Wireless Pers Commun (2013) 70:1001–1009 DOI 10.1007/s11277-012-0742-z Exact Outage Probability of Underlay Cognitive Cooperative Networks Over Rayleigh Fading Channels Khuong Ho-Van Published online: July 2012 © Springer Science+Business Media, LLC 2012 Abstract Our contribution in this paper is the derivation of an exact closed-form outage probability formula for underlay cognitive cooperative networks operated over Rayleigh fading channels The derivation considers the correlation among received signal-to-noise ratios, two critical constraints (interference power constraint and maximum transmit power constraint), and non identically distributed (i.d.) channels The derived formula is corroborated by Monte Carlo simulations and is served as an useful and effective tool to evaluate the performance behavior of underlay cognitive cooperative networks without time-consuming simulations under different operation parameters Numerical results illustrate that underlay cognitive cooperative networks suffer the outage saturation phenomenon for a given maximum interference power level Keywords Decode-and-forward · Cognitive radio · Underlay · Cooperative communications · Rayleigh fading channels Introduction Cognitive radio is an emerging technology attracting a great deal of attention due to its capability of improving spectrum utilization [1] In cognitive radio, unlicensed users/secondary users (SUs) are allowed to use the licensed band primarily allocated to licensed users/primary users (PUs) unless their operation does not degrade the performance of PUs in three modes: underlay, overlay, and interweave [2] In the underlay mode, SUs are allowed to use the spectrum when the interference caused by SUs on PUs is within the range tolerated by PUs In the overlay mode, SUs simultaneously share the same spectrum with PUs while maintaining or improving the transmission of PUs In the interweave mode, SUs are only permitted to use the empty spectrum left by PUs This paper considers the underlay mode for low implementation complexity K Ho-Van (B) Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam e-mail: khuong.hovan@yahoo.ca 123 1002 K Ho-Van In order to constrain the interference not to exceed a certain level that PUs can tolerate in the underlay mode, SUs must adaptively limit their transmit power, significantly reducing their transmission range Cooperative communications in which users share their own antennas to form a virtual antenna array for achieving the potentials of the space diversity without the need of co-located antenna arrays brings many benefits such as improved performance, increased system capacity, extended coverage, etc [3] As such, it can complement and overcome the above shortage of underlay cognitive networks Among various cooperative communications schemes, decode-and-forward (DF) and amplify-and-forward (AF) have been extensively investigated [4] In DF, each relay decodes information from the source, re-encodes it, and then forwards it to the destination In AF, each relay simply amplifies the received signal and forwards it to the destination Due to its capability of regenerating noise-free relayed signals, DF is selected in this paper The performance analysis of underlay cognitive cooperative networks has been extensively studied in [5–13] However, the most recently works closely related to our research are [12,13] In [12], the outage probability of underlay cognitive cooperative networks with DF is analyzed in consideration of the correlation among received SNRs This work is highly regarded as the first one that discovered and took such correlation into account Nevertheless, the authors just obtain a tight lower bound of the outage probability and consider only the interference power constraint Generally, two constraints (interference power constraint and maximum transmit power constraint) are imposed on the underlay cognitive networks An exact closed-form outage probability formula of underlay cognitive cooperative networks taking into account factors such as the correlation among received SNRs and two above constraints was reported in [13].1 Notwithstanding, [13] assumes all channels are i.d Our contribution is to generalize the derived formula in [13] by relaxing the assumption on i.d channels Our new formula is corroborated by Monte Carlo simulations and is served as an useful and effective tool to evaluate the performance behavior of underlay cognitive cooperative networks without time-consuming simulations under different operation parameters Numerical results illustrate that underlay cognitive cooperative networks suffer the outage saturation phenomenon for a given maximum interference power level The rest of this paper is organized as follows The next section describes the system model and analyzes the outage probability of underlay cognitive cooperative networks Simulated and analytical results are presented in Sect for performance evaluation Finally, the paper is concluded in Sect Outage Probability Analysis The underlay cognitive cooperative system model2 under consideration is depicted in Fig where a secondary relay R assists the transmission of a secondary source S to a secondary destination D, and both S and R use the same spectrum as a primary user P Assume the channel between the transmitter t and the receiver r experiences independent slowly varying flat Rayleigh fading with variance 1/λtr Hence, the channel gain gtr = |h tr |2 is an exponentially distributed random variable with the probability density function (pdf) f gtr (x) = λtr e−λtr x The authors would like to thank one of reviewers for pointing out the useful and missed reference [13] Since it appeared after our manuscript submission, there is a coincidence in the problem statement between our work and [13] However, our result is different from and more generalized than [13] Although this paper only considers the case of a single relay, it is straightforward to extend our result to the case of multiple relay nodes with best relay selection studied in [12,13] 123 Underlay Cognitive Cooperative Networks 1003 Fig System model P hSP hSR S Phase hRP R hSD hRD Phase D for x ≥ Different from [13] where all λtr ’s are assumed to be the same (i.e., i.d channels), we relax this assumption by considering non i.d channels In the underlay cognitive networks, the transmit power Pt of the transmitter t ∈ {S, R} is imposed by two constraints [2]: interference power constraint, Pt ≤ gItTP , and maximum transmit power constraint, Pt ≤ Pm , where IT is the maximum interference power level that PU still operates reliably, and Pm is the maximum transmit power In other words, Pt ≤ gItTP , Pm Consequently, the actual transmit power can be lower than the maximum one (i.e., when gItTP ≤ Pm ), resulting in the coverage range of the secondary transmitter in underlay cognitive networks less than that in other ones (e.g., interweave cognitive networks) The received SNR at the receiver r is given by [2] γtr = Pt gtr = N0 IT , Pm gt P gtr , N0 (1) where N0 is the noise variance at the receivers Let U be the transmission rate of secondary network According to the Shannon information theory, the outage (or unsuccessful information decoding) occurs at the receiver r if the received SNR γr meets the inequality U ≥ 21 log2 (1 + γr ) or γr ≤ k with k = 22U −1 where 21 indicates the whole transmission process spends two phases In the first phase, S broadcasts its signal which is received and processed by R and temporarily stored by D If R successfully recovers the source information (i.e., γ S R ≥ k), it will forward the processed signal to D in the second phase Then, D combines the signals from S and R with the maximum ratio combining (MRC) for restoring the source information In this case, the received SNR at D is γ S D +γ R D and an outage occurs if γ S D +γ R D < k Otherwise (i.e., γ S R ≤ k), R keeps silent in the second phase and D bases on the only signal received from S for decoding the source information In this case, the received SNR at D is γ S D and an outage occurs if γ S D < k According to the total probability law, the outage probability is defined as Po = Pr {γ S D < k, γ S R < k} + Pr {γ S D + γ R D < k, γ S R > k} T1 (2) T2 It is noted from (1) that the received SNRs at R and D in the first phase, γ S R = gISTP , Pm gNS0R and γ S D = gISTP , Pm gNS 0D , are correlated since they are related to g S P This correlation among the received SNRs is first discovered in [12] However, [12] only provides a tight lower bound of the outage probability and considers the interference power constraint, i.e., γtr = NIT0 ggttrP In [13], the authors investigate both constraints making the received SNRs of the form in (1) and derive the exact closed-form outage probability expression Nevertheless, the assumption on i.d channels is made there In this paper, we relax this assumption for more generalized 123 1004 K Ho-Van To derive the closed forms of two terms, T1 and T2 in (2), we note that Pr {gtr < m} = ∞ λtr e−λtr x d x = − e−λtr m , Pr {gtr > m} = m λtr e−λtr x d x = e−λtr m and the cumula- m tive density function of x R D = ⎛ IT gR P , − Pm g R D can be borrowed from [2] ⎞ βλ I RD T Pm xe Fx R D (x) = ⎝ − 1⎠ e − x + β IT λR D x Pm + 1, (3) where β = λ R P /λ R D The derivative of Fx R D (x) results in the pdf of x R D as ⎞ ⎛ βλ D I T − RPm βλ I λ x λR D x R D T R D β IT λ R D ⎝ xe e− Pm e− Pm − − 1⎠ e− Pm f x R D (x) = Pm x + β IT (x + β IT ) (4) Due to γ R D = x R D /N0 , the pdf of γ R D is f γ R D (x) = N0 f x R D (N0 x) = Ge−Gρ R D (x + G) e−ρ R D x + Gρ R D e−Gρ R D −ρ R D x e x+G −ρ R D e−Gρ R D − e−ρ R D x , where ρ R D = λ R D N0 Pm and G = (5) λ R P IT λ R D N0 2.1 Derivation of T1 With γ S D and γ S R having the form in (1), T1 is written as T1 = Pr {γ S D < k, γ S R < k} IT gS D gS R IT = Pr , Pm < k, , Pm = ⎝ Pr g S D < f g S P (x) d x⎠ f γ R D (y) dy ⎩ IT , P IT , P ⎭ 0 ⎛ k ∞ ⎝ = k (y) dy = 0 ⎢ ⎣ ⎡ k = ⎡ ⎧ ⎨ Pr ∞ ⎩ m x (k − y) N0 gS D < ⎛ ⎜ ⎝1 − e − ⎫ ⎬ IT x λ S D N0 (k−y) IT x ,Pm , Pm ⎭ ⎞ λ ⎟ ⎠e − ⎧ ⎨ Pr ⎩ x N0 k gS R > S R N0 k IT x ,Pm m IT x ⎢ ⎢ IT ⎢ Pm ⎢ ⎢ ⎣ + 1−e IT Pm − λ S D N0 (k−y)x IT 1−e − λ S D N0 (k−y) Pm e − e λ S R N0 kx IT − , Pm ⎭ ⎤ ⎞ f g S P (x) d x ⎠ f γ R D (y)dy ⎥ λ S P e−λ S P x d x ⎦ f γ R D ∞ ⎫ ⎬ ⎤ λS P λ S R N0 k Pm e−λ S P x d x λ S P e−λ S P x d x ⎥ ⎥ ⎥ ⎥ f γ R D (y) dy ⎥ ⎦ (7) For simplicity, we denote A= ρS D k + ρS R k + χ , ρS D (8) B = χe−(ρ S R k+χ) + e−ρ S R k − e−χ , ρS R k + χ (9) C = χe−(ρ S D k+ρ S R k+χ) , ρS D (10) D = e−(ρ S D +ρ S R )k − e−χ (11) Case 1:λ R D = λ S D (λ −λ )N y RD SD When inserting (5) into (7), there appear some integrals in the form of k e− Pm dy Therefore, in order to compute this integral, two cases (λ R D = λ S D and λ R D = λ S D ) must be considered 123 1006 K Ho-Van Inserting f γ R D (x) in (5) into (7), we have e−ρ R D k − G k+G T2 = BGe−Gρ R D + Dλ R D e−Gρ R D − (λ R D − λ S D ) − B e−Gρ R D − 1 − e−ρ R D k − e−(ρ R D −ρ S D )k e−(ρ R D −ρ S D )k − G k+G ⎞ RD C Gρ R D e C Ge−Gρ R D + + DGρ R D e−Gρ R D A+G (A+G) ⎠ I2 (k, ρ R D − ρ S D , G) −⎝ −Gρ − DGe−Gρ R D + C GeA+G R D (ρ R D − ρ S D ) − DGe−Gρ R D + ⎛ −Gρ + C Ge−Gρ R D (A+G)2 −Cρ R D C Ge−Gρ R D A+G C Gρ R D e−Gρ R D A+G e−Gρ R D − + I1 (k, ρ R D − ρ S D , A) , (12) where k I2 (k, ρ, G) = k I1 (k, ρ, A) = e−ρy dy = eρG (Ei (−ρk − ρG) − Ei (−ρG)) , y+G (13) e−ρy dy = e−ρ A (Ei (ρ (A − k)) − Ei (ρ A)) y−A (14) Here the exponential integral function Ei(x) is defined in [14] as Ei (x) = − ∫∞ −x is a built-in function in most computation softwares (e.g., Matlab) Case 2: λ R D = λ S D Again inserting f γ R D (x) in (5) into (7) and considering λ R D = λ S D , we have e−ρ R D k − G k+G T2 = BGe−Gρ R D − + + DGe−Gρ R D + C Ge−Gρ R D A+G e−t t dt which 1 − k+G G k+G C Ge−Gρ R D C Gρ R D e−Gρ R D + + DGρ R D e−Gρ R D ln A+G G (A + G)2 C Ge−Gρ R D (A + G) + −B e−Gρ R D − C Gρ R D e−Gρ R D − Cρ R D e−Gρ R D − A+G − e−ρ R D k + Dρ R D e−Gρ R D − k, ln A−k A (15) where ln(x) is the natural logarithm of x Illustrative Results For illustration purpose, we randomly select λ S P = 0.6366, λ S R = 0.2530, λ R P = 0.0316, λ R D = 0.0894 Two cases are considered: Case (λ S D = = λ R D ) and Case (λ S D = λ R D ) We assume the noise variance is normalized such that N0 = dB and the required transmission rate U = bps/Hz Figure investigate the effect of IT on the outage performance We fix Pm at 25 dB It is shown that analytical and simulated results4 are perfectly matched for both cases, confirming the accuracy of the derived formula in (2) Additionally, the outage performance is improved with respect to the 108 channel realizations are generated to obtain simulated results 123 Underlay Cognitive Cooperative Networks 1007 10 Case 1: Analysis Case 1: Simulation Case 2: Analysis Case 2: Simulation −1 Outage probability 10 −2 10 −3 10 −4 10 10 15 20 IT (dB) Fig Outage probability versus I T (Pm = 25 dB) 10 Outage probability Case 1: Analysis Case 1: Simulation Case 2: Analysis Case 2: Simulation 10 −1 10 −2 10 −3 10 15 20 25 30 Pm (dB) Fig Outage probability versus Pm (I T = 15 dB) increase in IT This is obvious since IT imposes a constraint on the transmit power and the higher is IT , the higher can the transmit power be, eventually enhancing communication reliability Figure compares simulated and analytical results when Pm varies from to 30 dB while IT is fixed at 15 dB It is seen that both analytical and simulated results are in the good agreement, again validating the proposed formula Additionally, the results show that underlay cognitive cooperative networks are quickly stable at high Pm This saturation phenomenon comes from the fact that the transmit power is limited by the minimum of the maximum interference power level, IT , and the maximum transmit power, Pm As such, when Pm exceeds a certain value (e.g., around 15 dB in Fig 3), the transmit power is completely controlled by IT , resulting in the same outage probability for any increase in Pm 123 1008 K Ho-Van Conclusions In this paper, the exact closed-form outage probability expression for underlay cognitive cooperative networks under the general conditions such as the correlation among the received SNRs, two constraints (interference power constraint and maximum transmit power constraint), and non i.d channels is derived and validated by simulated results Numerical results show that underlay cognitive cooperative networks suffer the outage saturation phenomenon for a certain maximum interference power level, and their performance is better with respect to the increase in the maximum interference power level References Yucek, T., & Arslan, H (2009) A survey of spectrum sensing algorithms for cognitive radio applications IEEE Communications Surveys & Tutorials, 11, 116–130 Lee, J., Wang, H., Andrews, J G., & Hong, D (2011) Outage probability of cognitive relay networks with interference constraints IEEE Transactions on Wireless 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Networks 1009 Author Biography Khuong Ho-Van received the B.E (with the first-rank honor) and the M.S degrees in Electronics and Telecommunications Engineering from Ho Chi Minh 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 From April 2001 to September 2004, he was a lecturer at Telecommunications Department, Ho Chi Minh City University of Technology During 2007–2011, he joined McGill University, Canada as a postdoctoral fellow Currently, he is an assistant professor at Ho Chi Minh City University of Technology His major research interests are modulation and coding techniques, MIMO system, digital signal processing, cooperative communications 123 ... imposed on the underlay cognitive networks An exact closed-form outage probability formula of underlay cognitive cooperative networks taking into account factors such as the correlation among received... evaluate the performance behavior of underlay cognitive cooperative networks without time-consuming simulations under different operation parameters Numerical results illustrate that underlay cognitive. .. works closely related to our research are [12,13] In [12], the outage probability of underlay cognitive cooperative networks with DF is analyzed in consideration of the correlation among received