Outage Analysis in SWIPT Enabled Cooperative AF/DF Relay Assisted Two-Way Spectrum Sharing Communication
11 Outage Analysis in SWIPT EnabledCooperative AF/DF Relay Assisted Two-WaySpectrum Sharing Communication
Sutanu Ghosh,
Student Member, IEEE , Tamaghna Acharya,
Member, IEEE , Santi P. Maity,
Member,IEEE
Abstract
This paper reports relative performance of decode-and-forward (DF) and amplify-and-forward (AF) relayingin a multi-antenna cooperative cognitive radio network (CCRN) that supports device-to-device (D2D) communica-tions using spectrum sharing technique in cellular network. In this work, cellular system is considered as primaryand internet of things devices (IoDs), engaged in D2D communications, are considered to be secondary system.The devices access the licensed spectrum by means of the cooperation in two-way primary communications.Furthermore, IoDs are energized by harvesting the energy from radio frequency (RF) signals, using simultaneouswireless information and power transfer (SWIPT) protocol. Closed form expressions of outage probability forboth cellular and D2D communications are derived and the impact of various design parameters for both AFand DF relaying techniques are studied. Based on the simulation results, it is found that the proposed spectrumsharing protocol, for both DF relaying and AF relaying schemes, outperform another similar network architecturein terms of spectrum efficiency. It is also observed that the performance of the proposed system using DF relayingis better than AF relaying scheme in terms of energy efficiency at same transmit power.
Keywords :
Spectrum sharing, simultaneous wireless information and power transfer, two-way relay network,cooperative cognitive radio network, energy efficiency.I. I
NTRODUCTION
Internet-of-things (IoT) is now under extensive development phase to support the needs of short packet (fewinformation bytes) delivery (through frequency access or sharing) at ultra-high reliability (low outage) and lowlatency (through efficient routing). Device-to-Device (D2D) communication looks promising in IoT networksdue to enhanced battery life and service availability [1]. It also improves the proximity gain and pairing gainof radio spectrum without any involvement of cellular base station (BS), especially when the radio frequencyspectrum is shared between D2D and cellular users [2]. Congestion due to proliferation of wireless networksin unlicensed band urges the essence of sharing of the spectrum, originally licensed to cellular users, so as tosupport future IoT communication [3].Cooperative cognitive radio network (CCRN) refers to a network model that facilitates overlay mode ofcognitive radio enabled spectrum sharing [4] and could be applied in D2D communication in 5G heterogeneousnetworks (HetNets). Following this, the pair of IoT devices (IoDs), using D2D communication as unlicensedusers, may be viewed as secondary users (SUs), while cellular nodes (i.e. evolved Node-B (eNB) and userequipment (UE)) can be modelled as primary users (PUs). IoDs can access the licensed spectrum of cellularsystem while agreeing to relay the signals of the latter, thereby improving the reliability of their communicationover the fading wireless channel. In literature, two-way communications [5] based on CCRN is studied toachieve higher spectrum efficiency (SE) over the one-way relaying [6]. The outage performance of the two-wayCCRN is studied in [5] and the impact of different relaying schemes on spectrum efficient network operationis reported.Recently, radio frequency energy harvesting (RF-EH) based relaying using simultaneous wireless informationand power transfer (SWIPT) protocol has been under investigation to enhance network lifetime and reliabilityof wireless communication [7]-[9]. SU transmitter of CCRN may follow amplify-and-forward (AF) or, decode-and-forward (DF) relaying to support PU communication and send its own message to SU receiver using theenergy harvested from the received PU signal. Relay node harvests the energy from the PU signal by followingpower splitting (PS) or, time switching (TS) based SWIPT protocols [10]. Performances of unidirectional orbidirectional communication using AF and DF relay aided SWIPT network are analysed in [11]-[13], [15]. Thesystem outage performances of One-way DF relay assisted communication in CCRN is studied in [11] over aNakagami fading channel. In [12], PU and SU outage performances are studied in two-way CCRN using AFrelay assisted network. The authors of [12] also show the impact of energy conversion efficiency on the systemenergy efficiency (EE). The performance of similar network model is studied in [13] using DF relay. Basedon the study of [13], it is shown that DF relay-assisted two-way communication is significantly more spectrumefficient than one-way communication [11] using PS relaying (PSR) protocol. However, privacy and securityissues may be of concern as DF relay is required to decode PU message during the relaying process [14]. Onthe contrary, in AF relaying, PU signal does not need to be decoded at the relaying node, although, it is often
Sutanu Ghosh and Tamaghna Acharya are with the Department of Electronics and Telecommunication Engineering, Indian institute of Engi-neering Science and Technology, Shibpur, Howrah, West Bengal, 711103, India. E-mail: [email protected]; [email protected] P. Maity is with the Department of Information Technology, Indian institute of Engineering Science and Technology, Shibpur,Howrah, West Bengal, 711103, India. E-mail: [email protected] a r X i v : . [ c s . I T ] J u l TABLE IS
YMBOLS AND DEFINITIONS
Symbols Definitions X s Signal transmission between IoD (SU ) and IoD (SU ) P p , P p Transmission power of PU , PU h i Channel gain of link PU i → SU ( i ∈ , ) g i Channel gain of link PU i → SU ( i ∈ , ) D i Distance between PU i and SU ( i ∈ , ) D j Distance between PU i and SU (for i=1, j=4 and i=2,j=5 ) n su , n pu Received noise at SU, PU, respectively h Channel gain of link SU → SU ρ Power splitting factor of SU α Power sharing factor at SU L Distance between PU and PU D Distance between SU and SU m Path loss exponent η Energy conversion efficiency at SU R PU , R SU Target rate of PU and SU communication, respectively considered to be an energy inefficient approach as noise gets amplified by the relay node. The performanceimprovement of SU communication over [13] is studied in [15] using bidirectional SU communication. Needlessto mention that the presence of multiple antennas at the source and or the relay in any SWIPT enabled relayassisted communication over fading channel would not only improve the reliability of information transfer, butalso enhances in energy harvesting at the relay node. Motivated by this, a preliminary study on the performanceof the same system model [13] with multiple antenna PUs is presented in [16]. In this paper, a comparativestudy between DF and AF relaying scheme is presented using PS protocol in CCRN.
Scope and Contributions:
The work in [13] follows three-phase communication using DF relaying with a single-antenna in a CCRNframework. However, it does not explore the impact of multi-antenna on SE and EE aspects of the proposed D2Doperation in 5G HetNets. Two-phase communication is more preferable over three-phase to enhance the systemthroughput. This leads us to explore the problem of spectrum sharing for bidirectional cellular communicationand unidirectional D2D communication simultaneously. To offer improved SE, the present cellular system usemultiple antennas, the trend seen on 5G [17]. To make this study more general, non-identical number of antennasare considered at two ends of the PU system. The nodes engaged in D2D communication are equipped withnecessary hardware to harvest energy from the RF signals of the cellular users. EH meets the energy requirementof the transmitting node for D2D communication to relay (AF/DF) the signals of cellular users and also its ownmessage transmission over the cellular spectrum simultaneously. The main objective of this work is to highlightthe relative improvements in SE and EE performances of AF and DF relaying as the PU system moves fromsingle antenna to multi-antenna system. Our contributions can be summarized as follows. • A novel CCRN architecture with multi-antenna PU system is proposed where two-phase protocol supportstwo-way SWIPT enabled communications between the pair of cellular users (PUs) and also one-way D2D (SU-to-SU) communication. Two fold benefits, the first one is the improved SE due to multi-antenna and the otherone is the throughput improvement due to two-phase protocol are achieved. • Closed form expressions of both PU and SU outage probability are derived for both DF and AF relayassisted communication using PSR protocol for multi-antenna CCRN framework. Simulation results closelymatch the analytical expressions. • The exact dependence of the system performance on various parameters like power sharing factor, powersplitting factor, transmission power is also shown through the simulation results. DF relaying mostly performsbetter than AF relaying. As we have used two-phase spectrum sharing protocol, therefore both of the relayingmechanisms show almost equally spectrum efficiency compared to the similar network architecture [13]. How-ever, AF relaying is found to be better in terms of incremental improvement in SE with increase in the numberof antennas.The various symbols used are introduced in Table I. The remaining part of the paper is arranged as follows.The system model is described in Section II and communication protocol description is given in Section III. Theoutage performance of both PU and SU are analysed in Section IV. The necessary simulation and the numericalresults in terms of the various system parameters is presented in Section V. Finally, the paper is concluded inSection VI. II. S
YSTEM MODEL
A. Assumptions and Notations
The system model consists of a pair of eNB and UE in long term evolution (LTE) network architecture asdepicted in Fig. 1. The eNB and UE are equipped with multiple antennas N a and N b , respectively. In absence ofdirect communication link, the cellular nodes (eNB and UE) intend to exchange their information via a singleantenna IoT device (IoD) (denoted as IoD in Fig. 1.), aiming to achieve the target rate of R P U at each side.
Simultaneously, IoD sends its own message signal to IoD to meet a target rate of R SU . Both of them usesingle antenna for communication. Here eNB and UE are considered as PU and PU , respectively and IoD and IoD are modelled as SU and SU , respectively. Fig. 1.
System model
We assume that PU i , ( i ∈ , ) uses fixed power supply, i.e., P p i , SU is powered through harvested energyfrom PUs’ RF signals ,using SWIPT, for relaying PUs’ messages and transmitting to SU . We consider both DFand AF relaying mechanisms to support the two-way PU communications. h i = [ h i, , h i, , ..., h i,N p ] , (N p ∈ N a ,N b ) and g i = [ g i, , g i, , ..., g i,N p ] , (i ∈ i toSU and PU i to SU , respectively, where h i,n ∼ CN (0 , and g i,n ∼ CN (0 , . Channel between SU → SU is represented by h , where h ∼ CN (0 , . All the channel distribution between PU and SU links followindependent and identically distributed (i.i.d) Rayleigh fading. Due to short distance the link between SU andSU is considered as Nakagami-m fading. The instantaneous channel state information (CSI) is assumed tobe unavailable at PU i ( i ∈
1, 2). It is also considered that the full-diversity space-time codes (like GABBAcodes [18]) are used at the PU nodes. Normally, in space-time code, power is uniformly distributed amongthe transmitting antennas. The distances between the users PU − SU , PU − SU , PU − SU , PU − SU andSU − SU are given by D , D , D , D and D , respectively with ‘ m ’as the path-loss exponent. Here n su and n pu indicate the received noise at SU and PU, respectively. These noises are additive white Gaussian noise(AWGN) with zero (0) mean and the variance σ . B. Protocol description
The two-way communications between PU and PU via SU take place in two time phases viz., multipleaccess channel (MAC) and broadcast (BC) phase, as shown in Fig. 2. In the MAC phase, both PU and PU simultaneously transmit their information signals to SU . SU is able to harvest energy from part of the receivedsignal using PSR protocol. In the case of AF relaying, SU broadcasts an amplified PU signal superimposedwith the SU signal X s in the BC phase, using the total harvested energy. The PU receivers are able to receivethe signals using the maximal-ratio-combining (MRC) and separate the desired signal using the self-interferencecancellation (SIC). Now, decoding of SU ’s message at SU is performed in the two phases; first the strongerPU signal is decoded considering SU ’s desired signal as interference. Once the strong PU signal is separatedfrom the received signal in the the first phase, SU decodes the desired SU ’s signal in presence of channelnoise.In the case of DF relaying, SU first decodes the PUs’ message signals received in the MAC phase, and thenbroadcasts a network coded primary signal superimposed on the secondary signal X s in the BC phase. Both thePU nodes decode the network coded (XOR operation) PU signal in presence of SU interfering signal and noise.Thereafter, SU decodes SU signal by cancelling PU signal’s based on the PUs’ messages received previouslyin the MAC Phase. Fig. 2.
Transmission frame structure in the two-way PSR protocol
III. R
ATE AND O UTAGE A NALYSIS IN DF RELAYING A. Rate Analysis
Now the received power in MAC phase at SU from PU and PU can be expressed as follows [19]. P (1) SU = P p N a ( D ) m (cid:80) N a n =1 | h ,n | + P p N b ( D ) m (cid:80) N b p =1 | h ,p | (1)where P p and P p are the transmit powers of PU and PU , respectively.Similarly, the received power at SU from PU and PU can be expressed as P (1) SU = P p N a ( D ) m (cid:80) N a n =1 | g ,n | + P p N b ( D ) m (cid:80) N b p =1 | g ,p | (2)A part ρ (referred to as power splitting factor in the subsequent discussion) of the received signal power atSU is used for energy harvesting, and the rest (1- ρ ) portion is used for information processing. The harvestedenergy in MAC phase is given by E s = ηρ (cid:0) P p N a ( D ) m (cid:80) N a n =1 | h ,n | + P p N b ( D ) m (cid:80) N b p =1 | h ,p | (cid:1) T (3)where 0 < η < for information processing is given by P (1 ,IT ) SU = (1 − ρ ) (cid:2) P p N a ( D ) m (cid:80) N a n =1 | h ,n | + P p N b ( D ) m (cid:80) N b p =1 | h ,p | (cid:3) (4)Now following the linear EH model, the available power at SU for transmission in the BC phase is givenby P s = ηρ (cid:0) P p N a ( D ) m (cid:80) N a n =1 | h ,n | + P p N b ( D ) m (cid:80) N b p =1 | h ,p | (cid:1) (5)In the MAC phase, the successful decoding of the received signals from PU and PU is possible at SU , if R (1) SU ≥ R P U , R (2) SU ≥ R P U and R (cid:80) SU ≥ R P U [20], where Q = R (11) SU = T T log (cid:0) (1 − ρ ) P p N a ( D ) m σ (cid:80) N a n =1 | h ,n | (cid:1) ,R (12) SU = T T log (cid:0) (1 − ρ ) P p N b ( D ) m σ (cid:80) N b p =1 | h ,p | (cid:1) ,R (cid:80) SU = T T log (cid:0) (1 − ρ ) P p N a ( D ) m σ (cid:80) N a n =1 | h ,n | + (1 − ρ ) P p N b ( D ) m σ (cid:80) N b p =1 | h ,p | (cid:1) (6)Similarly, in the MAC phase, the successful decoding of the received signals from PU and PU is possibleat SU , if R (1) SU ≥ R P U , R (2) SU ≥ R P U and R (cid:80) SU ≥ R P U , where Q = R (11) SU = log (cid:0) P p N a ( D ) m σ (cid:80) N a n =1 | g ,n | (cid:1) ,R (12) SU = log (cid:0) P p N b ( D ) m σ (cid:80) N b p =1 | g ,p | (cid:1) ,R (cid:80) SU = log (cid:0) P p N a ( D ) m σ (cid:80) N a n =1 | g ,n | + P p N b ( D ) m σ (cid:80) N b p =1 | g ,p | (cid:1) (7)In DF relaying, SU uses α ( < α < ) fraction of its total transmit power P s to relay the network-codedprimary information and rest (1- α ) fraction of P s is used to send its own independent message X s to SU . In absence of CSI, PU and PU are assumed to use suitable space-time coding techniques, the details of which are not within thescope of our current study. After MRC and cancellation of self-interference terms by applying SIC technique, the received signal-to-interference-noise ratio (SINR) at PU i (i ∈ p ∈ N a , N b ) receivers in BC phase can be expressed as γ DFi = αPs ( Di ) m (cid:80) Npw =1 | h i,w | − α ) Ps ( Di ) m (cid:80) Npw =1 | h i,w | + σ (8)It is assumed that SU , like SU , succeeds in decoding the PUs’ signals received in MAC phase. Based onthese prior knowledges, SU is able to separate the desired SU signal from the PUs’ interference received inBC phase [13]. Therefore, the received signal at SU can be rewritten as Y (2 ,DF ) SU = (cid:115) (1 − α ) P s D m h X s (cid:124) (cid:123)(cid:122) (cid:125) Required signal + n su (cid:124)(cid:123)(cid:122)(cid:125) Noise (9)Now, the achievable rate at
P U i is given by (based on the SINR at P U i ) ( w ∈ n, p ) R (2 ,DF ) P U i = log { γ DFi } = log (cid:26) a (cid:48) i ( (cid:80) Nan =1 a | h ,n | + (cid:80) Nbp =1 b | h ,p | ) (cid:80) Npw =1 | h i,w | b (cid:48) i ( (cid:80) Nan =1 a | h ,n | + (cid:80) Nbp =1 b | h ,p | ) (cid:80) Npw =1 | h i,w | +1 (cid:27) (10)where a = ηρ P p N a ( D ) m σ , b = ηρ P p N b ( D ) m σ , a (cid:48) i = α ( D i ) m , b (cid:48) i = (1 − α )( D i ) m .Based on (9), the achievable rate at SU is expressed as R (2 ,DF ) SU = log (cid:8) (1 − α ) PsD m | h | σ (cid:9) = log (cid:8) c ( (cid:80) N a n =1 a | h ,n | + (cid:80) N b p =1 b | h ,p | ) | h | (cid:9) (11)where c = (1 − α ) D m . B. Outage Probability Analysis
An outage occurs when the achievable rate of data transmission on any transmission link falls below thetarget rate of data transmission.
1) Outage Probability Analysis of Primary System:
Based on the definition, PU outage probability usingDF relaying mechanism can be determined as follows [13]: P ( P U,DF ) out = 1 − Succes Probability (cid:122) (cid:125)(cid:124) (cid:123)(cid:20) P { Q } (cid:124) (cid:123)(cid:122) (cid:125) MAC phase × P (cid:8) min ( R (2 ,DF ) P U , R (2 ,DF ) P U ) ≥ R P U (cid:9)(cid:124) (cid:123)(cid:122) (cid:125)
BC phase (cid:21) = 1 − (cid:20) P { Q } × P (cid:8) R (2 ,DF ) P U ≥ R P U (cid:9) × P (cid:8) R (2 ,DF ) P U ≥ R P U (cid:9)(cid:21) (12)Applying (6), the success probability of data transmission between both of the PU nodes (PU and PU ) andSU is expressed by following three conditions satisfied together. Q = R (11) SU = log (cid:0) A X (cid:1) ≥ R P U ,R (12) SU = log (cid:0) A Y (cid:1) ≥ R P U ,R (cid:80) SU = log (cid:0) A X + A Y (cid:1) ≥ R P U (13)where A = (1 − ρ ) P p N a ( D ) m σ , A = (1 − ρ ) P p N b ( D ) m σ , , u = 2 (cid:0) R PU (cid:1) − , u = 2 (cid:0) R PU (cid:1) − , u = 2 (cid:0) R PU (cid:1) − . X = (cid:80) N a n =1 | h ,n | and Y = (cid:80) N b p =1 | h ,p | follow the same nature of gamma distribution with N p degree of freedom. The probability density function (PDF) can be expressed as f X ( x ) = x Np − exp (cid:0) − x /Np (cid:1) Γ( N p ) (cid:0) /N p (cid:1) Np . Basedon (13), P { Q } can be expressed as [21, Sec. 3.381.1, 3.381.3, 8.350.1, 8.352.1] P { Q } = ∞ (cid:82) u A x Na − exp (cid:0) − x /Na (cid:1) Γ( N a ) (cid:0) /N a (cid:1) Na (cid:40) ∞ (cid:82) u A y Nb − exp (cid:0) − y /Nb (cid:1) Γ( N b ) (cid:0) /N b (cid:1) Nb dy (cid:41) dx − u − u A (cid:82) u A x Na − exp (cid:0) − x /Na (cid:1) Γ( N a ) (cid:0) /N a (cid:1) Na (cid:40) u − A x A (cid:82) u A y Nb − exp (cid:0) − y /Nb (cid:1) Γ( N b ) (cid:0) /N b (cid:1) Nb dy (cid:41) dx = N a )Γ( N b ) Γ (cid:32) N a , N a u /A (cid:33) Γ (cid:32) N b , N b u /A (cid:33) − (cid:40) N a ) − γ (cid:16) N b ,N b u A (cid:17) Γ( N a )Γ( N b ) (cid:41)(cid:110) γ (cid:0) N a , N a u − u A (cid:1) − γ (cid:0) N a , N a u A (cid:1)(cid:111) + (cid:80) N b − q a =0 exp (cid:0) − Nbu A (cid:1) (cid:80) qaq =0 ( − q ( qaq ) u qa − q q a !Γ( N a )(1 /N a ) Na (cid:0) /N b (cid:1) qa A qa A q (cid:16) N a − A NbA (cid:17) ( Na + q ) (cid:110) γ (cid:2) N a + q, (cid:0) N a − A N b A (cid:1) u − u A (cid:3) − γ (cid:2) N a + q, (cid:0) N a − A N b A (cid:1) u A (cid:3)(cid:111) (14) Proof : See Appendix A.where Γ( ., . ) and γ ( ., . ) are the upper and the lower incomplete gamma function, respectively. Since SU is used as a relay to support PU communication, therefore the successful information transmission from both the PU nodesto SU is essential in MAC phase for relaying the information in BC phase. The probability of successful data transmission from SU to PU can be written as R (2 ,DF ) P U = log { γ DF } = log (cid:40) a (cid:48) ( (cid:80) Nan =1 a | h ,n | + (cid:80) Nbp =1 b | h ,p | ) (cid:80) Nan =1 | h ,n | b (cid:48) ( (cid:80) Nan =1 a | h ,n | + (cid:80) Nbp =1 b | h ,p | ) (cid:80) Nan =1 | h ,n | +1 (cid:41) (15) P (cid:40) R (2 ,DF ) PU ≥ R PU (cid:41) = P (cid:40) a (cid:48) ( a X + b Y ) X b (cid:48) ( a X + b Y ) X +1 ≥ u (cid:41) = − P (cid:40) X ≤ k (cid:48) ( a X + b Y ) (cid:41) , f or, u < α (1 − α ) − P (cid:40) X ≥ k (cid:48) ( a X + b Y ) (cid:41) = 0 , otherwise. (16)where k (cid:48) = u a (cid:48) − u b (cid:48) . Now, the solution to (16) is obtained as (17) [21, Sec. 8.350.1, 8.352.1]. P (cid:40) X ≤ k (cid:48) ( a X + b Y ) (cid:41) = P (cid:32) Y ≤ k (cid:48) b X − a X b (cid:33) = N a ) γ (cid:32) N a , (cid:114) k (cid:48) a /N a (cid:33) − (cid:34) (cid:80) N b − p a =0 N aNa N bpa p a !Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) k (cid:48) b (cid:19) p a − r (cid:18) a b (cid:19) r (cid:80) ∞ l =0 (cid:80) lt l =0 (cid:0) lt l (cid:1) ( − l l !(2 r + N a − p a − l +2 t l ) (cid:16) k (cid:48) N b b (cid:17) l − t l (cid:32) N a − a N b b (cid:33) t l (cid:32)(cid:113) k (cid:48) a (cid:33) r + N a − p a − l +2 t l (cid:35) (17) Proof : See Appendix B.Similarly, the probability of successful data transmission from SU to PU can be written as P (cid:40) Y ≤ k (cid:48)(cid:48) ( a X + b Y ) (cid:41) = P (cid:32) X ≤ k (cid:48)(cid:48) a Y − b Y a (cid:33) = N b ) γ (cid:32) N b , (cid:114) k (cid:48)(cid:48) b /N b (cid:33) − (cid:34) (cid:80) N a − p a =0 N bNb N apa p a !Γ( N b ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) k (cid:48)(cid:48) a (cid:19) p a − r (cid:18) b a (cid:19) r (cid:80) ∞ l =0 (cid:80) lt l =0 (cid:0) lt l (cid:1) ( − l l !(2 r + N b − p a − l +2 t l ) (cid:16) k (cid:48)(cid:48) N a a (cid:17) l − t l (cid:32) N b − b N a a (cid:33) t l (cid:32)(cid:113) k (cid:48)(cid:48) b (cid:33) r + N b − p a − l +2 t l (cid:35) (18)where k (cid:48)(cid:48) = u a (cid:48) − u b (cid:48) .The closed form solution to PU outage probability can be determined using (14), (17)-(18).Closed form expression of PU outage probability using DF relaying is given in Appendix F.
2) Outage Probability Analysis of Secondary System:
SU outage probability using DF relaying is shown as follows: P ( SU,DF ) out = 1 − Success Probability (cid:122) (cid:125)(cid:124) (cid:123)(cid:20) P { Q } × P { Q } (cid:124) (cid:123)(cid:122) (cid:125) MAC Phase × P (cid:8) R (2 ,DF ) SU ≥ R SU (cid:9)(cid:124) (cid:123)(cid:122) (cid:125) BC Phase (cid:21) (19)Similar to (13), the probability of success due to data transmission from PU i ( i ∈ , ) to SU is defined by the followingthree conditions together. Q = R (11) SU = log (cid:0) B X (cid:1) ≥ R PU ,R (12) SU = log (cid:0) B Y (cid:1) ≥ R PU ,R (cid:80) SU = log (cid:0) B X + B Y (cid:1) ≥ R PU (20)where B = P p N a ( D ) m σ , B = P p N b ( D ) m σ , . X = (cid:80) N a n =1 | g ,n | and Y = (cid:80) N b p =1 | g ,p | follow the same nature of gamma distribution like X and Y ,respectively.Based on (20), P { Q } can be expressed as follows P { Q } = N a )Γ( N b ) Γ (cid:32) N a , N a u /B (cid:33) Γ (cid:32) N b , N b u /B (cid:33) − (cid:40) N a ) − γ (cid:16) N b ,N b u B (cid:17) Γ( N a )Γ( N b ) (cid:41)(cid:110) γ (cid:0) N a , N a u − u B (cid:1) − γ (cid:0) N a , N a u B (cid:1)(cid:111) + (cid:80) N b − q a =0 exp (cid:0) − Nbu B (cid:1) (cid:80) qaq =0 ( − q ( qaq ) u qa − q q a !Γ( N a ) (cid:0) /N a (cid:1) Na (cid:0) /N b (cid:1) qa B qa B q (cid:16) N a − B NbB (cid:17) ( Na + q ) (cid:110) γ (cid:2) N a + q, (cid:0) N a − B N b B (cid:1) u − u B (cid:3) − γ (cid:2) N a + q, (cid:0) N a − B N b B (cid:1) u B (cid:3)(cid:111) (21)Following (11), the probability of successful data transmission between the link SU to SU is expressed as P ( R (2 ,DF ) SU ≥ R SU ) = P (cid:16) Z ≥ u c ( a X + b Y ) (cid:17) (22)where u = 2 (2 R SU ) − . Z = | h | and it follows the gamma distribution with Nakagami shaping parameter m k . ThePDF of Z can be described as f Z ( z ) = z mk − exp (cid:0) − z /mk (cid:1) Γ( m k ) (cid:0) /m k (cid:1) mk . Since SU needs to decode the message of SU by removing PUs message in BC phase, therefore successful information transmissionfrom both PUs to SU is necessary in MAC phase. Now (22) can be evaluated as follows [21, Sec. 3.471.9, 8.352.2]: P ( R (2 ,DF ) SU ≥ R SU ) = (cid:80) N b − p a =0 (cid:16) Nbb (cid:17) pa p a !(1 /N a ) Na Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) u c (cid:19) p a − r (cid:0) a (cid:1) r (cid:18) N a − a Nbb (cid:19) Na + r Γ( N a + r ) { (1 /m k ) mk } Γ( m k ) (cid:34) (cid:26) u N b cb m k (cid:27) ( m k − p a + r ) / K m k − p a + r (cid:26) (cid:113) u N b m k cb (cid:27) − (cid:80) N a + r − j =0 1 j ! (cid:20) u (cid:18) N a − a Nbb (cid:19) ca (cid:21) j (cid:26) u (cid:18) N a − a Nbb (cid:19) ca m k + u N b cb m k (cid:27) ( mk − pa + r − j )2 K m k − p a + r − j (cid:26) (cid:118)(cid:117)(cid:117)(cid:116) u m k (cid:18) N a − a Nbb (cid:19) ca + u N b m k cb (cid:27)(cid:35) + (cid:34) (cid:80) N a − p a =0 (cid:18) u Naca (cid:19) pa ( m k ) mk p a !Γ( m k ) (cid:32) u N a ca m k (cid:33) ( m k − p a ) / K m k − p a (cid:18) (cid:113) u N a m k ca (cid:19)(cid:35) (23) Proof : See Appendix C.Finally, the closed form SU outage expression using DF relaying can be determined using (14), (21) and (23).Closed form expression of SU outage probability using DF relaying is given in Appendix G.
IV. R
ATE & O
UTAGE A NALYSIS OF AF RELAYING A. Rate Analysis
In the BC phase of AF relaying, SU broadcasts an amplified version of the signal generated after combining the PUsignals, received in the MAC phase. To include this in the analysis, harvested power P s in (5) is normalized using the factor ξ , expressed as ξ = (cid:118)(cid:117)(cid:117)(cid:116) − ρ ) (cid:20) Pp Na ( D m (cid:80) Nan =1 | h ,n | + Pp Nb ( D m (cid:80) Nbp =1 | h ,p | (cid:21) + σ ≈ (cid:118)(cid:117)(cid:117)(cid:116) − ρ ) (cid:20) Pp Na ( D m (cid:80) Nan =1 | h ,n | + Pp Nb ( D m (cid:80) Nbp =1 | h ,p | (cid:21) (24)Based on the superposition coding and AF relaying principle SU uses α ( < α < ) the fraction of its total transmitpower P s to relay the combined signal of primary information and the rest (1- α ) portion is used to send its own independentmessage X s to SU . As both PUs know their individual transmitted signals, consequently, they are able to cancel theirself-interference terms. Therefore, the instantaneous end-to-end SINR at PU i can be expressed using (25). γ AFi = ξ αPs (1 − ρ ) PpjN (cid:48) p ( DiDj ) m (cid:80) Npw =1 (cid:80) N (cid:48) pw (cid:48) =1 | h i,w | | h j,w (cid:48) | − α ) Ps (cid:80) Npw =1 | hi,w | Di ) m + ξ αPs (cid:80) Npw =1 | hi,w | σ Di ) m + σ = C i (cid:80) Npw =1 (cid:80) N (cid:48) pw (cid:48) =1 | h i,w | | h j,w (cid:48) | H i (cid:80) Npw =1 | h i,w | + E i (cid:80) Npw =1 (cid:80) N (cid:48) pw (cid:48) =1 | h i,w | | h j,w (cid:48) | + F i (cid:80) Npw =1 | h i,w | +1 (25)where C = ηρP p αN b σ ( D ) m ( D ) m , C = ηρP p αN a σ ( D ) m ( D ) m , H = (1 − α ) ηρP p N a σ ( D ) m , H = (1 − α ) ηρP p N b σ ( D ) m , E = (1 − α ) ηρP p N b σ ( D D ) m , E = (1 − α ) ηρP p N a σ ( D D ) m , F = ραη (1 − ρ )( D ) m , F = ραη (1 − ρ )( D ) m .Similarly, SU is able to detect PU signal by considering SU signal as an interference [22]. The instantaneous SINR canbe expressed applying (26). γ AFs,PU = αηρ ( D mσ | h | (cid:18) (cid:80) Nan =1 Pp D mNa | h ,n | + (cid:80) Nbp =1 Pp D mNb | h ,p | (cid:19) (1 − α )( D mσ ηρ | h | (cid:18) (cid:80) Nan =1 Pp D mNa | h ,n | + (cid:80) Nbp =1 Pp D mNb | h ,p | (cid:19) + αηρ | h | − ρ )( D m +1 = | h | (cid:0) (cid:80) Nan =1 U | h ,n | + (cid:80) Nbp =1 V | h ,p | (cid:1) | h | (cid:0) (cid:80) Nan =1 U | h ,n | + (cid:80) Nbp =1 V | h ,p | (cid:1) + U | h | +1 (26)where U = (1 − α )( D ) m σ ηρ P p N a ( D ) m , V = (1 − α )( D ) m σ ηρ P p N b ( D ) m , U = α ( D ) m (1 − ρ ) ηρ , U = αηρP p ( D D ) m σ N a , V = αηρP p ( D D ) m σ N b .The PU signals being a strong one, SU first decodes PU signals considering SU signal as interference and removes itfrom the received signal. Then it decodes its own signal in the presence of noise. The instantaneous SNR at SU can beexpressed in (27). γ AFs = (1 − α )( D mσ ηρ | h | (cid:18) (cid:80) Nan =1 Pp D mNa | h ,n | + (cid:80) Nbp =1 Pp D mNb | h ,p | (cid:19) αηρ | h | − ρ )( D m +1 = | h | (cid:0) (cid:80) Nan =1 U | h ,n | + (cid:80) Nbp =1 V | h ,p | (cid:1) U | h | +1 (27)Achievable rate at P U i is given by (based on the SINR) R (2 ,AF ) PU i = log { γ AFi } (28)The achievable rate for PU information decoding at SU is as R (2 ,AF ) s,PU = log { γ AFs,PU } (29)The achievable rate at SU is given by (based on the SNR at SU ) R (2 ,AF ) SU = log { γ AFs } (30) As noise power is negligible with respect to the power received for information processing, therefore noise power is neglected. B. Outage Probability Analysis
1) Outage Probability of Primary System:
PU outage probability using AF relaying is determined as follows: P ( PU,AF ) out = 1 − Success Probability (cid:122) (cid:125)(cid:124) (cid:123)(cid:20) P (cid:8) R (2 ,AF ) PU ≥ R PU (cid:9) × P (cid:8) R (2 ,AF ) PU ≥ R PU (cid:9)(cid:21) (31)Following (25), the probability of success due to data transmission between the link SU to PU is expressed as [21,Sec. 3.471.9, 8.352.1] P { R (2 ,AF ) PU ≥ R PU } = P (cid:20) C (cid:80) Nan =1 | h ,n | (cid:80) Nbp =1 | h ,p | H (cid:80) Nan =1 | h ,n | + E (cid:80) Nan =1 | h ,n | (cid:80) Nbp =1 | h ,p | + F (cid:80) Nan =1 | h ,n | +1 ≥ u (cid:21) = P (cid:20) C XYH X + E XY + F X +1 ≥ u (cid:21) = 1 − P (cid:20) Y < F u ( C − E u ) + H u X ( C − E u ) + u ( C − E u ) X (cid:21) = (cid:80) N b − q a =0 ( N b u ) q a (cid:32) N b u / ( C − E u ) N a + NbH u C − E u (cid:33) ( N a + q − l ) / (cid:0) − NbF u C − E u (cid:1) (cid:80) qaq =0 ( qaq ) (cid:80) ql =0 ( ql ) F qa − q H q − l q a !Γ( N a ) (cid:0) /N a (cid:1) Na ( C − E u ) qa K N a + q − l (cid:18) (cid:113) N b u ( C − E u ) (cid:0) N a + N b H u C − E u (cid:1)(cid:19) (32) Proof : See Appendix D.where X = (cid:80) N a n =1 | h ,n | , Y = (cid:80) N b p =1 | h ,p | . The symbol K v ( . ) indicates the modified Bessel function. Similarly, it isalso possible to determine the probability of successful data transmission between the link SU to PU .Closed form expression of PU outage probability using AF relaying is given in Appendix H.
2) Outage Probability Analysis of Secondary System:
SU outage probability, using AF relaying mechanism, canbe determined as follows: P ( SU,AF ) out = 1 − (cid:20) P (cid:8) R (2 ,AF ) SU ≥ R SU | R (2 ,AF ) s,PU ≥ R PU (cid:9)(cid:124) (cid:123)(cid:122) (cid:125) Success of SU → SU × P (cid:8) R (2 ,AF ) s,PU ≥ R PU (cid:9)(cid:124) (cid:123)(cid:122) (cid:125) Successful decoding of PU signals at SU (cid:21) (33)Following (27), the probability of successful data transmission between the link SU to SU is expressed using (34) [21,Sec. 3.471.9, 8.352.2]. P ( R (2 ,AF ) SU ≥ R SU ) = P (cid:40) | h | (cid:32) U (cid:80) Nan =1 | h ,n | + V (cid:80) Nbp =1 | h ,p | (cid:33) U | h | +1 ≥ u (cid:41) = (cid:80) N b − p a =0 (cid:80) p a r =0 (cid:0) p a r (cid:1) (cid:80) rl =0 ( − l (cid:0) rl (cid:1)(cid:16) u U N b V (cid:17) p a − r (cid:18) u NbV (cid:19) r − l (cid:18) U NbV (cid:19) l Γ( N a + l ) p a ! (cid:8) (1 /N a ) Na (cid:9)(cid:18) N a − U NbV (cid:19) Na + l Γ( N a ) exp (cid:16) − u U N b V (cid:17)(cid:34) (cid:0) u NbV mk (cid:1) mk + l − r K mk + l − r (cid:18) (cid:114) u NbmkV (cid:19) { (1 /m k ) mk } Γ( m k ) − (cid:80) N a + l − j =0 exp (cid:26) − (cid:18) u U U (cid:19)(cid:18) N a − U N b V (cid:19)(cid:27)(cid:18) N a − U NbV (cid:19) j (cid:80) jtl =0 ( jtl ) (cid:18) u U U (cid:19) j − tl (cid:18) u U (cid:19) tl j ! { (1 /m k ) mk } Γ( m k ) (cid:18) u N a U m k (cid:19) mk + l − r − tl K m k + l − r − t l (cid:18) (cid:113) u N a m k U (cid:19)(cid:35) + (cid:80) N a − p a =0 (cid:18) u NaU (cid:19) pa ( m k ) mk exp (cid:16) − u U NaU (cid:17) p a !Γ( m k ) (cid:80) p a r =0 (cid:0) p a r (cid:1) ( U ) p a − r (cid:32) u N a U m k (cid:33) ( m k − r ) / K m k − r (cid:18) (cid:113) u N a m k U (cid:19) (34) Proof : See Appendix E.Similar to (34), the probability of success for the decoding of the PU information in presence of SU interference andnoise can be written as P (cid:8) R (2 ,AF ) s,PU ≥ R PU (cid:9) = P (cid:26) Z (cid:0) U X + V Y (cid:1) Z (cid:0) U X + V Y (cid:1) + U Z +1 ≥ u (cid:27) = (cid:40) P (cid:8) Z (cid:0) S X + S Y (cid:1) ≥ u s U z + u s } , f or, u s < α (1 − α ) , otherwise. = (cid:80) N b − p a =0 (cid:80) p a r =0 (cid:0) p a r (cid:1) (cid:80) rl =0 ( − l (cid:0) rl (cid:1)(cid:16) u s U N b S (cid:17) p a − r exp (cid:16) − u s U N b S (cid:17) (cid:18) usNbS (cid:19) r − l (cid:18) S NbS (cid:19) l Γ( N a + l ) p a ! (cid:8) (1 /N a ) Na (cid:9)(cid:18) N a − S NbS (cid:19) Na + l Γ( N a ) (cid:34) (cid:0) usNbS mk (cid:1) mk + l − r { (1 /m k ) mk } Γ( m k ) K m k + l − r (cid:18) (cid:113) u s N b m k S (cid:19) − (cid:80) N a + l − j =0 exp (cid:26) − (cid:18) u s U S (cid:19)(cid:18) N a − S N b S (cid:19)(cid:27) (cid:16) u s N a S m k (cid:17) mk + l − r − tl (cid:18) N a − S NbS (cid:19) j (cid:80) jtl =0 ( jtl ) (cid:18) usU S (cid:19) j − tl (cid:18) usS (cid:19) tl j ! { (1 /m k ) mk } Γ( m k ) K m k + l − r − t l (cid:18) (cid:113) u s N a m k S (cid:19)(cid:35) + (cid:80) N a − p a =0 (cid:18) usNaS (cid:19) pa ( m k ) mk exp (cid:16) − usU NaS (cid:17) p a !Γ( m k ) (cid:80) p a r =0 (cid:0) p a r (cid:1) ( U ) p a − r (cid:32) u s N a S m k (cid:33) ( m k − r ) / K m k − r (cid:18) (cid:113) u s N a m k S (cid:19) , f or, u s < α (1 − α ) , otherwise. (35) Fig. 3. Outage Probability vs. α : (a) DF relaying (b) AF relaying where S = D ) m σ ηρ P p N a ( D ) m , S = D ) m σ ηρ P p N a ( D ) m , u s = u α − u (1 − α ) .Finally, the closed form SU outage expression using AF relaying can be determined using (34) and (35).Closed form expression of SU outage probability using AF relaying is given in Appendix M. V. N
UMERICAL R ESULTS AND D ISCUSSIONS
Numerical values for the system parameters used are enlisted in Table II. The constraint of the maximum PU outagelimit (say 0.1) is found to be met at α =0.81 by both AF and DF relaying for N a =2. This value of α is used for the resultsshown in Fig. 4, Fig. 5 and Fig. 6. Considering the practicability of the number of transmitting antennas on mobile deviceswe set N b =1 in our simulation results.The outage performances of both PU and SU systems are shown as variation in power-sharing factor ( α ) for both DFand AF relaying in Fig. 3.a and Fig. 3.b. It is clearly seen that the analytical results match perfectly with the simulationresults. For the small value of α , the higher fraction of the harvesting power is allocated to SU information transmissionand small fraction of power is used for DF relaying of PU information. Thus SU outage is found to be low while PUoutage to be high. The reverse situation is to observed at high value of α . The small value of α (close to 0) causes PUoutage of AF relaying same to that of DF relaying and SU performance is found to be the worst by following (33)-(35).Since 50 % or more power is assigned to PU information transmission to detect PU signals perfectly in presence of SUinterference, therefore the minimum value of α is shown as 0.5 for AF relaying in Fig. 3.b. The threshold limit of 10 % PUoutage is achieved at α =0.29 and N a =1 for DF relaying and the same outage limit is achieved at α =0.89 and N a =1 for AFrelaying. If the number of antennas is increased then PU performance is improved accordingly. If N a is increased from 1to 2, about 46 % improvement (reduction) in PU outage is observed for DF relaying and about 39 % improvement of PUoutage is observed for AF relaying. If N a is increased from 1 to 2, about 58 % improvement of SU outage is observed at α =0.27 and R SU =1 bps for DF relaying and about 42 % improvement of SU outage is observed at α =0.81 and R SU =1 bpsfor AF relaying.Fig. 4. and 5. show the outage performances of PU and SU, respectively with respect to power splitting factor ( ρ ) forboth DF and AF relaying mechanisms. As depicted in the figures, the outage performances of both PU and SU are verypoor for ρ → and ρ → . Initially, the performances of PU and SU outage are improved with the increase in the value of ρ , and they attain their minimum values of outage at the optimal value of ρ ∗ . Thereafter, when the value of ρ is increasedfurther, it leads to an increase in both PU and SU outage. The reason behind the characteristics of this graphical plot canbe explained as follows. Initially, when ρ value is very low, the energy harvested at SU is insufficient to broadcast theinformation at the BC phase and effectively the outage performance of both PU and SU are very poor. When the valueof ρ increases, the harvesting energy at SU is also increased and consequently both PU and SU outage performancesimprove accordingly. Further degradation on the outage performances are found due to major power allocation for EH andless power allocation for decoding the information received from PU to SU signal transmission. Performance of PU outageis significantly improved for DF relaying as compared to AF relaying mechanism whereas the SU outage performances arealmost same for both the relaying mechanisms with respect to ρ . The overall outage performances for both PU and SU areimproved with the increase in the number of antennas used at PU nodes.Now SE and EE of the system can be defined as [13] η SE = 2 × (1 − P PUout ) × R PU × T T + (1 − P SUout ) × R SU × T T (36)EE can be defined as the ratio of SE to the total energy consumed for PUs transmission [23]. TABLE IIS
ET OF NUMERICAL VALUES OF NECESSARY PARAMETERS
Name ValuePU target rate of information transmission ( R PU ) 0.2 bps/HzSU target rate of information transmission ( R SU ) 1 bps/HzDistance between PU and PU (L) 20 mDistance between SU and SU (D ) 10 mPU transmit power ( P p = P p = P p ) -23 dBm (Fig. 3 to Fig. 5), -50 dBm to -15 dBm (Fig. 6)Average noise power ( n pu = n su = n p ) -100 dBm [11]Path loss exponent ( m ) 2.7Energy conversion efficiency ( η ) 0.9Power splitting factor ( ρ ) 0.9Power sharing factor ( α ) 0.81 D = D & D = D L/ Fig. 4. PU Outage Probability vs. ρ : (a) DF relaying, (b) AF relayingFig. 5. SU Outage Probability vs. ρ : (a) DF relaying, (b) AF relayingFig. 6. System SE and EE vs. total PU transmit power : (a) DF relaying (b) AF relaying The SE and EE performances of the DF and AF relaying protocols are shown with respect to the total PU transmitpower in Fig. 6.a and Fig. 6.b, respectively. The performance of the proposed system using AF and DF relaying schemesare compared with similar system supporting two-way PU and one-way SU communication simultaneously [13]. As shownin figure, SE is improved and EE is deteriorated with the increase in PU transmit power. The characteristics of this graphicalplot may be explained as follows. As transmission power of PUs are increased, the PU and SU outage probabilities aredecreased accordingly following (14)-(18), (21), (23), (32), (34) and (35) for DF and AF relaying mechanisms, respectively.As inverse relationship is maintained between SE and the outage probability of both PU and SU system, the SE is improvedwith the increase in the transmission power of PU. Less increment of SE as compared to more energy consumption leads toa degradation of EE with the increasing value of PUs transmit power. Our proposed system using DF relaying mechanism ismore efficient as compared to AF relaying mechanism and similar model of two-way PU and one-way SU communication.In terms of SE, our proposed system using DF relaying mechanism is performing 224 % , 196 % and 142 % better as comparedto [13] for N a =3, N a =2 and N a =1, respectively at 52 dB transmit power. Our system using AF relaying is performing 34 % ,31 % and 22 % better in terms of SE as compared to [13] for given value of N a =3, N a =2 and N a =1, respectively at 70dB transmit power. In terms of EE, our proposed system using DF relaying mechanism is performing 62 % , 48 % and 21 % better as compared to [13] and on the other hand, the performance of [13] is 56 % , 60 % and 69 % better as compared tothe proposed system using AF relaying for given value of N a =3, N a =2 and N a =1, respectively at 52 dB transmit power.However, the proposed model is less energy efficient at high transmit power as compared to [13]. VI. C
ONCLUSIONS
This paper has investigated the scope of SWIPT enabled IoT communication, on the licensed spectrum using overlayspectrum sharing mode of cognitive radio networks. Transmitting IoD node harvests energy from the information bearingRF signals of both eNB and UE transmission and this energy is used in relaying between eNB and UE. Apart from theclose match between the analytical and simulation results on outage experiences in IoD and cellular systems, it is also tobe noted that DF relaying mechanism is more efficient over AF relaying in terms of EE and AF relaying is more sensitiveto the impact of the number of antennas used by the cellular system. The proposed network architecture and the spectrumsharing model can be extended further (i) to analyse the outage for more realistic non-linear RF-EH model, (ii) to improvethe secrecy of the relay assisted cellular communication using the PUs’ signals for friendly jamming in addition to EH and(iii) to study game theoretic modelling of the possible negotiations between cellular users and multiple IoT device pairs forefficient trading of the available radio resources of the former.
VII. I
MPORTANT I NTEGRALS AND THEIR SOLUTIONS
VII.1: [21, Sec. 3.471.9] ∞ (cid:90) x v − exp (cid:18) − βx − γx (cid:19) dx = 2 (cid:32) βγ (cid:33) v K v (2 (cid:112) βγ ) , (cid:2) Re ( β ) > , Re ( γ ) > (cid:3) (37) VII.2: [21, Sec. 3.351.3] ∞ (cid:90) x N a exp ( − µx ) dx = N a ! µ − N a − , (cid:2) Re ( µ ) > (cid:3) (38) VII.3: [21, Sec. 3.381.1] u (cid:90) x v − exp ( − µx )Γ( µ ) dx = µ − v γ ( v, µu )Γ( µ ) , (cid:2) Re ( v ) > (cid:3) (39) VII.4: [21, Sec. 3.381.3] ∞ (cid:90) u x v − exp ( − µx )Γ( µ ) dx = µ − v Γ( v, µu )Γ( µ ) , (cid:2) u > , Re ( µ ) > (cid:3) (40) VII.5: [21, Sec. 8.352.1] γ (1 + N a , x ) = N a ! (cid:34) − exp( − x ) (cid:32) N a (cid:88) r =0 x r r ! (cid:33)(cid:35) , (cid:20) N a = 0 , , ... (cid:21) (41) VII.6: [21, Sec. 8.352.2]
Γ(1 + N a , x ) = N a ! exp( − x ) (cid:32) N a (cid:88) r =0 x r r ! (cid:33) , (cid:20) N a = 0 , , ... (cid:21) (42) VII.7: [Taylor Series Expansion] exp( − x ) = ∞ (cid:88) n =0 ( − n x n n ! (43) VII.8: [Taylor Series Expansion] exp( x ) = ∞ (cid:88) n =0 x n n ! (44) VII.9: [Binomial Series Expansion] ( x + a ) n = n (cid:88) r =0 (cid:32) nr (cid:33) x n − r a r (45) The transmit power is normalized with respect to average channel noise power.
1) Appendix A: Proof of (14): [21, Sec. 3.381.1, 3.381.3, 8.350.1, 8.352.1]. P { Q } = ∞ (cid:82) u A x Na − exp (cid:0) − x /N a (cid:1) Γ( N a ) (cid:0) /N a (cid:1) Na (cid:40) ∞ (cid:82) u A y Nb − exp (cid:0) − y /N b (cid:1) Γ( N b ) (cid:0) /N b (cid:1) Nb dy (cid:41) dx − u − u A (cid:82) u A x Na − exp (cid:0) − x /N a (cid:1) Γ( N a ) (cid:0) /N a (cid:1) Na (cid:40) u − A x A (cid:82) u A y Nb − exp (cid:0) − y /N b (cid:1) Γ( N b ) (cid:0) /N b (cid:1) Nb dy (cid:41) dx = ∞ (cid:82) u A x Na − exp (cid:0) − x /N a (cid:1) Γ( N a ) (cid:0) /N a (cid:1) Na (cid:40) ∞ (cid:82) u A y Nb − exp (cid:0) − y /N b (cid:1) Γ( N b ) (cid:0) /N b (cid:1) Nb dy (cid:41) dx − u − u A (cid:82) u A x Na − exp (cid:0) − x /N a (cid:1) Γ( N a ) (cid:0) /N a (cid:1) Na (cid:40) u − A x A (cid:82) y Nb − exp (cid:0) − y /N b (cid:1) Γ( N b ) (cid:0) /N b (cid:1) Nb dy (cid:41) dx + u − u A (cid:82) u A x Na − exp (cid:0) − x /N a (cid:1) Γ( N a ) (cid:0) /N a (cid:1) Na (cid:40) u A (cid:82) y Nb − exp (cid:0) − y /N b (cid:1) Γ( N b ) (cid:0) /N b (cid:1) Nb dy (cid:41) dx = 1Γ( N a )Γ( N b )Γ (cid:32) N a , N a u /A (cid:33) Γ (cid:32) N b , N b u /A (cid:33) − u − u A (cid:82) u A x Na − exp (cid:0) − x /N a (cid:1) Γ( N a )Γ( N b ) (cid:0) /N a (cid:1) Na γ (cid:32) N b , N b u − A x A (cid:33) dx + u − u A (cid:82) u A x Na − exp (cid:0) − x /N a (cid:1) Γ( N a )Γ( N b ) (cid:0) /N a (cid:1) Na γ (cid:32) N b , N b u A (cid:33) dx = 1Γ( N a )Γ( N b )Γ (cid:32) N a , N a u /A (cid:33) Γ (cid:32) N b , N b u /A (cid:33) − N a ) (cid:0) /N a (cid:1) N a u − u A (cid:82) u A x N a − exp (cid:0) − x /N a (cid:1)(cid:34) − exp (cid:18) − N b u − A x A (cid:19) q a ! (cid:80) N b − q a =0 N bq a (cid:0) u − A x A (cid:1) q a (cid:35) dx + u − u A (cid:82) u A x Na − exp (cid:0) − x /N a (cid:1) Γ( N a )Γ( N b ) (cid:0) /N a (cid:1) Na γ (cid:32) N b , N b u A (cid:33) dx (Using Binomial Series Expansion, we can write as follows) = 1Γ( N a )Γ( N b )Γ (cid:32) N a , N a u /A (cid:33) Γ (cid:32) N b , N b u /A (cid:33) − N a ) (cid:0) /N a (cid:1) N a (cid:40) u − u A (cid:82) x N a − exp (cid:0) − x /N a (cid:1) dx − u A (cid:82) x N a − exp (cid:0) − x /N a (cid:1) dx (cid:41) + (cid:80) N b − q a =0 exp (cid:0) − Nbu A (cid:1) (cid:80) qaq =0 ( − q ( qaq ) u qa − q A q q a !Γ( N a ) (cid:0) /N a (cid:1) Na (cid:0) /N b (cid:1) qa A qa (cid:40) u − u A (cid:82) x N a + q − exp (cid:8) − (cid:0) N a − A N b A (cid:1) x (cid:9) dx − u A (cid:82) x N a + q − exp (cid:8) − (cid:0) N a − A N b A (cid:1) x (cid:9) dx (cid:41) + γ (cid:16) N b , N b u A (cid:17) Γ( N a )Γ( N b ) (cid:0) /N a (cid:1) N a (cid:40) u − u A (cid:82) x N a − exp (cid:0) − x /N a (cid:1) dx − u A (cid:82) x N a − exp (cid:0) − x /N a (cid:1) dx (cid:41) = 1Γ( N a )Γ( N b ) Γ (cid:32) N a , N a u /A (cid:33) Γ (cid:32) N b , N b u /A (cid:33) − (cid:40) N a ) − γ (cid:16) N b , N b u A (cid:17) Γ( N a )Γ( N b ) (cid:41)(cid:110) γ (cid:0) N a , N a u − u A (cid:1) − γ (cid:0) N a , N a u A (cid:1)(cid:111) + (cid:80) N b − q a =0 exp (cid:0) − N b u A (cid:1) (cid:80) q a q =0 ( − q (cid:0) q a q (cid:1) u q a − q A q q a !Γ( N a ) (cid:0) /N a (cid:1) N a (cid:0) /N b (cid:1) q a (cid:16) N a − A N b A (cid:17) ( N a + q ) A q a (cid:110) γ (cid:2) N a + q, (cid:0) N a − A N b A (cid:1) u − u A (cid:3) − γ (cid:2) N a + q, (cid:0) N a − A N b A (cid:1) u A (cid:3)(cid:111) (46)
2) Appendix B: Proof of (17): [21, Sec. 8.350.1, 8.352.1]. P (cid:40) X ≤ k (cid:48) ( a X + b Y ) (cid:41) = P (cid:32) Y ≤ k (cid:48) b X − a X b (cid:33) = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) k (cid:48) a (cid:82) x Na − exp (cid:0) − x /N a (cid:1) Γ( N a ) (cid:0) /N a (cid:1) Na (cid:40) k (cid:48) b x − a x b (cid:82) y Nb − exp (cid:0) − y /N b (cid:1) Γ( N b ) (cid:0) /N b (cid:1) Nb dy (cid:41) dx = 1Γ( N b ) (cid:114) k (cid:48) a (cid:82) { x Na − } exp (cid:0) − x /Na (cid:1) { (1 /N a ) Na } Γ( N a ) γ (cid:32) N b , N b (cid:104) k (cid:48) b x − a x b (cid:105)(cid:33) dx = 1Γ( N b ) (cid:114) k (cid:48) a (cid:82) { x Na − } exp (cid:0) − x /Na (cid:1) { (1 /N a ) Na } Γ( N a ) Γ( N b ) (cid:34) − exp (cid:110) − N b (cid:16) k (cid:48) b x − a x b (cid:17)(cid:111) (cid:80) N b − p a =0 (cid:26) N b (cid:16) k (cid:48) b x − a x b (cid:17)(cid:27) p a p a ! (cid:35) dx Using Binomial Series Expansion, we can write as follows , = 1Γ( N a ) γ (cid:32) N a , (cid:115) k (cid:48) a /N a (cid:33) − (cid:34) (cid:80) N b − p a =0 (cid:0) N b (cid:1) p a p a !(1 /N a ) N a Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) k (cid:48) b (cid:19) p a − r (cid:18) a b (cid:19) r (cid:114) k (cid:48) a (cid:82) ( x ) r + N a − p a − exp (cid:40) − (cid:16) k (cid:48) N b b (cid:17) x − (cid:32) N a − a N b b (cid:33) x (cid:41) dx (cid:35) Using Taylor Series Expansion, we can write as follows , = 1Γ( N a ) γ (cid:32) N a , (cid:115) k (cid:48) a /N a (cid:33) − (cid:34) (cid:80) N b − p a =0 (cid:0) N b (cid:1) p a p a !(1 /N a ) N a Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) k (cid:48) b (cid:19) p a − r (cid:18) a b (cid:19) r (cid:80) ∞ l =0 ( − l l ! (cid:114) k (cid:48) a (cid:82) ( x ) r + N a − p a − (cid:40)(cid:16) k (cid:48) N b b (cid:17) x + (cid:32) N a − a N b b (cid:33) x (cid:41) l dx (cid:35) Using Binomial Series Expansion, we can write as follows , = 1Γ( N a ) γ (cid:32) N a , (cid:115) k (cid:48) a /N a (cid:33) − (cid:34) (cid:80) N b − p a =0 (cid:0) N b (cid:1) p a p a !(1 /N a ) N a Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) k (cid:48) b (cid:19) p a − r (cid:18) a b (cid:19) r (cid:80) ∞ l =0 ( − l l ! (cid:80) lt l =0 (cid:0) lt l (cid:1)(cid:16) k (cid:48) N b b (cid:17) l − t l (cid:32) N a − a N b b (cid:33) t l (cid:114) k (cid:48) a (cid:82) ( x ) r + N a − p a − l +2 t l − dx (cid:35) = 1Γ( N a ) γ (cid:32) N a , (cid:115) k (cid:48) a /N a (cid:33) − (cid:34) (cid:80) N b − p a =0 (cid:0) N a (cid:1) N a (cid:0) N b (cid:1) p a p a !Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) k (cid:48) b (cid:19) p a − r (cid:18) a b (cid:19) r (cid:80) ∞ l =0 ( − l l !(2 r + N a − p a − l + 2 t l ) (cid:80) lt l =0 (cid:0) lt l (cid:1)(cid:16) k (cid:48) N b b (cid:17) l − t l (cid:32) N a − a N b b (cid:33) t l (cid:32)(cid:115) k (cid:48) a (cid:33) r + N a − p a − l +2 t l (cid:35) (47)
3) Appendix C: Proof of (23): [21, Sec. 3.471.9, 8.352.2], [24]. P ( R (2 ,DF ) SU ≥ R SU ) = P (cid:16) Z ≥ u c ( a X + b Y ) (cid:17) = P (cid:18) Y ≥ u cb Z − a X b (cid:19) = ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:20) u ca z (cid:82) { x Na − } exp ( − x /Na ) { (1 /N a ) Na } Γ( N a ) (cid:26) ∞ (cid:82) u cb z − a x b { y Nb − } exp ( − y /Nb ) { (1 /N b ) Nb } Γ( N b ) dy (cid:27) dx (cid:21) dz + ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:20) ∞ (cid:82) u ca z { x Na − } exp ( − x /Na ) { (1 /N a ) Na } Γ( N a ) dx (cid:21) dz = ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:20) u ca z (cid:82) { x Na − } exp ( − x /Na ) { (1 /N a ) Na } Γ( N a ) Γ (cid:16) N b , u N b cb z − a x N b b (cid:17) Γ( N b ) dx (cid:21) dz + I = ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:20) u ca z (cid:82) { x Na − } exp ( − x /Na ) { (1 /N a ) Na } Γ( N a ) (cid:26) exp (cid:16) − u N b cb z + a x N b b (cid:17) (cid:80) N b − p a =0 (cid:18) u N b cb z − a x N b b (cid:19) p a p a ! (cid:27) dx (cid:21) dz + I = (cid:80) N b − p a =0 (cid:16) N b b (cid:17) p a p a !(1 /N a ) N a Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) u c (cid:19) p a − r (cid:0) a (cid:1) r ∞ (cid:82) { z ( mk − pa + r − } exp ( − z /mk − u Nbcb z ) { (1 /m k ) mk } Γ( m k ) u ca z (cid:82) x N a + r − exp (cid:26) − (cid:16) N a − a N b b (cid:17) x (cid:27) dx dz + I (Following Binomial Series Expansion) = (cid:80) N b − p a =0 (cid:16) N b b (cid:17) p a p a !(1 /N a ) N a Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) u c (cid:19) p a − r (cid:0) a (cid:1) r ∞ (cid:82) { z ( mk − pa + r − } exp (cid:18) − z /m k − u N b cb z (cid:19) { (1 /m k ) mk } Γ( m k ) (cid:18) N a − a N b b (cid:19) N a + r γ (cid:32) N a + r, u (cid:18) N a − a N b b (cid:19) cza (cid:33) dz + I = (cid:80) N b − p a =0 (cid:16) N b b (cid:17) p a p a !(1 /N a ) N a Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) u c (cid:19) p a − r (cid:0) a (cid:1) r ∞ (cid:82) { z ( mk − pa + r − } exp (cid:18) − z /m k − u N b cb z (cid:19) { (1 /m k ) mk } Γ( m k ) Γ( N a + r ) (cid:18) N a − a N b b (cid:19) N a + r (cid:40) − j ! exp (cid:18) − u (cid:18) N a − a N b b (cid:19) cza (cid:19) (cid:80) N a + r − j =0 (cid:20) u (cid:18) N a − a N b b (cid:19) cza (cid:21) j (cid:41) dz + I = (cid:80) N b − p a =0 (cid:16) N b b (cid:17) p a p a !(1 /N a ) N a Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) u c (cid:19) p a − r (cid:0) a (cid:1) r (cid:18) N a − a Nbb (cid:19) Na + r Γ( N a + r ) { (1 /m k ) m k } Γ( m k ) (cid:20) (cid:26) u N b cb m k (cid:27) ( m k − p a + r ) / K m k − p a + r (cid:26) (cid:113) u N b m k cb (cid:27) − (cid:80) N a + r − j =0 j ! (cid:20) u (cid:18) N a − a N b b (cid:19) ca (cid:21) j ∞ (cid:82) { z ( m k − p a + r − j − } exp (cid:26) − z /m k − u N b cb z − u (cid:18) N a − a N b b (cid:19) cza (cid:27) dz (cid:21) + I = (cid:80) N b − p a =0 (cid:16) N b b (cid:17) p a p a !(1 /N a ) N a Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) u c (cid:19) p a − r (cid:0) a (cid:1) r (cid:18) N a − a Nbb (cid:19) Na + r Γ( N a + r ) { (1 /m k ) m k } Γ( m k ) (cid:20) (cid:26) u N b cb m k (cid:27) ( m k − p a + r ) / K m k − p a + r (cid:26) (cid:113) u N b m k cb (cid:27) − (cid:80) N a + r − j =0 j ! (cid:20) u (cid:18) N a − a N b b (cid:19) ca (cid:21) j (cid:26) u (cid:18) N a − a Nbb (cid:19) ca m k + u N b cb m k (cid:27) ( mk − pa + r − j )2 K m k − p a + r − j (cid:26) (cid:118)(cid:117)(cid:117)(cid:116) u m k (cid:18) N a − a Nbb (cid:19) ca + u N b m k cb (cid:27)(cid:35) + (cid:80) N a − p a =0 (cid:18) u N a ca (cid:19) p a ( m k ) m k p a !Γ( m k ) 2 (cid:32) u N a ca m k (cid:33) ( m k − p a ) / K m k − p a (cid:18) (cid:114) u N a m k ca (cid:19) (48)I is possible to determine using (49) [21, Sec. 3.471.9, 8.352.2]. I = ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:20) ∞ (cid:82) u ca z { x Na − } exp ( − x /Na ) { (1 /N a ) Na } Γ( N a ) dx (cid:21) dz = 1Γ( N a ) ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) Γ (cid:32) N a , u N a ca z (cid:33) dz = ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:34) exp (cid:16) − u N a ca z (cid:17) (cid:80) N a − p a =0 (cid:18) u N a ca z (cid:19) p a p a ! (cid:35) dz = (cid:80) N a − p a =0 (cid:18) u N a ca (cid:19) p a ( m k ) m k p a !Γ( m k ) ∞ (cid:82) (cid:0) z (cid:1) m k − p a − exp (cid:32) − z /m k − z u N a ca (cid:33) dz = (cid:80) N a − p a =0 (cid:18) u N a ca (cid:19) p a ( m k ) m k p a !Γ( m k ) 2 (cid:32) u N a ca m k (cid:33) ( m k − p a ) / K m k − p a (cid:18) (cid:114) u N a m k ca (cid:19) (49)
4) Appendix D: Proof of (32): [21, Sec. 3.471.9, 8.352.1]. P (cid:20) Y < F u ( C − Eu ) + Hu X ( C − Eu ) + u ( C − Eu ) X (cid:21) = ∞ (cid:82) x Na − exp (cid:0) − x /N a (cid:1) Γ( N a ) (cid:0) /N a (cid:1) Na (cid:40) (cid:20) Fu C − Eu + Hu x ( C − Eu + u C − Eu x (cid:21)(cid:82) y Nb − exp (cid:0) − y /N b (cid:1) Γ( N b ) (cid:0) /N b (cid:1) Nb dy (cid:41) dx = ∞ (cid:82) x Na − exp (cid:0) − x /N a (cid:1) Γ( N a )Γ( N b ) (cid:0) /N a (cid:1) Na γ (cid:26) N b , F u N b ( C − Eu ) + Hu N b x ( C − Eu ) + u N b ( C − Eu ) x (cid:27) dx = 1Γ( N a ) (cid:0) /N a (cid:1) N a ∞ (cid:82) x N a − exp (cid:0) − x /N a (cid:1)(cid:34) − exp (cid:26) − N b (cid:18) Fu ( C − Eu ) + Hu x ( C − Eu ) + u ( C − Eu ) x (cid:19)(cid:27) q a ! (cid:80) N b − q a =0 N bq a (cid:18) Fu ( C − Eu ) + Hu x ( C − Eu ) + u ( C − Eu ) x (cid:19) q a (cid:35) dx (Using Binomial Series Expansion, we can write as follows) = 1 − (cid:80) N b − q a =0 ( N b u ) q a exp (cid:0) − NbFu C − Eu (cid:1) (cid:80) qaq =0 ( qaq ) (cid:80) ql =0 ( ql ) F qa − q H q − l q a !Γ( N a ) (cid:0) /N a (cid:1) Na ( C − Eu ) qa (cid:40) ∞ (cid:82) x N a + q − l − exp (cid:8) − N a x − N b Hu x ( C − Eu ) − N b u ( C − Eu ) x (cid:9) dx (cid:41) = 1 − (cid:80) N b − q a =0 ( N b u ) q a exp (cid:0) − NbFu C − Eu (cid:1) (cid:80) qaq =0 ( qaq ) (cid:80) ql =0 ( ql ) F qa − q H q − l q a !Γ( N a ) (cid:0) /N a (cid:1) Na ( C − Eu ) qa (cid:32) N b u / ( C − Eu ) N a + N b Hu ( C − Eu ) (cid:33) ( N a + q − l ) / K N a + q − l (cid:18) (cid:114) N b u ( C − Eu ) (cid:0) N a + N b Hu C − Eu (cid:1)(cid:19) (50)
5) Appendix E: Proof of (34): [21, Sec. 3.471.9, 8.352.2], [24]. P ( R (2 ,AF ) SU ≥ R SU ) = P (cid:40) Z (cid:32) U X + V Y (cid:33) U Z + 1 ≥ u (cid:41) = P (cid:18) Y ≥ u ZV + u U V − U X V (cid:19) = ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:20) u zU + u U U (cid:82) { x Na − } exp ( − x /Na ) { (1 /N a ) Na } Γ( N a ) (cid:26) ∞ (cid:82) u zV + u U V − U x V { y Nb − } exp ( − y /Nb ) { (1 /N b ) Nb } Γ( N b ) dy (cid:27) dx (cid:21) dz + ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:20) ∞ (cid:82) u zU + u U U { x Na − } exp ( − x /Na ) { (1 /N a ) Na } Γ( N a ) dx (cid:21) dz = ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:20) u zU + u U U (cid:82) { x Na − } exp ( − x /Na ) { (1 /N a ) Na } Γ( N a ) Γ (cid:16) N b , u N b zV + u U N b V − U x N b V (cid:17) Γ( N b ) dx (cid:21) dz + I (cid:48) = ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:20) u zU + u U U (cid:82) { x Na − } exp ( − x /Na ) { (1 /N a ) Na } Γ( N a ) (cid:26) exp (cid:16) − u N b zV − u U N b V + U x N b V (cid:17) (cid:80) N b − p a =0 (cid:18) u N b zV + u U N b V − U x N b V (cid:19) p a p a ! (cid:27) dx (cid:21) dz + I (cid:48) = exp (cid:16) − u U N b V (cid:17) ∞ (cid:82) { z ( m k − } exp (cid:16) − z /m k − u N b zV (cid:17) { (1 /m k ) m k } Γ( m k ) (cid:20) u zU + u U U (cid:82) { x Na − } exp (cid:16) − x /N a + U x N b V (cid:17) { (1 /N a ) Na } Γ( N a ) (cid:26) (cid:80) N b − p a =0 (cid:18) u N b zV + u U N b V − U x N b V (cid:19) p a p a ! (cid:27) dx (cid:21) dz + I (cid:48) (Using Binomial Series Expansion, we can write as follows) = (cid:80) N b − p a =0 (cid:80) p a r =0 (cid:0) p a r (cid:1) (cid:80) rl =0 ( − l (cid:0) rl (cid:1)(cid:16) u U N b V (cid:17) p a − r exp (cid:16) − u U N b V (cid:17)(cid:18) u N b V (cid:19) r − l (cid:18) U N b V (cid:19) l p a ! ∞ (cid:82) { z ( m k − r + l − } exp (cid:16) − z /m k − u N b zV (cid:17) { (1 /m k ) m k } Γ( m k ) (cid:20) u zU + u U U (cid:82) { x Na + l − } exp (cid:16) − x /N a + U x N b V (cid:17) { (1 /N a ) Na } Γ( N a ) dx (cid:21) dz + I (cid:48) = (cid:80) N b − p a =0 (cid:80) p a r =0 (cid:0) p a r (cid:1) (cid:80) rl =0 ( − l (cid:0) rl (cid:1)(cid:16) u U N b V (cid:17) p a − r exp (cid:16) − u U N b V (cid:17)(cid:18) u N b V (cid:19) r − l (cid:18) U N b V (cid:19) l p a ! ∞ (cid:82) { z ( m k − r + l − } exp (cid:16) − zm k − u N b zV (cid:17) { (1 /m k ) m k } Γ( m k ) γ (cid:110) N a + l, (cid:18) u zU u U U (cid:19)(cid:18) N a − U N b V (cid:19)(cid:111)(cid:8) (1 /N a ) Na (cid:9)(cid:18) N a − U N b V (cid:19) Na + l Γ( N a ) dz + I (cid:48) = (cid:80) N b − p a =0 (cid:80) p a r =0 (cid:0) p a r (cid:1) (cid:80) rl =0 ( − l (cid:0) rl (cid:1)(cid:16) u U N b V (cid:17) p a − r exp (cid:16) − u U N b V (cid:17) (cid:18) u N b V (cid:19) r − l (cid:18) U N b V (cid:19) l Γ( N a + l ) p a ! (cid:8) (1 /N a ) N a (cid:9)(cid:18) N a − U N b V (cid:19) N a + l Γ( N a ) (cid:34) − exp (cid:26) − (cid:18) u zU + u U U (cid:19)(cid:18) N a − U N b V (cid:19)(cid:27) (cid:80) N a + l − j =0 (cid:20)(cid:18) u zU + u U U (cid:19)(cid:18) N a − U N b V (cid:19)(cid:21) j j ! (cid:35) ∞ (cid:82) { z ( m k − r + l − } exp (cid:16) − m k z − u N b zV (cid:17) { (1 /m k ) m k } Γ( m k ) dz + I (cid:48) = (cid:80) N b − p a =0 (cid:80) p a r =0 (cid:0) p a r (cid:1) (cid:80) rl =0 ( − l (cid:0) rl (cid:1)(cid:16) u U N b V (cid:17) p a − r exp (cid:16) − u U N b V (cid:17) (cid:18) u N b V (cid:19) r − l (cid:18) U N b V (cid:19) l Γ( N a + l ) p a ! (cid:8) (1 /N a ) N a (cid:9)(cid:18) N a − U N b V (cid:19) N a + l Γ( N a ) (cid:34) (cid:0) u N b V m k (cid:1) m k + l − r K m k + l − r (cid:18) (cid:114) u N b m k V (cid:19) { (1 /m k ) m k } Γ( m k ) − (cid:80) N a + l − j =0 exp (cid:26) − (cid:18) u U U (cid:19)(cid:18) N a − U N b V (cid:19)(cid:27)(cid:18) N a − U N b V (cid:19) j (cid:80) jt l =0 (cid:0) jt l (cid:1)(cid:18) u U U (cid:19) j − t l (cid:18) u U (cid:19) t l j ! ∞ (cid:82) { z ( m k − r + l − t l − } exp (cid:40) − m k z − u N b zV − (cid:18) u zU (cid:19)(cid:18) N a − U N b V (cid:19)(cid:41) { (1 /m k ) m k } Γ( m k ) dz (cid:35) + I (cid:48) (51) Finally, above equation can be written as follows P (cid:18) Y ≥ u ZV + u U V − U X V (cid:19) = (cid:80) N b − p a =0 (cid:80) p a r =0 (cid:0) p a r (cid:1) (cid:80) rl =0 ( − l (cid:0) rl (cid:1)(cid:16) u U N b V (cid:17) p a − r exp (cid:16) − u U N b V (cid:17) (cid:18) u N b V (cid:19) r − l (cid:18) U N b V (cid:19) l Γ( N a + l ) p a ! (cid:8) (1 /N a ) N a (cid:9)(cid:18) N a − U N b V (cid:19) N a + l Γ( N a ) (cid:34) (cid:0) u N b V m k (cid:1) m k + l − r K m k + l − r (cid:18) (cid:114) u N b m k V (cid:19) { (1 /m k ) m k } Γ( m k ) − (cid:80) N a + l − j =0 exp (cid:26) − (cid:18) u U U (cid:19)(cid:18) N a − U N b V (cid:19)(cid:27)(cid:18) N a − U N b V (cid:19) j (cid:80) jt l =0 (cid:0) jt l (cid:1)(cid:18) u U U (cid:19) j − t l (cid:18) u U (cid:19) t l j ! { (1 /m k ) m k } Γ( m k ) (cid:0) u N a U m k (cid:1) m k + l − r − t l K m k + l − r − t l (cid:18) (cid:114) u N a m k U (cid:19)(cid:35) + (cid:80) N a − p a =0 (cid:18) u N a U (cid:19) p a ( m k ) m k exp (cid:16) − u U N a U (cid:17) (cid:80) p a r =0 (cid:0) p a r (cid:1) ( U ) p a − r p a !Γ( m k ) 2 (cid:32) u N a U m k (cid:33) ( m k − r ) / K m k − r (cid:18) (cid:114) u N a m k U (cid:19) (52)I (cid:48) can be determined as follows [21, Sec. 3.471.9, 8.352.2]. I (cid:48) = ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:20) ∞ (cid:82) u zU + u U U { x Na − } exp ( − x /Na ) { (1 /N a ) Na } Γ( N a ) dx (cid:21) dz = 1Γ( N a ) ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) Γ (cid:32) N a , u N a zU + u U N a U (cid:33) dz = ∞ (cid:82) { z ( mk − } exp ( − z /mk ) { (1 /m k ) mk } Γ( m k ) (cid:34) exp (cid:16) − u N a zU − u U N a U (cid:17) (cid:80) N a − p a =0 (cid:18) u N a zU + u U N a U (cid:19) p a p a ! (cid:35) dz = (cid:80) N a − p a =0 (cid:18) u N a U (cid:19) p a ( m k ) m k exp (cid:16) − u U N a U (cid:17) (cid:80) p a r =0 (cid:0) p a r (cid:1) ( U ) p a − r p a !Γ( m k ) ∞ (cid:82) (cid:0) z (cid:1) m k − r − exp (cid:32) − m k z − u N a zU (cid:33) dz = (cid:80) N a − p a =0 (cid:18) u N a U (cid:19) p a ( m k ) m k exp (cid:16) − u U N a U (cid:17) (cid:80) p a r =0 (cid:0) p a r (cid:1) ( U ) p a − r p a !Γ( m k ) 2 (cid:32) u N a U m k (cid:33) ( m k − r ) / K m k − r (cid:18) (cid:114) u N a m k U (cid:19) (53)
6) Appendix F: Closed form PU outage expression for DF relaying: P ( PU,DF ) out = 1 − (cid:20) P { Q } × P (cid:8) R (2 ,DF ) PU ≥ R PU (cid:9) × P (cid:8) R (2 ,DF ) PU ≥ R PU (cid:9)(cid:21) = 1 − (cid:34) N a )Γ( N b ) Γ (cid:32) N a , N a u /A (cid:33) Γ (cid:32) N b , N b u /A (cid:33) − (cid:40) N a ) − γ (cid:16) N b , N b u A (cid:17) Γ( N a )Γ( N b ) (cid:41)(cid:110) γ (cid:0) N a , N a u − u A (cid:1) − γ (cid:0) N a , N a u A (cid:1)(cid:111) + (cid:80) N b − q a =0 exp (cid:0) − N b u A (cid:1) (cid:80) q a q =0 ( − q (cid:0) q a q (cid:1) u q a − q A q q a !Γ( N a ) (cid:0) /N a (cid:1) N a (cid:0) /N b (cid:1) q a (cid:16) N a − A N b A (cid:17) ( N a + q ) A q a (cid:110) γ (cid:2) N a + q, (cid:0) N a − A N b A (cid:1) u − u A (cid:3) − γ (cid:2) N a + q, (cid:0) N a − A N b A (cid:1) u A (cid:3)(cid:111)(cid:35) × (cid:34) N a ) γ (cid:32) N a , (cid:115) k (cid:48) a /N a (cid:33) − N b − (cid:88) p a =0 N aN a N bp a p a !Γ( N a ) p a (cid:88) r =0 ( − r (cid:32) p a r (cid:33)(cid:18) k (cid:48) b (cid:19) p a − r (cid:18) a b (cid:19) r ∞ (cid:88) l =0 l (cid:88) t l =0 (cid:32) lt l (cid:33) ( − l l !(2 r + N a − p a − l + 2 t l ) (cid:16) k (cid:48) N b b (cid:17) l − t l (cid:32) N a − a N b b (cid:33) t l (cid:32)(cid:115) k (cid:48) a (cid:33) r + N a − p a − l +2 t l (cid:35) × (cid:34) N b ) γ (cid:32) N b , (cid:115) k (cid:48)(cid:48) b /N b (cid:33) − N a − (cid:88) p a =0 N bN b N ap a p a !Γ( N b ) p a (cid:88) r =0 ( − r (cid:32) p a r (cid:33)(cid:18) k (cid:48)(cid:48) a (cid:19) p a − r (cid:18) b a (cid:19) r ∞ (cid:88) l =0 l (cid:88) t l =0 (cid:32) lt l (cid:33) ( − l l !(2 r + N b − p a − l + 2 t l ) (cid:16) k (cid:48)(cid:48) N a a (cid:17) l − t l (cid:32) N b − b N a a (cid:33) t l (cid:32)(cid:115) k (cid:48)(cid:48) b (cid:33) r + N b − p a − l +2 t l (cid:35) (54)
7) Appendix G: Closed form SU outage expression for DF relaying: P ( SU,DF ) out = 1 − (cid:2) P { Q } × P { Q } × P (cid:8) R (2 ,DF ) SU ≥ R SU (cid:9)(cid:3) = 1 − (cid:34) N a )Γ( N b ) Γ (cid:32) N a , N a u /A (cid:33) Γ (cid:32) N b , N b u /A (cid:33) − (cid:40) N a ) − γ (cid:16) N b , N b u A (cid:17) Γ( N a )Γ( N b ) (cid:41) (cid:110) γ (cid:0) N a , N a u − u A (cid:1) − γ (cid:0) N a , N a u A (cid:1)(cid:111) + (cid:80) N b − q a =0 exp (cid:0) − Nbu A (cid:1) (cid:80) qaq =0 ( − q ( qaq ) u qa − q A q q a !Γ( N a )(1 /N a ) Na (cid:0) /N b (cid:1) qa (cid:16) N a − A NbA (cid:17) ( Na + q ) A qa (cid:110) γ (cid:2) N a + q, (cid:0) N a − A N b A (cid:1) u − u A (cid:3) − γ (cid:2) N a + q, (cid:0) N a − A N b A (cid:1) u A (cid:3)(cid:111)(cid:35) × (cid:34) N a )Γ( N b ) Γ (cid:32) N a , N a u /B (cid:33) Γ (cid:32) N b , N b u /B (cid:33) − (cid:40) N a ) − γ (cid:16) N b , N b u B (cid:17) Γ( N a )Γ( N b ) (cid:41) (cid:110) γ (cid:0) N a , N a u − u B (cid:1) − γ (cid:0) N a , N a u B (cid:1)(cid:111) + (cid:80) N b − q a =0 exp (cid:0) − Nbu B (cid:1) (cid:80) qaq =0 ( − q ( qaq ) u qa − q B q q a !Γ( N a ) (cid:0) /N a (cid:1) Na (cid:0) /N b (cid:1) qa (cid:16) N a − B NbB (cid:17) ( Na + q ) B qa (cid:110) γ (cid:2) N a + q, (cid:0) N a − B N b B (cid:1) u − u B (cid:3) − γ (cid:2) N a + q, (cid:0) N a − B N b B (cid:1) u B (cid:3)(cid:111)(cid:35) × (cid:34) (cid:80) N b − p a =0 (cid:16) N b b (cid:17) p a p a !(1 /N a ) N a Γ( N a ) (cid:80) p a r =0 ( − r (cid:0) p a r (cid:1)(cid:18) u c (cid:19) p a − r (cid:0) a (cid:1) r (cid:18) N a − a Nbb (cid:19) Na + r Γ( N a + r ) { (1 /m k ) m k } Γ( m k ) (cid:20) (cid:26) u N b cb m k (cid:27) ( m k − p a + r ) / K m k − p a + r (cid:26) (cid:113) u N b m k cb (cid:27) − (cid:80) N a + r − j =0 j ! (cid:20) u (cid:18) N a − a N b b (cid:19) ca (cid:21) j (cid:26) u (cid:18) N a − a Nbb (cid:19) ca m k + u N b cb m k (cid:27) ( mk − pa + r − j )2 K m k − p a + r − j (cid:26) (cid:118)(cid:117)(cid:117)(cid:116) u m k (cid:18) N a − a Nbb (cid:19) ca + u N b m k cb (cid:27)(cid:35) + (cid:80) N a − p a =0 (cid:18) u N a ca (cid:19) p a ( m k ) m k p a !Γ( m k ) 2 (cid:32) u N a ca m k (cid:33) ( m k − p a ) / K m k − p a (cid:18) (cid:114) u N a m k ca (cid:19)(cid:35) (55)
8) Appendix H: Closed form PU outage expression for AF relaying: P ( PU,AF ) out = 1 − (cid:20) P (cid:8) R (2 ,AF ) PU ≥ R PU (cid:9) × P (cid:8) R (2 ,AF ) PU ≥ R PU (cid:9)(cid:21) = 1 − (cid:34) N b − (cid:88) q a =0 ( N b u ) q a exp (cid:0) − N b F u ( C − E u ) (cid:1) (cid:80) q a q =0 (cid:0) q a q (cid:1) (cid:80) ql =0 (cid:0) ql (cid:1) F q a − q H q − l q a !Γ( N a ) (cid:0) /N a (cid:1) N a ( C − E u ) q a (cid:32) N b u / ( C − E u ) N a + N b H u ( C − E u ) (cid:33) ( N a + q − l ) / K N a + q − l (cid:18) (cid:115) N b u ( C − E u ) (cid:0) N a + N b H u C − E u (cid:1)(cid:19) × N a − (cid:88) q a =0 ( N a u ) q a exp (cid:0) − N a F u ( C − E u ) (cid:1) (cid:80) q a q =0 (cid:0) q a q (cid:1) (cid:80) ql =0 (cid:0) ql (cid:1) F q a − q H q − l q a !Γ( N b ) (cid:0) /N b (cid:1) N b ( C − E u ) q a (cid:32) N a u / ( C − E u ) N b + N a H u ( C − E u ) (cid:33) ( N b + q − l ) / K N b + q − l (cid:18) (cid:115) N a u ( C − E u ) (cid:0) N b + N a H u C − E u (cid:1)(cid:19)(cid:35) (56)
9) Appendix M: Closed form SU outage expression for AF relaying: P ( SU,AF ) out = − (cid:34) (cid:80) N b − p a =0 (cid:80) p a r =0 (cid:0) p a r (cid:1) (cid:80) rl =0 ( − l (cid:0) rl (cid:1)(cid:16) u U N b V (cid:17) p a − r exp (cid:16) − u U N b V (cid:17) (cid:18) u N b V (cid:19) r − l (cid:18) U N b V (cid:19) l Γ( N a + l ) p a ! (cid:8) (1 /N a ) N a (cid:9)(cid:18) N a − U N b V (cid:19) N a + l Γ( N a ) (cid:20) (cid:0) u N b V m k (cid:1) m k + l − r K m k + l − r (cid:18) (cid:114) u N b m k V (cid:19) { (1 /m k ) m k } Γ( m k ) − (cid:80) N a + l − j =0 exp (cid:26) − (cid:18) u U U (cid:19)(cid:18) N a − U N b V (cid:19)(cid:27)(cid:18) N a − U N b V (cid:19) j (cid:80) jt l =0 (cid:0) jt l (cid:1)(cid:18) u U U (cid:19) j − t l (cid:18) u U (cid:19) t l j ! { (1 /m k ) m k } Γ( m k ) 2 (cid:0) u N a U m k (cid:1) m k + l − r − t l K m k + l − r − t l (cid:18) (cid:114) u N a m k U (cid:19)(cid:21) + (cid:80) N a − p a =0 (cid:18) u N a U (cid:19) p a ( m k ) m k exp (cid:16) − u U N a U (cid:17) p a !Γ( m k ) (cid:80) p a r =0 (cid:0) p a r (cid:1) ( U ) p a − r (cid:32) u N a U m k (cid:33) ( m k − r ) / K m k − r (cid:18) (cid:114) u N a m k U (cid:19)(cid:35) × (cid:34) (cid:80) N b − p a =0 (cid:80) p a r =0 (cid:0) p a r (cid:1) (cid:80) rl =0 ( − l (cid:0) rl (cid:1)(cid:16) u s U N b S (cid:17) p a − r exp (cid:16) − u s U N b S (cid:17) (cid:18) u s N b S (cid:19) r − l (cid:18) S N b S (cid:19) l Γ( N a + l ) p a ! (cid:8) (1 /N a ) N a (cid:9)(cid:18) N a − S N b S (cid:19) N a + l Γ( N a ) (cid:20) (cid:0) u s N b S m k (cid:1) m k + l − r K m k + l − r (cid:18) (cid:114) u s N b m k S (cid:19) { (1 /m k ) m k } Γ( m k ) − (cid:80) N a + l − j =0 exp (cid:26) − (cid:18) u s U S (cid:19)(cid:18) N a − S N b S (cid:19)(cid:27) (cid:0) u s N a S m k (cid:1) m k + l − r − t l (cid:18) N a − S N b S (cid:19) j (cid:80) jt l =0 (cid:0) jt l (cid:1)(cid:18) u s U S (cid:19) j − t l (cid:18) u s S (cid:19) t l j ! { (1 /m k ) m k } Γ( m k ) K m k + l − r − t l (cid:18) (cid:114) u s N a m k S (cid:19)(cid:21) + (cid:80) N a − p a =0 (cid:18) u s N a S (cid:19) p a ( m k ) m k exp (cid:16) − u s U N a S (cid:17) p a !Γ( m k ) (cid:80) p a r =0 (cid:0) p a r (cid:1) ( U ) p a − r (cid:32) u s N a S m k (cid:33) ( m k − r ) / K m k − r (cid:18) (cid:114) u s N a m k S (cid:19)(cid:35) (57) R EFERENCES[1] J. Lianghai, B. Han, M. Liu and H. D. Schotten, “Applying Device-to-Device Communication to Enhance IoT Services,”
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