Solvent treatment induced interface dipole and defect passivation for efficient and bright red quantum dot light-emitting diodes

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Coverage and Rate Analysis in Downlink L-tier HetNets with Fluctuating Beckmann Fading Jingrui Chen and Chaowei Yuan Abstract—The fluctuating Beckmann (FB) fading is a general fading model, and it extends the κ-µ shadowed fading to a power imbalance scenario. However, the FB fading contains a generalized Lauricella confluent hypergeometric function, which makes the performance analysis in downlink stochastic geometrybased heterogeneous cellular networks (HetNets) face a big challenge. In this letter, we firstly approximate the general and complex FB fading as a Gamma distribution based on the second-order moment matching method. For the application of Laplace trick, we then propose a simple and novel Erlang distribution to approximate the Gamma distribution with noninteger parameters. Using the proposed distribution as well as the Rician approximation for a small parameter, we successfully obtain approximate expressions for the coverage probability and average rate in a downlink L-tier HetNet with FB fading and validate through simulations. Index Terms—Stochastic geometry, heterogeneous cellular networks, fluctuating Beckmann fading, coverage probability, average rate.

I. I NTRODUCTION Modeling and performance analysis of heterogeneous cellular networks (HetNets) based on stochastic geometry has become a hot topic [1], [2]. To investigate the impact of the lineof-sight (LoS) fading on the coverage of stochastic geometrybased HetNets, the exponential-series expansion that a constrained non-linear optimization problem needs to be solved for exploring feasible weights and abscissas was employed to approximate a Rician fading in [3]. For a κ-µ shadowed fading scenario, a truncated functional series representation concerning a confluent hypergeometric function and a Rician approximation (RA) were utilized for the coverage analysis in single-tier cellular network [4]. Unlike [4], the generalized Laguerre polynomial expansion was applied to approximate the κ-µ shadowed fading in K-tier HetNets in [5]. It is known that the fluctuating Beckmann (FB) fading is a more general fading model [6], which generalizes the κ-µ shadowed fading to accounting for the influence of power imbalance in LoS and non LoS (NLoS) components, and extends a Beckmann fading to containing the effects of clustering and LoS fluctuation. However, the chief probability functions of the FB fading such as the probability density function (PDF) and complementary cumulative distribution function (CCDF) contain generalized Lauricella confluent hypergeometric functions (GLCHFs), which makes the approximate ways [3]– [5] face complex evaluations for the performance analysis of stochastic geometry-based HetNets. Although a mixture Gamma distribution has been utilized to replace several complex fading models in [7], it is difficult to approximate the FB fading due to the lack of adequate matching parameters. Most Jingrui Chen and Chaowei Yuan are with the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing, 100876, China, (e-mail: [email protected]; [email protected]). Last updated: June 14, 2019.

importantly, the Laplace trick of interference in stochastic geometry-based HetNets cannot be applied since the CCDF of the mixture Gamma distribution is not an exponential function. Notably, the Gamma distribution as a simple tool has been used to approximate complex distributions based on the second-order moment matching (SMM) method, including the interference approximation in stochastic geometry-based HetNets [8] and the κ-µ shadowed fading approximation [9]. What is more, by using the Gamma distribution with integer shape parameters (i.e., an Erlang distribution) to approximate complex fading model in terms of the SMM approach (termed the Erlang approximation) can avoid the non-integer derivatives of Laplace transform of interference in stochastic geometry-based HetNets. Unfortunately, the Erlang approximation suffers from approximate error problems for non-integer parameters. Motivated by the simple property of the SMM-based Gamma approximation, in this paper we first use it to replace the FB fading, and then propose a simple and novel Erlang distribution for the application of the Laplace trick and reducing the approximate error. The proposed distribution extends the Erlang distribution to containing non-integer shape parameters. Further, it closely matches the CCDFs of the FB and Gamma distributions when the non-integer shape parameter is greater than 2. Utilizing the proposed distribution and the RA for a small shape parameter, we finally derive the approximate expressions for the coverage probability and average rate in a downlink L-tier HetNet with FB fading for the desired link and an arbitrary fading for all interfering links. Numerical results show that the FB fading with rich clusters, severity fluctuation of the LoS component, and the larger power imbalance for one component improves the coverage probability and average rate. II. S YSTEM M ODEL Consider a downlink L-tier HetNet [1], where base stations (BSs) across tiers may differ in terms of their transmit powers, spatial densities, and cell-association biasing factors. All BSs are in open access for users. In tier l, l = 1, · · · , L, the locations of BSs are modeled as an independent, homogeneous Poisson point process (PPP) Ψl = {x} with density λl , and the transmit power and biasing factor are Pl and Bl respectively. Thus,Sthe union of L point processes sets up the L-tier HetNet Ψ = l∈[L] Ψl . Without loss of generality, consider a typical user who is located at the origin and each BS has an infinitely backlogged queue. The received power at a typical user from −α a BS at x ∈ Ψl is P (x) = Pl Bl Hl kxk , where Hl is the independent channel power and α (α > 2) is the pathloss exponent. Let the notation rl stand for the distance of a typical user to its serving BS located at x∗ in the l-th tier. The maximum biased-received-power association rule is utilized, i.e., the association tier is l = arg maxi∈[L] Pi Bi ri−α .

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1

Based on [1], the per-tier association probability and the P 2/α PDF of serving distance are Al = λl / i∈[L] λi T¯i and P 2/α 2 ¯ fl (rl ) = 2πλl rl exp(−πrl i∈[L] λi Ti )/Al respectively, where T¯i = Pi Bi /Pl Bl . Assuming that the channel power of the desired signal experiences the FB fading (denoted by Hx∗ ) ¯ = E [Hx∗ ]. with parameters {κ, µ, m, η, ρ2 } and mean value h According to [6, Eq. (4), Eq. (6)], the CDF and moment generating function (MGF) of the FB fading are

Exa.:1.1,5.2,1.1,0.1,0.1;10,10 Exa.:1.1,5.2,9.3,0.1,0.1;10,10 Exa.:10.6,8.3,1.1,0.1,0.1;10,10 Exa.:10.6,8.3,9.3,0.1,0.1;10,10 Exa.:10.6,8.3,9.3,10,0.1;0.1,10 Gamma Approximation (t1 ) Erlang Approximation (bt1 c) RA (n) ME Approximation (bt1 c)

0.9

CCDF of FB fading

0.8 0.7 0.6 0.5 t =1.26, 1 0.4

n=4

t1=3.81,n=15

0.88 0.48 0.86 t1=8.13,n=28 0.46 0.84 0.44

t1=10.25,n=33

0.3 0.2

FHx∗

¯ κ, µ, m, η, ρ; hx∗ = h;


m−µ/2 µ (4) hx∗ Φ2 (˜ a; b; ˜ c) , ¯µ Γ (µ + 1) υ1m h

υ2

t1=2.07,n=7

(1)

0

0.5

1

1.5

2

0.6 0.65 0.7 1 1.1 2.5

3

h*

 i−µ/2 ¯ ¯ 2ηsh 2sh 1 − µ(1+η)(1+κ) mm 1 − µ(1+η)(1+κ)  2    m , M (s) =   ρ 1 ¯ ¯ µκsh(1+η) µκsh(1+η) 1+ρ2 1+ρ2 m − µ(1+η)(1+κ)−2ηsh¯ + µ(1+η)(1+κ)−2sh¯ h

(2)   where h˜ a = −m + µ2 , −m + µ2 , m,im , b = µ + 1, √ −hx∗ η −hx∗ c1 −hx∗ c2 √ x∗ , ¯ √ ˜ c = h¯−h , h¯ , h¯ , c1,2 are the roots ηυ2 h υ2 2κ(ρ2 +η ) of υ1 s2 + βs + 1 with υ1 = υ2 + mµ(1+η)(1+ρ2 )(1+κ)2 , h i (4) −1 2 κ υ2 = µ2 (1+η)4η2 (1+κ)2 , and β = 1+κ + µ m , and Φ2 (·) is the GLCHF [10, Eq. (7.2)] and it is evaluated by [11]. The orthogonal multiple access is applied for eliminating the intra-tier interference. Similar to [12], consider the interfering signals experience an independent, identical and arbitrary fading channel, denoted by Hi , i ∈ [L]. Therefore, the signal-to-interference ratio (SIR) at a typical user assoP B H r −α ciated to its serving BS is SIR (rl ) = l l Ix∗ l , where P P −α I = i∈[L] x∈Ψi \x∗ Pi Bi Hi kxk . III. A PPROXIMATION OF THE FB FADING In order to study the performance of stochastic geometrybased HetNets with FB fading, the SMM-based Gamma approximation is utilized. Lemma 1. Based on the SMM method, the FB fading can be approximated as a Gamma distribution with parameters 2 µm(1+κ) ¯ 1.   and t2 = h/t t1 = κ(ρ2 +η 2 ) 4m 4κm+2m+µκ2 − (1+η) 2 η(1+κ)+

0.1

1+ρ2

 2  ∂2 Proof: By calculating E Hx∗ = ∂s 2 [MHx∗ (s)]|s=0 and with some algebraic manipulations according to the SMMbased Gamma approximation completes the proof. Combining with [6, Table I], various fading models are approximated by Lemma 1, including the κ-µ shadowed fading (i.e., taking η = 1 in t1 . Note that the approximate result is identical to [9, Eq. (3)]), Beckmann fading (i.e., plugging κ = K, µ = 1, m → ∞, η = q, and ρ = γ in t1 ), etc. It is well-known that the CCDF of Gamma distribution G is F¯H (hx∗ ) = 1 − γ (t1 , t2 hx∗ ) /Γ (t1 ), where γ (a, b) = x∗ R b a−1 u e−u du. The Gamma distribution can be changed as an 0 E Erlang distribution with an exponential CCDF F¯H (hx∗ ) = x∗ Pt1 −1 j (h /t ) exp (−h /t ) /Γ (j + 1) when t takes an x∗ 2 x∗ 2 1 j=0 E integer. The F¯H (h ) makes the performance analysis in x∗ x∗ stochastic geometry-based HetNets tractable. Regrettably, the

Fig. 1.

CCDF of FB fading with parameters {κ, µ, m, η1 , ρ21 ; η2 , ρ22 }.

t1 in Lemma 1 is not a positive integer all the time. And thus, E G by directly using the F¯H (hx∗ ) to replace the F¯H (hx∗ ) x∗ x∗ gives rise to approximate errors. To conquer this problem, we E (hx∗ ) can largely find that by adding a fractional part to F¯H x∗ G (hx∗ ) decrease the approximate error. Specifically, the F¯H x∗ with non-integer t1 can be approximated as bt c

1 x∗ /t2 ) E FˆHx∗ (hx∗ ) = F¯H (hx∗ ) + Γ(bt∆(h x∗ 1 c+1) exp(hx∗ /t2 ) Pbt1 c j = j=0 Uj (hx∗ /t2 ) exp (−hx∗ /t2 ) /Γ (j + 1),

(3)

where Uj = 1 for j = 0, · · · , bt1 c−1, and Uj = ∆ = t1 −bt1 c for j = bt1 c. If t1 takes a positive integer (i.e., ∆ = 0), then E FˆHx∗ (hx∗ ) = F¯H (hx∗ ). Hence, (3) can be regarded as a x∗ generalization of the Erlang distribution by including the noninteger shape parameter. For this reason, we deem appropriate to name it as the modified Erlang (ME) distribution. The G F¯H (hx∗ ) with non-integer t1 is also replaced by a Rician x∗ distribution (i.e., the RA [4]) as F˜Hx∗ (hx∗ ) =

Xn i=0

Xi j=0

j

ti (hx∗ (1 + t)) , (4) i!j! exp (hx∗ (1 + t) + t)

p

where t = t1 − 1 + t1 (t1 − 1) and t1 ≥ 1. Combing with Lemma 1, (3), or (4), the CCDF of the FB fading is replaced by an exponential function. Fig.1 plots the G E curves of (1), F¯H (hx∗ ), F¯H (hx∗ ), (3), and (4) for a difx∗ x∗ ferent range of fading parameters, including κ = {1.1, 10.6} (weak or strong LoS), µ = {5.2, 8.3} (rich and non-integer clusters), m = {1.1, 9.3} (heavy or mild fluctuation of the LoS component), the power imbalance for the same component (the in-phase component {η, ρ2 } = {0.1, 0.1} or the quadrature component {η, ρ2 } = {10, 10}), and the power imbalance for one component (the LoS component {η, ρ2 } = {10, 0.1} or the NLoS component {η, ρ2 } = {0.1, 10}). It shows that the CCDF of FB fading  with the power imbalance for the same component (i.e., η, ρ2 = {0.1, 0.1; 10, 10}) or for one component (i.e., {η, ρ2 } = {10, 0.1; 0.1, 10}) respectively have identical values. In addition, the squared errors between (1) and (3), (1) and (4) are both lower than 2.5 × 10−4 as t1 ≥ 2, which indicates that the RA and ME approximation closely match the FB fading compared to the Erlang approximation. Since (3) needs a few summation terms compared to (4), it simplifies the theoretical evaluations in special cases.

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E0

IV. C OVERAGE PROBABILITY AND AVERAGE RATE First of all, we provide the closed-form expression of the j-th derivatives of the Laplace transform of interference. Lemma 2. For the downlink HetNet described in Section II, ∂j α the closed-form expression for ∂z j [LI (zsrl )]|z=1 is  2 N q j j! 2πrl Eq Z(s) q XY αq!(−1) ∂j = [LI (zsrlα )] , (5) ∂z j L−1 (srα ) N ! z=1

I

gj q=1

q

l

2/α P ¯2/α , the Laplace transwhere Z (s) = (sPl Bl ) i∈[L] λi Ti
α form of interference LI (srl ) = exp −πrl2 Z (s) E0 , gj = Pj j Pj { (N1 , · · · , Nj ) ∈ N(∪{0}) q=1 Nq q = j, q=1 Nq = q},

R∞ 1 − LHi rlα u−α/2 du, and E0 = (sPl Bl )−2/α 2/α

Eq = EHi [hi

=

2/α EHi [hi γ

R∞



1 − LHi u−α/2 du, and Eq

(t2 /θ)2/α

(q − 2/α, θhi /t2 )].

Proof: Substituting the SIR into the definition of the coverage probability [1, Eq. (12), Eq. (13)] yields P pc (θ) = l∈[L] Al Erl [P[ SIR (rl ) > θ| rl ]] P = l∈[L] Al E{rl ,I} [P( Hx∗ > θIrlα /Pl Bl | rl )] # "  j   bt 1c (a) P P θIrlα θIrlα Uj ≈ Al E{rl ,I} exp − Pl Bl t2 rl j! Pl Bl t2 j=0 l∈[L] i h h  i bt c 1 (b) P P Al Uj θzIrlα ∂j = E − rl j j EI exp r l ∂z P B t (−1) j! l l 2 l∈[L] j=0 j 1c P Q P (c) P bt

=

l∈[L] j=0 gj q=1

z=1

R∞ 0

2πλl rl Uj 2πrl2 vEq (−1)j Nq ! exp

(

q Nq

/αq!(−1) ) (πrl2 Bl )

drl , (9)

γ (q − 2/α, sPl Bl hi )].

Proof: According to the definition of the Laplace transform, we have LI (zsrlα ) = E [exp (−zsIrlα )] (a)

= E [exp (−zsrlα I1 )] E [exp (−zsrlα I2 )]
= exp −πrl2 Z (s) E0 (z) ,

(6)

(b)

where step (a) is enabled by the fact that Ψ and {Hi } are independent, where Hi follows fromPan independent, identical −α and arbitrary distribution, I1 = , x∈Ψl \x∗ Pl Bl Hl kxk P P −α and I2 = , (b) is obtained i∈[L]\l x∈Ψi Pi Bi Hi kxk by using the probability generating functional of PPP and the change of variables, where E0 and Z (z) are given in Lemma 2. The final expression is derived by utilizing ∂j the Fa`a di Bruno’s formula [13], i.e., ∂z [f (y (z))] =  N q j P Qj 1 ∂q 1 j!f (y (z)) gj q=1 Nq ! q! ∂zq [y (z)] , where gj is given in Lemma 2, the change of variables, plugging in z = 1, and some algebraic simplification. Lemma 2 can be extended to the generalized BS association 2/α policy [14] by using the λl E[Xl ] to replace λl , where Xl is a long-term shadowed fading. Combining with Lemma 1, Lemma 2, (3), and (4), the approximate expressions for the coverage probability and average rate are as follows. Theorem 1. For the downlink HetNet described in Section II, the approximate expressions of the coverage probability based on (3) and (4) are respectively pˆc (θ) ≈

bt1 c XX X λl Uj Clj Γ (Dj ) j

l∈[L] j=0 gj

Dj

,

(7)

(−1) Bl

and p˜c (θ) ≈

n X i X XX l∈[L] i=0 j=0 gj

λl ti C˜lj Γ (Dj ) , j ˜ Dj i! exp (t) (−1) B

(8)

l

where t1 and t2 are given by Lemma 1, Uj is given by (3),  Nq P 2/α j Q 2Eq i∈[L] λi T¯i 1 t is given by (4), Clj = , Nq ! αq!(−1)q (θ/t2 )−2/α q=1 Pj P 2/α 2/α Dj = 1 + q=1 Nq , Bl = (1 + (θ/t2 ) E0 ) i∈[L] λi T¯i , ˜l = Bl (t2 → 1/ (1 + t)), C˜lj = Clj (t2 → 1/ (1 + t)), B

=

where step (a) follows from the assumption that the desired signal experiences the FB fading which is approximated by Lemma 1 and (3) (or (4)), (b) follows from j ∂j the fact that (−α) exp (−α) = ∂z j [exp (−αz)] |z=1 , (c) is obtained by substituting (5), s = θ/Pl Bl t2 , Al , and 2/α P ¯2/α . Since fl (rl ) in it, where v = (θ/t2 ) i∈[L] λi Ti P j Qj Qj Nq = z q=1 Nq q=1 αNq , by calculating the inq=1 (αz) tegral through [15, Eq. (3.351.3)], and with some algebraic manipulations yields (7) (or (8)). This completes the proof. Theorem 2. By utilizing the derived coverage probability pc (θ) (i.e., pˆc (θ) or p˜c (θ)), the average rate of the downlink L-tier HetNet described in Section II is XnGCQ ak pc (bk ) log e 2 R≈ , (10) k=1 1 + bk 2

π sin ϕk where pc (·) is given by Theorem 1, ak = 4nGCQ cos2 (ek ) , bk = tan (ek ), ek = π (1 + cos ϕk ) /4, and ϕk = π (2k − 1) /2nGCQ for k = 1, 2, · · · , nGCQ .

Proof: Based on the definition of the average rate [1, Eq. (23), Eq.(24)], we get P R = l∈[L] Al Erl [ log2 (1 + SIR (rl ))| rl ] R (a) P Erl [ 0∞ P[ SIR(rl )>eτ −1|rl ]dτ ]log2 e = l∈[L] A−1 l R ∞ Erl [P[ SIR(rl )>u|rl ]]log2 e (b) P = l∈[L] Al 0 du 1+u (c) R ∞ pc (u)log e 2 = 0 du, 1+u

(11)

R where (a) follows the fact that E [X] = τ >0 P (X > τ )dτ for X > 0, (b) uses the change of variable and exchanges between the integral and the expectation, and (c) follows the definition of the coverage probability pc (θ) = P A l∈[L] l Erl [P[ SIR (rl ) > θ| rl ]]. The final expression is achieved by applying the Gaussian-Chebyshev quadrature (GCQ) [16, Eq.25.4.39] rule to remove the integral. V. N UMERICAL RESULTS The results are depicted for a two-tier HetNet with BS densities {λ1 , λ2 } = {1.10 × 10−6 , 1.28 × 10−6 }, transmit powers {P1 , P2 } = {43, 23}dBm, biasing factors {B1 , B2 } = {1, 2}, path-loss exponent α = 4, tier association rule

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1

Ana. ME:1.1,5.2,1.1,0.1,0.1;10,10 Ana. ME:1.1,5.2,9.3,0.1,0.1;10,10 Ana. ME:10.6,8.3,1.1,0.1,0.1;10,10 Ana. ME:10.6,8.3,9.3,0.1,0.1;10,10 Ana. ME:κ−µ shad.:10.6,8.3,9.3,1,1 Ana.ME:10.6,8.3,9.3,10,0.1;0.1,10 Ana. RA:1.1,5.2,1.1,0.1,0.1;10,10 Ana. RA:10.6,8.3,1.1,0.1,0.1;10,10 Ana. Erlang Approximation Simulations

t1=3.81,n=15 0.9

Coverage Probability

0.8 0.7

t1=2.07,n=7 t1=1.26,n=4

0.6 0.5

VI. C ONCLUSION

0.885 0.88

0.4

0.875

t1=10.25,n=33

0.87

0.3

t1=9.13,n=30

0.865

t1=8.13,n=28

0.2 −4.4 −10

−4.2

−8

−6

−4

−3.8

−4

−2

0

2

4

6

8

10

SIR Threshold in dB

Fig. 2. Coverage probability with a wide range of fading parameters {κ, µ, m, η1 , ρ21 ; η2 , ρ22 }. −3

7

x 10

ME(1.1,5.2,1.1,0.1,0.1;10,10) ME(1.1,5.2,9.3,0.1,0.1;10,10) ME(10.6,8.3,1.1,0.1,0.1;10,10) ME(10.6,8.3,9.3,0.1,0.1;10,10) ME(10.6,8.3,9.3,1,1) ME(10.6,8.3,9.3,10,0.1;0.1,10) RA(1.1,5.2,1.1,0.1,0.1;10,10) RA(10.6,8.3,1.1,0.1,0.1;10,10) Erlang(1.1,5.2,1.1,0.1,0.1;10,10) Erlang(1.1,5.2,9.3,0.1,0.1;10,10) Erlang(10.6,8.3,1.1,0.1,0.1;10,10) Erlang(10.6,8.3,9.3,0.1,0.1;10,10) Erlang(10.6,8.3,9.3,1,1) Erlang(10.6,8.3,9.3,10,0.1;0.1,10)

t1=2.07,n=7 −4

x 10

6

Squared error

2 5

0

4

t1=1.26,n=4

−5

0

5

10

t =3.81,n=15 1

−5

x 10

3 8 6 2

4

−6

0

t1>8,n>28 −5

0

5

5

10 0

0 −8

−6

R EFERENCES

x 10

2 1

−4

−2

0

2

−5 4

0 6

5

10 8

10

SIR Threshold in dB

Fig. 3. bility.

In this paper, we have first approximated the FB fading as a Gamma distribution according to the SMM method, and then proposed the ME distribution to replace the Gamma distribution for utilizing the Laplace trick of interference in stochastic geometry-based HetNets. The proposed distribution closely matches the CCDFs of Gamma and FB distributions as t1 ≥ 2. Applying the ME approximation and the RA for a small parameter, we have finally derived the coverage probability and average rate in the downlink L-tier HetNet when the desired signal experiences FB fading and the interfering signals undergo an arbitrary fading. Numerical results indicate that the FB fading with rich clusters, heavy fluctuation of the LoS component, and the larger power imbalance for one component increases the coverage probability and average rate.

Squared error between simulated and approximate coverage proba-

l = arg maxi∈[L] Pi Bi ri−α , and the FB fading for the desired signal and Rayleigh fading for all interfering signals. The values of non-integer parameters of FB fading (i.e., {κ, µ, m, η1 , ρ21 ; η2 , ρ22 }) are identical to Section III. A no power imbalance scenario {η, ρ2 } = {1, 1} (i.e., the κ-µ shadowed fading) is also considered. For simple calculations, we set nGCQ = 100 for Theorem 2. Fig. 2 plots the coverage probability when the FB fading takes a different range of parameter values. Compared to the Erlang distribution, it shows that the ME-based and RA-based results exactly match the simulation results as t1 ≥ 2. When t1 < 2, the RA is close to the simulated result. Besides, it illustrates that rich clusters µ, severity m, and the larger power imbalance for one component {η, ρ2 } = {10, 0.1; 0.1, 10} (i.e., t1 = 10.25) provide high coverage probability. Fig. 3 depicts the squared error between the simulated and approximate coverage probability. The squared errors between the simulated and ME, the simulated and RA are both lower than 4.0 × 10−4 when t1 ≥ 2. It demonstrates that the RA and ME both reduce the approximate errors compared to the Erlang approximation. Note that the summation term of RA is bigger than 7, which makes the theoretical evaluations intractable. When the parameters of the approximated FB fading change from t1 = 1.26 to t1 = 10.25, the simulated and theoretical average rate are close to each other, and both of them increase from 2.24bit/Hz to 2.43bit/Hz. Moreover, it shows similar insights as to the influence of FB fading on coverage probability.

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