LF = GDF = GADF = 2 are the iversity orer of LF, DF, ADF relaying. This is consistent to the ecays of the cooperation protocols shown in the age curves in Fig. 4 i.e., those with ρ s = 0. G LF c =, G DF 2 ln4 2 c =, ln4 G ADF c = are the coing gains of LF, DF, ADF ln4 relaying, respectively [24]. Since the coing gain ression oes not inclue the term E s /N 0, the gain appears in the form of the parallel shift of the age curves. It is easy to fin increases monotonically as increases limite to -.3069. Therefore, it proves that 25 always has negative value for any given SNR, which inicates that the P is smaller than P. Theorem 2: The age probability of LF is larger than that of ADF, i.e., P > P. Proof: The ifference between P in 6 P in 8 is P P = γ 2 SD ln4 4 ln2 3 ln4 4 ln2 3. 2 28 We assume 4 ln2 3 4 ln2 3 SD with γ SD > 2. Since ln4 4 ln2 3 > 0, we have SD γ 2 SD γ ln4 4 ln2 3 SD < ln4 4 ln2 3 SD ln4 4 ln2 3, 29 2 inicating the right h sie of 28 is greater than 0. This proves P > P. 2 Note that the approximate age ressions 6, 7, 8 are obtaine at high SNR regime, therefore it is reasonable to give proof for > >. 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 7 IV. OUTAGE ANALYSIS IN CORRELATED FADING Let the correlation of two complex faing gains be efine as ρ = h h 2. The joint pf of two signal amplitues of correlate Rayleigh faing channels is given by [37]. It is not ifficult to convert the amplitue joint pf into that of the SNRs, γ γ 2, as pγ, γ 2 = γ γ 2 ρ 2 I 2 ρ γ γ 2 0 γ γ 2 ρ 2 [ ] γ γ γ2 γ 2 ρ 2, 30 where I 0 is the zero-th orer moifie Bessel s function of the first kin. A. Outage Behavior of LF in Correlate Faing We follow the same technique as that use for analyzing the age in inepenent faing. The multiple integral with the constraint 8, 9 0 over the joint pf 30 leas to the age probability ressions of LF relaying in spatially correlate S-D R-D links, as P = Θ =Θ 0 =Θ 0 ρ 2 s I 2 ρ s 0 γsd ρ s 2 [ ] Θ γrd ρ s 2 =Θ 0, Θ Θ Θ Θ Θ =Θ 0 γ I 0 [ ρ 2 s Θ Θ =Θ 0 γ I 0 [ ρ 2 s =Θ 0 2 ρ s γsd ρ s 2 ] γrd ρ s 2 =Θ Θ, Θ [ξ,] =Θ 0 2 ρ s γsd ρ s 2 ] γrd ρ s 2 B. Outage Behavior of DF in Correlate Faing. 3 Similarly, the age probability ression of DF relaying in the correlate S-D R-D link variation can be resse as, P = Θ =Θ 0 ρ 2 s I 2 ρ s 0 γsd ρ s 2 [ ] γrd ρ s 2 Θ, Θ Θ =Θ 0 Θ. 32 C. Outage Behavior of ADF in Correlate Faing When analyzing the ADF age probability, two correlation terms, ρ s ρ t, correlations of the complex faing gains of the S-D R-D links, those of the two ajacent S- D transmissions, respectively, have to be taken into account. After several mathematical manipulations, we have the age probability ression of ADF relaying over the links with correlate faing, as A P = Θ Θ Θ ρ 2 s I 0 [ ] γrd =Θ 0 2 ρ s γsd ρ s 2 ρ s 2, Θ =Θ 0 =Θ 0 Θ Θ γ S D=Θ 0 γ S D ρ 2 t I 2 ρ t γ S D 0 γsd γ S D ρ t 2 [ ] γs D γ S D ρ t 2 γ S D. 33 Note that, the age ressions 3, 32, 33 in correlate faing reuce to those over inepenent links 3, 4, 5 by setting ρ s = 0 ρ t = 0. In Fig. 4, we can observe that with the high spatial faing correlation, the age performance ifference among the LF, DF, ADF relaying schemes vanishes. The impact of spatial correlation ρ s on the age probabilities of the LF, DF ADF systems are emonstrate in Fig. 5. As can be seen from the figure, over the entire ρ s value region, LF exhibits superior age performance to DF. This is because with LF, the relay system can be seen as a istribute turbo coe 3. It is also foun that ADF achieves lower age probability than LF. This is because the inamissible rate region with ADF is mae smaller by the feeback information than LF, as shown in Fig. 2 Fig. 3. It is observe that for all the three relaying schemes, the age probabilities increase 3 Note that the iterative processing for LF may cause transmit elay /or excessive power computation than DF. However, the increase in transmit elay or energy consumption is negligible, especially when the SNR is high. 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 8 Outage Probablity 0 2 0 3 0 4 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 D. Approximations The Bessel function of the first kin can be resse with its series ansion by Frobenius metho [38, 9..0], as: I 0 x = m=0 m x 2m. 34 m!γm 2 Then, 3, 32, 33 can be, respectively, approximate as P ln4 3 2 γ 2 RD ρ 2 s ln4 2 ln4 ρ 2 s 2 ln2 3 2 SD ρ 2 s 2 ln4 3 2 γ 2 2 ln4 3 2 SD 4 ln6 4γ 2, 35 SD γ2 Fig. 5. Outage probability versus the spatial correlation ρ s, where ρ t = 0. ρ s P ln4 3 2 γ 2 RD ρ 2 s A P ln4 3 2 γ 2 RD ρ 2 s ln4 ρ 2 s 2 ln2 3 2 SD ρ 2 s, 36 ln4 ρ 2 s 2 ln2 3 2 γ S D ρ 2 s ln4 ρ 2 t 4 ln2 3 2 γ S D ρ 2 t ln4 3 2 γ 2 RD ρ 2 t. 37 The accuracy of the approximations is emonstrate in Fig. 7 in next section. ρt ρs Fig. 6. Impacts of time correlation ρ t spatial correlation ρ s on age probability of ADF. E. Diversity Orer The iversity orer is efine as [39] logp = lim γ logγ. 38 Accoring to 38, the iversity orer of LF, DF, ADF over correlate channels can be resse as as the spatial correlation ρ s between the S-D R-D links becomes larger. Fig. 6 shows the age performance of ADF relaying with the time correlation ρ t spatial correlation ρ s as parameters. As can be seen from the figure, the effect of ρ t is not as significant as ρ s. The larger the ρ t value, the bigger the age probability of ADF but slightly. lnp LF = lim γ lnγ { [ 5 ln4 7 4 ln6 = lim ln γ 3 4γ 4 γ ] } ln4 2 ln2 2 ln4 6 γ 2 ρ 2 s 3 / lnγ = 2, ρ 2 s ρ s, 39 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 9 lnp DF = lim γ lnγ { [ 2 ln4 3 = lim ln γ γ 3 ln4 4 ln6 2 4γ 4 ] ln4 2 ln2 2 ln4 6 γ γ 2 ρ 2 s 3 ρ 2 s / lnγ} = 2, ρ s, 40 A lnp ADF = lim γ lnγ { [ ln4 = lim ln γ γ 3 ρ 2 t ln4 γ 4 ln2 2 ln4 6 4 ρ 2 t γ 2 ρ 2 s 2 ln2 3 3 ρ 2 s ln4 3 ] 2 γ 3 ρ 2 s / lnγ} = 2, ρ s, ρ t, 4 where,, are represente by a generic symbol γ uner the equilateral triangle noes location assumption. It is shown that the full iversity gain can be achieve by LF, DF, ADF relaying as long as ρ s for LF DF, ρ s ρ t for ADF. When ρ s = ρ t =, i.e., in fully correlate faing, the iversity orer reuces to one. V. OPTIMAL RELAY LOCATIONS FOR MINIMIZING THE OUTAGE PROBABILITY In this section, we investigate the optimal relay locations which minimize the age probability of the LF, DF, ADF relaying schemes. The age ressions can be rewritten so that they are functions of position of relay noe by taking the geometric gain into consieration. It is shown that, the relay location optimization can be formulate as a convex optimization problem. A. Optimal Relay Locations in Inepenent Faing Let SD, RD, enote the istances between S D, R D, S R respectively. With G SD being normalize to unity, G G RD can be efine as α α G = SD GRD = SD RD [40], respectively, where α is the path loss onent. Then, the average SNRs of the S-R R-D links can be given as = 0lgG B, 42 = 0lgG RD B. 43 With the normalize S-D link length SD =, substituting 42 43 into 6, 7, 8, yiels the age probability ressions of LF, DF, ADF relaying over the links suffering from inepenent faing with respect to the position of R as P = P = 0 lg ln4 4 ln2 3 P = ln4 α 2 4 ln2 3 2 0 lg α 0 lg ln4 4 ln2 3 2 α 0 lg ln4 4 ln2 3 α 2 ln4 4 ln2 3 2 SD 0 lg α γsd 0 lg α, 44 0 lg α γsd 0 lg α, 45 0 lg α γ 2 SD 0 lg α, 46 uner the assumption that R moves along the line between S D, with which = RD =. The general optimization problem with regar to can be formulate as = arg min P subject to: 0,. 47 Proposition : Outage probability ressions of LF, DF, ADF relaying in 44, 45, 46, respectively, are convex with respect to 0,. Proof: See Appenix A. Taking the first-orer erivative of P, P, P, respectively, in 44, 45, 46 with respect to setting the erivative equal to zero, we have P 0 lg α = ln4 4 ln2 3 2 0 lg α 4 ln2 3 ln4 0 lg α 2 0 lg α = 0, 48 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 0 TABLE II. OPTIMAL RELAY LOCATION B 0 5 20 25 30, ρs = 0 0.332 0.3322 0.3323 0.3324 0.3328, ρs = 0 0.507 0.508 0.509 0.5020 0.5020, ρ s = 0 0.996 0.9937 0.9944 0.9957 0.9978, ρs = 0.99 0.60 0.647 0.6253 0.677 0.682, ρs = 0.99 0.789 0.7822 0.7829 0.7839 0.784, ρ s = 0.99 0.999 0.9993 0.9995 0.9998 0.9999 P P 0 lg α = ln4 4 ln2 3 2 0 lg α 0 lg α = ln4 4 ln2 3 2 0 lg α 0 lg α = 0, 49 ln4 4 ln2 3 γ 2 SD 0 lg α 2 SD = 0, 50 It may be excessively complex to erive the licit ression for from 48, 49, 50. Hence, the Newton- Raphson metho [4, Chapter 9] is use to numerically calculate the solution of. B. Optimal Relay Locations in Correlate Faing Similarly to the inepenent faing case, the age probability ressions of LF, DF ADF relaying in correlate faing with respect to the position of R can be resse as 5, 52, 53. The optimization problem for the correlate faing channels case can also be formulate by 47 for the LF, DF, ADF relaying techniques. Proposition 2: Outage probability ressions of LF, DF, ADF relaying, respectively, given by 5, 52, 53 respectively, are convex with respect to 0,. Proof: The convexity of 5, 52, 53 can be proven by taking secon-orer partial erivative of P, P, P in 5, 52, 53 with respect to, showing that the erivative results are positive in the range 0, 4. Take the first-orer erivative of P, P, A P, respectively, in 5, 52, 53 with respect P to set the erivative result equal to zero, = 0, Outage Probablity 0 2 0 3, Approx, Approx ρs=0, Approx ρs=0.99 0 4 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x coorinate of relay Fig. 7. Outage probabilities versus relay locations for ifferent relaying schemes over inepenent correlate channels, where = 5 B ρ t = 0. P = 0, P A = 0. As in the inepenent faing case, the optimal relay location for each relaying schemes over correlate faing channels can also be obtaine by utilizing the Newton-Raphson metho [4, Chapter 9]. The optimal relay location for ifferent relaying schemes over spatially inepenent ρ s = 0 as well as spatially correlate ρ s = 0.99 channels are shown in Table II 5. Obviously, with the correlate S-D R-D links, the optimal relay positions for achieving the smallest age probability shift closer to D than the inepenent faing case. However, for specific relay scheme the optimal relay positions stay the same e.g., for LF in spatially inepenent faing ρ s = 0 the optimal relay position is at the mipoint, = 0.5 regarless of the average SNR values. The eviations of are ue to the numerical calculation errors Fig. 7 epicts the impact of the relay location on age performances of LF, DF ADF relaying in spatially inepenent ρ s = 0 correlate ρ s = 0.99 cases. The relay is assume to move along the line between S x = 0 D x =. Both the numerically calculate age probabilities approximate age probabilities are plotte. It is foun from the figure that the approximate numerically calculate age probability curves are consistent with each other. This observation inicates the accuracy of the approximations. It is foun from Fig. 7 that in statistically inepenent faing the optimal relay locations for LF, DF, ADF that achieve the lowest probabilities are = 0.5, < 0.5, > 0.5, respectively. On the contrary, in correlate faing, the optimal relay location for all the relaying schemes are closer to D. This is because the faing correlation reuces the coing gain achieve by the transmission over the S-D R-D links. 4 The etails of the proof are straightforwar but lengthy, therefore are omitte here for brevity. 5 We only focus on the spatial correlation ρ s which have impact on age of DF, LF, ADF relaying. 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access P = γsd 0 lg α ln4 0 lg α ρ 2 s 2 SD 2 ln2 3 0 lg α ρ 2 s ln4 3 2 2 0 lg α ρ 2 s 2 ln4 3 2 γsd 0 lg α 2 2 ln4 3 γsd 0 lg α ln4 SD γsd 0 lg α 4 ln6 γsd 0 lg α 2, 5 2 SD 4γ 2 SD P = γsd 0 lg α ln4 0 lg α ρ 2 s 2 SD 2 ln2 3 0 lg α ρ 2 s ln4 3 2 2, 52 0 lg α ρ 2 s γsd 0 lg α A P = γsd 0 lg α ln4 0 lg α ρ 2 s ln4 3 2 2 0 lg α ρ 2 s 2 SD 4 ln2 3 γsd 0 lg α 0 lg α ρ 2 t 2 SD 2 ln2 3 0 lg α ρ 2 s ln4 γsd 0 lg α 0 lg α ρ 2 t ln4 3 2 2 0 lg α γsd 0 lg α ρ 2 t. 53 The impact of the time correlation ρ t of the complex faing gains between the two transmission slots over the S-D link as well as the spatial correlation ρ s of the complex faing gains between the S-D R-D links on optimal relay location with LF, DF, ADF relaying is epicte in Fig. 8. With LF DF, the value of increases as ρ s increases, which inicate that the optimal relay locations shift closer to D as the spatial correlation increases. Obviously, ρ t has no impact on with LF DF relaying. With ADF system, interestingly, it is foun that oes not change regarless of the ρ s value, when ρ t is small. As the time correlation ρ t becomes large, with ADF exhibits the same tenency as that with LF DF. In general, the optimal with LF is larger than that with DF smaller than that with ADF. VI. OPTIMAL POWER ALLOCATION FOR MINIMIZING THE OUTAGE PROBABILITY In this section, our aim is to minimize the age probabilities for the LF, DF, ADF relaying techniques by ajusting powers allocate to S R. Let the powers allocate to S R be enote as E T k E T k, respectively, where E T represents the total transmit power k 0 k is the power allocation ratio. With the noise variance of each link being normalize to the unity, the geometric gain times transmit power is equivalent to their corresponing average SNR. Then the average SNRs of each link can be resse as functions of k as = E T kg SD, = E T kg RD, = E T kg. A. Optimal Power Allocation in Inepenent Faing The age ressions of LF, DF, ADF in inepenent faing with respect to k can be written as P = P = E T 2 k kg SDG RD ln4 4 ln2 3 ET 2 k2 G SD G E T kg ln4 4 ln2 3 2E T kg, 54 ET 2 k kg SDG RD E T kg ln4 4 ln2 3 ET 2, 55 k2 G SD G 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 2 TABLE III. OPTIMAL POWER ALLOCATION RATIO k ρs ADF Fig. 8. Impact of time correlation ρ t spatial correlation ρ s on optimal relay location, where = 5 B. P = DF LF ET 2 k kg SDG RD E T kg ln4 4 ln2 3 ET 3 k3 G G SD ρt ln4 4 ln2 3, 56 respectively. The optimization problem with regars to k can be formulate as k = arg min P k k 57 subject to: k 0,. Proposition 3: Outage probability ressions of LF, DF, ADF relaying, respectively, in 54, 55, 56 are convex with respect to k 0,. Proof: See Appenix B. Taking the first-orer erivative of P, P, P, respectively in 54, 55, 56 with respect to k setting the erivative result equal to zero, we have, P ET 2 k kgsdgrd E T kg = k ln4 4 ln2 3 ET 2 k2 G SDG ln4 4 ln2 3 2E T kg = 0, 58 k E T B 0 5 20 25 30, ρs = 0 0.7532 0.7444 0.747 0.7408 0.740, ρs = 0 0.6989 0.679 0.6698 0.6688 0.6685, ρ s = 0 0.506 0.5053 0.5046 0.5039 0.5022, ρs = 0.99 0.686 0.6309 0.5302 0.5298 0.5258, ρs = 0.99 0.6266 0.5530 0.593 0.53 0.59, ρ s = 0.99 0.509 0.507 0.505 0.504 0.503 P P ET 2 k kgsdgrd E T kg = k ln4 4 ln2 3 E 2 T k2 G SDG k ET 2 k kgsdgrd E T kg = k ln4 4 ln2 3 ET 3 k3 G G SD ln4 4 ln2 3 k = 0, = 0. 59 60 The optimal power allocation ratio k can be obtaine by numerically solving 58, 59, 60. B. Optimal Power Allocation in Correlate Faing The age ressions of LF, DF, ADF in correlate faing can be written as 6, 62, 63, respectively. The optimization problem in correlate faing can also be formulate by 57 for LF, DF, ADF relaying. Proposition 4: Outage probability ressions of LF, DF, ADF relaying, respectively, in 6, 62, 63, are convex with respect to k 0,. Proof: The convexity of 6, 62, 63 can be proven by taking secon-orer partial erivative of P, P, P in 6, 62, 63 with respect to k, showing that the erivative results are positive in the range k 0, 6. By taking the first-orer erivative of P, P, A P, respectively, in 6, 62, 63 with respect to P k setting the erivative result equal to zero, k = 0, P k = 0, P A k = 0, similarly to the inepenent faing case, the optimal power allocation ratio k of each relaying schemes over correlate faing channels can be numerically obtaine. Optimal k for the ifferent relaying schemes are shown in Table III 7. Obviously, the larger the total transmit power 6 The etails of the proof are straightforwar but lengthy, therefore are omitte here for brevity. 7 We only focus on the spatial correlation ρ s which have impact on age of DF, LF, ADF relaying. 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 3 ln4 P = E T kg SD E T kg SDE T kg RD ρ 2 s 2 ln2 3 2E T kg 2 SD ET kgrd ρ2 s ln4 3 2 2 ln4 3 2 ln4 3 ln4 E T kg SDE T kg 2 RD ρ2 s ET 2 k2 G 2 2ET 2 k2 G 2 SD ET kgsd E T kg SD 4 ln6, 4ET 2 k2 G 2 SD E2 T k2 G 2 6 P = ln4 E T kg SD E T kg SDE T kg RD ρ 2 s 2 ln2 3 2E T kg 2 SD ET kgrd ρ2 s ln4 3 2 E T kg SDE T kg 2 RD ρ2 s E T kg SDE T kg SD, 62 A P = ln4 E T kg SD E T kg SDE T kg RD ρ 2 s 2 ln2 3 2E T kg 2 SD ET kgrd ρ2 s ln4 3 2 E T kg SDE T kg 2 RD ρ2 s ln4 E T kg SDE T kg RDE T kg SD ρ 2 t 4 ln2 3 2E 2 T k2 G 2 SD ET kgrdet kgsd ρ2 t ln4 3 2 E T kg SDE 2 T k2 G 2 RD ET kgsd ρ2 t. 63 0 3, Approx PDF, Approx PADF, Approx DF ρs=0.99 Outage Probablity 0 4 0 5 ρs=0 LF ADF 0 6 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Transmit Power Ratio k ρt ρs Fig. 9. Outage probabilities versus power allocations for ifferent relying schemes over inepenent correlate channels, where E T = 20 B ρ t = 0. E T, the more transmit power shoul be allocate to R both in the case of faing variation on the S-D R-D links are inepenent spatially correlate. It is also foun that for achieving minimum age, R nees more power with LF than with DF, nees less power than with ADF. Moreover, with large spatial channel correlation ρ s, more power shoul be allocate to R for all the relaying techniques. Fig. 9 presents the impact of the power allocation ratios to S R on age probabilities. In numerical results, we normalize the geometric gain of each link to unity uner the Fig. 0. Impact of time correlation ρ t spatial correlation ρ s on optimal power allocation ratio k, where E T = 20 B. equilateral triangle noes location assumption, investigate the impact of power allocation ratio k on age performance. The total transmit power is fixe at 20 B. Note that the approximate age curves well match with the numerically calculate curves. It can be observe from Fig. 9 that, with high spatial correlation ρ s, nearly equal power allocation k 0.5 yiels the lowest age probabilities for all LF, DF, ADF relaying. On the contrary, in statistically inepenent faing, the equal power allocation is optimal for ADF, while DF LF nee more power for S, which is consistent with the results shown in Table III. 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/ACCESS.207.2699785, IEEE Access 4 Fig. 0 shows the impact of ρ t ρ s on optimal power allocation ratio k for LF, DF, ADF relaying, where the total transmit power for S R is set at E T = 20 B. It can be seen from the figure that the larger the spatial correlation ρ s, the smaller the optimal k value require with DF LF, inicating that more power nees to be allocate to R. It is also foun that, with DF, more power nees to be allocate to source noe than with LF, in orer to achieve the smallest age probability. This is because DF forwars only the error-free information sequences from relay noe. Note that the equal power allocation strategy is always optimal for ADF regarless of the values of ρ t ρ s. VII. CONCLUSIONS We have analyze the age probabilities with LF, DF, ADF relaying in statistically inepenent correlate faing. The iversity coing gains have been erive from high-snr, yet accurate approximations in inepenent faing. The iversity orers for LF, DF, ADF relaying have also been obtaine in correlate faing. Our analysis emonstrates that the full iversity gain can be achieve by either of LF, DF, ADF relaying, as long as the faing variations are not fully correlate. It is proven that coing gain with the LF strategy is larger than with DF relaying but smaller than with the ADF metho, in inepenent faing. Furthermore, the optimal relay locations optimal power allocations between the source relay have been investigate. The observations suggest that compare to the inepenent faing case, the relay shoul be locate closer to the estination, or more transmit power be allocate to the relay for achieving the lowest age probabilities, in the correlate faing case. An important extension is to take into consieration of ifferent channel moels, ifferent topologies, ynamic network change, which is left as future work as summarize in Table IV. Future Work TABLE IV. Complex Structure Multi-relay Channel Moels No Rayleigh FUTURE WORK AND CHALLENGES Challenges Multiple helper problem is one of the open questions in Network Information Theory. In many cases, joint pf for more than one channel realization is still unknown. APPENDIX A PROOF OF CONVEXITY OF 44, 45, AND 46 By taking secon-orer partial erivative of P, P, P in 44, 45, 46 with respect to, respectively, we have 2 P 2 = 9α α α 4 α2 0 lg 3 α 90α 2 2α 2α2 0 lg α 4, 64 2 P 2 = 2 P 2 = 20α 2 2α 2α2 0 lg 3 α 0α 2 α α 0 lg α 2, 65 20α 2 0 lg α 2α γsd 0 lg 3 2α2 α α 2 α γsd 0 lg α 2 α. 66 Obviously, 64, 65, 66 are positive in the range 0 which inicates that the objective functions are convex with respect to 0. APPENDIX B PROOF OF CONVEXITY OF 54, 55, AND 56 Taking the secon-orer erivative of P, P, A P in 54, 55, 56 with respect to k, respectively, we have 2 P E k 2 = T G k 6 5ET 3 G2 SD G RDk 3 k 2 0E T G 9 k 50 ET 2 G SDG RD k 4 7 9 25ET 3 G SDG 2 E T G k k5 20ET 3 G2 SD G RDG k 4 k, 2 P k 2 = 9 E T G k 5 2ET 3 G2 SD G RDk 4 k 67 E T G k ET 3 G2 SD G RDk 3 k 2 6 ET 2 G SDG k 4 5 2ET 4 G2 SD G RDG k 5 k, 2 A P E k 2 = T G k 2 5ET 3 G2 SD G RDk 3 k 2 68 0E T G k 9 50 ET 3 G SDG RD k 5 9 9 0ET 4 G2 SD G k 6 E T G k 20ET 3 G2 SD G RDG k 4 k. 69 It is not ifficult to fin that 68, 67, 69 are positive in the range 0 k which inicates that the objective functions are convex with respect to 0 k. REFERENCES [] K. Anwar T. Matsumoto, Accumulator-assiste istribute turbo coes for relay systems loiting source-relay correlation, IEEE Commun. Lett., vol. 6, no. 7, pp. 4 7, July 202. 269-3536 c 206 IEEE. Translations content mining are permitte for acaemic research only. Personal use is also permitte, but republication/reistribution requires IEEE permission. See
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