v Abstract In the past decade, cooperative communications has been emerging as a pertinent technology for the current and upcoming generations of mobile communication infrastructure The indispensable benefits of this technology have motivated numerous studies from both academia and industry on this area In particular, cooperative communications has been developed as a means of alleviating the effect of fading and hence improve the reliability of wireless communications The key idea behind this technique is that communication between the source and destination can be assisted by several intermediate nodes, so-called relay nodes As a result, cooperative communication networks can enhance the reliability of wireless communications where the transmitted signals are severely impaired because of fading In addition, through relaying transmission, communication range can be extended and transmit power of each radio terminal can be reduced as well The objective of this thesis is to analyze the system performance of cooperative relay networks integrating advanced radio transmission techniques and using the two major relaying protocols, i.e., decodeand-forward (DF) and amplify-and-forward (AF) In particular, the radio transmission techniques that are considered in this thesis include multiple-input multiple-output (MIMO) systems and orthogonal spacetime block coding (OSTBC) transmission, adaptive transmission, beamforming transmission, coded cooperation, and cognitive radio transmission The thesis is divided into an introduction section and six parts based on peer-reviewed journal articles and conference papers The introduction provides the readers with some fundamental background on cooperative communications along with several key concepts of cognitive radio systems In the first part, performance analysis of cooperative single and multiple relay networks using MIMO and OSTBC transmission is presented wherein the diversity gain, coding gain, outage probability, symbol error rate, and channel capacity are assessed It is shown that integrating MIMO and OSTBC transmission into cooperative relay networks provides full diversity gain In the second part, the performance benefits of MIMO relay networks with OSTBC and adaptive transmission strategies are investigated In the third part, the performance improvement with respect to outage probability of coded cooperation applied to opportunistic DF relay networks over conventional cooperative networks is shown In the fourth part, the effects of delay of channel state information feedback from the destination to the source and co-channel interference on system performance is analyzed for beamforming AF relay networks In the fifth part, cooperative diversity is investigated in the context of an underlay cognitive AF relay network with beamforming In the sixth part, finally, the impact of the interference power constraint on the system performance of multi-hop vi cognitive AF relay networks is investigated vii Preface This thesis summarizes my research within the fields of cooperative communications and cognitive radio networks The work has been carried out at the School of Engineering and School of Computing, Blekinge Institute of Technology, Karlskrona, Sweden The thesis comprises an introduction section followed by six publication parts, as follows Part I MIMO Cooperative Relay Networks with OSTBCs A Performance Analysis of Decouple-and-Forward MIMO Relaying in Nakagami-m Fading B MIMO Cooperative Multiple-Relay Networks with OSTBCs over Nakagami-m Fading Part II Adaptive Transmission in MIMO AF Relay Networks with Orthogonal Space-time Block Codes over Nakagami-m Fading Part III Outage Performance for Opportunistic Decode-and-Forward Relaying Coded Cooperation Networks over Nakagami-m Fading Part IV Beamforming Amplify-and-Forward Relay Networks with Feedback Delay and Interference Part V Cognitive AF Relay Networks with Beamforming under Primary User Power Constraint over Nakagami-m Fading Channels Part VI Impact of Interference Power Constraint on Multi-hop Cognitive AF Relay Networks over Nakagami-m Fading ix Acknowledgements With the few words saying here, it is impossible to adequately express all my appreciation to people who have granted support to me during this study My principle advisor, Professor Hans-Jă urgen Zepernick, deserves significant appreciation and many, many thanks He has granted unlimited support to help me to achieve many academic milestones as well as to prepare me to obtain many more I would also like to express deep gratitude to his professional work, expertise, inspiration, and encouragement His priceless efforts, valuable dedication, and enthusiasm has left an unforgettable mark on my academic career Many special thanks to my previous co-advisor Professor Mats Pettersson and my co-advisor Dr Patrik Arlos for their valuable support and advices Also, many thanks go to the Vietnam International Education Development (VIED) for funding this research Of many of my colleagues and friends, I like to thank Duong Quang Trung, Lei Shu, Maged Elkashlan, Tran Hung, Chu Thi My Chinh, Hoang Le Nam, ă Erik Ostlin, and Ngo Quoc Hien for their cooperation I am also grateful to my friends, Muhammad Imran Iqbal, Charles Kabiri, Louis Sibomana, Ulrich Engelke, Thomas Sjăogren, and Vu Viet Thuy, for their friendship Many thanks also go to all my colleagues and friends at the Blekinge Institute of Technology for their help and support Many profound thanks go to Staffan Andersson and David Erman, for teaching me Win Tsun Freely attending their class, I had the opportunity to enjoy excellent time out of study with wonderful practice, knowledge, and philosophy of martial art Special thanks go to my father, my mother, my brothers and my sisters for their support, love, and encouragement, which have meant to me much more than what I can ever express Phan Hoc Karlskrona, December 2012 xi Publication List Part I is published as: H Phan, T Q Duong, and H.-J Zepernick, “Performance analysis of decoupleand-forward MIMO relaying in Nakagami-m fading,” IEICE Transactions on Communications, vol E95-B, no 09, pp 3003–3006, Sep 2012 H Phan, T Q Duong, and H.-J Zepernick, “MIMO cooperative multiplerelay networks with OSTBCs over Nakagami-m fading,” in Proc IEEE Wireless Communications and Networking Conference, Paris, France, Apr 2012 Part II is published as: H Phan, T Q Duong, H.-J Zepernick, and L Shu, “Adaptive transmission in MIMO AF relay networks with orthogonal space-time block codes over Nakagami-m fading,” EURASIP Journal on Wireless Communications and Networking, vol 2012, pp 1-13, Jan 2012, DOI:10.1186/1687-14992012-11 Part III is published as: H Phan, T Q Duong, and H.-J Zepernick, “Outage performance for opportunistic decode-and-forward relaying coded cooperation networks over Nakagami−m fading,” in Proc International Symposium on Wireless Communication Systems, Aachen, Germany, Nov 2011 Part IV is published as: H Phan, T Q Duong, M Elkashlan, and H.-J Zepernick, “Beamforming amplify-and-forward relay networks with feedback delay and interference,” IEEE Signal Process Lett., vol 19, no 1, pp 16–19, Jan 2012 xii Part V is published as: H Phan, H.-J Zepernick, T Q Duong, H Tran, and T M C Chu, “Cognitive AF relay networks with beamforming under primary user power constraint over Nakagami-m fading channels,” Wireless Communications and Mobile Computing, Nov 2012, DOI: 10.1002/wcm.2317 Part VI is published as: H Phan, H.-J Zepernick, and H Tran, “Impact of interference power constraint on multi-hop cognitive AF relay networks over Nakagami-m fading,” IET Communications, 2013, accepted for publication with minor revision Publications in conjunction with this thesis: T M C Chu, H Phan, and H.-J Zepernick, “Opportunistic spectrum access for cognitive amplify-and-forward relay networks,” in Proc IEEE Vehicular Technology Conference, Dresden, Germany, Jun 2013, accepted for publication T M C Chu, H Phan, and H.-J Zepernick, “On the performance of underlay cognitive radio networks using M/G/1/K queueing model,” in IEEE Commun Lett., Jan 2013, accepted for publication H Phan, T M C Chu, H.-J Zepernick, and P Arlos, “Queueing analysis of opportunistic decode-and-forward relay networks,” in Proc International Conference on Computing, Management and Telecommunications, Ho Chi Minh City, Vietnam, Jan 2013 T M C Chu, H Phan, T Q Duong, M Elkashlan, and H.-J Zepernick, “Beamforming transmission in cognitive AF relay networks with feedback delay,” in Proc International Conference on Computing, Management and Telecommunications, Ho Chi Minh City, Vietnam, Jan 2013 T M C Chu, H Phan, and H.-J Zepernick, “Amplify-and-forward relay assisting both primary and secondary transmissions in cognitive radio networks over Nakagami-m fading,” in Proc IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, Sydney, Australia, Sep 2012 xiii H Tran, H.-J Zepernick, M Fiedler, and H Phan, “Outage probability, average transmission time and quality of experience for cognitive radio networks over general fading channels,” in Euro-NF Conference on Next Generation Internet, Karlskrona, Sweden, Jun 2012 H Tran, H.-J Zepernick, and H Phan, “Impact of the number of antennas and distances among users on cognitive radio networks,” in Proc Advanced Technologies for Communications, Ha Noi, Vietnam, Oct 2012 H Phan, T Q Duong, and H.-J Zepernick, ”MIMO AF semi-blind relay networks with OSTBC transmission over Nakagami-m fading,” in Proc International Conference on Signal Processing and Communication Systems, Honolulu, Hawaii, USA, Dec 2011 H Phan, T Q Duong, and H.-J Zepernick, ”SER of amplify-and-forward cooperative networks with OSTBC transmission in Nakagami-m fading,” in Proc IEEE Vehicular Technology Conference, San Francisco, USA, Sep 2011 H Phan, T Q Duong, and H.-J Zepernick, “Full-rate distributed space-time coding for bi-directional cooperative communications,” in Proc International Symposium on Wireless Pervasive Computing, Modena, Italy, May 2010 186 Part VI where X1 = X2 = 1≤k≤K−M (γk ) K−M +1≤k≤K (9) (γk ) (10) and ≤ M ≤ K In fact, the bound of γ given in (8) is represented in terms of the harmonic mean of the minimum SNR of the first M hops and of the next K − M hops To get the most tight bound, it is shown in [14] that M = K End-to-End Performance Analysis In this section, we present an analysis for the outage probability and SER This requires the statistical distributions of the RVs Xi , i ∈ {1, 2}, appearing in the SNR in (8) The following lemma facilitates the performance investigation throughout our study and is used to reach Theorem Lemma Without loss of generality, the parameters of i.n.i.d Nakagami−1 m fading channels are assumed to satisfy β1 α1−1 = β2 α2−1 = = βK αK Then, the CDF of Xi , i ∈ {1, 2}, can be given by mK−M −1 K−M nk +ik m1 −1 FX1 (γ) = − ηkjk iK−M =0 i1 =0 K−M × k=1 k=1 jk =1 (γ + δk ) jk Γ(nk + ik )ρnk βknk ik !Γ(nk )αknk mK−M +1 −1 FX2 (γ) = − mK −1 (11) K nl +il iK−M +1 =0 K × l=K−M +1 iK =0 l=K−M +1 jl =1 ηljl (γ + δl )jl Γ(nl + il )ρnl βlnl il !Γ(nl )αlnl (12) where the coefficients ηkjk and ηljl are defined as ηkjk = (nk + ik − jk )!  dnk +ik −jk × n +i −j  dγ k k k K−M u=1,u=k  γ iu  (γ + δu )nu +iu (13) γ=−δk Impact of Interference Power Constraint on Multi-hop Cognitive (nl + il − jl )!  ηljl = dnl +il −jl × n +i −j  dγ l l l K u=K−M +1,u=l  γ iu  (γ + δu )nu +iu 187 (14) γ=−δl where δk = ρβk αk−1 Proof See Appendix A −1 Theorem Assuming that β1 α1−1 = β2 α2−1 = = βK αK , the CDF of the SNR γU B can be written as m1 −1 FγU B (γ) = − mK−M −1 K−M iK−M =0 k=1 i1 =0 K−M nk +ik mK−M +1 −1 mK −1 K Γ(nk + ik )ρnk βknk ik !Γ(nk )αknk nl +il ηkjk jl ηljl × k=1 iK =0 l=K−M +1 jl =1 jk =1 iK−M +1 =0 K × l=K−M +1 Γ(nl + il )ρnl βlnl (γ + δl )jk −jl B(jk + 1, jl ) il !Γ(nl )αlnl × γ −2jk F1 jk , jk + 1; jk + jl + 1, − (γ + δl )(γ + δk ) γ2 (15) Proof See Appendix B 3.1 Outage Probability Outage probability is defined as the probability that the instantaneous SNR falls below a target threshold γth Hence, a lower bound on the outage probability of the multi-hop cognitive AF relay networks over i.n.i.d Nakagami-m fading can be determined as Po = FγU B (γth ) 3.2 Symbol Error Rate Theoretically, the SER can be derived based on the CDF of the instantaneous SNR, γU B , as [15] √ a b Pe = √ π ∞ FγU B (γ)γ − exp(−bγ)dγ (16) 188 Part VI where a and b stand for modulation parameters In (16), it can be challenging to get a tractable closed-form solution to the SER directly from the CDF of the instantaneous SNR, γU B Hence, we utilize the CDF of the tight upper bound on γU B , referred to as γU B2 , for the analysis In particular, this bound can be expressed as [16, eq (25)] γU B2 = min(X1 , X2 ) (17) In view of the order statistic theory, it is straightforward to see that the CDF of γU B2 , FγU B2 (γ), can be formulated as m1 −1 FγU B2 (γ) = − mK−M −1 K−M iK−M =0 k=1 i1 =0 K−M nk +ik mK−M +1 −1 mK −1 Γ(nk + ik )ρnk βknk ik !Γ(nk )αknk K nl +il × k=1 jk =1 iK−M +1 =0 K × l=K−M +1 iK =0 l=K−M +1 jl =1 Γ(nl + il )ρ βlnl il !Γ(nl )αlnl nl ηkjk ηljl (γ + δk )jk (γ + δl )jl (18) Substituting (18) in (16) as well as applying the partial fraction decomposition for the term inside the remaining integral [13, eq (3.326.2)], we have √ m −1 mK−M −1 K−M Γ(nk + ik )ρnk βknk a a b Pe = − √ 2 π i =0 ik !Γ(nk )αknk i =0 k=1 K−M K−M nk +ik mK−M +1 −1 mK −1 K nl +il × k=1 jk =1 iK−M +1 =0 ηkjk ηljl iK =0 l=K−M +1 jl =1  ∞ jk Γ(nl + il )ρnl βlnl  γ − e−bγ ξp dγ × nl il !Γ(nl )αl (γ + δk )p p=1 l=K−M +1  ∞ jl − 12 −bγ γ e dγ  + ζq q (γ + δ ) l q=1 K (19) where ξp = ζq = djk −p (jk − p)! dγ jk −p (γ + δl )jl γ=−δk djl −q (jl − q)! dγ jl −q γ=−δl (γ + δk )jk (20) (21) Impact of Interference Power Constraint on Multi-hop Cognitive 189 A solution to the remaining integral in (19) can be found with the help of [17, eq (2.3.6.9)] which results in Pe = √ m −1 mK−M −1 a a b − 2 i =0 i =0 K−M K−M nk +ik mK−M +1 −1 k=1 mK −1 K Γ(nk + ik )ρnk βknk ik !Γ(nk )αknk nl +il × k=1 jk =1 iK−M +1 =0 K × l=K−M +1 × ξp δk2 jl −p ηkjk ηljl iK =0 l=K−M +1 jl =1 Γ(nl + il )ρnl βlnl il !Γ(nl )αlnl + U q=1 K−M jk U p=1 1 , + − p, bδk 2 1 −q , + − q, bδl ζq δl2 2 (22) Asymptotic Performance Analysis In this section, an asymptotic analysis for the multi-hop cognitive AF relay network is developed to study its cooperative behavior at high SNR To quantify the diversity advantage of the investigated systems at high SNR, diversity gain is employed as a helpful measure Usually, the approximation of the CDF of the instantaneous SNR FγU B (γ) at zero, obtained from its MacLaurin series, is adopted to examine the asymptotic performance Nevertheless, due to the complex mathematical expression of FγU B (γ), it is not feasible to derive its MacLaurin expansion Therefore, we invoke the approach in [18, Lemma 1] which proves that the MacLaurin series of the CDF of the SNR γU B has the same first nonzero coefficient as the MacLaurin series of the CDF of its upper bound γU B2 The asymptotic outage probability can be given in the following theorem Theorem Under Nakagami-m fading, the asymptotic outage probability of a multi-hop cognitive AF relay network can be quantified as I Po∞ = i=1 mr α ri i Γ(mri + nri ) Γ(mri + 1)Γ(nri ) ρmri βrmi ri m γth (23) where mr1 = = mrI = m = min(m1 , , mK ) r1 , , rI ∈ {1, , K} (24) (25) 190 Part VI 10 -1 10 K=5, M=3 -2 Outage Probability 10 -3 K=4, M=2 10 -4 10 K=2, M=1 -5 10 -6 Analysis 10 Simulation Asymptotic -7 10 -10 -5 Q/N 10 15 20 (dB) Figure 2: Outage probability versus peak interference power-to-noise ratio of multi-hop cognitive AF relay networks for different number of hops K Proof See Appendix C In addition, the SER at high SNR of the multi-hop cognitive AF relay network can be evaluated using the following corollary Corollary An asymptotic expression for the SER of the multi-hop cognitive AF relay network over Nakagami-m fading can be given by Pe∞ = aΓ(m + 1/2) √ πbm I i=1 mr Γ(mri + nri ) α ri i Γ(mri + 1)Γ(nri ) ρmri βrmi ri (26) where mr1 = = mrI = m = min(m1 , , mK ) r1 , , rI ∈ {1, , K} (27) (28) 191 Impact of Interference Power Constraint on Multi-hop Cognitive 10 -1 10 K=5, M=3 -2 Symbol Error Rate 10 -3 10 -4 10 K=2, M=1 K=4, M=2 -5 10 -6 10 -7 10 Analysis Simulation Asymptotic -8 10 -10 -5 Q/N 10 15 20 (dB) Figure 3: SER versus peak interference power-to-noise ratio of multi-hop cognitive AF relay networks for different number of hops K Proof Substituting (23) into (16) and using [13, eq (3.381.4)] to solve the remaining integral, we get (26) Numerical Results and Discussion In this section, we provide numerical examples to reveal the impact of the value of fading severity parameters, the number of hops, and the distances from PURX to SUTX and to SRk on the system performance In all scenarios, the outage threshold is selected as γth = dB, the modulation scheme is binary phase shift keying (BPSK) with a = and b = In addition, the path-loss exponent is set as ν = for a highly shadowed urban area, the normalized distances among the terminals in the secondary relay network are {dsk }5k=1 ={0.3; 0.5; 0.4; 0.6; 0.8}, and the fading severity parameters of the channels from PURX to the terminals of the secondary relay network are 192 Part VI -1 10 -2 10 Case -3 10 Outage Probability Case -4 10 -5 10 -6 10 Case -7 10 -8 10 Analysis Simulation Asymptotic -9 10 -10 -5 Q/N 10 15 20 (dB) Figure 4: Outage probability versus peak interference power-to-noise ratio of multi-hop cognitive AF relay networks for different fading severity parameters m {nk }5k=1 ={5; 2; 3; 4; 1} For verification of the analysis, Monte-Carlo simulations are provided showing very close agreement with analytical results Figs and illustrate the outage probability and SER versus peak interference power-to-noise ratio Q/N0 , respectively The fading severity parameters of the multi-hop secondary relay network are set as {mk }5k=1 ={4; 3; 2; 2; 1}, and the normalized distances from PURX to SUTX and to SRk are chosen as {dpk }5k=1 ={1.2; 1.4; 1.6; 1.8; 1.5} As can be seen from these figures, the asymptotic curves tightly converge to both the analytical and simulated ones for sufficiently high values of the peak interference power-to-noise ratio Q/N0 We can observe from Figs and that diversity gains for the cases of {K = 2, M = 1}, {K = 4, M = 2}, and {K = 5, M = 3} are 3, 2, and 1, respectively These observations agree with the analytical derivation given in (23) Also, under Nakagami-m fading, increasing the number of hops, which leads to a high possibility of decreasing the minimum of fading severity 193 Impact of Interference Power Constraint on Multi-hop Cognitive -1 10 -2 10 Case -3 10 Symbol Error Rate -4 10 -5 Case 10 -6 10 -7 10 -8 10 Analysis -9 10 Case Simulation Asymptotic -10 10 -10 -5 Q/N 10 15 20 (dB) Figure 5: SER versus peak interference power-to-noise ratio of multi-hop cognitive AF relay networks for different fading severity parameters m parameters m, may result in lower diversity gain Figs and depict the outage probability and SER versus peak interference power-to-noise ratio with different fading severity parameters m for three cases, respectively We set the number of hops as K = 4, and the normalized distances from PURX to SUTX and to SRk as {dpk }4k=1 ={1.2; 1.4; 1.6; 1.8} The fading severity parameters for each case are selected as • Case 1: {mcase }k=1 ={1; 2; 1; 2} k i 4 • Case i: {mcase }k=1 ={mcase }k=1 + (i − 1), i = 2, k k It is found that for the same number of relaying hops, simultaneously increasing the fading severity parameters m of all relaying hops provides improvement in system performance due to the increased diversity gain Figs and show the outage probability and SER versus peak interference power-to-noise ratio Q/N0 , respectively, by increasing the distances from PURX to the terminals of the secondary relay network for the following cases: 194 Part VI -1 10 -2 10 Case Outage Probability -3 10 Case -4 10 Case -5 10 -6 10 Analysis Simulation Asymptotic -7 10 -10 -5 Q/N 10 15 20 (dB) Figure 6: Outage probability versus peak interference power-to-noise ratio of multi-hop cognitive AF relay networks for different distances from the PU to the secondary relay network • Case 1: {dpcase }k=1 ={1.2; 1.4; 1.6; 1.8} k i 4 • Case i: {dpcase }k=1 ={dpcase }k=1 + 0.4(i − 1), i = 2, k k The number of relaying hops is selected as K = 4, and the fading severity parameters of the relaying hops are set as {mk }4k=1 ={4; 3; 2; 2} As shown in Fig and Fig 7, the larger distances from PURX to the terminals of the secondary relay network are, the better system performance in terms of coding gain can be achieved It is also observed from these figures that the diversity gains for the three examined cases are identical which indicates that distances from PURX to the terminals of the secondary relay network only affect the coding gain Impact of Interference Power Constraint on Multi-hop Cognitive 195 -1 10 -2 10 Case Symbol Error Rate -3 10 Case -4 10 -5 Case 10 -6 10 Analysis Simulation Asymptotic -7 10 -10 -5 Q/N 10 15 20 (dB) Figure 7: SER versus peak interference power-to-noise ratio of multi-hop cognitive AF relay networks for different distances from the PU to the secondary relay network Conclusions In this paper, we have studied the impact of the fading channels, the interference power constraint, and the number of hops on the performance of multi-hop cognitive AF relay networks Closed-form expressions for tightly bounded outage probability and SER, which are utilized for examining the system performance, are presented In addition, an asymptotic analysis has been established to reveal the cooperative diversity of the considered network Our presented analysis shows that the diversity gain is quantified as the minimum of the fading severity parameters of all the hops of the secondary relay network 196 Part VI Appendices A Proof of Lemma In this appendix, we derive the CDF of the minimum of the SNR of the first K − M hops X1 To this end, in (5), we denote T1 = |hk |2 and T2 = |gk |2 for the sake of brevity and to calculate the CDF of γk = ρT with T = TT21 By definition, the CDF of RV T can be written as ∞ FT (γ) = FT1 (γt2 )fT2 (t2 )dt2 (A.1) mk −1 The CDF FT1 (γ), obtained from fT1 (γ) as FT1 (x) = 1−exp(−αk x) i=0 (αk x)i i! , and the PDF fT2 (t2 )dt2 in (7) are substituted in (A.1), yielding β nk FT (γ) = − k Γ(nk ) mk −1 i=0 (αk γ)i i! ∞ t2nk +i−1 e−(αk γ+βk )t2 dt2 (A.2) We apply [13, eq (3.381.4)] to solve the remaining integral in (A.2) and the relationship, γk = ρT , followed by algebraic manipulations, resulting in mk −1 FT (γ) = − i=0 βknk Γ(nk + i)ρnk γi nk i!Γ(nk )αk (γ + δk )nk +i (A.3) In view of the order statistic theory, the CDF of RV X1 in (9) can be determined as K−M mk −1 k=1 ik =0 FX1 (γ) = − Γ(nk + ik )ρnk βknk γ ik ik !Γ(nk )αknk (γ + δk )nk +ik (A.4) Using the identity product to manipulate (A.4), we obtain (11) The CDF of X2 given in (12) can be derived similarly as the CDF of X1 B Proof of Theorem In this appendix, to prove Theorem 1, we will derive the CDF of γU B given in (8) From (8) together with the total probability theorem, the CDF FγU B (γ) Impact of Interference Power Constraint on Multi-hop Cognitive 197 can be written as ∞ FγU B (γ) = X1 x ≤ γ|X2 = x2 fX2 (x2 )dx2 X1 + x Pr ∞ = FX2 (γ) + x2 γ x2 − γ FX γ fX2 (x2 )dx2 (B.1) Changing the variable for the second integral in (B.1) and substituting (11) and fX2 (γ) obtained by differentiating FX2 (γ) in (12) into (B.1), we obtain m1 −1 FγU B (γ) = − mK−M −1 K−M iK−M =0 k=1 i1 =0 K−M nk +ik mK−M +1 −1 mK −1 Γ(nk + ik )ρnk βknk ik !Γ(nk )αknk K nl +il × k=1 jk =1 iK−M +1 =0 K × l=K−M +1 ∞ × iK =0 l=K−M +1 jl =1 ηkjk jl ηljl (γ + δk )jk Γ(nl + il )ρnl βlnl il !Γ(nl )αlnl xj2k dx2 (x2 + γ (γ + δk )−1 )jk (x2 + γ + δl )jl +1 (B.2) Using [13, eq (3.197.1)] to solve the integral in (B.2) along with some manipulations yields (15) C Proof of Theorem In this appendix, we obtain the approximation of the CDF of γU B around zero To this end, it is first required to find the MacLaurin expansion of γk = ρT Taking the n-th order derivatives of the two sides of (A.1) with respect to γ, we have ∞ (n) FT (γ)|γ=0 = ∂ n [FT1 (φ)] ∂γ n fT2 (t2 )dt2 γ=0 (C.1) 198 Part VI where φ = γt2 According to [13, eq (0.430.1)], we get ∂ n [FT1 (φ)] ∂γ n n u−1 = u=1 v=0 γ=0 u (−1)v φv ∂ u FT1 (φ) ∂ n φu−v v Γ(u + 1) ∂φu ∂γ n (C.2) γ=0 Based on the fact that φv |γ=0 = if and only if v = 0; otherwise φv |γ=0 = ∂ u FT1 (φ) ∂ u F (φ) |γ=0 = αkmk if along with the fact that ∂φT1u |γ=0 = if u < mk , ∂φu u = mk , we reach the following conclusion: Fγk (γ)|γ→0 = FT γ ρ |γ→0 = χk γ mk + o(γ mk ) (C.3) m αk k Γ(mk +nk ) Γ(mk +1)Γ(nk ) ρmk βkmk Furthermore, K−M FX1 (γ) = − k=1 (1 − Fγk (γ)), the where χk = using the identity product to expand FX1 (γ) can be simplified as MacLaurin approximation of I I Fγri (γ)|γ→0 = FX1 (γ)|γ→0 = i=1 χ ri γ m ri (C.4) i=1 mr1 = = mrI = min(m1 , , mK−M ) (C.5) Similarly, the MacLaurin approximation of FX2 (γ) is J FX2 (γ)|γ→0 = χ si γ m si (C.6) i=1 ms1 = = msJ = min(mK−M +1 , , mK ) (C.7) By applying a similar treatment as with obtaining the MacLaurin approximation for Fγk (γ) in (C.3), which is omitted here due to the space limitation, we obtain (23) References [1] A Sendonaris, E Erkip, and B Aazhang, “User cooperation diversity Part I: System description,” IEEE Trans Commun., vol 51, no 11, pp 1927–1938, Nov 2003 [2] M O Hasna and M.-S Alouini, “Outage probability of multihop transmission over Nakagami fading channels,” IEEE Commun Lett., vol 7, no 5, pp 216–218, May 2003 Impact of Interference Power Constraint on Multi-hop Cognitive 199 [3] J Boyer, D D Falconer, and H Yanikomeroglu, “Multihop diversity in wireless relaying channels,” IEEE Trans Commun., vol 52, no 10, pp 1820–1830, Oct 2004 [4] J Si, Z Li, X Chen, B Hao, and Z Liu, “On the performance of cognitive relay networks under 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