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Wireless Pers Commun DOI 10.1007/s11277-016-3464-9 On the Performance of Opportunistic Relay Selection in Cognitive Radio Networks with Primary User’s Interference and Direct Channel Khuong Ho-Van1 • Lien Pham-Hong2 • Son Vo-Que1 Tra Luu-Thanh1 • Ĩ Springer Science+Business Media New York 2016 Abstract This paper analyzes outage performance of opportunistic relay selection in cognitive radio networks with primary user’s interference and direct channel over independent non-identically distributed fading channels and under maximum transmit power constraint and interference power constraint Exact analysis is firstly proposed and then extended to asymptotic analysis to have insights into important performance metrics such as coding gain and diversity order The proposed analysis can also be applied to corresponding analysis for opportunistic relay selection in cognitive radio networks with PU’s interference and without direct channel to highlight the usefulness of direct channel in relaying communications without any sacrifice of power and bandwidth A multitude of results illustrate an achievable full diversity order, a considerable system performance deterioration due to primary user’s interference but this deterioration can be drastically remedied by increasing the number of involved relays, and the superiority of opportunistic relay selection with direct channel to that without it Keywords Opportunistic relay selection Á Cognitive radio Á Interference Á Direct channel Introduction Currently, inefficient traditional spectrum allocation by means of fixed primary users (PUs), emergence of several new wireless applications (e.g., video calling, file transferring, high-definition video streaming, and high-speed internet access), and a severe spectrum & Khuong Ho-Van khuong.hovan@yahoo.ca Department of Telecommunications Engineering, HoChiMinh City University of Technology, 268 Ly Thuong Kiet Str., District 10, HoChiMinh City, Vietnam Department of Computer and Communication Engineering, HoChiMinh City University of Education Technology, Vo Van Ngan Str., Thu Duc District, HoChiMinh City, Vietnam 123 K Ho-Van et al under-utilization as reported in an extensive survey on frequency spectrum utilization carried out by the Federal Communications Commission induce radio spectrum to become more and more scarce [1] Consequently, devising novel technologies to relieve the pressure of the spectrum scarcity is urgent and essential The cognitive radio technology (e.g., [2]) is one among them, which efficiently resolves this problem by allowing secondary users (SUs) to opportunistically utilize the spectrum inherently allotted to PUs as long as communications of PUs is not damaged Communications of PUs is guaranteed at an acceptable degree as long as the interference from SUs is controlled To this end, SUs usually operates in three modes: interweave, overlay, and underlay [3] Due to its distinct feature of low implementation complexity, the underlay mode has recently gained much interest, e.g., [4–19] and citations therein This mode properly allocates the transmit power of SUs to control the interference to PUs in either short-term or long-term manner According to the long-term mechanism, the transmit power of SUs is allocated to meet the pre-determined outage probability of PUs [4–10] while according to the short-term mechanism, the transmit power of SUs must satisfy either interference power constraint [11–13] or both interference power constraint and maximum transmit power constraint [14–19] It is the limitation of transmit power of SUs in the underlay mode that significantly shortens the transmission coverage of SUs To overcome this drawback, relaying communications techniques has recently been integrated into SUs to exploit short-range point-to-point communications for low path-loss [20] In relaying communications, communications between the source and the destination can be assisted by multiple relays operated in either amplify-and-forward (AF) or decode-andforward (DF) manner [21] for high performance but low bandwidth efficiency due to the requirement of orthogonal channels for different relays in order to avoid mutual interference Therefore, the relay selection in which a single relay among all potential candidates is selected is preferred to optimize system resource utilization, such as power and bandwidth, in comparison with multi-relay assisted transmission while remaining the same diversity order [18, 22] Several relay selection schemes in cognitive radio networks1 were proposed in [9–14, 16, 18, 19, 33–41] To be more specific, the opportunistic relay selection was proposed in [9–11, 18, 33–35] in which the relay with the maximum end-to-end signal-tonoise ratio (SNR) is selected; the reactive relay selection scheme, which chooses the relay among all potential candidates (i.e., all relays are presumed to correctly recover source information) with the largest SNR to the destination, was investigated in [12–14, 16, 34, 36]; the Nth best-relay selection and the Lth-worst relay selection were proposed in [19] and [37], respectively; the relay selection scheme for maximizing secrecy capacity was studied in [38]; the relay selection scheme with the compromise between the gain for SUs and the loss for PUs was suggested in [39]; the partial relay selection, which simply selects the relay with the largest SNR from the source, was analyzed in [40, 41] However, several assumptions have been imposed on these works for analysis simplicity: (1) independent partially-identical (i.p.i) [9–11, 13, 18, 31, 33, 34, 37, 39–41] or independent identical (i.i.) fading distributions [14, 16, 19, 36]; (2) no interference from PUs; (3) no direct channel; (4) un-correlation among received SNRs Among relay selection schemes, the opportunistic relay selection is theoretically proved to be optimal (e.g., [18]), and hence, it is interesting to evaluate its information-theoretic This paper focuses on the DF relays in cooperative cognitive networks and hence, relay selection schemes with the AF relays (e.g., [23–28]) or in dual-hop cognitive networks (e.g., [29–32]) are not necessarily surveyed 123 On the Performance of Opportunistic Relay Selection in… performance limit (i.e., outage probability) under practical conditions such as both interference and maximum transmit power constraints, i.n.i.d fading channels, presence of PU’s interference and direct channel, correlation among Signal-to-Interference plus Noise Ratios (SINRs)2 This is the objective of this paper To the best of our knowledge, no analysis accounts for all these practical conditions The current work presents the following contributions: • Propose an exact closed-form outage probability expression for the opportunistic relay selection in cognitive radio networks over i.n.i.d Rayleigh fading channels and under both maximum transmit power and interference power constraints, the presence of PU’s interference and direct channel, and correlation among received SINRs The proposed expression is helpful in fast evaluating the system performance without timeconsuming simulations • Extend the proposed exact analysis to asymptotic analysis to obtain important performance metrics (e.g., diversity order and coding gain), which proves that the opportunistic relay selection achieves the full diversity order offered by all involved relays and the source • Propose a corresponding analysis for the opportunistic relay selection with PU’s interference and without direct channel for the convenience of comparing the opportunistic relay selection with and without direct channel as well as emphasizing the importance of the direct channel in relaying communications It is recalled that no matter whether the direct channel is utilized for signal combining at the destination, the system resource utilization such as power and bandwidth is almost unchanged As such, it is essential to investigate how much gain the direct channel can contribute to performance improvement of the opportunistic relay selection • Provide numerous results to expose useful insights into the system performance such as diversity order, coding gain, substantial performance enhancement with respect to the increase in the number of relays, considerable system performance degradation owing to the PU’s interference, and superiority of cooperative relaying (i.e., relaying communications with direct channel) to its dual-hop counterpart (i.e., relaying communications without direct channel) This paper is structured as follows The next section presents the system model under investigation Exact and asymptotic analysis for the opportunistic relay selection with/ without direct channel is elaborately discussed in Sect Section provides numerous results to corroborate the proposed analysis and illustrates the outage behavior of the opportunistic relay selection in key system parameters Finally, useful conclusions close the paper in Sect System Model Figure demonstrates a system model for the opportunistic relay selection in cognitive radio networks with PU’s interference and direct channel In the secondary network, the source Ss communicates the destination Sd with the assistance of the selected relay Sb in the group of K relays, S ¼ fS1 ; S2 ; ; SK g We assume that secondary transmitters operate in the underlay mode (e.g., [5, 15, 17, 18]), and hence, the mutual interference between the As will be shown in the next section, both PU’s interference and direct channel induce received SINRs to be correlated, making the analysis more complicated but more general and practical 123 K Ho-Van et al Primary transmitter Primary receiver Pt Pr Transmission Interference S1 Sb Sd Stage Ss SK Stage Secondary network Fig System model primary network and the secondary network is available In other words, Ss and Sb interfere communications between the primary transmitter Pt and the primary receiver Pr , and Pt also causes interference to the received signals at relays and Sd It is recalled that the interference from the primary network to the secondary network was neglected for analysis simplicity (e.g., [6, 8, 10, 12–19, 24, 31, 33, 40, 42–47] and citations therein) It is the interference from PUs that makes the performance analysis complicated but general and practical Additionally, it is apparent that two stages of the opportunistic relay selection in the secondary network can take place instantaneously with communications of two different primary transmitter-receiver pairs Nevertheless, in order to have a compact figure, Fig only illustrates one transmitter-receiver pair However, this paper still reflects this general case by assuming two different primary transmitter-receiver pairs throughout the following analysis (i.e., there are two different channel coefficients from Pt to Sd for two corresponding stages, namely htd1 and htd2 ) Towards this end, channel coefficients are shown in Table Wireless channels are assumed to be independent, frequency-flat, and Rayleigh-distributed Therefore, the channel coefficient, hpq , between the transmitter p and the receiver q can be modelled as a circular symmetric complex Gaussian random variable with zero mean and 1=kpq -variance, i.e., hpq $ CN ð0; 1=kpq Þ In contrast to existing works in relay selection where i.p.i.3 or i.i.4 fading distributions are assumed for simplicity of performance analysis, this paper investigates i.n.i.d fading channels, and so, all kpq ’s, 8fp; qg are not necessarily equal, making our work more general and practical That means that kpq ’s are partitioned into groups of equal value For instance, ksk ’s, kkd ’s, kkr ’s, ktk ’s with k R are assumed to be equal in [9–11, 13, 18, 31, 40–43] That means that kpq ’s, 8fp; qg are equal in [14, 16, 19, 33] 123 On the Performance of Opportunistic Relay Selection in… Table Notations for channel coefficients R ¼ f1; ; Kg Notation Channel coefficient between hsd $ CN ð0; 1=ksd Þ Ss and Sd in the stage hsk $ CN ð0; 1=ksk Þ Ss and Sk in the stage 1, k R hsr $ CN ð0; 1=ksr Þ Ss and Pr in the stage htk $ CN ð0; 1=ktk Þ Pt and Sk in the stage 1, k R htd1 $ CN ð0; 1=ktd1 Þ Pt and Sd in the stage hkr $ CN ð0; 1=kkr Þ Sk and Pr in the stage 2, k R hkd $ CN ð0; 1=kkd Þ Sk and Sd in the stage 2, k R htd2 $ CN ð0; 1=ktd2 Þ Pt and Sd in the stage As demonstrated in Fig 1, the opportunistic relay selection takes place in two stages In the stage 1, Ss broadcasts the signal ws with the power P s (i.e., P s ¼ E ws fjws j2 g where E X fxg denotes the expectation operator) while Pt is concurrently transmitting the signal wt1 with the power P t1 ¼ E wt1 fjwt1 j2 g The signals from Ss and Pt cause the mutual interference between the primary network and the secondary network To this effect, the received signals at Sk and Sd , correspondingly, can be modeled as zsk ẳ hsk ws ỵ htk wt1 ỵ nsk ; 1ị zsd ẳ hsd ws ỵ htd1 wt1 ỵ nsd ; 2ị where zpq is the signal received at Sq from Sp and npq $ CN ð0; N0 Þ is the additive white Gaussian noise (AWGN) at the receiver Sq It immediately follows from (1) and (2) that the SINRs at Sk and Sd in the stage can be represented, correspondingly, as csk ¼ csd ¼ P s jhsk j2 P t1 jhtk j2 þ N0 P s jhsd j2 P t1 jhtd1 j2 þ N0 ; ð3Þ : ð4Þ In the stage 2, the selected relay, namely Sb , decodes the source signal and re-encodes the decoded information before forwarding to Sd Generally, the received signal at Sd from the relay Sk in the stage can be expressed as zkd ¼ hkd wk ỵ htd2 wt2 ỵ nkd ; 5ị where wk is the signal transmitted by Sk with the power P k ¼ E wk fjwk j2 g and interfered by the signal wt2 transmitted by Pt with the power P t2 ¼ E wt2 fjwt2 j2 g (i.e., two different primary transmitters are assumed for two stages to remain the system model general) This paper considers the opportunistic relay selection (e.g., [18, 42]), which selects the relay Sb with the largest SINR over the relaying channel Ss À Sb À Sd Mathematically, the SINR generated from this relaying channel can be represented as csbd ¼ maxðminðcsk ; ckd ÞÞ; k2R ð6Þ where ckd is computed from (5) as 123 K Ho-Van et al ckd ¼ P k jhkd j2 P t2 jhtd2 j2 ỵ N0 : 7ị It is recalled that the opportunistic relay selection can be implemented in a distributed manner using the timer method in [22] where each relay Sk sets its timer with the value that is inversely proportional to minðcsk ; ckd Þ and the relay with the timer that runs out first is selected For further performance enhancement at almost no cost of power and bandwidth, we take advantage of the direct channel between Ss and Sd , which is different from existing works (e.g., [18, 42]) where this channel is neglected for analysis simplicity To combine both signals from the relaying channel and the direct channel at Sd , the selection combining is assumed for low complexity as compared to the maximum ratio combining [48] To this end, the end-to-end SINR at Sd is expressed as ce2e ẳ maxcsd ; csbd ị: ð8Þ In the underlay mode, the secondary transmitter Sp must control its transmit power such that the interference induced at PUs does not exceed the maximum interference power, I, that PUs can tolerate, i.e., P p I =jhpr j Additionally, each SU is designed with the As such, in order to meet both the interference power maximum transmit power, P constraint and the maximum transmit power constraint as well as to maximize the trans mission coverage, the transmit power of Sp should be set as P p ẳ minI=jhpr j2 ; Pị The system model under consideration is more general and practical than existing works (e.g., [18, 42]) by accounting for both direct channel and PU’s interference, and i.n.i.d fading channels These additional factors also create correlation among received SINRs, which was neglected in the analysis of [18, 42], and it is this correlation that makes the analysis of this paper more complicated Specifically, the proposed system model induces the following correlations: they are correlated This • Since csd and csk have a common term P s ẳ minI=jhsr j2 ; Pị, leads to the correlation among quantities minðcsk ; ckd Þ in (6), and the correlation between csd and csbd in (8) It is noted that both [18] and [42] assumed un-correlation among quantities mincsk ; ckd ị ã Correlation among mincsk ; ckd Þ in (6) is also caused by the fact that ckd ’s in (7) have a common term htd2 This correlation is apparently present due to the PU’s interference, which is not available in [18, 42] These correlations can be broken down by using the conditional probability concept, which is very useful for the analysis of the next section Performance Analysis The derivation of the exact closed-form outage probability expression at Sd for the opportunistic relay selection in cognitive radio networks with PU’s interference and direct channel is firstly proposed in this section, which is then used for asymptotic analysis to expose important performance metrics such as diversity order and coding gain Since the proposed analysis framework is relatively general, it is straightforwardly extended to corresponding analysis for the opportunistic relay selection with PU’s interference and without direct channel to illustrate the advantage of utilizing the direct channel in relaying communications without exhaustive simulations 123 On the Performance of Opportunistic Relay Selection in… 3.1 Exact Analysis The outage probability is defined as the probability that ce2e is below a threshold c0 , i.e., PCC c0 g where c0 ¼ 22s À with s being the required transmission rate and o ¼ Prfce2e PrfX g is the probability of the event X Since ce2e contains two common quantities, x ¼ jhsr j2 and y ¼ jhtd2 j2 , which cause correlation among received SINRs as discussed in Sect 2, PCC o must be evaluated in terms of conditional probabilities, i.e., PCC o ¼ E x;y fPrfce2e c0 jx; ygg ẳ E x;y fPrfmaxcsd ; csbd ị c0 jx; ygg j > < = zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{> ¼ E x;y Prfcsd c0 jxg Prfcsbd c0 jx; yg : > > :|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} ; ð9Þ g À Á Since hpq $ CN 0; 1=kpq , the probability density function (pdf) and the cumulative dis 2 tribution function (cdf) of hpq are represented as f h zị ẳ kpq ekpq z and F h zị ẳ j pq j j pq j À eÀkpq z for z ! 0, respectively As such, it immediately follows that ( ) c0 x P s jhsd j2 g ẳ Pr P t1 jhtd1 j2 ỵ N0 < P t1 jhtd1 j2 þ N0 c0 = x ¼ Pr jhsd j2 ; : Ps 99 8 == < < P t1 jhtd1 j ỵ N0 c0 2 jhtd1 j ; x ¼ E jhtd1 j2 Pr jhsd j ;; : : Ps 0 < = P t1 jhtd1 j2 ỵ N0 c0 jhtd1 j2 ; xA ¼ E jhtd1 j2 Fjhsd j2 @ : ; Ps ( ) ðPt1 jhtd1 j ỵN0 ịksd c0 Ps ẳ E jhtd1 j2 À eÀ jhtd1 j ; x ð10Þ ¼ À Td ; where ) ðPt1 jhtd1 j2 ỵN0 ịksd c0 Ps e jhtd1 j ; x ( À Td ¼ E jhtd1 j2 ẳ Z1 e Pt1 zỵN0 ịksd c0 Ps fjhtd1 j2 zịdz ẳ 11ị Z1 e Pt1 zỵN0 ịksd c0 Ps Àktd1 z ktd1 e dz ktd1 eÀ ¼ P t1 ksd c0 Ps N0 ksd c0 Ps þ ktd1 : 123 K Ho-Van et al Similarly, the j term in (9) can be written as ' & j ẳ Pr maxmincsk ; ckd ịị c0 x; y k2R Y ẳ Prfmincsk ; ckd ị c0 jx; yg k2R ẳ Y k2R ẳ 12ị Prfminðcsk ; ckd Þ ! c0 jx; ygÞ YB C @1 À Prfcsk ! c0 jxg Prfckd ! c0 jyg A: |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} k2R ak bk The ak term in (12) can be computed in the same manner as (10), i.e., ak ẳ Tk ; 13ị where ktk eÀ Tk ¼ P k c t1 sk Ps N0 ksk c0 Ps ỵ ktk 14ị : Meanwhile, after using (7), the bk term in (12) should be rewritten as ' & P t2 y ỵ N0 ịc0 bk ¼ Pr jhkd j2 ! y : Pk Since P k ¼ yIk ; P with yk ¼ jhkr j2 , the bk term is further simplified as ' & ðP t2 y ỵ N0 ịc0 bk ẳ Pr jhkd j2 y Pk & & '' ðP t2 y ỵ N0 ịc0 ; y y ẳ E yk Pr jhkd j2 k Pk & ' P t2 y ỵ N0 ịc0 ¼ À E yk Fjhkd j2 yk ; y Pk & k P yỵN ịc ' kd t2P 0 k ¼ E yk e y k ; y ( ) Àkkd Pt2 yỵN0 ịc0 = yI ;P k ẳ E yk e y k ; y ¼ Z1 e kkd Pt2 yỵN0 ịc0 yk I fjhkr j2 yk ịdyk ỵ l ẳ e kkd Pt2 yỵN0 ịc0 P ð16Þ fjhkr j2 ðyk Þdyk Z1 e À kkd Pt2 yỵN0 ịc0 yk I l ẳ Ek eDk y 123 Zl Àkkr yk kkr e Bk ; 1ỵ y ỵ Ck dyk ỵ Zl e 15ị kkd Pt2 yỵN0 ịc0 P kkr ekkr yk dyk On the Performance of Opportunistic Relay Selection in… where Ak ¼ À eÀkkr l kkr IeÀkkr l Ak P t2 kkd c0 kkr I ỵ kkd c0 Ck ẳ P t2 kkd c0 P t2 kkd c0 Dk ¼ P Bk ¼ E k ¼ Ak eÀ I I¼ N0 P P¼ N0 P t2 P t2 ¼ N0 I lẳ : P kkd c0 P 17ị Inserting (13) and (16) into (12) and then (10) and (12) into (9), one obtains the compact form of PCC o as ( ) Y CC Po ¼ E x;y ð1 À Td Þ ð1 À bk Tk Þ : ð18Þ k2R Using the fact that Y ak ị ẳ ỵ k2R K X 1ịu uẳ1 Kuỵ1 X Kuỵ2 X s1 ẳ1 s2 ẳs1 ỵ1 K X Y su ẳsu1 ỵ1 k2A ak ỵ 1ịK Y ak ; 19ị k2R where A ẳ fRẵs1 ; Rẵs2 ; ; R½su g,5 to expand the product in (18), one obtains PCC o ẳ Y ; G; ỵ K1 X uẳ1 1ịu Kuỵ1 X Kuỵ2 X K X s1 ẳ1 s2 ẳs1 ỵ1 Y A GA ỵ 1ịK Y R GR ; 20ị su ẳsu1 ỵ1 where B ¼ f;; A; Rg with ; denoting the empty set, and ( ) Y Y B ¼ E x ð1 À Td Þ Tk ; ð21Þ k2B ( GB ¼ E y Y ) bk : ð22Þ k2B It is apparent that the derivation of the exact closed-form expression of PCC o is completed after solving Y B and GB R½j denotes the jth element of the set R 123 K Ho-Van et al Theorem The exact closed-form representation of Y B is given by Y B ẳ Y^B Y^fB;dg ; 23ị where L ẳ B or L ¼ fB; dg, and ! k c X Y ktk PeÀ skP Y ktk I ksr mk nl; ML ; lk ị ỵ As ; Y^L ẳ P k c P k c ỵ ktk P k2L t1 sk k2L k2L t1 sk ð24Þ with As ¼ À eÀksr l being defined in (17) and lk ¼ ktk I P t1 ksk c0 c0 X ksk I k2L ML ẳ ksr ỵ Y mk ẳ 25ị lj lk 26ị 27ị ; j2Lnk na; b; cị ẳ ebc Eibẵa ỵ cị: P t1 ẳ 28ị P t1 : N0 29ị In (28), EiðÁÞ is the exponential integral function in [49, eq (8.211.1)], which is a built-in function in most computation software (e.g., Matlab) Proof We further decompose Y B in (21) as ( ) ( ) Y Y YB ¼ Ex Tk À E x Td Tk ; k2B ð30Þ k2B where each term in (30) has a common form as ( ) Y ^ Tk : YL ¼ Ex ð31Þ k2L By inserting (31) into (30), we see that (30) perfectly matches (23) Consequently, in order to complete the proof, we should prove that (31) is represented in closed-form as (24) Towards this end, we firstly plug (14) into (31), and then perform some basic manipulations to simplify (31) as Y^L ¼ Z1 ksr e l ¼ Àksr x Y ktk eÀ k2L Pt1 ksk c0 I N0 ksk c0 x I x ỵ ktk dx ỵ Zl ksr x ksr e k2L Pt1 ksk c0 P N0 ksk c0 P ỵ ktk dx ! Z1 k c Y ktk I Y Y ktk PeÀ skP ML x ksr e dx ỵ As ; P k c x ỵ lk P k c ỵ ktk P k2L t1 sk k2L k2L t1 sk l where lk and ML are defined in (25) and (26), respectively 123 Y ktk eÀ ð32Þ On the Performance of Opportunistic Relay Selection in… Q Finally, using the partial fraction expansion to decompose the k2L xỵl product in (32), k one obtains ! k c X Z eÀML x Y ktk PeÀ skP Y ktk I ^ ksr ; 33ị YL ẳ mk dx ỵ As P k c x ỵ lk P k c ỵ ktk P k2L t1 sk k2L k2L t1 sk l where mk is defined in (27) The last integral in (33) are represented in closed-form as (28) by firstly changing the integral variable and then using the definition of EiðÁÞ Given (28), (33) matches (24), completing the proof h Theorem GB can be represented in closed-form as ! ! Bj1 jBX jX jcỵ1 jBX jcỵ2 jBj Y X U; ỵ Ek UC ỵ UB ; GB ẳ cẳ1 k2B j1 ẳ1 j2 ẳj1 ỵ1 where C ẳ fB½j1 ; B½j2 ; ; B½jc g, D ¼ f;; C; Bg, and ! X Y Bk ktd2 vk eCk HB EiCk HB ị; UD ẳ k2D 34ị jc ẳjc1 ỵ1 35ị k2D with vk ẳ Y Cj Ck j2Dnk HB ẳ ktd2 ỵ X ÁÀ1 ; ð36Þ Dk : k2B and jBj denoting the cardinality of the set B Proof Inserting (16) into (22) and after some algebraic manipulations, one obtains ! ( P ) Ày Dk Y Y B k GB ¼ Ek E y e k2B 1ỵ : 37ị y ỵ Ck k2B k2B Q Bk product, it immediately Applying (19) once again to decompose the k2B ỵ yỵC k follows that (37) coincides (34) where ! ( P Ày Dk Y Y UD ¼ Bk E y e k2B k2D ) : y ỵ Ck k2D ð38Þ Consequently, in order to complete the proof, we firstly need to evaluate (38) and then prove that the result of this evaluation is exactly (35) To this effect, using the partial Q fraction expansion for the k2D yỵC product in (38), one obtains k 123 K Ho-Van et al UD ¼ Y ! ( Ày P Bk E y e k2D ¼ Y ! Bk k2D ¼ Y ! Bk ¼ k2D X Bk ktd2 ) > : y ỵ Ck > ; P Z1 vk k2D ! vk k2B y ỵ Ck k2D P y D > = vk E y k2D k2D Y X X Dk Ày Dk ð39Þ e k2B ktd2 ektd2 y dy y ỵ Ck X Z1 vk k2D eHB y dy; y ỵ Ck where vk and HB are defined in (36) With the aid of [49, Eq (358.4)], the last integral in (39) is solved in closed-form as R1 eÀHB y C k HB EiðÀCk HB Þ Inserting this result into (39), one can see that (39) is in a yỵCk dy ẳ Àe h perfect agreement with (35), completing the proof Substituting (23) and (34) into (20), one obtains the exact closed-form outage probability of the opportunistic relay selection in cognitive radio networks with PU’s interference and direct channel It is worth emphasizing that although the opportunistic relay selection was discussed (e.g., [18, 42]), its exact closed-form outage probability expression in (20) has not been reported in any open literature for the general case of i.n.i.d fading channels, selection combining, PU’s interference, direct channel, and correlation among received SINRs On contrary, [18] and [42] consider i.p.i fading channels, no direct channel, no PU’s interference, and un-correlation among received SNRs To the best of the authors’ knowledge, (20) is totally novel and represented in a very convenient and compact form for the analytical evaluation In addition, derivations in this section are relatively general, and thus, can be used for the corresponding analysis of the opportunistic relay selection in cognitive radio networks with PU’s interference and without direct channel Indeed, the outage probability of such a system model can be represented as PDH o ẳ Prfcsbd 40ị c0 g: By following the same procedure of deriving PCC o in (9) with setting csd ¼ 0, it immediately follows that ^ PDH o ¼ Y ; G; ỵ K1 X uẳ1 1ịu Kuỵ1 X Kuỵ2 X s1 ẳ1 s2 ẳs1 ỵ1 K X Y^A GA ỵ 1ịK Y^R GR : 41ị su ẳsu1 þ1 It is noted that (41) is still novel since it accounts for PU’s interference, i.n.i.d fading channels, and correlation among received SINRs Therefore, the analysis in [18] and [42] where no PU’s interference, i.p.i fading distributions, and un-correlation among received SNRs are assumed is only a special case of (41) Additionally, it is interesting to discover DU that since Y^B [ Y B , PCC o \Po In other words, utilizing the direct channel is always beneficial without any significant sacrifice of system resources such as power and bandwidth 123 On the Performance of Opportunistic Relay Selection in… 3.2 Asymptotic Analysis The coding gain and the diversity order are important performance metrics characterizing a relay selection scheme The former demonstrates the performance gap between the coded system and the uncoded one while the latter shows how fast the outage probability reduces with respect to the SNR To analyze them, we should investigate the outage performance in the high SNR regime, i.e., P ! x!0 Using the that eÀqx % À qx, where q is a positive constant, and Àl fact Á P s ¼ P x ; , one can approximate the third equality in (10) as 8 = < P t1 jhtd1 j2 ỵ N0 ksd c0 P!1 jhtd1 j2 ; x g % E jhtd1 j2 ; : Ps ẳ Z1 P t1 z ỵ N0 Þksd c0 ktd1 eÀktd1 z dz Ps ð42Þ ¼ Qsd c0 À Á; P lx ; where Qsd ẳ P t1 ỵ ksd : ktd1 ð43Þ Imitating (42) with P k ¼ P ylk ; , one can also approximate (13) and (16) as P!1 ak % À Rsk c0 À Á; P lx ; ð44Þ and & ' kkd P t2 y ỵ N0 Þc0 P!1 yk ; y bk % E yk À Pk 1 Z Zl t2 y ỵ N0 ịc0 k P P!1 C B kkd P t2 y ỵ N0 ịc0 yk kd kkr ekkr yk dyk ỵ kkr ekkr yk dyk A % 1À@ I P l ð45Þ Mkd ðP t2 y ỵ 1ịc0 ; % P P!1 where P t1 ỵ ksk ; ktk kkr l e ỵ kkd : ẳ kkr l Rsk ẳ Mkd 46ị Plugging (44) and (45) into (12), and then (42) and (12) into (9), one obtains the approximation of PCC o in the high SNR regime as 123 K Ho-Van et al P!1 PCC % E x;y o P!1 % E x;y ( ( " # !!) Qsd c0 Y Rsk c0 Mkd P t2 y ỵ 1Þc0 À Á À Á 1À 1À 1À P P lx ;1 k2R P lx ;1 Qsd c0 Y Rsk c0 M P y ỵ 1ịc0 Rsk Mkd P t2 y ỵ 1ịc20 l l ỵ kd t2 À Á À P P x ;1 k2R P x ;1 P lx ;1 !) : ð47Þ By eliminating the high-order term in the above product, one can further approximate (47) as ( Kỵ1 ) Y Y Y y CC P!1 c0 K P t2 Qsd E x;y Gk 1ỵ Mkd ; ð48Þ Po % Gk P lx ; k2R k2R k2R where ! Rsk ỵ1 : Gk ẳ 49ị P t2 Mkd lx ; Q Utilizing (19) once again to decompose the k2R ỵ Gyk product in (48), one can represent (48) in closed-form as P!1 PCC % o ! Kỵ1 K1 Kuỵ1 K Y X Kuỵ2 X X X c0 K P t2 Qsd W ; V ; þ ÁÁÁ WAVA þ WRVR Mkd ; P u¼1 s1 ẳ1 s2 ẳs1 ỵ1 su ẳsu1 ỵ1 k2R 50ị where B ¼ f;;A;Rg and È jB j WB ¼ Ey y É ¼ Z1 Àj B j yjBj ktd2 eÀktd2 y dy ẳ jBj!ktd2 ; 51ị Q Gk > > = < k2B Àl Á ; VB ¼ Ex > ; :min x ; > ð52Þ with B ¼ RnB To complete the approximation of PCC o in the high SNR regime, we must evaluate (52) Towards this end, inserting (49) into (52), one obtains ( !) Y Rsk ÀjBj À Á À Á V B ẳ P t2 E x : 53ị 1ỵ lx ; k2B Mkd lx ; Using (19) to decompose the above product, one can express (53) in closed-form as Bj1 jBjcỵ1 jBjcỵ2 jX jBj X X X ÀjBj V B ¼ P t2 @M; ỵ MC ỵ MBA: 54ị cẳ1 j1 ẳ1 j2 ẳj1 ỵ1 ẩ ẫ ẩ ẫ B , and ẳ ;; C; where C ẳ Bẵj1 ; Bẵj2 ; ; B½jc , D 123 jc ẳjc1 ỵ1 On the Performance of Opportunistic Relay Selection in ( ) Rsk Á À Á MD ¼ E x x ; k2D Mkd lx ; &h l iÀjDjÀ1 ' Y R sk ¼ E x ; x M kd k2D Z1 jDjỵ1 Zl x C Y Rsk B ẳ@ ksr eksr x dx ỵ ksr eksr x dxA l Mkd Àl Y l ð55Þ k2D jỵ1 D jX ksr l D ỵ ! Y Rsk e A ẳ @As ỵ : k Mkd ksr lịjDjỵ1 kẳ0 k!ksr lÞ k2D It is noted that PCC o in (50) is very compact to determine the diversity order and the coding gain of the opportunistic relay selection in cognitive radio networks with PU’s interference and direct channel Indeed, according to [42], PCC o in the high SNR regime can be represented in terms of the diversity order, JdCC , and the coding gain, JcCC , as P!1 À CC ÁÀJdCC PCC Consequently, after performing the parameter matching, we can % Jc P o infer from (50) that the opportunistic relay selection achieves the full diversity order of JdCC ¼ K ỵ offered by all available secondary relays and the source Furthermore, the coding gain is offered by JcCC " ! #Kỵ1 K1 Kuỵ1 K Y X Kuỵ2 X X X K ¼ Qsd P t2 W ; V ; ỵ WAVA ỵ WRVR Mkd : c0 uẳ1 s1 ẳ1 s2 ẳs1 ỵ1 su ẳsu1 ỵ1 k2R ð56Þ Following the same procedure of deriving (50), one can achieve the approximation of PDH o in the high SNR regime as ! K K1 Kuỵ1 K Y X Kuỵ2 X X X DH P!1 c0 P t2 W ; K; ỵ W A KA ỵ W R KR Mkd ; Po % P u¼1 s1 ¼1 s2 ẳs1 ỵ1 su ẳsu1 ỵ1 k2R 57ị where B ¼ f;; A; Rg, W B is given by (51), and Bj1 jBjcỵ1 jBjcỵ2 jX jBj X X X jBj KB ẳ P t2 @H; ỵ HC ỵ HBA; cẳ1 j1 ẳ1 j2 ẳj1 ỵ1 58ị jc ẳjc1 ỵ1 with jDj ksr l X Y Rsk D! e ðksr lÞk A : HD ẳ @As ỵ D Mkd k lịj j kẳ0 k! sr 59ị k2D From (57), one can infer the diversity order and the coding gain of the opportunistic relay selection in cognitive radio networks with PU’s interference and without direct channel, respectively, as 123 K Ho-Van et al JdDH ẳ K; 60ị and JcDH ẳ c0 P t2 " W ; K; ỵ K Kuỵ1 X Kuỵ2 X X uẳ1 s1 ẳ1 s2 ẳs1 þ1 ! K X W A KA þ W R KR su ẳsu1 ỵ1 Y #K1 Mkd : k2R 61ị The asymptotic analysis proves that utilizing the direct channel increases the diversity order, making the opportunistic relay selection with direct channel always better than that without direct channel at almost no cost of bandwidth and power Illustrative Results This section provides numerous results to demonstrate the outage performance of the opportunistic relay selection in cognitive radio networks with PU’s interference and with/ without direct channel as well as to corroborate the proposed analytical expressions For illustration purpose, we only investigate a primary transmitter-receiver pair (i.e., P t1 ¼ P t2 ¼ P t and ktd1 ¼ ktd2 ¼ ktd ) and arbitrarily choose user positions as shown in Fig The variance of the channel coefficient, hpq , between the transmitter p and the receiver q is Àf modeled as 1=kpq ¼ dpq (e.g., [50]) where dpq is the distance between the transmitter p and the receiver q while f is the path-loss exponent To limit case-studies, we assume f ¼ In the sequel, three different relay sets fS1 g, fSk g3k¼1 , fSk g5k¼1 are illustrated for K ¼ 1; 3; 5, correspondingly 0.7 0.6 Pr 0.5 Pt 0.4 y 0.3 0.2 S4 0.1 S3 S S S1 −0.2 0.2 0.4 0.6 x 123 d −0.1 Fig User positions S S s 0.8 On the Performance of Opportunistic Relay Selection in… Figure demonstrates the outage performance of the opportunistic relay selection with respect to the variation of maximum transmit power-to-noise variance ratio, P, for the proportional factor l ¼ 0:2, the PU’s power-to-noise variance ratio P t ¼ 20 dB, and the required transmission rate s ¼ bit/s/Hz It is seen that the exact analysis (i.e., (20) and (41)) is in an excellent agreement with the simulation while the asymptotic analysis (i.e., (50) and (57)) perfectly matches the simulation at large values of P, verifying the accuracy of the proposed expressions Additionally, the system performance is drastically enhanced with respect to the increase in the number of involved relays K This is reasonable since the larger K, the higher diversity order is achievable, and so, the smaller the outage probability Moreover, the increase in P also improves the system performance This comes from the fact that P upper bounds the transmit power of SUs and thus, the larger P, the larger the transmit power, which ultimately reduces the corresponding outage probability However, due to the PU’s interference, the system is always in outage at small values of P (e.g., P\20 dB) For all system parameters under consideration, the opportunistic relay selection with direct channel is significantly better that without direct channel, as expected and analyzed in Sect This encourages utilizing the direct channel in relaying communications Figure illustrates the impact of the outage threshold c0 (or the required transmission rate s) on the performance of the opportunistic relay selection for P ¼ 20, P t ¼ 15 dB, l ¼ 0:4 It is seen that the analysis excellently matches the simulation, validating the accuracy of the derived expressions Additionally, the outage threshold c0 ¼ 22s À significantly deteriorates the outage performance of the opportunistic relay selection with/ without direct channel This comes from the fact that given operation conditions, the more stringent the system performance requirement (i.e., the larger outage threshold), the higher the probability that the system is in outage Moreover, similar to Fig 3, increasing the number of relays can considerably improve system performance (e.g., the outage probability is reduced more than 10 times when K increases from to for s ¼ 0:1 bits/s/Hz) 10 K=1 −2 Outage probability 10 −4 10 K=3 Sim.: no DC Exact: no DC Asym.: no DC Sim.: DC Exact: DC Asym.: DC −6 10 −8 10 K=5 −10 10 15 20 25 30 35 40 45 50 55 Maximum transmit power−to−noise variance ratio (dB) Fig Outage probability versus P ’Sim.’, ’Exact’, ’Asym.’, and ’DC’ stand for ’Simulation’, ’Exact analysis’, ’Asymptotic analysis’, and ’Direct channel’, respectively 123 K Ho-Van et al Outage probability 10 Sim.: no DC & K = Exact: no DC & K = Sim.: DC & K = Exact: DC & K = Sim.: no DC & K = Exact: no DC & K = Sim.: DC & K = Exact: DC & K = Sim.: no DC & K = Exact: no DC & K = Sim.: DC & K = Exact: DC & K = −1 10 −2 10 −3 10 0.2 0.4 0.6 0.8 1.2 1.4 Required transmission rate (bits/s/Hz) Fig Outage probability versus the required transmission rate ’Sim.’, ’Exact’, and ’DC’ stand for ’Simulation’, ’Exact analysis’, and ’Direct channel’, respectively 10 K=1 −1 Outage probability 10 K=3 −2 10 Sim.: no DC Exact: no DC Sim.: DC Exact: DC −3 10 K=5 −4 10 0.2 0.4 0.6 0.8 Proportional factor µ Fig Outage probability versus the proportional factor l ’Sim.’, ’Exact’, and ’DC’ stand for ’Simulation’, ’Exact analysis’, and ’Direct channel’, respectively Furthermore, for all operation parameters under investigation, cooperative relaying always outperforms dual-hop relaying Figure illustrates the effect of the proportional factor l on the outage performance of the opportunistic relay selection with/without direct channel for P ¼ 20, P t ¼ 15 dB, 123 On the Performance of Opportunistic Relay Selection in… 10 K=1 −1 Outage probability 10 K=3 −2 10 Sim.: no DC Exact: no DC Sim.: DC Exact: DC −3 10 K=5 −4 10 10 15 20 25 30 P (dB) t Fig Outage probability versus P t s ¼ 0:2 bits/s/Hz It is observed that the simulation perfectly agrees the analysis, again corroborating the validity of the proposed expressions Moreover, the increase in l considerably enhances the outage performance This is reasonable since the increase in l ¼ I =P is equivalent to the increase in I and hence, inducing PUs more tolerable with the interference from SUs As a result, SUs can operate with high transmit powers, eventually mitigating their outage probability Furthermore, the performance of the opportunistic relay selection is dramatically improved with the higher number of involved relays In addition, cooperative relaying is always superior to dual-hop relaying for all operation parameters under consideration Figure illustrates the effect of the PU’s interference on the outage behavior of the opportunistic relay selection for P ¼ 20 dB, l ¼ 0:2, and s ¼ 0:5 bits/s/Hz Results illustrate the excellent agreement between the analysis and the simulation, again confirming the validity of the proposed expressions In addition, the increase in the PU’s transmit power, which is equivalent to the increase in the PU’s interference, drastically degrades the performance of SUs Fortunately, the impact of the PU’s interference can be considerably remedied by increasing the number of relays (e.g., the outage probability is reduced more than 10 times when K increases from to at P t ¼ dB for both cases of with and without direct channel) Moreover, utilizing the direct channel can significantly improve the performance of the opportunistic relay selection for any considered system parameters Conclusions This paper firstly proposes an exact outage analysis framework for the opportunistic relay selection in cognitive radio networks with PU’s interference and with/without direct channel under a general scenario: i.n.i.d Rayleigh fading channels, the mutual interference 123 K Ho-Van et al between the secondary network and the primary network, maximum transmit power constraint and interference power constraint, and correlation among received SINRs Then the proposed framework is extended to asymptotic analysis to have insights into important performance metrics such as coding gain and diversity order Simulations corroborate the validity of the proposed analysis Also, numerous results illustrate that (1) the relay selection is essential in cognitive radio networks not only because of performance improvement with respect to the increase in the number of relays but also because of low total transmit power and small transmission bandwidth; (2) the PU’s interference significantly degrades the performance of the opportunistic relay 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San Diego, CA: Academic Press 50 Goldsmith, A (2005) Wireless communications Cambridge: Cambridge Universtiy Press Khuong Ho-Van received the B.E (with the first-rank honor) and the M.S degrees in Electronics and Telecommunications Engineering from HoChiMinh City University of Technology, Vietnam, in 2001 and 2003, respectively, and the Ph.D degree in Electrical Engineering from University of Ulsan, Korea in 2006 During 2007–2011, he joined McGill University, Canada as a postdoctoral fellow Currently, he is an Associate professor at HoChiMinh City University of Technology His major research interests are modulation and coding techniques, diversity technique, digital signal processing, and cognitive radio Lien Pham-Hong received the B.E degree in Telecommunications Engineering from HaNoi City University of Technology, Vietnam, in 1957 and the Ph.D degree in Electrical Engineering from University of Slovakia, Slovakia, in 1993 Currently, she is an Associate professor at Ho Chi Minh City University of Technical Education Her major research interests are modulation and coding techniques, diversity technique, digital communication systems, wireless systems, and cognitive radio 123 On the Performance of Opportunistic Relay Selection in… Son Vo-Que was born in Quang Ngai, Vietnam He received the B.E degree in Electrical and Electronics Engineering from Ho Chi Minh City University of Technology, Vietnam in 2003 Two years later, he got the M.E degree in Telecommunications Engineering from the same university He received the Ph.D degree in Electrical Engineering from University of Bremen, Germany in 2011 Since 2004, Dr Son has been a faculty member of Department of Electronic Engineering at Ho Chi Minh City University of Technology His research interests lie in Mobile and Wireless Communications, Wireless Sensor Networks, Internet of Things, autonomous/M2M communications He is an author of several journals and tens of conference papers Tra Luu-Thanh received the B.E degree in Electrical Engineering from Ho Chi Minh City University of Technology, Vietnam in 2001, and the Ph.D degree in Telecommunication Engineering from Telecom ParisTech, France in 2006 His research interests are telecommunications engineering, computer network, computer & network security, and embedded system 123 ... corresponding analysis for the opportunistic relay selection with PU’s interference and without direct channel to illustrate the advantage of utilizing the direct channel in relaying communications... relay selection with and without direct channel as well as emphasizing the importance of the direct channel in relaying communications It is recalled that no matter whether the direct channel. .. noted that PCC o in (50) is very compact to determine the diversity order and the coding gain of the opportunistic relay selection in cognitive radio networks with PU’s interference and direct channel