Outage Probability and Capacity Scaling Law of Multiple RIS-Aided Cooperative Networks
Liang Yang, Yin Yang, Daniel Benevides da Costa, Imene Trigui
aa r X i v : . [ c s . I T ] J u l Outage Probability and Capacity Scaling Law ofMultiple RIS-Aided Cooperative Networks
Liang Yang, Yin Yang, Daniel Benevides da Costa, and Imene Trigui
Abstract —In this letter, we consider a dual-hop cooperativenetwork assisted by multiple reconfigurable intelligent surfaces(RISs). Assuming that the RIS with the highest instantaneousend-to-end signal-to-noise ratio (SNR) is selected to aid thecommunication, the outage probability (OP) and average sum-rate are investigated. Specifically, an exact analysis for the OPis developed. In addition, relying on the extreme value theory,closed-form expressions for the asymptotic OP and asymptoticsum-rate are derived, based on which the capacity scaling lawis established. Our results are corroborated through simulationsand insightful discussions are provided. In particular, our analysisshows that the number of RISs as well as the number of reflectingelements play a crucial role in the capacity scaling law of multipleRIS-aided cooperative networks. Also, comparisons with relay-aided systems are carried out to demonstrate that the proposedsystem setup outperforms relaying schemes both in terms of theOP and average sum-rate.
Index Terms —Average sum-rate, capacity scaling law, outageprobability, reconfigurable intelligent surfaces (RISs).
I. I
NTRODUCTION
Reconfigurable intelligent surface (RIS) has been regardedas an emerging cost-effective technology for future wirelesscommunication systems. These artificial surfaces are com-posed of reconfigurable electromagnetic materials that can becontrolled and programmed by integrated electronic devices,providing potential gains in terms of spectrum and energyefficiencies [1]. By comparing with traditional relaying tech-nology, RISs do not need high hardware complexity and costoverhead during their operation. In addition, RISs are able tocustomize the wireless environment through the use of nearly-passive reflecting elements, enabling the system designers tofully control the electromagnetic response of the impingingsignals into the environmental objects [2]. Therefore, dueto their unique properties, RISs arise as one of the keytechnologies to realize the futuristic concept of smart radioenvironment.Very recently, RISs have been extensively studied in theliterature. Specifically, the authors in [3] formulated a beam-forming optimization problem of RIS-aided wireless commu-nication systems under discrete phase shift constraints. In [4],the authors applied RISs in downlink multi-user communica-tions. The authors in [5] proposed a deep learning methodfor deploying RISs in an indoor environment. In [6], the
L. Yang and Y. Yang are with the College of Information Sci-ence and Engineering, Hunan University, Changsha 410082, China, (e-mail:[email protected], [email protected]).D. B. da Costa is with the Department of Computer Engineering, FederalUniversity of Cear´a, Sobral, CE, Brazil (email: [email protected]).I. Trigui is with the University of Quebec, Montreal, QC, Canada (e-mail:[email protected]). authors quantitatively analyzed the coverage of RIS-assistedcommunication systems through an outage probability (OP)analysis. The authors in [7] conducted a performance analysisof the application of RISs in mixed free-space optical (FSO)and radio-frequency (RF) dual-hop communication systems. In[8], it was shown that the application of RISs can effectivelyimprove the coverage and reliability of unmanned aerial ve-hicles (UAV) communication systems. In [9], the use of RISsfor improving physical layer security was examined. On theother hand, along the years, the study of capacity scaling lawof wireless communication systems has been of paramountimportance. For example, in [10], it was studied the capacityof multi-user multi-antenna relay networks with co-channelinterference, while the authors in [11] analyzed the scalingrates for the OP and average sum-rate assuming a single RIS-assisted communication system.Although the aforementioned works have provided inter-esting contributions, the research field on RIS is still at itsinfancy. This paper aims to fill out an important gap whichexists in the literature, which is the investigation of multipleRISs in dual-hop cooperative networks. To the best of theauthors’ knowledge, such kind of system setup has not beeninvestigated in the literature yet. Specifically, assuming amultiple RIS-aided system setup where the RIS with thehighest instantaneous signal-to-noise ratio (SNR) is selectedto assist the communication, an exact analysis for the OP isdeveloped. In addition, relying on the extreme value theory,closed-form expressions for the asymptotic OP and asymptoticsum-rate are derived, based on which new capacity scalinglaws are established. Our results are corroborated through sim-ulations and insightful discussions are provided. In particular,our analysis shows that the number of RISs as well as thenumber of reflecting elements play a crucial role in capacityscaling law of multiple RIS-aided cooperative networks. Also,comparisons with relay-aided systems are carried out to showthat the proposed system setup outperforms relaying schemesboth in terms of the OP and average sum-rate.The remainder of this paper is structured as follows. InSection II, the system and channel models are introduced.Section III carries out a detailed performance analysis interms of the OP and average sum-rate, while Section IVpresents illustrative numerical results which are followed byinsightful discussions and corroborated through simulations.Finally, Section V concludes the paper.II. S
YSTEM AND C HANNEL M ODELS
As shown in Fig. 1, we consider a wireless communicationsystem consisted of one source (S), one destination (D), and K RIS 1 SS ki h D D
RIS kRIS K ki g ...... Fig. 1. System model.
RISs, where each RIS is composed by N reflecting elements.Similar to [1], it is assumed full channel state information(CSI). The communication process is briefly described next.Firstly, S sends the signal to the k -th RIS ( k = 1 , . . . , K ) and then it passively reflects the signal to D . With this aim,the k -th RIS (denoted, for simplicity, by RIS k ) optimizesthe phase reflection coefficient to maximize the receivedSNR at D, therefore improving the end-to-end quality of thecommunication system. The channels are assumed to undergoindependent Rayleigh fading. Thus, the signal received at Dcan be expressed as y Dk = p E s " N X i =1 h ki e jφ ki g ki x + n, (1)where E s denotes the average transmitted energy per symbol, x is the transmitted signal, n ∼ CN (0 , N ) stands for theadditive white Gaussian noise (AWGN), φ ki represents theadjustable phase produced by the i -th reflecting element ofthe RIS k , h ki = d − v/ SR k α ki e − jθ ki and g ki = d − v/ R k D β ki e − jϕ ki are the channel gains of the S- RIS k and RIS k -D links, re-spectively, where d SR k and d R k D denote the distances ofthe S- RIS k and RIS k -D links, respectively, and v denotesthe path loss coefficient. In addition, α ki and β ki representthe respective channels’ amplitudes, which are independentdistributed Rayleigh random variables (RVs) with mean √ π/ and variance (4 − π ) / , and θ ki and ϕ ki refer to the phases ofthe respective fading channel gains. As considered in previousworks, we assume that RIS k has perfect knowledge of θ ki and ϕ ki . From (1), the instantaneous end-to-end SNR at D can beexpressed as γ k = E s (cid:12)(cid:12)(cid:12)P Ni =1 α ki β ki e j ( φ ki − θ ki − ϕ ki ) (cid:12)(cid:12)(cid:12) N d vSR k d vR k D . (2)From [1], in order to maximize γ k , RIS k can be smartlyconfigured to fully eliminate the phase shifts by setting φ ki = As will be shown later, we consider that the RIS which provides thehighest instantaneous end-to-end SNR is selected to assist the communication.However, at this moment, assume that the communication is carried outthrough the k -th RIS. It is worth noting that RISs work similar to relaying systems, howevertheir working principles are rather different and more detailed explanationscan be found in [12]. θ ki + ϕ ki . Therefore, the maximized γ k can be written as γ k = E s (cid:16)P Ni =1 α ki β ki (cid:17) N d vSR k d vR k D = ¯ γ k A , (3)where A = P Ni =1 α ki β ki and ¯ γ k = E s N d vSRk d vRkD stands forthe average SNR.III. P ERFORMANCE A NALYSIS
In this section, we analyze the OP and average sum-rate ofthe considered system setup. Along the analytical derivations,it will be considered that one out of K RIS is selected to aidthe communication. Specifically, the choice of the suitable RISis performed to maximize the received signal at the destination.Therefore, the selection principle of the RIS can be expressedas k ∗ = arg max k ∈{ , ,...,K } γ k ˜ γ k , (4)where ˜ γ k is the average channel SNR measured by RIS k inthe past window of length s t . Note that ˜ γ k is introduced in thedenominator to maintain the long-term fairness. Next, similarto [13] ,we focus on the investigation of small-scale channelfading. Thus, the selection principle in (4) simplifies to k ∗ = arg max k ∈{ , ,...,K } γ k . (5)In what follows, the analysis is developed assuming aclustered configuration for the deployment of the RISs, whichimplies that the end-to-end links undergo independent identi-cally distributed (i.i.d.) Rayleigh fading . A. Outage Probability Analysis1) Exact Analysis:
Based on the clustered deployment ofthe RISs, it follows that γ = γ = . . . = γ k = γ , which yields A = P Ni =1 α i β i . Let B i = α i β i , then the probability densityfunction (PDF) of B i can be readily obtained as f B i ( γ ) =4 γK (2 γ ) , where K ( · ) is the modified Bessel function ofthe second kind with zero order [14]. According to [15], onecan attest that the PDF of B i is a special case of the K G distribution. In [15], the authors stated that the PDF of thesum of multiple K G RVs can be well-approximated by thePDF of √ W with W = P Ni =1 B i , in which the PDF of W is approximated by a squared K G distribution. Therefore, thePDF of A can be represented by a squared K G distribution.Thus, the PDF of γ can be written as [16] f γ ( γ ) = 2Ξ l + m Γ( l )Γ( m )¯ γ l + m γ ( l + m − K l − m (2Ξ p γ/ ¯ γ ) , (6)where l and m are the shaping parameters, Γ( · ) denotes thegamma function, Ω △ = E [ A ] is the mean power, and Ξ = p lm/ Ω . Non-clustered configuration analysis arises as an interesting study forfuture works. Therefore, the results obtained in this paper can serve as abenchmark for these new investigations.
Relying on the idea presented in [17], we make use of themixed gamma (MG) distribution to rewrite the PDF given in(6) as f γ ( γ ) = M X i =1 w i γ ρ i − ¯ γ − ρ i e − ε i γ/ ¯ γ , (7)where M denotes the number of terms of the sum, w i = χ i P Mj =1 χ j Γ( ρ j ) ε − ρjj , ρ i = m , ε i = Ξ /t i , and χ i = Ξ m y i t l − m − i Γ( m )Γ( l ) , with y i and t i representing, respectively, theweight factor and the abscissas of the Gaussian-Laguerreintegration [18]. From probability theory, the cumulative dis-tribution function (CDF) of γ can be derived as F γ ( γ ) = M X i =1 w i ε − ρ i i Υ (cid:18) ρ i , ε i γ ¯ γ (cid:19) , (8)where Υ( · , · ) represents the lower incomplete gamma function[14].According to the order statistics theory, the CDF of γ k ∗ ,which is given in (5), can be formulated as F γ k ∗ ( γ ) = ( F γ ( γ )) K , (9)which yields the following PDF f γ k ∗ ( γ ) = Kf γ ( γ )[ F γ ( γ )] K − . (10)From [19], the OP can be defined as the probability that theeffective received SNR γ k ∗ is less than a given threshold γ th ,which is mathematically written as P out = Pr( γ k ∗ < γ th ) .Thus, by replacing (8) into (9), the system OP can be expressedas P out = " M X i =1 w i ε − ρ i i Υ (cid:18) ρ i , ε i γ th ¯ γ (cid:19) K . (11)
2) Asymptotic Analysis:
According to [14], Υ( a, b ) canbe rewritten in the form Υ( a, b ) = e − b P ∞ n =0 ( b a + n /a ( a +1) . . . ( a + n )) . Based on this identity, Υ( ρ i , ε i γ/ ¯ γ ) can beasymptotically expressed as Υ (cid:18) ρ i , ε i γ ¯ γ (cid:19) ≃ e − εiγ ¯ γ (cid:18) ρ − i (cid:18) ε i γ ¯ γ (cid:19) ρ i + o ( γ ρ i +1 ) (cid:19) . (12)At SNR regime, one can ignore the high order term o ( γ ρ i +1 ) .Thus, an asymptotic outage expression for (11) can be ex-pressed as P out ≃ γ ρ i ] K " M X i =1 w i e − εiγ th¯ γ ρ − i γ ρ i th K . (13)The above expression indicates that the achievable diversityorder of the proposed system setup is Kρ i , which can alsobe written as KN since ρ i equals to m , and this latter isdetermined by N . B. Asymptotic Sum-Rate Analysis
To analyze the asymptotic sum-rate, we depart from theidea presented in [20, Lemma 2]. More specifically, let { z ,. . . , z K } i.i.d. RVs with a common CDF F Z ( · ) and PDF f Z ( · ) ,satisfying the property that F Z ( · ) is less than one for all finite z and is twice differentiable for all z , which implies that lim z →∞ − F Z ( z ) f Z ( z ) = C > . (14)for some constant C . Then, the expression max ≤ k ≤ K z k − h K converges in distribution to a limiting RV with CFD given by exp( − e − x / C ) . It is worth noting that the CDF of h K is givenby − K .The above result allows to say that the maximum of K i.i.d. RVs grows like h K . Therefore, in the sequel we derivethe asymptotic sum-rate assuming a high number of RISs, i.e.,as K −→ ∞ . Due to the intricacy in carrying out such kind ofanalysis departing from the MG distribution presented in theprevious section, here, for sake of tractability, the CDF of γ will be written by using the non-central chi-square distribution[6], i.e., F γ ( γ ) = 1 − Q √ λσ , √ γ √ ¯ γσ ! , (15)where Q ν ( c, d ) denotes the Marcum Q -function, λ = ( Nπ ) ,and σ = N (1 − π ) . At high SNR regime, based on [22], theMarcum Q -function Q τ ( x, y ) can be asymptotically expressedas Q τ ( x, y ) ≃ (1 − ϑ ) − τ exp( − ϑ y ) exp (cid:18) τ ϑx − ϑ (cid:19) , (16)where ϑ stands for the Chernoff parameter (0 < ϑ < ) . Byreplacing (16) into (15), the CDF of γ can be asymptoticallyexpressed as F γ ( γ ) ≃ − (1 − ϑ ) − exp (cid:18) − ϑγ ¯ γ N (16 − π ) (cid:19) × exp (cid:18) ϑ N π − ϑ )(16 − π ) (cid:19) , (17)and its corresponding PDF is f γ ( γ ) ≃ (1 − ϑ ) − ϑ ¯ γN (16 − π ) exp (cid:18) − ϑγ ¯ γ N (16 − π ) (cid:19) × exp (cid:18) ϑ N π − ϑ )(16 − π ) (cid:19) . (18)Then, we can show that lim γ →∞ − F γ ( γ ) f γ ( γ ) = ¯ γN (16 − π )16 ϑ = C > . (19)Also, by solving F ( h K ) = 1 − K , it follows that h K = (cid:20) ln K −
12 ln(1 − ϑ ) + ϑ N π − ϑ )(16 − π ) (cid:21) × ¯ γN (16 − π )16 ϑ . (20)Therefore, for a large number of RISs, the maximum SNR γ grows as in (20), which is the function of K and N for fixed γ [dB]
15 18 21 24 27 30 O u t age P r obab ili t y -10 -8 -6 -4 -2 Simulation,RIS-aided systemsAnalysis,RIS-aided systemsAsymptoticSimulation,relaying systems
K=1, 2, 3
Fig. 2. Outage probability versus ¯ γ for different number of RISs/relays andassuming N = 3 . ¯ γ . Accordingly, the asymptotic sum-rate can be approximatedby C K ≃ log (1 + h K ) ≃ log (cid:18) ln K −
12 ln(1 − ϑ ) + ϑ N π − ϑ )(16 − π ) (cid:19) + log (cid:18) ¯ γN (16 − π )16 ϑ (cid:19) ≃ log (ln K ) + log (cid:18) − π ϑ (cid:19) + log ¯ γ + log N . (21)The above expression indicates that K and N play asignificant role in increasing the sum-rate.IV. N UMERICAL R ESULTS AND D ISCUSSIONS
In this section, illustrative numerical examples are presentedto verify the impact of the key system parameters on theoverall performance. Our analysis is corroborated throughMonte Carlo simulations, in which simulation points aregenerated. Moreover, comparisons with relay-aided systemsare carried out to demonstrate that the proposed system setupoutperforms relaying schemes both in terms of the OP andaverage sum-rate.In Fig. 2, the OP of RIS-aided systems and relaying systemsis plotted assuming γ th = 20 dB , K = 1 , , , and N = 3 .From this figure, one can see that RIS-aided systems havebetter performance than relaying ones. Moreover, the systemperformance improves as K increases, in which the slopeof the curves changes according to K , corroborating thepresented asymptotic analysis. In Fig. 3, the OP of RIS-aidedsystems for different values is plotted for different values of K and N . It can be clearly seen that increasing the valueof N can significantly improve the system performance. Inboth figures, one can observe that the analytical results matchperfectly with the simulation results. Also, at high SNRs, theasymptotic results are close to the exact values.Fig. 4 depicts the OP for different combinations of N and K , but keeping the product KN with the same value. It can be γ [dB] O u t age P r obab ili t y -10 -8 -6 -4 -2 SimulationAnalysis
N=10,K=1,2,3 N=5,K=1,2,3
Fig. 3. Outage probability versus ¯ γ for different values of K and N . γ [dB]
20 21 22 23 24 25 26 27 28 29 30 O u t age P r obab ili t y -22 -20 -18 -16 -14 -12 -10 -8 -6 AnalysisAsymptotic (N,K)=(2,6),(3,4),(4,3),(6,2)
Fig. 4. Outage probability for different combinations pairs of ( N , K ). clearly seen that the four curves have the same slopes, whichverifies the correctness of the derived diversity order, i.e., KN .Furthermore, the slight change of N and K can lead to animproved system performance, however such performance ismostly dependent on the value of N .In Fig. 5, the asymptotic sum-rate is plotted for differentvalues of N and assuming K = 5 . From this figure, one canattest that N has a great impact on the system performance.Our results are also compared with the relay schemes (as-suming the same number as considered for RIS-aided case,i.e., K = 5 ) to show the performance gain of the proposedsystem setup. In Fig. 6, the average sum-rate versus K isdepicted by setting N = 10 , and γ th = 10 dB . It canbe clearly seen that the asymptotic curves well reflect thescaling law of the considered RIS-aided system, althoughthey do not coincide with simulation results. However, theasymptotic value becomes tighter to the simulation valueswhen N increases. Finally, it can be observed that the proposedsystem setup is significantly better than the relay schemes interms of average sum-rate. γ [dB] -10 -5 0 5 10 15 A v e r age s u m -r a t e [ b i t s / s / H z ] SimulationAsymptotic
RIS, N=5,10,20 Relay
Fig. 5. Average capacity of RIS-aided systems for different values of N andassuming K = 5 RISs/relays. K A v e r age s u m -r a t e [ b i t s / s / H z ] SimulationAsymptotic
RIS, N=10, 15 Relay
Fig. 6. Average capacity versus K for different values of N . V. C
ONCLUSION
In this work, we investigated dual-hop cooperative networkassisted by multiple RISs. Assuming that one of out K RISs (which one having N elements) is selected to add thecommunication process, an exact analysis for the OP waspresented and closed-form expressions for the asymptotic OPand asymptotic sum-rate were derived, based on which thecapacity scaling law was determined. Our results showed thatthe achievable diversity order of the considered system equalsto KN . In addition, we have compared the proposed schemewith a DF relay scenario and it is shown that the former oneoutperforms considerably the latter one both in terms of OPand average sum-rate. Our results are not only novel but canbe used as a benchmark for future studies. Potential new worksinclude non-clustered configuration for the RISs’ deploymentas well as the proposal of other techniques of RIS selection.R EFERENCES[1] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M. S. Alouini andR. Zhang, “Wireless communications through reconfigurable intelligent surfaces,”
IEEE Access , vol. 7, pp. 116753-116773, Aug. 2019.[2] C. Liaskos, S. Nie, A. Tsioliaridou, A. Pitsillides, S. Ioannidis, and I.Akyildiz, “A new wireless communication paradigm through software-controlled metasurfaces,”
IEEE Commun. Mag. , vol. 56, no. 9, pp. 162-169, Sep. 2018.[3] Q. Wu and R. Zhan, “Beamforming optimization for wireless networkaided by intelligent reflecting surface with discrete phase shifts,”
IEEETrans. Wireless Commun. , vol. 68, no. 3, pp. 1838-1851, Mar. 2020.[4] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah and C. Yuen,“Reconfigurable intelligent surfaces for energy efficiency in wirelesscommunication,”
IEEE Trans. Wireless Commun. , vol. 18, no. 8, pp.4157-4170, Aug. 2019.[5] C. Huang, G. C. Alexandropoulos, C. Yuen, and M. Debbah, “Indoorsignal focusing with deep learning designed reconfigurable intelligentsurfaces,” in
Proc. IEEE IWSPAWC , Cannes, France, 2019, pp. 1-5.[6] L. Yang, Y. Yang, M. O. Hasna, and M. Alouini, “Coverage, prob-ability of SNR gain, and DOR analysis of RIS-aided communicationsystems,”
IEEE Wireless Commun. Lett. , Early Access, DOI: 10.1109/L-WC.2020.2987798.[7] L. Yang, W. Guo, and I. S. Ansari, “Mixed dual-hop FSO-RF communi-cation systems through reconfigurable intelligent surface,”
IEEE Com-mun. Lett. , vol. 24, no. 7, pp. 1558-1562, Jul. 2020,[8] L. Yang, F. Meng, J. Zhang, M. O. Hasna, and M. Di Renzo, “On the per-formance of RIS-assisted dual-hop UAV communication systems,”
IEEETrans. Veh. Technol. , Early Access, DOI:10.1109/TVT.2020.3004598.[9] L. Yang, J. Yang, W. Xie, M. O. Hasna, T. Tsiftsis, M. DiRenzo, “Secrecy performance analysis of RIS-aided wireless commu-nication systems,”
IEEE Trans. Veh. Technol. , Early Access, DOI:10.1109/TVT.2020.3007521.[10] I. Trigui, S. Affes and A. Stphenne, “Capacity scaling laws ininterference-limited multiple-antenna AF relay networks with userscheduling,”
IEEE Trans. Commun. , vol. 64, no. 8, pp. 3284-3295, Aug.2016.[11] I. Trigui, W. Ajib and W. Zhu, “A comprehensive study of reconfig-urable intelligent surfaces in generalized fading,” [Online]. Available:https://arxiv.org/abs/2004.02922.[12] K. Ntontin, M. Di Renzo, J. Song, F. Lazarakis, J. D. Rosny, D.T. Phan Huy, O. Simeone, R. Zhang, M. Debbah, G. Lerosey, M.Fink, S. Tretyakov, S. Shamai, “Reconfigurable intelligent surfacesvs. relaying: Differences, similarities, and performance comparison,”[Online]. Available: https://arxiv.org/abs/1908.08747.[13] C. Chen and L. Wang, “A unified capacity analysis for wireless systemswith joint multiuser scheduling and antenna diversity in Nakagamifading channels,”
IEEE Trans. Commun. , vol. 54, no. 3, pp. 469-478,Mar. 2006.[14] I. S. Gradshteyn, and I. M. Ryzhik,
Table of integrals, series, andproducts , 7th ed. San Diego, CA, USA: Academic, 2007.[15] K. P. Peppas, “Accurate closed-form approximations to generalised- K sum distributions and applications in the performance analysis of equal-gain combining receivers,” IET Commun. , vol.5, no.7, pp. 982-989, May2011.[16] L. Yang, F. Meng, Q. Wu, D. B. da Costa and M. Alouini, “Ac-curate closed-form approximations to channel distributions of RIS-aided wireless systems,”
IEEE Wireless Commun. Lett. , Early Access,DOI:10.1109/LWC.2020.3010512.[17] S. Atapattu, C. Tellambura, and H. Jiang, “A mixture gamma distributionto model the SNR of wireless channels,”
IEEE Trans. Wireless Commun. ,vol. 10, no. 12, pp. 4193-4203, Dec. 2011.[18] M. Abramowitz and I. A. Stegun,
Handbook of mathematical functions:With formulas, graphs, mathematical tables.
New York, USA: Dover,1965.[19] M. K. Simon and M.S. Alouini,
Digital communication over fadingchannels: A unified approach to performance analysis , 1st Edition. JohnWiley, 2000.[20] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamformingusing dumb antennas,”
IEEE Trans. Inform. Theo. , vol. 48, pp. 1277-1294, Jun. 2002.[21] V. M. Kapinas, S. K. Mihos, and G. K. Karagiannidis, “On themonotonicity of the meneralized marcum and nuttall Q-functions,”
IEEETrans. Inform. Theo. , vol. 55, no. 8, pp. 3701-3710, Aug. 2009.[22] M. K. Simon and M. S. Alouini, “Exponential-type bounds on thegeneralized marcum Q-function with application to error probabilityanalysis over fading channels,”