Exact Ergodic Capacity Analysis for Cognitive Underlay Amplify-And-Forward Relay Networks over Rayleigh Fading Channels

Research and Development on Information and Communication Technology Exact Ergodic Capacity Analysis for Cognitive Underlay Amplify-and-Forward Relay Networks over Rayleigh Fading Channels Vo Nguyen Quoc Bao and Vu Van San Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam E-mail: baovnq@ptithcm.edu.vn, sanvv@ptit.edu.vn Correspondence: Vo Nguyen Quoc Bao Communication: received 20 August 2017, revised 1 September 2017, accepted 1 September 2017 Abstract: In this paper, we propose a novel derivation approach to obtain the exact closed form expression of ergodic capacity for cognitive underlay amplify-and-forward (AF) relay networks over Rayleigh fading channels. Simulation results are performed to verify the analysis results. Numerical results are provided to compare the system performance of cognitive underlay amplify-and-forward relay networks under both cases of AF and decode-and-forward (DF) confirming that the system with DF provides better performance as compared with that with AF. Keywords: Ergodic capacity, Rayleigh fading channels, cogni- tive radio, underlay relay networks, amplify-and-forward, decode- and-forward. I. INTRODUCTION Cognitive radio has been widely considered as a promis- ing technology for next generation mobile networks due to its superior spectral efficiency [1, 2]. There are three con- ventional spectrum sharing approaches including underlay, overlay, and interweave, where the underlay approach is often of interest in practical implementation. The key idea of the cognitive underlay approach is to allow unlicensed networks to transmit/receive data on the same frequency band licensed to primary (licensed) networks if their inter- ference to the primary users is below a certain level. An important issue in cognitive underlay networks that has attracted much attention recently is to guarantee the secondary network performance and coverage due to the constraint of the transmit powers for secondary networks, i.e., the maximum allowable interference at the primary receiver [3]. As a result, many advantaged techniques for the physical layer are proposed for cognitive underlay networks, e.g., see [4–11]. Although all above-mentioned research works have derived the system performance in terms of outage probability and ergodic capacity, the er- godic capacity is still expressed under asymptotic or upper- bound expression due to the complicated form of the end- to-end signal-to-noise ratio (SNR) of dual-hop amplify- and-forward (AF) relaying [12]. As an alternative, ergodic capacity of underlay decode-and-forward (DF) systems is usually employed to estimate that of underlay AF systems at high SNR regime leading to the fact that we cannot understand the performance gap between AF and DF. To the best knowledge of the authors, exact and general ergodic capacity analysis of cognitive underlay dual-hop relaying over Rayleigh fading channels remains an open problem. This paper is to fill this important gap, i.e., providing the exact closed-form expression of the system ergodic capacity in terms of dilogarithm functions [13, 14]. All analytical results developed in this paper are corrobo- rated by MATLAB-based simulation results, verifying the accuracy of the proposed derivation approach and the pro- vided results. Numerical results are provided to investigate the effect of channel and system settings on the system ergodic capacity. The main contributions of this paper are summarized as follows. First, we propose a novel approach to derive the ex- act closed form for system ergodic capacity over Rayleigh fading channels. Second, we compare the performance of cognitive underlay dual-hop relaying networks in terms of ergodic capacity under amplify-and-forward and decode- and-forward. It should be noted that the proposed approach is not only applicable for Rayleigh fading channels but also for other generalized fading channels, i.e., Nakagami-m and Rician. The rest of this paper is organized as follows. In Sec- tion II, we introduce the model under study and describe the proposed protocol. Section III shows the formulas allowing evaluation of the system ergodic capacity over Rayleigh fading channels. In Section IV, we compare simulation and theoretical results. The paper is concluded in Section V. 12 Vol. E–3, No. 14, Sep. 2017 s r d PU-Tx PU-Rx Figure 1. Cognitive underlay AF relay network. II. SYSTEM MODEL We consider a cognitive underlay dual hop relaying system with single amplify-and-forward relay, which is illustrated in Figure 1. Assuming no direct link existed, the communication between the source (s) and the desti- nation (d) is performed with the help of the relay (r) via a time-division multiple access. In particular, the source broadcast its signal in the first time slot with transmit power Ps, which is received by the relay. After amplifying with the variable gain, the scaled signal is forwarded to the destination with transmit power Pr. Denote hsp and hrp as the channel coefficients of the link from the source and the relay to the primary node, respectively, the transmit powers for the source and the relay under underlay mode of cognitive networks can be set as follows [15]: Ps = Ip |hsp |2 , (1) Pr = Ip |hrp |2 , (2) where Ip denotes the interference temperature constraint at the primary receiver (PU-Rx). The instantaneous signal-to-noise ratios (SNRs) of the first and second hops are respectively given by γ1 = Ip N0 |hsr |2 |hsp |2 , (3) γ2 = Ip N0 |hrd |2 |hrp |2 , (4) where hsp and hrp are the channel coefficients of the link from s→ r and r→ d, respectively. Here, we denote N0 as the additive Gaussian white noise power. Under Rayleigh fading channels, hAB with A ∈ {s, r} and B ∈ {p, r, d} is a complex Gaussian random variable, so the channel power |hAB |2 is an exponential distributed random variable with expected value λAB = E{|hAB |2} with E{·} being the expectation operator. Assuming that the path loss effect is taken into account, we have λAB = d −η AB , where η and dAB denote the path-loss exponent and the distance between node A and node B, respectively [16]. In dual-hop relaying networks with non-regenerative variable gain relays, the instantaneous Channel State Infor- mation (CSI) of the first hop is utilized to ensure the fixed power at the output of the relay. Accordingly, the effective SNR received at d can be expressed as [17] γAF = γ1γ2 γ1 + γ2 + 1 . (5) III. CAPACITY ANALYSIS 1. Amplify-and-Forward Throughout the paper, we assume that the data channel is unknown at the transmitter but perfectly known at the receiver. Considered as an important metric in designing wireless networks, ergodic capacity reveals the upper bound on the amount of information, which can be reliably transmitted over noisy wireless channels with a certain probability of error. It is well-known that if the probability density function (PDF) of the end-to-end SNR is available, the ergodic capacity (in bits/second) per unit bandwidth can be calculated by evaluating the following integral, which is of the form CAF = 12EγAF {log(1 + γAF)} = 1 2 ∞∫ 0 log(1 + γ) fγAF (γ)dγ, (6) where the pre-factor 1/2 is included to reflect the fact that the source-destination communication occurs in two orthogonal time slots. However, with the current form of γAF in (5), finding an exact closed-form evaluation of (6) is a challenging mathematical problem due to the complexity of statistical distribution of γAF along with the presence of the nonlinear log function. To deal with such problem, (6) should be expressed in a more mathematically tractable form. To achieve this, the basic properties of the log function is employed after a some manipulations, yielding CAF = 12 ln 2EγAF [ ln ( 1 + γ1γ2 γ1 + γ2 + 1 )] = 1 2 ln 2 Eγ1 [ln (1 + γ1)]︸ ︷︷ ︸ C1 + 1 2 ln 2 Eγ2 [ln (1 + γ2)]︸ ︷︷ ︸ C2 − 1 2 ln 2 Eγ1+γ2 [ln (1 + γ1 + γ2)]︸ ︷︷ ︸ C3 . (7) 13 Research and Development on Information and Communication Technology Theorem 1: The exact closed-form expression of ergodic capacity of cognitive underlay AF networks over Rayleigh fading channels is CAF = 12 ln 2 [ α1 lnα1 α1 − 1 + α2 lnα2 α2 − 1 − (α1 + α2) ln (α1 + α2) α1 + α2 − 1 ] + α1α2 log(α1α2) 2 ln 2(α1 + α2)(α1 + α2 − 1)2 × [1 − (α1 + α2) + (α1 + α2) ln(α1 + α2)] − α1α2 [log(1 + α1) + log(1 + α2)] 2 ln 2(1 − α1 − α2)(α1 + α2) − α1α2 [K (α1, α2) +K (α2, α1)] 2 ln 2(1 − α1 − α2)2 , (8) where K (α1, α2) is given in (9) shown at the top of the next page with Li2(z) = 0∫ z log(1−t)dt t denoting the dilogarithm function [18, Eq. 27.7.1]. Proof: To derive (1), we need to know the PDFs of γ1, γ2, and γ1+γ2. For γ1 and γ2, recalling the results in [19], their PDFs can be given by fγi (γ) = αi (γ + αi)2 , (10) where α1 = Ip N0 λsr λsp and α2 = Ip N0 λrd λrp . For γ1+γ2, since the moment generating function (MGF) approach [20]1, could not be used due to its high derivation complexity, we start from the definition of the cumulative distribution function (CDF) of γ1 + γ2 and invoke the concept of conditional probability, namely Fγ1+γ2 (γ) = Pr(γ1 + γ2 < γ) = γ∫ 0 Fγ1 (γ − x) fγ2 (x)dx. (11) In (11), Fγ1 (·) is the CDF of γi , which is obtained by integrating (10) from 0 to γ as follows: Fγ1 (γ) = γ γ + α1 . (12) Substituting (12) and (10) into (11), after some manipula- tions, we have Fγ1+γ2 (γ) = γ γ + α1 + α2 (13) + α1α2 (γ + α1 + α2)2 [ log α1 γ + α1 + log α2 γ + α2 ] . 1The PDF can be yielded by taking the inverse Laplace transform of the MGF of γ1 + γ2, which can be obtained as the product of the MGF of its summands. Employing the relationship between the PDF and the CDF, we can obtain the PDF of γ1 + γ2 as fγ1+γ2 (γ) = α1 + α2 (γ + α1 + α2)2 − 2α1α2 log(α1α2)(γ + α1 + α2)3 − α1α2(γ + α1 + α2)2 ( 1 γ + α1 + 1 γ + α2 ) (14) + 2α1α2 (γ + α1 + α2)3 [log(γ + α1) + log(γ + α2)] . Having the PDFs of γ1, γ2 and γ1 + γ2 in hands allows us to derive C1, C2 and C3. We easily recognize that C1 and C2 take the general form, which is written as J(a) = ∞∫ 0 ln(1 + γ) a(γ + a)2 dγ. (15) With the help of the identity [18, Eq. (4.291.17)], we have J(a) =  a ln a a − 1 , a , 1 1, a = 1 . (16) We are now in a position to derive C3. Rewriting C3 in an explicit form, we obtain C3 = ∞∫ 0 ln(1 + γ) fγ1+γ2 (γ)dγ =(α1 + α2) ∞∫ 0 ln(1 + γ) (γ + α1 + α2)2 dγ︸ ︷︷ ︸ I1 − 2α1α2 log(α1α2) ∞∫ 0 ln(1 + γ) (γ + α1 + α2)3 dγ︸ ︷︷ ︸ I2 (17) − α1α2 ∞∫ 0 ln(1 + γ) (γ + α1 + α2)2 ( 1 γ + α1 + 1 γ + α2 ) dγ + 2α1α2 ∞∫ 0 ln(1 + γ) [log(γ + α1) + log(γ + α2)] (γ + α1 + α2)3 dγ︸ ︷︷ ︸ I3 , where Ii with i = 1, 2, 3 are auxiliary functions, which are derived next. Starting with I1 and using the identity [18, Eq. (4.291.17)], we have I1 =  ln (α1 + α2) α1 + α2 − 1 , α1 + α2 , 1 1, α1 + α2 = 1 . (18) 14 Vol. E–3, No. 14, Sep. 2017 K (α1, α2) =  − ln2(1−α1)2 + ln 2α2 2 + lnα1 ln [ (1−α1)(α1+α2) α2 ] − Li2 ( α1 α1−1 ) + Li2 ( −α1α2 ) , α1 < 1 pi2 6 + ln2α2 2 + Li2 ( − 1α2 ) , α1 = 1 pi2 2 − ln 2(α1−1) 2 + ln2α2 2 + lnα1 ln [ (α1−1)(α1+α2) α2 ] −< { Li2 ( α1 α1−1 )} + Li2 ( −α1α2 ) , α1 > 1 (9) To solve I2, based on integration by parts, we obtain I2 = − ln(1 + γ) 2(γ + α1 + α2)2 ∞ γ=0 + 1 2 ∞∫ 0 dγ (γ + 1)(γ + α1 + α2)2 = { 1−(α1+α2)+(α1+α2) ln(α1+α2) 2(α1+α2)(α1+α2−1)2 , α1 + α2 , 1, 1 4, α1 + α2 , 1. (19) Using integration by parts again, I3 is re-expressed as I3 = − ln(1 + γ) [log(γ + α1) + log(γ + α2)] 2(γ + α1 + α2)2 ∞ γ=0︸ ︷︷ ︸ →0 + 1 2 ∞∫ 0 ln(1 + γ) (γ + α1 + α2)2 [ 1 γ + α1 + 1 γ + α2 ] dγ + 1 2 ∞∫ 0 log(γ + α1) + log(γ + α2) (1 + γ)(γ + α1 + α2)2 dγ︸ ︷︷ ︸ I4 . (20) Plugging (20) into (17) and then canceling the like terms, C3 is simplified as C3 = (α1 + α2)I1 − 2α1α2 log(α1α2)I2 + α1α2I4. (21) For I4, by employing partial fraction decomposition, we have J4 = 11 − α1 − α2 ∞∫ 0 log(γ + α1) + log(γ + α2) (γ + α1 + α2)2 dγ + 1 (1 − α1 − α2)2 ∞∫ 0 ( log(γ + α1) γ + 1 − log(γ + α1) γ + α1 + α2 ) dγ︸ ︷︷ ︸ K(α1,α2) + 1 (1 − α1 − α2)2 ∞∫ 0 ( log(γ + α2) γ + 1 − log(γ + α2) γ + α1 + α2 ) dγ︸ ︷︷ ︸ K(α2,α1) . (22) In (22), we can see that the second and third integrals take from the general form, given as follows: K (a, b) = ∞∫ 0 ( log(γ + a) γ + 1 − log(γ + a) γ + a + b ) dγ. (23) By recognizing the integral representation of the diloga- rithm function [21, Eq. (27.7.1)], i.e., Li2(−x) = ∫ x 1 ln t/(t− 1)dt, after several manipulations, we find out that K (a, b) can be derived as shown in (24) at the top of the next page. Furthermore, making use the fact that = {Li2[a/(a − 1)]} = 2piarc cot(1 − 2a) for a > 1, (24) can be rewritten as (9), where <(·) and =(·) represent the real and imaginary parts of a complex number, respectively. For the case of α1+α2 = 1, it is straightforward to derive K (a, b) = ∞∫ 0 log(γ + α1) + log(γ + α2) (1 + γ)3 dγ = α1 − 1 + (α1 − 2)α1 logα1 2(α1 − 1)2 + α2 − 1 + (α2 − 2)α2 logα2 2(α2 − 1)2 . (25)  Pulling everything together, i.e., (6), (16), and (17), we can obtain the exact closed-form expression for CAF. It should be noted that the dilogarithm function is available as a built-in function in most well-known mathematical software tools such as MATLAB and MATHEMATICA. Besides, there exist efficient approaches to directly calculate the dilogarithm, e.g., see [13, 14]. 2. Amplify-and-Forward versus Decode-and-Forward In this section, we provide the ergodic capacity for dual hop DF networks, which are considered as a counterpart of AF networks. The ergodic capacity of the underlay DF networks is defined as CDF = 12 ∞∫ 0 log2(1 + γ) fγDF (γ)dγ, (26) where γDF is the equivalent end-to-end SNR of the system. It is well-known that the exact form of γDF in terms of γ1 and γ2 is not mathematically visible. To proceed further, we adopt the mathematical tractability approximation approach suggested by Wang et al. [22]. In particular, the equivalent end-to-end SNR of re-generative relaying systems, γDF, can be tightly approximated irrespective of the employed modulation scheme as γDF = min(γ1, γ2). (27) 15 Research and Development on Information and Communication Technology K (a, b) =  − ln2(1−a)2 + ln 2b 2 + ln a ln [ (a−1)(a+b) b ] − Li2 ( a a−1 ) + Li2 (− ab ) , a < 1 pi2 6 + ln2b 2 + Li2 ( − 1b ) , a = 1 pi2 2 + 2ipiarc cot(1 − 2a) − ln2(a − 1) + ln2b − 2 ln a ln [ b (a−1)(a+b) ] − Li2 ( a a−1 ) + Li2 (− ab ) , a > 1 (24) The PDF of γDF is given by fγDF (γ) = d dγ [ 1 − (1 − Fγ1 (γ)) (1 − Fγ2 (γ)) ] = α1α2 (γ + α1)(γ + α2)2 + α1α2 (γ + α2)(γ + α1)2 (a) = α1α2 α2 − α1 [ 1 (γ + α1)2 − 1(γ + α2)2 ] , (28) where (a) immediately follows after using partial fraction technique with some simple manipulations. Theorem 2: The ergodic capacity of cognitive underlay DF relay networks is tightly approximated by CDF = α1α22 ln 2(α2 − α1) ( logα1 α1 − 1 − logα2 α2 − 1 ) . (29) Proof: Plugging (28) into (26) and taking the integral, we have (after simplification steps). For α1 = α2 = α, CDF is simplified as CDF =  α(1 − α + α logα) 2 ln 2(α − 1)2 , α , 1 1 2 ln 2 , α = 1 (30)  Having CAF and CDF in hands allows us to numerically evaluate the system ergodic capacity for a given network and channel condition. IV. NUMERICAL RESULTS AND DISCUSSION This section is to verify the proposed derivation approach and to study effects of the system and channel settings on the system ergodic capacity. We consider the two- dimensional system model, where the source, the relay, the destinations, and the primary receiver are placed at coordinate (0,0), (d, 0), (1,0), and (xp, yp), respectively. Recalling that here we adopt the simplified path loss model, we can model λAB = d −η AB , where dAB denotes the distance between node A and node B, and η is the path- loss exponent. In Figure 2, we plot the system ergodic capacity versus average SNRs in dB for three different cases of the primary user location: (i) Case A: (0.3, 0.3), (ii) Case B: (0.6, 0.6), and (iii) Case C: (0.9, 0.9). It can be seen that the simulation results are in excellent alignment with the 0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 I p/No[dB] Er go di c Ca pa cit y Analysis − AF Analysis − DF Simulation case C case B case A Figure 2. Ergodic capacity versus average SNRs. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.8 1 1.2 1.4 1.6 1.8 2 d Er go di c Ca pa cit y Analysis Simulation I p = 5 dB I p = 0 dB Amplify−and−Forward Decode−and−Forward Figure 3. Effect of relay locations on the system ergodic capacity. analysis results for both cases of AF and DF, confirming the correctness of the proposed derivation approach. We also see that case A outperforms case B, which, in turns, outperforms case C suggesting that the system capacity can improve when the secondary networks are located far away from the primary receivers, as expected. For the same channel and system settings, the system using AF provides better capacity as compared with that using DF. It can be explained by considering the fact that DF can eliminate noise when making right decoding, while AF amplifies the noise when forwarding. In Figure 3, we investigate the effect of secondary 16 Vol. E–3, No. 14, Sep. 2017 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR [dB] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Er go di c Ca pa cit y AF DF Simulation PU-RX (0; 0.1) PU-RX (0.5; , 0.1) PU-RX (1.0, 0.1) Figure 4. Effect of relay locations on the system ergodic capacity. relay position on the system ergodic capacity for a fixed coordinate of the primary receiver. We can see that the best location for secondary relays is a complicated function of Ip and used relaying technique. In Figure 4, we investigate the effect of the location of the primary receiver on the system performance. We consider three extreme cases for the primary receiver, i.e., very close to the secondary source, very close to the secondary relay, and very close to the secondary destination. Among three cases, the last case provides the best performance while the first case gives the worst performance. It can be explained by the fact that the primary receiver locations in relation to the secondary transmitter significantly affect the secondary system performance. V. CONCLUSION This paper has proposed a novel derivation approach to study the performance of cognitive underlay amplify- and-forward relay networks. In addition, our approach is applicable for other fading channel models. 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Giannakis, “Smart regenerative relays for link-adaptive cooperative communications,” in Proceedings of the 40th Annual Conference on Information Sciences and Systems, 2006, pp. 1038–1043. Vo Nguyen Quoc Bao (SMIEEE) is an associate professor of Wireless Communi- cations at Posts and Telecommunications Institute of Technology (PTIT), Vietnam. He is serving as the Dean of Faculty of Telecommunications and a Director of the Wireless Communication Laboratory (WCOMM). His research interests include wireless communications and information theory with current emphasis on MIMO systems, cooperative and cognitive commu- nications, physical layer security, and energy harvesting. He has published more than 150 journal and conference articles that have 1700+ citations and H-index of 22. He is the Technical Editor in Chief of REV Journal on Electronics and Communications. He is also serving as an Associate Editor of EURASIP Journal on Wireless Communications and Networking, an Editor of Trans- actions on Emerging Telecommunications Technologies (Wiley ETT), and VNU Journal of Computer Science and Communication Engineering. He served as a Technical Program co-chair for ATC (2013, 2014), NAFOSTED-NICS (2014, 2015, 2016), REV- ECIT 2015, ComManTel (2014, 2015), and SigComTel 2017. He is a Member of the Executive Board of the Radio-Electronics Association of Vietnam (REV) and the Electronics Information and Communications Association Ho Chi Minh City (EIC). He is currently serving as a scientific secretary of the Vietnam National Foundation for Science and Technology Development (NAFOSTED) scientific Committee in Information Technology and Computer Science (2014-2016). Vu Van San received Ph.D. degree in 2000. In 1983, he joined in the Research Institute of Posts and Telecommunications (RIPT). He is now working at Posts and Telecommunications Institute of Technol- ogy (PTIT). His research interests are in the areas of high-speed optical communi- cations, access networks, and digital trans- mission systems, wireless communications systems and signal processing. He is currently the Editor in Chief of Journal of Science and Technology on Information and Communications. He is also serving as a member of science and technology committee of the Ministry of Information and Communications. 18