Exploiting Multiple Access in Clustered Millimeter Wave Networks: NOMA or OMA?
Wenqiang Yi Yuanwei Liu Arumugam Nallanathan
cs.IT math.IT





















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Exploiting Multiple Access in Clustered Millimeter Wave Networks: NOMA or OMA?
    
Wenqiang Yi, Yuanwei Liu and Arumugam Nallanathan  Queen Mary University of London, London, UK  
    December 30, 2023
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  In this paper, we introduce a clustered millimeter wave network with non-orthogonal multiple access (NOMA), where the base station (BS) is located at the center of each cluster and all users follow a Poisson Cluster Process. To provide a realistic directional beamforming, an actual antenna pattern is deployed at all BSs. We provide a nearest-random scheme, in which near user is the closest node to the corresponding BS and far user is selected at random, to appraise the coverage performance and universal throughput of our system. Novel closed-form expressions are derived under a loose network assumption. Moreover, we present several Monte Carlo simulations and numerical results, which show that: 1) NOMA outperforms orthogonal multiple access regarding the system rate; 2) the coverage probability is proportional to the number of possible NOMA users and a negative relationship with the variance of intra-cluster receivers; and 3) an optimal number of the antenna elements is existed for maximizing the system throughput.





§ INTRODUCTION



The ever-increasing requirements of Internet-enabled applications and services have exhaustively strained the capacity of conventional cellular networks. One promising technology for augmenting the throughput of fifth generation (5G) wireless systems is exploiting new spectrum resources, e.g. millimeter wave (mmWave) <cit.>. Comparing with the traditional sub-6 GHz carrier frequency, mmWave has two distinguishing properties. One is the sensitivity to blockage effects, which increases the path loss of non-line-of-sight (NLOS) signals <cit.>. The other is the small wavelength, which shortens the size of antenna elements so that large antenna arrays can be employed at mmWave devices for enhancing the directional array gain <cit.>. Accordingly, several works have paid attention to these distinctive features when analyzing mmWave networks. The primary article <cit.> proposed an line-of-sight (LOS) disc model to reflect the impact of blockages. However, the directional beamforming method in this work was over-simplified and hence failed to depict the exact properties of a practical antenna. Then, a realistic antenna pattern was introduced in <cit.>. To capture the randomness of networks, stochastic geometry has been widely applied in numerous researches <cit.>. More specifically, the locations of base stations (BSs) follow a Poisson Point Process (PPP). Since mmWave is able to support ultra-high throughput in short-distance communications <cit.>, a recent work <cit.> considered a Poisson Cluster Process (PCP) instead of PPP model to evaluate the short-range mmWave network, which obtains a close characterization to the real world.

In addition to expanding the available spectrum range, another significant objective of 5G cellular networks is improving the spectral efficiency. Recently, non-orthogonal multiple access (NOMA) has kindled the attention of academia since it realizes multiple access in the power domain rather than the traditional frequency domain <cit.>. The main advantage of such approach is that NOMA possesses a perfect balance between coverage fairness and universal throughput <cit.>. In contrast to the conventional orthogonal multiple access (OMA), a new technique named successive interference cancellation (SIC) is applied at the NOMA user who has a robust channel condition to extract the requested information <cit.>. The power allocation strategies for NOMA networks were introduced in <cit.> to assure the fairness for all users. In a single cell scenario, the downlink sum-rate and outage probability were analyzed in <cit.> and the uplink NOMA performance with a power back-off method was investigated in <cit.>. However, the aforementioned articles focus on the noise-limited system and inter-cell interference is ignored for tractability of the analysis. In fact, the interference is an important factor when studying the coverage performance, especially in the sub-6 GHz networks. Authors in <cit.> offered a dense NOMA network with multiple inter-cells. Under this model, both uplink and downlink transmissions were evaluated. Like mmWave communications, stochastic geometry has also been utilized in NOMA networks <cit.> to model the positions of the primary and secondary NOMA users.

As mentioned above, although traditional NOMA networks can improve the spectral efficiency, the limited bandwidth resources below 6 GHz is substantially restrict the development of future wireless networks. Note that mmWave obtains a large amount of free spectrum. Applying NOMA into mmWave networks is an ideal way to satisfy the challenging demand of 5G cellular networks. Moveover, when the transmission distance is long, the path loss of mmWave communications is severer than that of conversional networks, so the inter-cell interference in mmWave networks is weak compared with sub-6 GHz scenarios'. Particularly,  in a loose mmWave network, such interference can be ignored due to its negligible received power, which will dramatically perfect the system coverage. With the aid of the PCP model as discussed in <cit.>, we proposed a practical user selection scheme to evaluate the performance of clustered mmWave networks with NOMA. The actual antenna pattern <cit.> is also employed to enhance the analytical accuracy. The main contributions of this treatise are as follows: 1) we derive the closed-form equation for Laplace transform of inter-cluster interferences; 2) novel expressions for coverage probabilities of near user and far user are deduced and closed-form algorithms under a loose network assumption are provided as well; 3) NOMA performs better than OMA regarding the system rate in our mmWave networks due to the adjustable power allocation coefficients; and 4) there exists an optimal number of antenna elements for obtaining the maximum system throughput due to the inverse feedback of two paired users.




§ NETWORK MODEL





 §.§ Spatial Model




As shown in Fig. <ref>, we consider a clustered downlink mmWave NOMA transmission scenario. All BSs are deployed at cluster centers and the distribution of NOMA users follows one of typical PCPs, which is a tractable variant of Thomas cluster process <cit.>. More specifically, the BSs constituting parent points are distributed as a homogeneous PPP Φ with density λ_c. The union of NOMA receivers in a cluster with one parent at y∈Φ represents offspring points, which is denoted by ℕ_y. Such NOMA users are independent and identically distributed (i.i.d.) following symmetric normal distributions with variance σ^2 and mean 0. Therefore, the user at x∈ℕ_y in reference to the corresponding BS at y is given by

    f_X( x ) = 1/2πσ ^2exp(  -  x ^2/2σ ^2).


In each cluster, the number of NOMA users is fixed as 2K. These 2K users are divided into K orthogonal pairs and each pair utilizes a single orthogonal beam. All BSs are assumed to serve the whole intra-cluster users at a same time. As a result, there is no intra-cluster interferences except the paired user's signal, but inter-cluster interferences from other BSs using the equivalent orthogonal beam are still existed. Additionally, a typical cluster is randomly chosen to enhance the generality. The typical BS included in the typical cluster is located at the origin of the considered plane. In this paper, we focus on a typical pair of users from the typical cluster, where the paired User k and User j represent the near user and far user, respectively. A tractable user selection scheme named Nearest-Random (NR) is introduced for analyzing the performance. More specifically, under NR scheme, User k is the nearest receiver to the typical BS and User j is randomly chosen from the rest of the NOMA users.




 §.§ Blockage Effects



One remarkable characteristic of mmWave networks is that mmWave signals are sensitive to obstacles. Therefore, the LOS links have a distinctive path loss law with NLOS transmissions. Note that each cluster can be visualized as a dense mmWave network due to the small variance σ^2 of NOMA users. Under this condition, one obstacle may block all receivers behind it, so we adopt the LOS disc to model the blockage effect <cit.>. This blockage model fits the practical scenarios better than other patterns <cit.>. Accordingly, the LOS probability inside the LOS disc with a radius R_L is one, while the NLOS probability outside the disc is one. With the aid of LOS disc model, we provide the path loss law of our proposed networks with a distance ṙ as below

    L_p(ṙ) = 𝐔( R_L - ṙ)C_Lṙ^ - α _L + 𝐔( ṙ- R_L)C_Nṙ^ - α _N,

where C_κ is the intercept and α_κ is the path loss exponent. κ=L and N represent the LOS and NLOS links, respectively. 𝐔(.) is the unit step function.




 §.§ Antenna Beamforming



Another constraint for mmWave networks is the high cost and power consumption for signal processing components. We adopt analog beamforming in this work for achieving a low complexity beamforming design. More specifically, the directions of beams are controlled by phase shifters. We invoke the optimal analog precoding which implies that BSs try to align the direction of beams with the angel-of-departure (AoD) of channels. Hence high beamforming gains can be obtained. Assuming uniform linear array composed of M antenna elements is deployed at all BSs, then based on this precoding design, the effective channel gain at User k aligning with the optimal analog beamforming is given by

    | h_k^Hw_k|^2 = M| g_k|^2,

where h_k and w_k are the channel vector and beamforming vector, respectively. |g_k | represents the independent Nakagami-N_κ fading for User k <cit.>.

Regarding any other User k̂, the effective channel gain is expressed as below

    | h_k̂^Hw_k|^2=   | g_k̂|^2sin^2( π M( θ _k - θ _k̂))/Msin^2( π( θ _k - θ _k̂))
    
    =    M| g_k̂|^2G_F( θ _k - θ _k̂).

where G_F(  ·) denotes the normalized Fejér kernel with parameter 1/M and ( θ _k - θ _k̂) is uniformly distributed over [-q/λ,q/λ] <cit.>.



 §.§ Signal Model

We assume that in the typical cluster, the typical BS is located at y_0∈Φ. Then, User k at x_k and User j at x_j are paired and served by the same beam. The distances of them obey d_k≤d_j. Moreover, the power allocation coefficients satisfy the conditions that a_k < a_j and a_k + a_j = 1, which is for fairness considerations <cit.>. In terms of other clusters, the interfering BS located at y ∈Φ/y_0 provides an optimal analog beamforming for User ξ_y, which is chosen uniformly at random. As a consequence, the received signal is given by

    y_k =   h_k^Hw_k√(a_kP_tL_p( x_k))s_k_Desired Signal + h_k^Hw_k√(a_jP_tL_p( x_k))s_j_SIC Signal
        + ∑_y ∈Φ /y_0h_y → k^Hw_ξ _y√(P_tL_p( x_k + y))s_ξ _y_Inter - Cluster + n_0_Noise

and

    y_j =   h_j^Hw_k√(a_jP_tL_p( x_j))s_j_Desired Signal + h_j^Hw_k√(a_kP_tL_p( x_j))s_k_Intra - Cluster 
        + ∑_y ∈Φ /y_0h_y → j^Hw_ξ _y√(P_tL_p( x_j + y))s_ξ _y_Inter - Cluster + n_0_Noise,

where P_t is the transmit power of BSs. h_y →ϖ represents the channel vector from BS at y to User ϖ and ϖ∈{k,j}.

Note that SIC is carried out at User k, and hence User k will first decode the signal of User j with the following signal-to-interference-plus-noise-radio (SINR)

    γ _k → j = a_j| h_k^Hw_k|^2L_p( x_k)/a_k| h_k^Hw_k|^2L_p( x_k) + I_inter,k + σ _n^2,

where I_inter,ϖ = ∑_y ∈Φ /y_0| h_y →ϖ^Hw_ξ _y|^2L_p( x_ϖ + y) and σ _n^2 is the noise power normalized by P_t. Therefore, 1/σ _n^2 is the signal-to-noise-ratio (SNR) of our system.

If this decoding is successful, User k then decodes the signal of itself. Based on (<ref>), the SINR of  User k to decode its own message can be expressed as

    γ _k = a_k| h_k^Hw_k|^2L_p( x_k)/I_inter,k + σ _n^2.


Regarding User j, it directly decodes its own message by treating the signal of User k as the interference. Based on (<ref>), the SINR of  User j is given by

    γ _j = a_j| h_j^Hw_k|^2L_p( x_j)/a_k| h_j^Hw_k|^2L_p( x_j) + I_inter,j + σ _n^2.





§ DISTANCE DISTRIBUTIONS



In this section, we will discuss the distance distributions of the typical paired users. These distributions are the basis for analyzing the performance of our system. To simplify the notation, we first introduce a typical distribution named Rayleigh Distribution as below <cit.>.

Rayleigh Distribution: Under Rayleigh distribution, the probability density function (PDF) is given by R_p( v, σ) = v/σ ^2exp(  - v^2/2σ ^2),v > 0 and the cumulative distribution function is given by R_c( v,σ) = 1 - exp(  - v^2/2σ ^2),v > 0, where σ^2 is the variance parameter as mentioned in (<ref>).




 §.§ Distance Distribution of Near User



In the typical cluster, we assume that the distances between NOMA users and the typical BS form a set {R_i}_i=1:2K which can be denoted by ℝ_y_0. The realization of R_i is defined as r_i = x_i, where x_i ∈ℕ_y_0. Note that x_i is i.i.d. as a Gaussian random variable with σ^2. If the considered NOMA user is selected at random, we are able to drop the index i from r_i as every r_i follows the same distribution. Under this condition, r is a Gaussian random variable with variance σ^2, so the PDF of distance r is as follows <cit.>

    f_r(r)=R_p( v, σ).


Under NR scheme, User k is the nearest NOMA receiver. We assume User k is located at x_1 with a distance r_1 to the typical BS, so the PDF of r_1 can be expressed as below.

In NF scheme, the near user is the nearest NOMA user with a distance r_1=x_1. Therefore, the PDF of such distance r_1 is as follows

    f_d^1( r_1) = 2Kr_1/σ ^2exp(  - Kr_1^2/σ ^2).


Note that there are (2K-1) NOMA users distributed farther than the considered distance r_1. Therefore, the PDF of distance r_1 is f_d^1( r_1) = 2K( 1 - R_c( r_1,σ))^2K - 1R_p( r_1,σ). With the aid of Rayleigh distribution, we are able to derive the expression as above.






 §.§ Distance Distribution of Far User



In contrast to the near user, the far user in NR scheme is randomly chosen from the rest NOMA users in the typical cluster. Assuming the possible User j is located at x_f^r ∈ℕ_y_0 / x_1 with a distance r_f = x_f^r, the distribution of distance r_f is expressed as below.

The randomly selected far user in NF scheme at x_f^r has a distance r_f to the typical BS and r_f>r_1, so the conditional PDF of distance r_f is given by

    f_d^f( r_f|r_1) = {[ R_p( r_f,σ)/1 - R_c( r_1,σ),                    r_f > r_1;                           0,                      r_f≤r_1;                              ]..


When r_f≤r_1, the probability is zero as all far users are farther than r_1. When r_f>r_1, the possible User j follows Rayleigh distribution over the rang (r_1,∞ ]. Therefore, such distance distribution can be summarized as above.






§ PERFORMANCE EVALUATION



In this section, we characterize the coverage performance and system throughput of proposed NR strategy depending on the distributions of distances.

In NR scheme, we choose the nearest NOMA receiver as near user due to the best channel condition, which contributes to minimizing the coefficient a_k so that more power can be allocated to far user. On the other side, selecting User j at random from the rest NOMA users aims to provide a fair selection law. To make the tractable analysis, we first deduce the Laplace Transform of Interferences in our system.




 §.§ Laplace Transform of Interferences



We only concentrate on the Laplace transform of inter-cluster interferences because there is no interfering devices located in the typical cluster.

The inter-cluster interferences are provided from all BSs except the typical BS, then the closed-form Laplace transform of such interferences is given by

    ℒ_I( s )  = exp(  - π ^2λ _cR_L^2/2n_1∑_i_1 = 1^n_1𝒢_F^I( s,ζ _i_1q/λ)√(1 - ζ _i_1^2)),

where

    𝒢_F^I( s,g ) = ρ _N( sMC_NG_F( g )/N_NR_L^α _N) - ρ _L( N_LR_L^α _L/sMG_F( g )C_L), 
       ρ _L( v ) = _2F_1( N_L,N_L + 2/α _L;N_L + 2/α _L + 1; - v)2v^N_L/( α _LN_L + 2),
       ρ _N( v ) = _2F_1(  - 2/α _N,N_N;1 - 2/α _N; - v),(α_N>2),

_2F_1(.) is Gauss hypergeometric function. ζ _i_1 = cos( 2i_1 - 1/2n_1π) over [-1,1] denotes the Gauss-Chebyshev node and i_1=1,2,...,n_1. The parameter n_1 has a function to balance the complexity and accuracy <cit.>. Only if the n_1→∞, the equality is established.

See Appendix A.






 §.§ Coverage Probability for Near User



We pre-decide SINR thresholds τ_k and τ_j for User k and User j, respectively. These two thresholds should satisfy the condition (a_j - τ _ja_k > 0) to ensure the success of NOMA protocols <cit.>. Note that the near user has SIC procedure, which means the decoding will be success only when (γ _k → j>τ_j). Under this condition, the coverage probability is the percentage of the received SINR γ_k that excesses τ_k. Accompanying with fact that the near user under NR scheme is the nearest NOMA receiver, the coverage probability for near user User k can be defined as below

    P_k( τ _k) = P[ γ _k > τ _k|γ _k → j > τ _j,r_1 = x_1],

where ℙ[.] is the probability function.

The expressions for coverage probability of near user can be divided into two case : a_kτ _j<a_j≤a_kτ _j( 1 + 1/τ _k ) and a_j > a_kτ _j( 1 + 1/τ _k ). In most mmWave scenarios, the power allocation coefficient a_j is far larger than a_k because of the severe path loss for far user. Therefore, we focus on the more practical case with a_j > a_kτ _j( 1 + 1/τ _k ) in this paper.

When a_j > a_kτ _j( 1 + 1/τ _k ), the coverage probability of near user under NR scheme is given by

    P_k( τ _k) ≈   ∫_0^R_LΘ _L( r_1,τ _k,a_k)f_d^1( r_1)dr_1
       + ∫_R_L^∞Θ _N( r_1,τ _k,a_k)f_d^1( r_1)dr_1 .

where

    Θ _κ( r,τ ,β) =   ∑_n_κ = 1^N_κ(  - 1) ^n_κ + 1N_κ n_κexp(  - n_κψ _κτr^α _κσ _n^2/β MC_κ)
       ×ℒ_I( n_κψ _κτr^α _κ/β MC_κ),

and ψ _κ = N_κ( N_κ!)^ - 1/N_κ.

With the similar proof procedure of Theorem 1 in <cit.>, we obtain the expression as above.



In the reality, the coverage radius of the macro BS is always farther than R_L, which means the majority of BSs communicates with the considered user via an NLOS link. Note that the received power from NLOS signals is negligible <cit.> . We propose a special case as below.

Special Case 1: In a loose network, the density of BSs is small enough to ensure the majority of BSs utilizing NLOS links to provide the inter-cluster interferences. Therefore, we ignore all inter-cluster interferences and coverage probability from NLOS links, namely, ℒ_I(.)=0 and Θ _N(.)=0. Moreover, we assume α_L=2 as it is the practical value for numerous frequencies <cit.>.


Under special case 1, the closed-form coverage probability for near user is changed into

    P̂_k( τ _k) ≈K/σ ^2∑_n_L = 1^N_L(  - 1) ^n_LN_L  n_Lexp(  - A( τ _k)R_L^2) - 1/A( τ _k),

where A( τ) = n_Lψ _Lτσ _n^2/a_kMC_L + K/σ ^2.

With the fact ∫_0^B vexp(  - Av^2)dv  = 1 - exp(  - AB^2)/2A, (<ref>) can be simplified as above.




  §.§.§ Coverage Probability for Far User

In contrast to the near user, the coverage probability for User j at x_j only depends on τ_j. However, as the directional beamforming of the typical BS is aligned towards the near user, the effective channel gain for User j will fit (<ref>) rather than (<ref>). Note that the far user is randomly selected from the intra-cluster NOMA receivers, except the near user. We define the coverage probability of far user as follows

    P_j( τ _j) = ℙ[ γ _j > τ _j| r_f = x_j.].

As discussed in Lemma <ref> and Laplace transform of interferences, we obtain the coverage probability of User j as below.

The coverage probability for User j at x_j with a distance r_f is given by

    P_j( τ _j) ≈π/2n_2∑_i_2 = 1^n_2𝒢_j( τ _j,ζ _i_2q/λ)√(1 - ζ _i_2^2),

where

    G_j( τ _j,g) ≈   ∫_0^R_L∫_r_1^R_LΔ _L( τ _j,g) dr_f f_d^1(r_1)dr_1
       + ∫_R_L^∞∫_r_1^∞Δ _N( τ _j,g) dr_ff_d^1(r_1) dr_1, 
    Δ _κ( τ _j,g) =    Θ _κ( r_f,τ _j,( a_j - τ _ja_k)G_F( g ))f_d^f( r_f |r_1).


With the aid of Lemma 2, we are able to obtain the above expressions via the similar method as discussed in Theorem 1.



Under special case 1, in a loose network, the closed-form coverage probability for far user is given by

    P̂_j( τ _j) ≈π/2n_2∑_i_2 = 1^n_2𝒢̂_j( τ _j,ζ _i_2q/λ)√(1 - ζ _i_2^2),

where

    𝒢̂_j( τ _j,g) = ∑_n_L = 1^N_L(  - 1)^n_L + 1N_L  n_LK/2σ ^4Q( τ _j,g)
       ×( 1/Q( τ _j,g) + χ + Q( τ _j,g)exp(  - ( Q( τ _j,g) + χ)R_L^2)/( Q( τ _j,g) + χ)χ
        - exp(  - Q( τ _j,g)R_L^2)),

and Q( τ _j,g) = n_Lψ _Lτ _jσ _n^2/( a_j - τ _ja_k)G_F( g )MC_L + 1/2σ ^2 and χ  = 2K - 1/2σ ^2.

With the similar proof method as discussed in Corollary <ref>, we obtain Corollary <ref>.






 §.§ System Throughput



To compare with the traditional OMA method, we provide the system throughput in this part. Assuming the bandwidth B is separated equally into two parts for transferring information to User k and User j under OMA. We have the system throughput expressions for NOMA and OMA as below.

If the rate requirement for User k and User j are R_k and R_j, respectively, the equations of system throughput for NOMA and OMA are given by

    R_s^NOMA = R_kP_k( 1 - 2^R_k/B ) + R_jP_j( 1 - 2^R_j/B ), 
       R_s^OMA = R_kP_k( 1 - 2^2R_k/B )| _a_k = 1 + R_jP_j( 1 - 2^2R_j/B )| _a_j = 1.






§ NUMERICAL RESULTS



In this section, we first present the general network settings in Table. <ref> <cit.>. The reference distance is assumed to be one meter. Then, numerical results with Monte Carlo simulations are provided for analyzing the performance of our networks.

As shown in Fig. <ref>, there is a negligible difference between the theoretical results and the simulations, thereby corroborating our analysis. In the loose network, namely λ_c=1/250^2π, closed-form expressions in Corollary 1 and Corollary 2 can be the replacement of the exact analytical equations due to the easy-operation and high-accuracy. Moreover, these closed-form expressions are suitable for numerous practical scenarios, where the density of macro BSs is around 1/250^2π. On the other side, the coverage probabilities for two typical paired users are proportional to SNR and near user outperforms far user under our assumptions.





In terms of the antenna beamforming, two paired users have inverse performances as illustrated in Fig. <ref>. In general, the coverage probability of near user is a increasing function with the antenna scale M, while that of far user is opposite. Due to the randomness of the beamforming vector w_j, the coverage probability of User j is fluctuant. In our case, M=2 becomes the best choice for far user. Lastly, the large number of NOMA users K and small variance σ^2 are able to improve the coverage performance for both users.

Comparing with the OMA method, NOMA scenario with a_k=0.6 performs better in the low SNR region, while in the high SNR region, the best option is NOMA method with a_k=0.1. Accordingly, by modifying the power allocation coefficients in NOMA protocol, we are able to achieve a higher system rate than utilizing the traditional OMA technique. Fig. <ref> also demonstrates that the impact of antenna scale is various across the considered SNR range. More elements deployed at the antenna will weaken the throughput at the high SNR region. As a result, there should be an optimal M for maximizing the system rate.




§ CONCLUSION



In this paper, we propose a NR scheme in clustered mmWave networks with NOMA technique. With the aid of the stochastic geometry, the novel analytical expressions for coverage and system throughput are presented, especially we derive closed-form equations for a loose network, which can be utilized in numerous practical noise-limited scenarios. As analyzed in previous sections, the coverage probability is proportional to SNR and the number of NOMA users K. Large variance σ^2 will impair the received SINR. Lastly, our NOMA system beats the traditional OMA case regarding the system rate by adjusting the power allocation coefficients.


equationsection


§ APPENDIX A: PROOF OF LEMMA <REF>
 




For User k, the Laplace transform of interferences is

    ℒ_I^k( s ) =   𝔼[ exp(  - sI_inter,k)]
    
     =   𝔼[ e^ - s∑_y ∈Φ /y_0M| g_y → k|^2G_F( θ _k - θ _ξ _y)L_p( x_k + y)]
    
     =   ℒ_L^k(s)ℒ_N^k(s).


By applying the probability generating functional of PPP <cit.> and calculating the expectation of Gamma random variable | g_y → k|^2 and the antenna beamforming G_F(.), we obtain

    L_L^k( s ) = e^ - πλ _cλ/q∫_ - q/λ^q/λ( ∫_0^R_L( 1 - ( 1 + sMG_F( g )C_L/N_Lv^α _L)^ - N_L)vdv) dg.


Using the same proof procedure for NLOS group, we obtain

    L_N^k( s ) = e^ - πλ _cλ/q∫_ - q/λ^q/λ( ∫_R_L^∞( 1 - ( 1 + sMG_F( g )C_N/N_Nv^α _N)^ - N_N)vdv) dg.


By substituting (<ref>), (<ref>) and the definition of Gauss hypergeometric function into (<ref>), we obtain

    ℒ_I^k( s ) = exp(  - πλ _cλ R_L^2/2q∫_ - q/λ^q/λ𝒢_F^I( s,g ) dg).


Note that the Laplace transform of interferences for User j has the similar deducing procedure, so two paired users share the same expressions. Therefore we are able to drop the index k from (<ref>). After that, applying Gaussian-Chebyshev quadrature equation into (<ref>), the proof is complete.


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