On the Performance of NOMA-based Cooperative Relaying with Receive Diversity
OOn the Performance of NOMA-based CooperativeRelaying with Receive Diversity
Vaibhav Kumar, Barry Cardiff, and Mark F. Flanagan
School of Electrical and Electronic Engineering, University College Dublin, Belfield, Dublin 4, IrelandEmail: [email protected], [email protected], mark.fl[email protected]
Abstract —Non-orthogonal multiple access (NOMA) is widelyrecognized as a potential multiple access (MA) technology forefficient spectrum utilization in the fifth-generation (5G) wirelessstandard. In this paper, we present the achievable sum rate analy-sis of a cooperative relaying system (CRS) using NOMA with twodifferent receive diversity schemes – selection combining (SC),where the antenna with highest instantaneous signal-to-noiseratio (SNR) is selected, and maximal-ratio combining (MRC).We also present the outage probability and diversity analysisfor the CRS-NOMA system. Analytical results confirm that theCRS-NOMA system outperforms the CRS with conventionalorthogonal multiple access (OMA) by achieving higher spectralefficiency at high transmit SNR and achieves a full diversityorder.
I. I
NTRODUCTION
NOMA has recently been recognized as a promising multi-ple access technology for 5G wireless networks and beyond,as it can meet the ubiquitous and heterogeneous demandson low latency and high reliability, and can support massiveconnectivity by providing high throughput and better spectralefficiency [1]. It enables multiple users to simultaneously sharea time slot, a frequency channel and/or a spreading code, viamultiplexing them in the power domain at the transmitter andusing successive interference cancellation (SIC) at the receiverto remove messages intended for other users.An interesting application of NOMA for a power-domainmultiplexed system using cooperative relaying in Rayleigh dis-tributed block fading channels was proposed in [2], where thesource was able to deliver two data symbols to the destinationin two time slots with the help of a relay. The advantageof such a system can easily be seen in terms of throughput,compared to the conventional OMA relaying system wherea single symbol is delivered to the destination in two timeslots. In particular, closed-form expressions for the averageachievable sum rate and for near-optimal power allocationwere derived in [2]. A performance analysis of the CRS-NOMA system over Rician fading channels was presentedin [3], where the authors developed an analytical frameworkfor the average achievable sum-rate and also proposed amethod to calculate the approximate achievable rate by usingGauss-Chebyshev integration.In this paper, we investigate the performance of the CRS-NOMA system for the case when the relay and the destinationare equipped with multiple receive antennas. We considertwo different diversity combining techniques at the relay anddestination receivers – SC and MRC. We derive closed-form expressions for the average achievable sum-rate and outageprobability for the CRS-NOMA system for the cases of SCand MRC receivers. For the purpose of comparison, we presentnumerical results for the achievable rate of the CRS-OMAsystem with SC and MRC. In order to have a better insightinto the system performance, we present the diversity analysisfor the CRS-NOMA system and prove analytically that thesystem achieves full diversity order for both SC and MRCschemes. II. S
YSTEM M ODEL
Consider the CRS-NOMA model shown in Fig. 1, whichconsists of a source S with a single transmit antenna, a relay R with N r receive antennas and a single transmit antenna,and a destination D with N d receive antennas. All nodes areassumed to be operating in half-duplex mode and all wirelesslinks are assumed to be independent and Rayleigh distributed.The channel coefficient between the source and the i th relayantenna (1 ≤ i ≤ N r ) is denoted by h sr,i and has mean-squarevalue Ω sr for any value of i , while that between the sourceand the j th destination antenna (1 ≤ j ≤ N d ) is denoted by h sd,j and has the mean-square value Ω sd for any value of j .Similarly, the channel coefficient between the relay and the k th destination antenna (1 ≤ k ≤ N d ) is denoted by h rd,k andhas mean-square value Ω rd for any value of k . Furthermore,it is assumed that the channels between the source and thedestination are on average weaker than those between thesource and the relay, i.e., Ω sd < Ω sr . Source S Relay R (combiner) Destination D (combiner)... ... N r N d Fig. 1: System model for CRS-NOMA with multiple receiveantennas. .In the CRS-NOMA scheme, the source broadcasts √ a P t s + √ a P t s to both relay and destination, where s and s are the data-bearing constellation symbols which aremultiplexed in the power domain ( E {| s i | } = 1 for i = 1 , , P t is the the total power transmitted from the source, and a and a are power weighting coefficients satisfying theconstraints a + a = 1 and a > a . Upon reception, thedestination decodes symbol s treating interference from s as additional noise, while the relay first decodes symbol s a r X i v : . [ c s . I T ] F e b nd then applies SIC to decode symbol s . In the second timeslot, the source remains silent and only the relay transmits itsestimate of symbol s , denoted by ˆ s , to the destination withfull transmit power P t . In this manner, two different symbolsare delivered to the destination in two time slots.In contrast to this, in the conventional OMA scheme, thesource broadcasts symbol s with power P t to both relay anddestination in the first time slot and the relay retransmits theestimate of symbol s , denoted by ˆ s , to the destination in thesecond time slot. The destination then combines both copies ofsymbol s and in this manner only a single symbol is deliveredto the destination in two time slots.III. P ERFORMANCE A NALYSIS
In this section, we present the achievable sum-rate, outageprobability and diversity analysis of the CRS-NOMA systemwith two different receive diversity combining techniques,namely SC and MRC.
A. Reception using SC for CRS-NOMA
The signal received at the relay (resp. destination) in thefirst time slot is given by y sµ, SC = h sµ,i ∗ (cid:16)(cid:112) a P t s + (cid:112) a P t s (cid:17) + n sµ , where µ = r (resp. µ = d ) and i ∗ = argmax ≤ i ≤ N µ ( | h sµ,i | ) .Moreover, n sµ denotes complex additive white Gaussian noise(AWGN) with zero mean and variance σ . The receivedinstantaneous signal-to-interference-plus-noise ratio (SINR) atthe relay for decoding symbol s and the instantaneous signal-to-noise ratio (SNR) for decoding symbol s (assuming thesymbol s is decoded correctly) are γ (1) sr, SC = δ sr a P t δ sr a P t + σ and γ (2) sr, SC = δ sr a P t σ , respectively, where δ sr = | h sr,i ∗ | . Simi-larly, the received instantaneous SINR at the destination forthe decoding of symbol s is given by γ sd, SC = δ sd a P t δ sd a P t + σ ,where δ sd = | h sd,j ∗ | . In the next time slot, the relay transmitsthe decoded symbol ˆ s to the destination with power P t . Thereceived signal at the destination is given by y rd, SC = h rd,k ∗ (cid:112) P t ˆ s + n rd , where k ∗ = argmax ≤ k ≤ N d ( | h rd,k | ) and n rd is zero-meancomplex AWGN with variance σ . The received instantaneousSNR at the destination while decoding the symbol s is givenby γ rd, SC = δ rd P t σ , where δ rd = | h rd,k ∗ | . Since the symbol s should be correctly decoded at the destination as well as atthe relay for SIC, the average achievable rate for the symbol s is given by (c.f. [3]) ¯ C s , SC = 12 ln(2) (cid:20) ρ (cid:90) ∞ − F X ( x )1 + ρx dx − ρa (cid:90) ∞ − F X ( x )1 + ρa x dx (cid:21) = 12 ln(2) ( I − I ) , (1)where ρ = P t /σ is the transmit SNR, X (cid:44) min { δ sr , δ sd } and F X ( x ) denotes the cumulative distribution function (CDF)of the random variable X . Theorem 1.
A closed-form expression for the average achiev-able rate for symbol s in Rayleigh fading using SC in CRS-NOMA is given by ¯ C s , SC = 12 ln(2) N r (cid:88) k =1 N d (cid:88) j =1 ( − k + j (cid:18) N r k (cid:19)(cid:18) N d j (cid:19) × (cid:20) exp (cid:18) χ k,j ρ (cid:19) Γ (cid:18) , χ k,j ρ (cid:19) − exp (cid:18) χ k,j ρa (cid:19) Γ (cid:18) , χ k,j ρa (cid:19)(cid:21) , (2) where χ k,j = ( k/ Ω sr )+( j/ Ω sd ) and Γ( · , · ) denotes the upper-incomplete Gamma function.Proof : See Appendix A.The average achievable rate for symbol s is givenby (c.f. [3]) ¯ C s , SC = ρ (cid:90) ∞ − F Y ( x )1 + ρx dx, (3)where Y (cid:44) min { δ sr a , δ rd } . Proposition 1.
A closed-form expression for the averageachievable rate for symbol s in Rayleigh fading using SCin CRS-NOMA is given by ¯ C s , SC = N r (cid:88) k =1 N d (cid:88) j =1 ( − k + j (cid:18) N r k (cid:19)(cid:18) N d j (cid:19) exp (cid:18) θ k,j ρ (cid:19) Γ (cid:18) , θ k,j ρ (cid:19) , (4) where θ k,j = k Ω sr a + j Ω rd .Proof. Analogous to the arguments in Appendix A and usinga transformation of random variables, we have − F Y ( x ) = N r (cid:88) k =1 N d (cid:88) j =1 ( − k + j (cid:18) N r k (cid:19)(cid:18) N d j (cid:19) exp( − θ k,j x ) , (5)Substituting the expression of − F Y ( x ) from (5) into (3) andsolving the integration using [4, eqn. (3.352-4), p. 341], theclosed-form expression for ¯ C s , SC reduces to (4). (cid:4) The average achievable sum-rate for the CRS-NOMA sys-tem using SC in Rayleigh fading is therefore given using (2)and (4) as ¯ C sum , SC = ¯ C s , SC + ¯ C s , SC . (6)It is interesting to note that for N r = N d = 1 , (6) reducesto [2, eqn. (14)]. B. Reception using SC for CRS-OMA
The signal received at the relay (resp. destination) in thefirst time slot is given by y sµ, SC − OMA = h sµ,i ∗ (cid:112) P t s + n sµ , where µ = r (resp. µ = d ) and i ∗ = argmax ≤ i ≤ N µ ( | h sµ,i | ) .In the next time slot, the relay forwards its estimate of s ,denoted by ˆ s , to the destination. The signal received at thedestination is given by y rd, SC − OMA = h rd,k ∗ (cid:112) P t ˆ s + n rd . he average achievable rate for the symbol s is givenby (c.f. [5]) ¯ C SC − OMA = 0 . E W [log (1 + W ρ )] , (7)where W (cid:44) min { δ sr , δ sd + δ rd } and E Z [ · ] denotes theexpectation with respect to the random variable Z . Since thefocus of this paper is on the NOMA-based systems, we do notpresent a closed-form analysis for CRS-OMA. C. Outage probability for CRS-NOMA using SC
In this subsection, we will characterize the outage probabil-ity of symbols s and s for the CRS-NOMA using selectioncombining in Rayleigh fading. We define O , SC as the outageevent for symbol s using SC, i.e., the event where either therelay or the destination fails to decode s successfully. Hencethe outage probability for the symbol s is given by Pr( O , SC ) = Pr( C s , SC < R )= Pr (cid:20)
12 log (cid:18) a ρX a ρX (cid:19) < R (cid:21) = Pr( X < Θ )= F δ sr (Θ ) + F δ sd (Θ ) − F δ sr (Θ ) F δ sd (Θ ) , (8)where C s , SC is the instantaneous achievable rate of symbol s in CRS-NOMA using SC in Rayleigh fading, R isthe target data rate for the symbol s , (cid:15) = 2 R − and Θ = (cid:15) ρ ( a − (cid:15) a ) . The system design must ensure that a > (cid:15) a , otherwise the outage probability for symbol s will always be 1 as noted in [6]. The closed-form expressionsfor F δ sr (Θ ) and F δ sd (Θ ) are given in Appendix A. Next, wedefine O , SC as the outage event for symbol s using SC. Thisoutage event can be decomposed as the union of the followingdisjoint events: (i) symbol s cannot be successfully decodedat the relay; (ii) symbol s is successfully decoded at the relay,but symbol s cannot be successfully decoded at the relay; and(iii) both symbols are successfully decoded at the relay, butsymbol s cannot be successfully decoded at the destination.Therefore, the outage probability for the symbol s may beexpressed as Pr( O , SC ) = Pr( δ sr < Θ ) + Pr( δ sr ≥ Θ , δ sr < Θ )+ Pr( δ sr > Θ , δ rd < (cid:15) /ρ ); if Θ < Θ Pr( δ sr < Θ ) + Pr( δ sr > Θ , δ rd < (cid:15) /ρ );otherwise= F δ sr (Θ)+ F δ rd ( (cid:15) /ρ ) − F δ sr (Θ) F δ rd ( (cid:15) /ρ ) , (9)where R is the target data rate for the symbol s , (cid:15) = 2 R − , Θ = (cid:15) a ρ and Θ = max { Θ , Θ } . Theclosed-form expressions for F δ sr (Θ) and F δ rd ( (cid:15) /ρ ) are givenin Appendix A. The rate calculation is based on the assumption that the destinationperforms SC in the first and the second time slots and then applies MRC onthe resulting signals from the two time slots. In the case where the destinationapplies SC on the resulting signals instead of MRC, W will instead bedefined as min { δ sr , max { δ sd , δ rd }} , and this will result in a performancedegradation with respect to the system described here. D. Diversity analysis for CRS-NOMA using SC
From (23), we have F δ sr (Θ ) = N r (cid:88) k =1 ( − k − (cid:18) N r k (cid:19) (cid:20) − exp (cid:18) − k Θ Ω sr (cid:19)(cid:21) = N r (cid:88) k =1 ∞ (cid:88) l =1 ( − k + l l ! (cid:18) N r k (cid:19) (cid:18) k Θ Ω sr (cid:19) l = ∞ (cid:88) l = N r ( − l Θ l l !Ω lsr × N r (cid:88) k =1 (cid:18) N r k (cid:19) ( − k k l (Using [4, eqn. (0.154-3), p. 4]) = ( − N r (cid:15) N r ( a − (cid:15) a ) N r N r !Ω N r sr × N r (cid:88) k =1 (cid:18) N r k (cid:19) ( − k k N r ρ − N r + O (cid:104) ρ − ( N r +1) (cid:105) , (10)where O is the Landau symbol. Hence it is clear from (10) that F δ sr (Θ ) decays as ρ − N r as ρ → ∞ . Similarly, it can be easilyshown that F δ sd (Θ ) decays as ρ − N d and F δ sr (Θ ) F δ sd (Θ ) decays as ρ − ( Nr + N d ) as ρ → ∞ . Therefore it is straight-forward to conclude using (8) that the diversity order of thesymbol s is min { N r , N d , N r N d } = min { N r , N d } . Follow-ing similar arguments, it can be shown that the diversity orderof the symbol s is min { N r , N d } . E. Reception using MRC for CRS-NOMA
The signal received at the relay (resp. destination) in thefirst time slot is given by y sµ, MRC = h Hsµ (cid:16) h sµ (cid:16)(cid:112) a P t s + (cid:112) a P t s (cid:17) + n sµ (cid:17) , where µ = r (resp. µ = d ), h sµ = [ h sµ, h sµ, · · · h sµ,N µ ] T ∈ C N µ × , n sµ = [ n sµ, n sµ, · · · n sµ,N µ ] T ∈ C N µ × , ( · ) H isthe Hermitian operator and ( · ) T is the transpose operator. Theelements in the vector h sµ are independent and distributedas CN (0 , Ω sµ ) and the elements in n sµ are independent anddistributed according to CN (0 , σ ) .The received instantaneous SINR at the relay for decodingsymbol s and instantaneous SNR for decoding symbol s (assuming the symbol s is decoded correctly) are obtained as γ (1) sr, MRC = λ sr a P t λ sr a P t + σ and γ (2) sr, MRC = λ sr a P t σ , respectively,where λ sr = (cid:80) N r i =1 | h sr,i | . Similarly, the received instan-taneous SINR at the destination while decoding s is givenby γ sd, MRC = λ sd a P t λ sd a P t + σ , where λ sd = (cid:80) N d i =1 | h sd,i | . Inthe next time slot, the relay transmits the decoded symbol ˆ s to the destination with power P t . The received signal at thedestination (after applying MRC) is given by y rd, MRC = h Hrd (cid:16) h rd (cid:112) P t ˆ s + n rd (cid:17) , where h rd = [ h rd, h rd, · · · h rd,N d ] T ∈ C N d × with inde-pendent elements each distributed as CN (0 , Ω rd ) and n rd =[ n rd, n rd, · · · n rd,N d ] T ∈ C N d × with independent ele-ments each distributed according to CN (0 , σ ) . The receivednstantaneous SNR at the destination while decoding the sym-bol s is γ rd, MRC = λ rd P t σ , where λ rd = (cid:80) N d i =1 | h rd,i | . Theaverage achievable rate for the symbol s is given by (c.f. [3]) ¯ C s , MRC = 12 ln(2) (cid:20) ρ (cid:90) ∞ − F X ( x )1 + xρ dx − ρa (cid:90) ∞ − F X ( x )1 + xρa dx (cid:21) = 12 ln(2) ( I − I ) , (11)where X = min { λ sr , λ sd } . Theorem 2.
A closed-form expression for the average achiev-able rate for symbol s for CRS-NOMA using MRC inRayleigh fading is given by ¯ C s , MRC = 12 ln(2) N r − (cid:88) i =0 N d − (cid:88) j =0 Γ(1 + i + j ) i ! j !Ω isr Ω jsd ρ i + j (cid:34) exp (cid:18) φρ (cid:19) × Γ (cid:18) − i − j, φρ (cid:19) − a i + j exp (cid:18) φρa (cid:19) Γ (cid:18) − i − j, φρa (cid:19)(cid:35) , (12) where φ = Ω − sr + Ω − sd and Γ( · ) denotes the Gamma function.Proof : See Appendix B.The average achievable rate for the symbol s is givenby (c.f. [3]) ¯ C s , MRC = ρ (cid:90) ∞ − F Y ( x )1 + xρ dx, (13)where Y (cid:44) min { λ sr a , λ rd } . Proposition 2.
The closed-form expression for the averageachievable rate for symbol s for CRS-NOMA using MRC inRayleigh fading is obtained as ¯ C s , MRC = 12 ln(2) N r − (cid:88) i =0 N d − (cid:88) j =0 Γ(1 + i + j ) i ! j ! a i Ω isr Ω jrd ρ ( i + j ) × exp (cid:18) ξρ (cid:19) Γ (cid:18) − i − j, ξρ (cid:19) , (14) where ξ = (Ω sr a ) − + Ω − rd .Proof. Similar to the arguments in Appendix B and using atransformation of random variables, we have, − F Y ( x ) = exp( − xξ ) N r − (cid:88) i =0 N d − (cid:88) j =0 x i + j i ! j ! a i Ω isr Ω jrd . (15)Substituting − F Y ( x ) from (15) into (13) and solving theintegral using [4, eqn. (3.383-10), p. 348], the closed-formexpression for ¯ C s , MRC becomes equal to (14). (cid:4)
Hence, the average achievable sum-rate for the CRS-NOMAusing MRC in Rayleigh fading is obtained using (12) and (14)as ¯ C sum , MRC = ¯ C s , MRC + ¯ C s , MRC . (16)It is important to note that for N r = N d = 1 , (16) reducesto [2, eqn. (14)]. F. Reception using MRC for CRS-OMA
The signals received in the first time slot at the relay (resp.destination) is given by y sµ, MRC − OMA = h Hsµ (cid:16) h sµ (cid:112) P t s + n sµ (cid:17) , where µ = r (resp. µ = d ). In the next time slot, the relayforwards its estimate of s , denoted by ˆ s , to the destination.The signal received at the destination is given by y rd, MRC − OMA = h Hrd (cid:16) h rd (cid:112) P t ˆ s + n rd (cid:17) . Similar to the case of CRS-OMA using SC, the averageachievable rate for symbol s in CRS-OMA using MRC isgiven by (c.f. [5]) ¯ C MRC − OMA = 0 . E Z [log (1 + Z ρ )] , (17)where Z (cid:44) min( λ sr , λ sd + λ rd ) . G. Outage probability for CRS-NOMA using MRC
Similar to the CRS-NOMA using SC, we define O , MRC asthe event that the symbol s is in outage in the CRS-NOMAusing MRC in Rayleigh fading. Hence, Pr( O , MRC ) = Pr( C s , MRC < R ) = F X (Θ )= F λ sr (Θ )+ F λ sd (Θ ) − F λ sr (Θ ) F λ sd (Θ ) , (18)where C s , MRC is the instantaneous achievable rate for symbol s in CRS-NOMA using MRC in Rayleigh fading. Similarly,we define O , MRC as the event that the symbol s is in outagein the CRS-NOMA using MRC in Rayleigh fading. Therefore, Pr( O , MRC ) = F λ sr (Θ)+ F λ rd (cid:18) (cid:15) ρ (cid:19) − F λ sr (Θ) F λ rd (cid:18) (cid:15) ρ (cid:19) . (19)The closed-form expressions for F λ sr (Θ ) , F λ sd (Θ ) , F λ sr (Θ) and F λ rd ( (cid:15) /ρ ) can be found using the fact that λ sr , λ sd and λ rd are Gamma distributed random variables. H. Diversity analysis of CRS-NOMA using MRC
Since λ sr is Gamma distributed with shape N r and scale Ω sr we have F λ sr (Θ ) = 1Γ( N r ) γ (cid:18) N r , Θ Ω sr (cid:19) , (20)where γ ( · , · ) is lower-incomplete Gamma function. Using theseries expansion of the lower-incomplete Gamma function asgiven in [7, eqn. 8.11.4, p. 180], F λ sr (Θ ) = 1Γ( N r ) (cid:18) Θ Ω sr (cid:19) N r exp (cid:18) − Θ Ω sr (cid:19) ∞ (cid:88) k =0 Θ k Γ( N r )Ω ksr Γ( N r + k + 1) . Using the series expansion of the exponential function andreplacing Θ by (cid:15) ρ ( a − (cid:15) a ) yields F λ sr (Θ ) = ∞ (cid:88) l =0 ∞ (cid:88) k =0 ( − l Θ N r + l + k Ω N r + l + ksr Γ( N r + l + k )= (cid:15) N r ρ − N r ( a − (cid:15) a ) N r Ω N r sr Γ( N r ) + O (cid:16) ρ − ( N r +1) (cid:17) . (21)t is clear from (21) that F λ sr (Θ ) decays as ρ − N r as ρ → ∞ .Similarly, it can be proved that F λ sd (Θ ) decays as ρ − N d as ρ → ∞ . Also, using the series expansion of the lower-incomplete Gamma function and the exponential function, wehave F λ sr (Θ ) F λ sd (Θ ) = 1Γ( N r )Γ( N d ) γ (cid:18) N r , Θ Ω sr (cid:19) γ (cid:18) N d , Θ Ω sd (cid:19) = ∞ (cid:88) l =0 ∞ (cid:88) k =0 ∞ (cid:88) i =0 ∞ (cid:88) j =0 ( − l + i Θ N r + N d + l + k + i + j Ω N r + l + ksr Ω N d + i + jsd × N r + l + k )Γ( N d + i + j )= (cid:15) N r + N d ρ − ( N r + N d ) ( a − (cid:15) a ) N r + N d Ω N r sr Ω N d sd Γ( N r )Γ( N d ) + O (cid:16) ρ − ( N r + N d +1) (cid:17) . (22)Hence it is straightforward to conclude using (18), (21)and (22) that the diversity order for the symbol s is min { N r , N d , N r N d } = min { N r , N d } . Analogously, by rep-resenting F λ sr (Θ) and F λ rd ( (cid:15) /ρ ) in (19) in terms of thelower-incomplete gamma function, it can be shown that thediversity order for the symbol s is min( N r , N d , N r N d ) =min( N r , N d ) . IV. R ESULTS AND D ISCUSSION
In this section we present analytical and numerical resultsfor the average achievable rate and the outage probabilityfor the cooperative relaying system. We consider the CRSsystem where Ω sd = 1 , Ω sr = 10 and Ω rd = 2 . . For allNOMA-based systems, we consider a = 0 . and R = R =1 bps/Hz. Fig. 2 shows a comparison of the average achiev-able rate for the CRS-NOMA (both numerical and analyticalresults) and CRS-OMA (numerical results) systems. It is clearfrom the figure that for low transmit SNR ρ , the CRS-NOMAsystem performs worse compared to the conventional CRS-OMA system in terms of achievable rate, but as the transmitSNR ρ becomes large, the CRS-NOMA system outperformsits OMA counterpart for both SC and MRC schemes. It isevident from Fig. 2a that the CRS-NOMA using SC with N r = N d = 1 achieves the same spectral efficiency as that ofthe CRS-OMA using SC with N r = N d = 2 at high transmitSNR. Also, the CRS-NOMA using SC with N r = N d = 2 achieves higher spectral efficiency as compared to CRS-OMAusing SC with N R = N d = 4 at high SNR. From Fig. 2b, itis clear that the CRS-NOMA using MRC with N r = N d = 2 achieves the same spectral efficiency as that of the CRS-OMAusing MRC with N r = N d = 4 at high transmit SNR. It canalso be noted that the CRS-NOMA system using MRC resultsin a higher average achievable sum-rate as compared to theCRS-NOMA system using SC.Fig. 3 shows the outage probability of the symbols s and s with varying transmit SNR ρ for the CRS-NOMA systemusing SC. It is clear that the diversity order for both symbolsis min( N r , N d ) as derived in Section III-D. We do not realize the actual scenario for numerical computation, but rathergenerate the random variables and then evaluate (6), (7), (16) and (17). (a) Using SC. (b) Using MRC.
Fig. 2: Average achievable rate for the CRS. -20 -15 -10 -5 (a) Symbol s -20 -15 -10 -5 (b) Symbol s Fig. 3: Outage probability for CRS-NOMA using SC. -20 -15 -10 -5 (a) Symbol s -20 -15 -10 -5 (b) Symbol s Fig. 4: Outage probability for CRS-NOMA using MRC.Fig. 4 shows the outage probability of the symbols s and s with varying transmit SNR ρ for the CRS-NOMA systemusing MRC. It is evident from the figure that the diversityorder for both symbols is min( N r , N d ) as proved analytically.It can also be noted that the outage probabilities for symbols s and s are lower for the CRS-NOMA system using MRCas compared to the corresponding probabilities for the CRS-NOMA system using SC.V. C ONCLUSION
In this paper, we provided a comprehensive achievable sum-rate analysis of a CRS-NOMA system with receive diversity.We considered two different diversity combining schemes –SC and MRC. It was shown that the CRS-NOMA systemoutperforms its OMA-based counterpart by achieving higherspectral efficiency. Our analysis also confirms that the CRS-NOMA can achieve the same rate as CRS-OMA, but witha smaller number of receive antennas. We also presentedthe outage probability analysis of the CRS-NOMA system.Diversity analysis of the CRS-NOMA system confirms thathe system achieves full diversity order of min( N r , N d ) forboth SC and MRC schemes.A PPENDIX AP ROOF OF T HEOREM | h sr,i | is Rayleigh distributed for every i ∈{ , , . . . , N r } , the CDF of | h sr,i ∗ | is given by F | h sr,i ∗ | ( x ) = (cid:20) − exp (cid:18) − x Ω sr (cid:19)(cid:21) N r = 1 + N r (cid:88) k =1 ( − k (cid:18) N r k (cid:19) exp (cid:18) − kx Ω sr (cid:19) . Therefore, the CDF of δ sr can be obtained as F δ sr ( x ) = Pr( | h sr,i ∗ | ≤ x ) = Pr( | h sr,i ∗ | ≤ √ x )= 1 + N r (cid:88) k =1 ( − k (cid:18) N r k (cid:19) exp (cid:18) − kx Ω sr (cid:19) . (23)The CDF of δ sd (resp. δ rd ) can be found by replacing Ω sr by Ω sd (resp. Ω rd ), while also replacing N r by N d , in (23). TheCDF of X = min { δ sr , δ sd } can be found as F X ( x ) = F δ sr ( x ) + F δ sd ( x ) − F δ sr ( x ) F δ sd ( x ) . Therefore, − F X ( x ) = N r (cid:88) k =1 N d (cid:88) j =1 ( − k + j (cid:18) N r k (cid:19)(cid:18) N d j (cid:19) exp ( − χ k,j x ) , (24)where χ k,j = ( k/ Ω sr )+( j/ Ω sd ) . Using (1) and (24), we have I = ρ N r (cid:88) k =1 N d (cid:88) j =1 ( − k + j (cid:18) N r k (cid:19)(cid:18) N d j (cid:19) (cid:90) ∞ exp( − χ k,j x )1 + ρx dx = N r (cid:88) k =1 N d (cid:88) j =1 ( − k + j (cid:18) N r k (cid:19)(cid:18) N d j (cid:19) exp (cid:18) χ k,j ρ (cid:19) Γ (cid:18) , χ k,j ρ (cid:19) , (25)where the integral above is solved using [4, eqn. (3.352-4), p. 341] and the fact that − Ei( − x ) = Γ(0 , x ) . Here Ei( · ) denotes the exponential integral. Similarly, using (1) and (24),we have I = N r (cid:88) k =1 N d (cid:88) j =1 ( − k + j (cid:18) N r k (cid:19)(cid:18) N d j (cid:19) exp (cid:18) χ k,j ρa (cid:19) Γ (cid:18) , χ k,j ρa (cid:19) . (26)Using (1), (25) and (26), the closed-form expression for theaverage achievable rate for symbol s in Rayleigh fading usingSC in CRS-NOMA reduces to (2); this completes the proof. Given two independent random variables U and V with probability densityfunctions (PDFs) f U ( x ) and f V ( x ) respectively, and CDFs F U ( x ) and F V ( x ) respectively, the PDF of W (cid:44) min {U , V} is given by f W ( x ) = f U ( x )[1 − F V ( x )] + f V ( x )[1 − F U ( x )] and the CDF of W is given by F W ( x ) = F U ( x ) + F V ( x ) − F U ( x ) F V ( x ) . A PPENDIX BP ROOF OF T HEOREM | h sr,i | (1 ≤ i ≤ N r ) and | h sd,i | (1 ≤ i ≤ N d ) areRayleigh distributed, the random variables λ sr and λ sd areGamma distributed with shape N r and N d respectively, andscale Ω sr and Ω sd respectively. Since the shape parameters arepositive integers, the corresponding CDFs for λ sr and λ sd caneach be represented as a special case of the Erlang distribution.It follows that the CDF of X can be written as F X ( x ) = 1 − exp( xφ ) N r − (cid:88) i =0 N d − (cid:88) j =0 x i + j i ! j !Ω isr Ω jsd , where φ = Ω − sr + Ω − sd . Hence I in (11) can be solvedusing [4, eqn. (3.383-10), p. 348] as I = N r − (cid:88) i =0 N d − (cid:88) j =0 i ! j ! Ω isr Ω jsd (cid:90) ∞ exp( − xφ ) x ( i + j ) xρ dx = N r − (cid:88) i =0 N d − (cid:88) j =0 exp (cid:16) φρ (cid:17) Γ(1 + i + j ) i ! j ! Ω isr Ω jsd ρ (1+ i + j ) Γ (cid:18) − i − j, φρ (cid:19) . (27)Similarly, I can be solved as I = N r − (cid:88) i =0 N d − (cid:88) j =0 exp (cid:16) φρa (cid:17) Γ(1 + i + j ) i ! j ! Ω isr Ω jsd ( ρa ) (1+ i + j ) Γ (cid:18) − i − j, φρa (cid:19) . (28)Using (11), (27) and (28), the closed-form expression for theaverage achievable rate of symbol s for CRS-NOMA usingMRC in Rayleigh fading reduces to (12); this completes theproof. A CKNOWLEDGMENT
This publication has emanated from research conductedwith the financial support of Science Foundation Ireland (SFI)and is co-funded under the European Regional DevelopmentFund under Grant Number 13/RC/2077.R
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