On the Distribution Function of the Generalized Beckmann Random ...

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCOMM.2017.2786665, IEEE Transactions on Communications 1

On the Distribution Function of the Generalized Beckmann Random Variable and Its Applications in Communications Bingcheng Zhu, Zhaoquan Zeng, Julian Cheng, Senior Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE

Abstract—The Beckmann distribution has a wide range of applications in radio-frequency (RF) communications, free-space optical (FSO) communications and underwater wireless optical communications (UWOC). However, the cumulative distribution function (CDF) of the Beckmann random variable (RV) does not have a closed-form expression, which makes it challenging to derive analytical solutions for the outage probability of systems involving Beckmann RVs. In this work, we study the generalized Beckmann distribution, which includes the Beckmann, Rayleigh, Rician, Nakagami-m, Hoyt, κ-µ, η-µ, single-sided Gaussian and the Beaulieu-Xie distributions as special cases. Three approaches are proposed to estimate the CDF of the generalized Beckmann distribution, including closed-form upper and lower CDF bounds, single-fold integration based on the closed-form characteristic function, and a left-tail CDF approximation. We compare the three approaches in terms of the ranges of applications and the computation time complexity. Based on the new CDF estimation techniques, one can efficiently evaluate the outage probabilities of pointing-error-limited FSO systems, UWOC systems, and maximum-ratio combining over arbitrarily correlated generalized Beckmann channels. Index Terms—Communication system performance, Fading channels, Optical communication, Rayleigh channels, Rician channels, Underwater communication.

I. I NTRODUCTION The Beckmann distribution is a versatile mathematical tool in communication theory. It generalizes the Rayleigh, Rician, Hoyt and single-sided Gaussian distributions [1], [2], [3, Chap. 2], thus it is also referred to as the “generalized Rician distribution”. A Beckmann random variable (RV) is the 2-norm of a vector comprising two independent Gaussian RVs, where the Gaussian RVs can have different means and variances. The Beckmann distribution has been used to model the radial displacement of laser light on the receiving plane in free-space optical communications (FSO) [4], and it is also related to the pointing error of underwater wireless optical Bingcheng Zhu is with College of Telecommunication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu, China. (e-mail: [email protected]) Zhaoquan Zeng and Julian Cheng are with School of Engineering, The University of British Columbia, Kelowna, BC, Canada. (e-mail: [email protected], [email protected]) Norman C. Beaulieu is with the School of Information and Communication Engineering, Beijing University of Posts and Telecomunications (BUPT), Beijing, P.R. China, 100876. (e-mail: [email protected]) This work is supported by National Science Foundation of China (61322112, 61531166004), NUPTSF(NY216008), Young Elite Scientist Sponsorship Program by CAST (YESS20160042), and an NSERC Discovery Grant.

communications (UWOC) [5] as well as the fading amplitude of radio frequency wireless communications when the rough scattering surface is of finite conductance [6]–[8]. Despite the wide range of applications, the Beckmann distribution has not been as popular as its special cases, such as the Rician and the Rayleigh distributions, and an essential reason is the mathematical intractability of using the Beckmann distribution in theoretical analyses. To this end, it remains as an open problem to derive closed-form expressions for the probability density function (PDF) or cumulative distribution function (CDF) of the Beckmann RV [9]–[12]. The difficulty arises because the variances of the independent Gaussian components of the Beckmann radial amplitude have unequal variances and nonzero means [12], [13]. The PDF of the Beckmann distribution can be expressed in form of a single integral [6, eq. (9)], [10] or infinite series [10], [11], and the CDF was recently obtained in form of a single integral in [12, eq. (8)]. A closed-form outage probability expression for maximum-ratio combining (MRC) over Beckmann fading channels was derived in [14] based on the generalized moment-generating function of the Beckmann distribution, but the result is only valid for interferencelimited scenarios where the additive Gaussian noise is assumed negligible. The symbol error rate of MRC over Beckmann fading channels was expressed in the form of infinite series in [1, eq. (14)], and the outage probability of orthogonal spacetime coding system was obtained in [15, eq. (18)] in the form of infinite series as well. Recently, the symbol error rate and outage probability bounds for diversity receptions over correlated Beckmann fading channels were derived in [16], but the derived bounds are not tight in low to medium signal-tonoise ratio (SNR) regions, and the derivations circumvented the problem of solving for the CDF and the PDF of the Beckmann RV. Another recent work also studied bounds on the Beckmann CDF [17], but the discussion was limited in 2dimensional space, which cannot provide solutions for systems with multiple links. In this work, closed-form CDF bounds on the Beckmann distribution are derived, and the bounds can be made arbitrarily tight by adjusting an integer parameter N . The derivation applies a geometrical bounding technique for the integral region, which was used to bound the Marcum Q-functions in [18]–[20]. Then we study the generalized Beckmann RV, which is the 2-norm of a vector comprising multiple correlated Gaussian RVs. The generalized Beckmann distribution

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includes Beckmann, Nakagami-m, κ-µ, η-µ and Beaulieu-Xie distributions as special cases [13], [21, eqs. (6), (22)], [22]. The aforementioned bounding technique for the Beckmann distribution is generalized to higher dimensions to derive the arbitrarily tight CDF bounds on the generalized Beckmann distribution. In addition, another two CDF estimation methods are proposed as complements to the proposed bounds, where the first method is based on the closed-form characteristic function (CF) of gamma RVs, and the second method is based on asymptotic analysis techniques. The three approaches are compared in terms of accuracy and computation time complexity. We also demonstrate the applications of the CDF estimation of the generalized Beckmann distribution in evaluating the outage probabilities of pointing-error-limited FSO systems in weak turbulence scenarios, UWOC systems, and MRC over arbitrarily correlated generalized Beckmann fading channels. The remainder of this paper is organized as follows. In Section II, we review the construction of the Beckmann RVs and their PDF and CDF expressions. In Section III, we derive the Beckmann CDF bounds. Subsequently, we study the generalized Beckmann distribution, derive its CDF upper and lower bounds, and introduce another two CDF estimation schemes in Section IV. Tightness of the bounds is verfied in Section V. Applications of the CDF bounds are shown in Section VI, and numerical results are given in Section VII. Finally, Section VIII makes several concluding remarks.

II. T HE B ECKMANN R ANDOM VARIABLE

A Beckmann RV R is generated by two Gaussian RVs X and Y as R=

√ X2 + Y 2

(1)

2 where X and Y are independent and X ∼ N (µX , σX ), 2 Y ∼ N (µY , σY ). The Beckmann distribution is versatile as it specializes to the Rician distribution when σX = σY , the Rayleigh distribution when µX = µY = 0 and σX = σY , the Hoyt distribution when µX = µY = 0 and σX ̸= σY , the single-sided Gaussian when X = 0 or Y = 0.

The PDF of R can be expressed in an integral form as [6, eq. (9)] r 2πσX σY [ ] ∫ 2π 2 2 (r cos θ − µX ) (r sin θ − µY ) × exp − − dθ 2 2σX 2σY2 0 (2)

y

F (r ) Q n- ( r )

x

Q n+ ( r )

Fig. 1.

The integral region in (4) and its approximations.

[12, eq. (8)] FR (r) = ∫2π

( ) 1 exp −A2 γ (θ0 ) 2πσX σY

( )) 1 ( 1 − exp −γ (θ) r2 + ρ (θ) r 2γ (θ) 0 (3) ( 2 ) √ ρ (θ) ρ (θ) π exp + 3 4γ (θ) 4γ 2 (θ) ( [ ] [ ]) ρ (θ) 2γ (θ) r − ρ (θ) √ × erf √ + erf dθ 2 γ (θ) 2 γ (θ) ∫x 2 θ where erf (x) = √2π 0 exp(−t2 )dt, γ(θ) = cos + 2 2σX ( ) 2 θ0 θ0 sin θ and ρ(θ) = A cos θσcos + sin θσsin , where θ0 = 2 2 2 2σY Y √X 2 2 arctan(µY /µX ) and A = µX + µY . Eqs. (2) and (3) cannot be further simplified, thus numerical integration or estimation has to be used to estimate the outage probability and error rates involving the Beckmann distribution. Based on the definition of the Beckmann RV in (1), we can also obtain the CDF of the Beckmann RV as (√ ) FR (r) = Pr X2 + Y 2 ≤ r ∫∫ = fX (x) fY (y) dxdy √ 2 2 x +y ≤r ] [ ∫∫ 2 2 (y − µY ) 1 (x − µX ) − dxdy = exp − 2 2πσX σY 2σX 2σY2 √ ×

x2 +y 2 ≤r

(4) where fX (x) and fY (y) are, respectively, the PDFs of X and Y . The integral in (4) is more intuitively and geometrically illustrated than that in (3), and we shall use it as the starting point in bounding FR (r).

fR (r) =

and its CDF can also be expressed as a single-fold integral as

III. B OUNDING THE C UMULATIVE D ISTRIBUTION F UNCTION OF THE B ECKMANN D ISTRIBUTION A. Bounding the Integral Region We define the integral region in (4) as } { √ Φ (r) = (x, y) : x2 + y 2 ≤ r

(5)



which is a disk with radius r and is centered at the origin, and it is shown in Fig. 1. The integral region can be split into N subsections as

According to (6) and (10), we obtain

Φ (r) =Φ (r) ∩ {(x, y) : |y| ≤ r} [{ } 1 =Φ (r) ∩ (x, y) : |y| ≤ r N } { 1 2 ∪ (x, y) : r < |y| ≤ r ∪ · · · N N }] { N −1 N r < |y| ≤ r ∪ (x, y) : N N =Θ1 (r) ∪ Θ2 (r) ∪ · · · ∪ ΘN (r)

and according to (6) and (13), we obtain

− Φ(r) ⊇ Θ− 1 (r) ∪ · · · ∪ ΘN (r)

+ Φ(r) ⊆ Θ+ 1 (r) ∪ · · · ∪ ΘN (r) .

(6)

(16)

B. Bounds on the Cumulative Distribution Function 1) Upper Bound to the CDF: According to (14) and (16), we can obtain

where | · | is the modulus sign, N is a positive integer and { √ r r } Θn (r) = (x, y) : x2 + y 2 ≤ r, (n − 1) ≤ |y| ≤ n N N (7) for n = 1, · · · , N , which is a slice of the disk Φ (r), and obviously, Θk (r) ∩ Θl (r) = ϕ, ∀k ̸= l (8) where ϕ is the empty set, 1 ≤ l ≤ N and 1 ≤ k ≤ N . According to [23], for two regions Ω1 = {(x, y) : f (x, y) ≤ 0} and Ω2 = {(x, y) : g(x, y) ≤ 0)}, if f (x, y) ≤ 0 ⇒ g(x, y) ≤ 0 where “⇒” means “results in”, it can be obtained that Ω1 ⊆ Ω2 . Noting that √ { √ { ( )2 x2 + y 2 ≤ r |x| ≤ r2 − Nr n ⇒ r r r r N (n − 1) ≤ |y| ≤ N n N (n − 1) ≤ |y| ≤ N n (9) where the left-hand side of “⇒” is the inequality set defining Θn . Therefore, we can lower bound the integral region Θn (r) as √ { ( )2 } ∆ (x, y) : |x| ≤ r2 − Nr n , − Θn (r) ⊇ Θn (r) = r r N (n − 1) ≤ |y| ≤ N n (10)

FR (r) ∫∫ = Φ(r)

≤

[ ] 2 2 1 (x − µX ) (y − µY ) exp − − dxdy 2 2πσX σY 2σX 2σY2

N ∫∫ ∑

n=1 + Θn (r)

[ ] 2 2 (x − µX ) 1 (y − µY ) exp − − dxdy 2 2πσX σY 2σX 2σY2 (17)

where the inequality holds because the integrand is positive. Therefore, we can obtain a closed-form bound for FR (r) if we can derive closed-form expressions for the summands in (17). The double integrals in (17) can be factorized as [ ] ∫∫ 2 2 1 (x − µX ) (y − µY ) exp − − dxdy 2 2πσX σY 2σX 2σY2 Θ+ n (r) √ 2 r r 2 −( N (n−1))

∫

= −

√ 2 r r 2 −( N (n−1))



r

∫N n

 × r N

(n−1)

r −N

∫(n−1)

+

and obviously, − Θ− k (r) ∩ Θl (r) = ϕ, ∀k ̸= l

(15)

(11)

r n −N

[ ] 2 (x − µX ) 1 √ exp − dx 2 2σX 2πσX

[ ] 2 (y − µY ) 1 √ exp − dy 2σY2 2πσY [

1 (y − µY ) √ exp − 2σY2 2πσY

2

]

  dy 

(18) (r) where 1 ≤ l ≤ N and 1 ≤ k ≤ N . An illustration of Θ− n where each integral can be expressed as a Gaussian Q∫∞ is shown in Fig. 1. Similarly, it can be proved that function, which is defined as Q(x) = √12π exp(−t2 /2)dt,  √ { √ x  |x| ≤ r2 − ( r (n − 1))2 x2 + y 2 ≤ r and thus we obtain N ⇒ [ ] r r ∫∫  r (n − 1) ≤ |y| ≤ r n 2 2 N (n − 1) ≤ |y| ≤ N n N N 1 (x − µX ) (y − µY ) exp − − dxdy (12) 2 2πσX σY 2σX 2σY2 where the right side of “⇒” defines Θn (r). Therefore, we can Θ+ n (r)  ( √ )  obtain 2 r − r 2 −( N (n−1)) −µX √ { } ( )2  Q  σX   ∆ (x, y) : |x| ≤ r2 − Nr (n − 1) ,  Θn (r) ⊆ Θ+ (r) = (√ )  = n  r r 2 r r 2 −( N (n−1)) −µX (19)   N (n − 1) ≤ |y| ≤ N n −Q σX (13) ) (r )  (r  (n−1)−µ n−µ and Q N σY Y − Q N σY Y + + ( r ) ( r )  Θk (r) ∩ Θl (r) = ϕ, ∀k ̸= l (14) × − N n−µY − N (n−1)−µY +Q − Q σY σY where 1 ≤ l ≤ N and 1 ≤ k ≤ N . An illustration of Θ+ n (r) ∆ + is shown in Fig. 1. = Gn (r). 0090-6778 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


Substituting (19) into (17), we obtain FR (r) ≤

N ∑

and

G+ n (r).

(20)

n=1

2) Lower Bound to the CDF: According to (11) and (15), we obtain ( ) ∫ ∫ exp − (x−µ2X )2 − (y−µ2Y )2 2σX 2σY FR (r) = dxdy 2πσX σY Φ(r) ( ) (21) 2 2 N ∫ ∫ exp − (x−µ2X ) − (y−µ2Y ) ∑ 2σX 2σY ≥ dxdy. 2πσ σ X Y n=1 Θ− n (r)

Following similar procedures for (18) and (19), we obtain [ ] ∫∫ 2 2 (x − µX ) 1 (y − µY ) exp − − dxdy 2 2πσX σY 2σX 2σY2 Θ− n (r)

(

=

( Q

 ×

(

) √ 2 r n) −µX − r 2 −( N σX r N

2

) )

r r 2 −( N n) −µX σX

−Q

)

(n−1)−µY ( rσY ) − N n−µY +Q σY

Q

(√

(r )  n−µ − Q N σY Y ( r )  − N (n−1)−µY −Q σY

) (r )) ( (r N (n − 1) − µY N n − µY −Q lim Q σY →0 σY σY { 1, µY ∈ U + (n) = 0, else.

We can substitute (26) and (27) into (19) and obtain lim G+ n (r)  ( √ ) 2 r  − r 2 −( N (n−1)) −µX   Q  σX    (√ ) 2 r = r 2 −( N (n−1)) −µX  −Q , µY ∈ U + (n) ∪ U − (n)  σX      0, else. (28)

σY →0

Therefore, when N → ∞, we have lim G+ n (r)  ( √ ) 2 2   Q − r −µY −µX  σX   (√ ) r 2 −µ2Y −µX = −Q , µY ∈ U + (n) ∪ U − (n)  σX     0, else (29)

σY →0,N →∞

since the bounds in (24) and (25) all approach µY as N → ∞. Therefore, eq. (29) leads to

= G− n (r). ∆

(22) Substituting (22) into (21), and according to (20) we can obtain N ∑ n=1

G+ n (r) ≥ FR (r) ≥

N ∑

G− n (r).

(27)

(23)

σY →0,N →∞

=

 

Q



0, else.

n=1

The upper and lower bounds in (23) can be made arbitrarily tight by choosing a sufficiently large N . An intuitive explanation for the tightness is presented in Fig. 1, which shows that the difference between the bounding integral regions in (15) and (16) becomes quite small when the number of slices is large. A rigorous proof of the tightness will be given in Section V, which covers a more general case. C. Special Cases When σY → 0 or σX → 0 To simplify the notations, we define two sets { } r r U − (n) = µY : − n ≤ µY ≤ − (n − 1) (24) N N and { r } r U + (n) = µY : (n − 1) ≤ µY ≤ n . (25) N N Therefore, when σY → 0, we have ( ( r ) ( r )) − N n − µY − N (n − 1) − µY lim Q −Q σY →0 σY σY { − 1, µY ∈ U (n) = 0, else (26)

N ∑

lim

G+ n (r)

( n=1 √2 −

r −µ2Y −µX σX

) −Q

(√

r 2 −µ2Y −µX σX

) , r2 > µ2Y (30)

Analogously, it can also be proved that N ∑

lim

σY →0,N →∞

=

 

Q



0, else.

( n=1 √2 −

G− n (r)

r −µ2Y −µX σX

) −Q

(√

r 2 −µ2Y −µX σX

) , r2 > µ2Y (31)

Comparing (23) with (30) and (31), we obtain FR (r)  ( √ ) (√ ) − r 2 −µ2Y −µX r 2 −µ2Y −µX  Q − Q , r2 > µ2Y σX σX =  0, else (32) as σY → 0 according to the squeeze theorem. Considering the symmetry of x and y in (4), we can also obtain FR (r)  ( √ ) (√ ) r 2 −µ2X −µY − r 2 −µ2X −µY  −Q , r2 > µ2X Q σY σY =  0, else (33)



Integral region of the exact CDF

as σX → 0.

Integral region of the lower bound

IV. T HE G ENERALIZED B ECKMANN R ANDOM VARIABLE A. Bounds on the CDF A generalized Beckmann RV1 can be defined as the square root of the quadratic sum of M correlated Gaussian random variables, which can be generated as v u M u∑ S = ∥g∥ = t G2m (34) m=1

where ∥·∥ denotes the 2-norm of a vector, and g = [G1 , G2 , · · · , GM ]T is a Gaussian random vector whose covariance matrix is [ ] Rg = E (g − µg )(g − µg )T (35) where µg is the mean vector of g. The CDF of S in (34) can be expressed as ( ) 2 FS (s) = Pr (S ≤ s) = Pr ∥g∥ ≤ s2 . (36) Since the covariance matrix Rg is positive definite, we can decompose Rg as Rg = ST ΛS, where S is an M × M orthogonal matrix, and Λ = diag(λ1 , λ2 , · · · , λM ) is an M × M diagonal matrix where the λm ’s are the eigenvalues of Rg . The correlated Gaussian vector g can be decorrelated to a random vector c by the following mapping rule c = Sg

(37) T

where S is an orthogonal matrix and c = [C1 , C2 , · · · , CM ] , which has mean vector µc = Sµg , and it can be proved that c has a diagonal covariance matrix Λ. Therefore, based on (36) and (37), we can obtain an M -fold integral CDF expression as ( ) 2 FS (s) = Pr ∥c∥ ≤ s2 ( ) M ∑ 2 1 −1 ∫ exp − 2 λm (cm − µc,m ) (38) √m=1 = dcM M M ∏ 2 cT (2π) |λm | M cM ≤s m=1

where cM = [c1 , · · · , cM ]T , cm is the mth element of vector cM , and µc,m is the mth element of vector µc , and dcM , dc1 · · · dcM . Integrands in the form of (38) can be denoted as ) ( k ∑ 2 exp − 21 λ−1 (c − µ ) m c,m m ∆ √m=1 fck (ck ) = . (39) k k ∏ (2π) |λm | m=1

Integral regions in the form of that in (38) can be expressed as { } M ∑ M 2 2 Φ (s) = (c1 , · · · , cM ) : cm ≤ s (40) m=1 1 In

some works, the generalized Beckmann fading is also referred to as “generalized Rician fading” [1].

3− Fig. 2. The integral region for the lower bound Θ3− 1 (s) ∪ Θ2 (s) ∪ · · · ∪ 3− 3 ΘN (s) and the exact integral region Φ (s) shown in a three-dimensional space.

which can also be expressed as { } M −1 ∑ ΦM (s) = (c1 , · · · , cM ) : c2m ≤ s2 − c2M .

(41)

m=1

Following similar procedures to those in (6), we can segment ΦM (s) into N pieces as M M ΦM (s) = ΘM 1 (s) ∪ Θ2 (s) ∪ · · · ∪ ΘN (s)

where ∆ ΘM n =

  (c , · · · , cM ) :  1

M −1 ∑

c2m ≤ s2 − c2M ,

m=1 n−1 N s

≤ |cM | ≤

n Ns

(42)

  

.

(43)

The integral region ΘM n (s) can then be bounded as   M −1 ( n )2  ∑  2 2 c ≤ s − s , ∆ − N ΘM (c , · · · , cM ) : m=1 m n (s) =  1  n n−1 N s ≤ |cM | ≤ N s + ⊂ ΘM (s) ⊂ ΘM n (s) n  M −1 ( )2  ∑  c2m ≤ s2 − n−1 s , ∆ N = (c1 , · · · , cM ) : m=1 .   n−1 n s ≤ |c | ≤ s M N N (44) − (s)’s in a 3-dimensional space is shown The union of the ΘM n in Fig. 2. According to (42) and (44), eq. (38) can be bounded as ∫ N ∑ fcM (cM )dcM n=1M −1 ∑

∫

m=1

2

n n c2m ≤s2 −( N s) , n−1 N s≤|cM |≤ N s

≤

fcM (cM )dcM 2 cT M cM ≤s

≤

N ∑

∫ fcM (cM )dcM .

n=1M −1 ∑ m=1

2

n−1 n c2m ≤s2 −( n−1 N s) , N s≤|cM |≤ N s

(45)



where the eigenvalues in the kth group are all identical to a value λk , and the number of elements in the kth group is K ∑ denoted by Mk , so Mk = M . The elements in c can be

After some simplification, we can obtain ) (n )  ( n−1  s−µ s−µ N ∑ Q N √λ c,M − Q N √λ c,M M  ( nM ) ( n−1 )  − N s−µc,M − N s−µc,M √ √ +Q − Q n=1 λM λM ∫ fcM −1 (cM −1 )dcM −1 × M −1 ∑ m=1

2

n s) c2m ≤s2 −( N

≤ Pr (S ≤ s) ≤ ) (n )  ( n−1  s−µ s−µ N ∑ Q N √λ c,M − Q N √λ c,M M  ( nM ) ( n−1 )  − N s−µc,M − N s−µc,M √ √ +Q − Q n=1 λM λM ∫ fcM −1 (cM −1 )dcM −1 × M −1 ∑ m=1

k=1

reordered according to their variances in (49), and we can obtain ]T ∆ [ ˜c = cT1 , cT2 , · · · , cTK (49) (46)

∫

N ( )∑ ( ) rM n1 , s− rM −1 n2 , s− M M −1

(

r2 nM −1 , s− 2

k=1 m=1

nM −1 =1

n1 =1 N ∑

(

k=1

(

− 12 λ−1 k

λ1,1 , λ1,2 , · · · , λ1,M1 , λ2,1 , · · · , λ2,M2 , · · · , λK,1 , · · · , λK,MK | {z } | {z } {z } | =λ1

=λ2

=λK

2

(52)

which denotes the joint PDF of the RVs in ck . When M1 = M2 = · · · = MK = 1, eq. (51) specializes to (38). By splitting the integral region as we did in (42), the CDF in (51) can be expressed as FS (s) =

∫

N ∑

k=1 m=1

) ( + )] [ ( + s − µc,1 −s1 − µc,1 √ −Q 1√ . × Q λ1 λ1 where the functions rm (n, s)’s are defined as ( n−1 ) (n ) s − µc,m N s − µc,m √ rm (n, s) = Q −Q N √ λm λm (48) ( n ) ( n−1 ) − N s − µc,m − N s − µc,m √ √ +Q −Q λm λm √ ( )2 + 2 − nM −m −1 s+ where s+ = (s ) and s− m m = m+1 m+1 N √ ( ) 2 nM −m − − 2 (s− and s+ m+1 ) − M = sM = s. Therefore, N sm+1 − s+ m ’s and sm ’s can be expressed as functions of s by deduction. Eq. (47) specializes to (23) when M = 2. If some of the eigenvalues of Rg are identical, we can obtain bounds with fewer summands. Suppose the eigenvalues of Rg , or the variances of c, can be classified into K groups as

)

M ∑k

(ck,m − µc,k,m ) √ m=1 M k (2π) k λM k

exp

) +

r2 nM −1 , s2

(51)

∆

K−1 ∑k ∑ M

nM −1 =1

fcik (ck )dck,Mk

c2k,m ≤s2

n=1

n2 =1

(50)

where dck,Mk = dck,1 dck,2 · · · dck,Mk and

∆

) ( − )] [ ( − s1 − µc,1 −s1 − µc,1 √ √ −Q × Q λ1 λ1 (47) N N ∑ ∑ ) ( ( ) + + ≤ FS (s) ≤ rM n1 , sM rM −1 n2 , sM −1 ···

K M ∑ ∑k

fcik (ck ) =

)

K ∏

FS (s) =

n2 =1 N ∑

T

whose elements have identical variances λk , and the mean value of Ck,m is denoted by µc,k,m . The CDF in (38) can be expressed as

2

n1 =1

···

∆

ck = [Ck,1 , Ck,2 , · · · , Ck,Mk ]

c2m ≤s2 −( n−1 N s)

where the (M − 1)-fold integrals can be expressed as the (M − 1)th-order Marcum Q-functions if µc,1 = µc,2 = · · · = µc,M −1 = 0 and λ1 = λ2 = · · · = λM −1 [19, eq. (3)]. By comparing (38) and (46), it can be observed that the M -fold integral can be bounded by the weighted sum of the (M − 1)-fold integral. By repeating this process M − 1 times, we obtain closed-form bounds for Pr (S ≤ s) as N ∑

where the subvector ck is

K ∏ (

c2k,m ≤s2 −

fcik (ck ) dck,Mk

M K ∑ m=1

√ c2K,m , n−1 N s≤

M K ∑

m=1

n c2K,m ≤ N s

)

k=1

(53) which can be bounded as (54) on the top of the next page. Note that ∫ fcik (ck ) dcK,MK √

n−1 N s≤



M K ∑

m=1

n c2K,m ≤ N s

(√

M ∑K

)



√ λ−1 µ2c,K,m , (n−1)s  Q MK  K N λK 2   m=1  √ ( ) =   MK   −1 ∑ ns −Q MK µ2c,K,m , N √ λK λ 2

where QM (a, b) =

m=1

(55)

K

( 2 2) ∫∞ ( x )M −1 exp − x +a IM −1 (ax) dx x a 2 b

is the Marcum Q-function [19], which is an embedded function in Matlab and Wolfram Alpha. We can apply (55) to



∫

N ∑

K ( ∏

v k=1 uM u ∑ K−1 K ∑k 2 ∑ M 2 n−1 n 2 ck,m ≤s −( N s) , N s≤t c2 ≤ns K,m N

fcik (ck ) dck,Mk

)

≤ FS (s)

n=1

k=1 m=1

m=1

∫

N ∑

≤

n=1 K−1 ∑ M ∑k k=1 m=1

K ( ∏

v k=1 uM )2 ( u ∑ n−1 t K c2 ns 2 − n−1 s , c2 ≤s s≤ ≤ K,m N k,m N N

process in (56) K − 1 times, we obtain N ∑

N ( )∑ ( ) qK n1 , s− , M qK−1 n2 , s− K K K−1 , MK−1

n1 =1

n2 =1 N ∑

···

( ) q2 nK−1 , s− 2 , M2

nK−1 =1



× 1 − Q M1 2

(√

N  ∑    n=1 

λ−1 K

Q MK 2

(√

M ∑K m=1

λ−1 K

−Q MK 2

∫

k=1 m=1

c2k,m ≤(s− K−1 )

  )   

Q MK 2

−Q MK

fcik



)

(ck ) dck,Mk ≤ FS (s) )

× 1 − Q M1

(

where

v  u Mk u ∑ (n − 1) s  √ qk (n, s, Mk ) = Q Mk tλ−1 µ2c,k,m , k 2 N λk m=1 (58) v  u Mk u ∑ ns − Q Mk tλ−1 µ2c,k,m , √  . k 2 N λk m=1

K

fcik (ck ) dck,Mk

 v u M1 + u ∑ s tλ−1 µ2c,1,m , √1  1 λ1 m=1 (57)



)

k=1 c2k,m ≤(s+ K−1 )

( ) q2 nK−1 , s+ 2 , M2

2

m=1

×

n2 =1

nK−1 =1

(

√ λ−1 µ2c,K,m , (n−1)s  K N λK  m=1 (√ )   M ∑K 2  ns √ λ−1 , µ K c,K,m N λ K−1 ∏

N ∑

···

v  u M1 − u ∑ s tλ−1 µ2c,1,m , √1  ≤ FS (s) 1 λ1 m=1

N ( )∑ ( ) q K n1 , s + , M qK−1 n2 , s+ K K K−1 , MK−1

n1 =1

K

M ∑K

∫

k=1 m=1

m=1

ns µ2c,K,m , N √ λ

N ∑

≤

2

(√

2

K−1 ∑ M ∑k

M ∑K



k=1



N  ∑   ≤  n=1 

) √ µ2c,K,m , (n−1)s N λK

K−1 ∏

× K−1 ∑ M ∑k

(54)

m=1

simplify (54) and obtain



) fcik (ck ) dck,Mk .

2

(56)

Note that the sum in (47) has N M −1 summands while the sum in (57) only has N K−1 summands, thus evaluation using (57) is more efficient if some of the eigenvalues are identical. Based on [19, eq. (10)], we can obtain a relationship between the 0.5th-order Marcum Q-function and the Gaussian Q-function as Q 12 (a, b) = Q(a + b) + Q(b − a). (59) Therefore, when M1 = M2 = · · · = MK = 1, we can express the 0.5th-order Marcum Q-functions in (58) with the Gaussian Q-function, and (57) specializes to (47).

where the bounds are a weighted sum of + − (M − MK )-fold integrals, where s = s = s K K √ ( )2 n −1 + + + and (sk+1 )2 − K−k sk+1 and sk = N √ ( ) 2 nK−k − 2 s− (s− k = k+1 ) − N sk+1 . Repeating the bounding

B. Single-Fold-Integral Expressions for the CDFs and the PDFs The square of the decorrelated Gaussian RV in (37), i.e. 2 |Cm |, follows a noncentral chi-square distribution with 1



degree of freedom. Thus it has a closed-form CF expression as [3, eq. (2.3-41)] ( 2 ) jµc,m t exp 1−2jλ mt √ 2 (t) = φCm (60) 1 − 2jλm t

C. Approximating the Left Tail of the CDF When s → 0, the CDF FS (s) can be approximated as ∫ FS (s) = fcM (cM )dcM 2 cT M cM ≤s

2

where j 2 = −1. Therefore, the CF of ∥c∥ can be obtained as [1, eq. (9)] ( 2 ) jµc,m t M exp ∏ 1−2jλm t √ φ∥c∥2 (t) = (61) 1 − 2jλm t m=1 which specializes to the CF of the noncentral chi-square distribution when λm = 1, ∀m = 1, 2, · · · , M [3, eq. (2.32 2 41)]. Based on (61), we obtain the PDF of S 2 = ∥c∥ = ∥g∥ as ) ( 2 jµc,m t ∫∞ ∏ M exp 1−2jλm t 1 (62) √ fS 2 (s) = e−jts dt 2π 1 − 2jλ t m m=1

−∞

which leads to the CDF of S as ( ) ( ) FS (s) = Pr (S < s) = Pr S 2 < s2 = FS 2 s2 ( 2 ) jµc,m t ∫∞ M (64) 1 1 1 ∏ exp 1−2jλm t −jts2 √ = − e dt 2 2π jt m=1 1 − 2jλm t −∞

and the PDF of S is fS (s) =

dFS (s) 1 = ds π

∫∞ −∞

( 2 ) jµc,m t M exp ∏ 2 1−2jλm t √ e−jts dt. s 1 − 2jλ t m m=1 (65)

Equations (62) to (65) can be evaluated using numerical integration by truncating the infinite integral regions. There are two main factors that can hamper the accuracy of the numerical integration, where the first is the truncation error and the second is the oscillatory behavior of the integrand. The truncation error could be larger than the exact integral value by several orders of magnitude when s → 0, as the exact integral values of (63) and (64) approach zero. The exact value of the CDF grows with s, thus the truncation error becomes negligible compared to the exact CDF value when s is large. Therefore, eqs. (63) and (64) are not suitable for evaluating the CDF left tail of S or S 2 . Although it is valid to further increase the numerical integral region in order to reduce the truncation error, the numerical integration could quickly become unacceptably time-consuming. When s grows too large, the integrands in (62) to (65) oscillate wildly, which could make the numerical integration unreliable [24]. Therefore, the CF-based approach is not suitable for evaluating the CDF right tail.

( ) dcM + o sM

(66)

2 cT M cM ≤s

where fcM (0) is the first-order Taylor series expansion of fcM (cM ) at the origin, and g(s) is o(f (s)) if lim g (s) /f (s) = 0. We comment that the Taylor series s→a expansion can be arbitrarily precise in a sufficiently small neighbourhood, thus the infinitesimal in (66) becomes negligible when s → 0. Noting that the M -sphere with radius s has volume ∫ π M/2 sM (67) V (s) = dcM = Γ(M/2 + 1) 2 cT M cM ≤s

−∞

which is in the form of the Fourier transform. According to the property of integration of Fourier transforms, we have ( 2 ) jµc,m t ∫∞ M 1 1 1 ∏ exp 1−2jλm t −jts √ e dt (63) FS 2 (s) = − 2 2π jt m=1 1 − 2jλm t

∫

= fcM (0)

∆

where Γ(z) =

∫∞

xz−1 exp(−x)dx, we can substitute (67) into

0

(66) and obtain

FS (s) = fcM (0)

( ) π M/2 sM + o sM Γ(M/2 + 1)

(68)

when s → 0. Based on (39), we can also express (68) as ( ) M ∑ 2 exp − 12 λ−1 µ M m c,m ( ) π2 m=1 √ FS (s) = sM + o sM . M Γ( 2 + 1) M M ∏ (2π) |λm | m=1

(69) Noting that Rg = ST ΛS and µc = Sµg , eq. (69) can be rewritten as ) ( M ( ) exp − 12 µTg R−1 π2 g µg sM + o sM (70) FS (s) = √ M M (2π) det (Rg ) Γ( 2 + 1) for s → 0, where det (Rg ) is the determinant of matrix Rg . V. T IGHTNESS OF THE CDF B OUNDS Based on (45), the difference between the CDF upper bound and the lower bound can be expressed as ∫ N ∑ ∆ fcM (cM ) dcM ∆FS (s) = (71) n=1 M + − Θn (s)\ΘM (s) n + − where ΘM (s) \ ΘM (s) denotes the region difference n n M+ − between Θn (s) and ΘM (s). Using the global maximum n /√ M M ∏ value of fcM (cM ), which is 1 (2π) λm , to replace m=1

the positive integrand in (71), we obtain an upper bound of the difference as (72) shown at the top of the next page where the second equality is based on the volume formula for the multidimensional sphere, and the last equality in (72) holds because most of the summands are canceled by each other. Noting that the upper bound of △FS (s) is a decreasing function of N , we



∆FS (s) =

∫

N ∑

fcM (cM ) dcM

n=1 ( ) M −1 n s 2 ≤ ∑ c2 ≤s2 − n−1 s 2 , n−1 s≤|c |≤ n s s 2 −( N ) M m N N N m=1

1

≤ √ (2π)

dcM −1

M ∏

n=1 ( ) M −1 λm n s 2 ≤ ∑ c2 ≤s2 − n−1 s 2 s 2 −( N ) m N m=1

M

M ∏

(2π)M

λm

m=1



N ∑

1

= √

∫

∫

N ∑

π

M −1 2

(

 ( )  s2 − Γ M2−1 + 1

n=1

(

dcM n−1 n s≤|cM |≤ N N

s

 ) ) M2−1 ( ( n )2 ) M2−1 n−1 2 2  s s − s − s N N N

(72)

m=1

1

= √

M ∏

(2π)M

π ( λm

NΓ

M −1 2

M −1 2

) sM . +1

m=1

conclude that the bounds can be arbitrarily tightened by setting a sufficiently large N . Since (47) is obtained by repeating the bounding procedures M − 1 times, the closed-form bounds in (47) can be made arbitrarily tight as long as N is sufficiently large. Following similar procedures to those leading to (72), the tightness of (57) can also be verified.

When the link distance is less than 300 meters and the parameter Cn2 is less than 10−15 m−2/3 , we have fha (t) ≈ δ(t − 1), where δ(·) is the Dirac delta function, which is shown in Fig. 3. Therefore, by substituting fha (t) ≈ δ(t − 1) into (75), we obtain ( ) 1 y f hl hp ha Pt x|hp (y) ≈ δ −1 (76) hl hp Pt x hl hp Pt x

VI. A PPLICATIONS OF THE CDF E STIMATION A PPROACHES

which implies that the probability mass concentrates at y = hl hp Pt x, thus hl hp ha Pt x ≈ hl hp Pt x. This usually happens when the turbulence level is weak, for example, at highaltitudes or in cold areas. Therefore, when the turbulence is weak, eq. (73) can be approximated as

A. Free-Space Optical Communications In point-to-point FSO communications, the received signal can be expressed as [4, eq. (1)] y = hl hp ha Pt x + n

(73)

where x is the power-normalized transmitted signal; Pt is the average transmitting power; hl represents the constant path loss determined by the link distance and weather condition; hp is the pointing error loss factor, and ha is the atmospheric fading loss factor; n represents the additive white Gaussian noise. In short-distance FSO communications ( −2σ0 (dl ) ln D2 Pt √ ( ( 2 )) 8σ (d ) l 0 = 1 − Pr Rf < −2σ02 (dl ) ln exp (cdl ) γth . D2 Pt (84)

30 2

-15

m

-2/3

,d=200 m

2 -15 Cn =10 2 -16 Cn =10

m

-2/3

,d=300 m

m

-2/3

,d=300 m

Cn =10

25

a

f h (h)

20

15

10

5

0 0

0.5

1

1.5

2

h

Fig. 3.

According to (20) and (23), eq. (84) can be bounded as (√ ( 2 )) N ∑ 8σ (d ) l 0 1− G− −2σ02 (dl ) ln exp (cdl ) γth n D2 Pt n=1

PDFs of ha in weak turbulence scenarios.

where A0 Pt hl is assumed to be above γth ; otherwise, outage happens with probability 1. According to (23), eq. (79) can be bounded as  √ N 2 ∑ w γ z th eq F SO  ≥ Pout  − (γth ) 1− ln G− n 2 A P h 0 t l n=1  √ (80) N 2 ∑ w γth  zeq + ≥1− − ln . Gn 2 A0 Pt hl n=1 When the pointing error distributes on one dimension, such as σY = 0, we can simplify (79) based on (32) and obtain  wz2eq γth 2   1, −  2 ln A0 Pt hl − µY ≤ 0    √   2 wz  γ  − − 2eq ln A Pthh −µ2Y −µX  0 t l  1 − Q  σX F SO (γth ) = Pout   √ 2   wz  γ  − 2eq ln A Pthh −µ2Y −µX  0 t l  +Q   , else.   σX  (81) B. Underwater Wireless Optical Communications In UWOC systems, the received power can be expressed as [26] πD2 B(dl , Rf )x + n (82) y= 4 where D is the√diameter of the aperture; dl is the link distance; Rf = Xf2 + Yf2 is the radial displacement mod2 eled by a Beckmann RV, where Xf ∼ N (µX , σX ) and Yf ∼ N (µY , σY2 ); B(·, ·) is called the beam-spread function for the nonscattered light defined as [27, eq. (12)] ) ( Pt r2 B(L, r) = exp(−cdl ) (83) exp − 2πσ02 (dl ) 2σ02 (dl ) where σ02 (dl ) is the variance of the Gaussian beam; dl denotes the link distance; c presents the light attenuation coefficient in

UW ≥ Pout (γth )

≥1−

N ∑

G+ n

(√

(

−2σ02 (dl ) ln

n=1

8σ02 (dl ) exp (cdl ) γth D 2 Pt

)) . (85)

When the pointing error distributes on one dimension, such as σY = 0, we can simplify (84) based on (32) and obtain UW Pout (γth ) ) ( 2  8σ0 (dl ) 2 2  (cd ) γ exp (d ) ln 1, −2σ  l th − µY ≤ 0 0 l D 2 Pt  √    ( 2 )    2 (d ) ln 8σ0 (dl ) exp(cd )γ 2 −µ  − −2σ −µ X l l th 0 Y  D 2 Pt      1 − Q  σX =  √   ( 2 )  8σ (d )   −2σ02 (dl ) ln D02 P l exp(cdl )γth −µ2Y −µX  t      +Q   , else.  σX  

(86) C. Outage Probability of MRC Over Correlated Wireless Fading Channels In a wireless diversity reception system with L antennas, the received L × 1 signal vector y can be expressed as √ y = Pt hx + n (87) where h = [h1 , h2 , · · · , hL ]T is the vector of the complex fading channel coefficients, and n denotes [ the] noise vector. Without loss of generality, we assume E nnH = IL where IL is an L × L identity matrix. For correlated generalized Beckmann fading channels, each fading amplitude of h in (87) can be expressed as a norm of the accompanying Gaussian vectors. Specifically, the fading amplitude of the lth branch is v uK u∑ G2 (88) |h | = t l

k,l

k=1



where the Gk,l ’s are the kth accompanying Gaussian RVs of the lth channel coefficients2 ; K = 2 for the Rayleigh, Rician, Beckmann and Hoyt fading channels, and K = 2m for Nakagami-m fading channels, and K = M for the generalized Beckmann fading channels. Therefore, the output instantaneous SNR of MRC can be expressed as γM RC = Pt h h = Pt

l=1

2

|hl | = Pt

L ∑ K ∑

0.8 10 -1 0.6 R

L ∑

Zoom in 1

F (r)

H

10 0

G2k,l .

×10 -3 Zoom in

(89)

l=1 k=1

10 -2

Fig. 4 compares the CDF bounds and its numerical estimation for the Beckmann distribution. The CDF is estimated using two numerical approaches including the trapezoidal method and the Gauss-Legendre quadrature, which are based on (3). The number of panels for the trapezoidal integration is r/1000 and the number of points for Gauss-Legendre quadrature is 5. It can be observed from Fig. 4 that the bounds become tighter when N grows, as expected, and that N = 20 leads to quite tight bounds. This indicates that N does not need to be large in order to achieve satisfactory bounds. It can also be observed that the bounds are tighter at the the left and the right tails, i.e. when s → 0 and s → ∞. When s → 0, the integral region in (4) becomes smaller and smaller, thus the continuous integrand in the small integral region has almost identical values, and then the ratio between the upper and the lower bounds in (17) and (21) is the ratio between the − areas of the bounding regions, i.e. Θ− 1 (r) ∪ · · · ∪ ΘN (r) and + + Θ1 (r) ∪ · · · ∪ ΘN (r). This area ratio of the bounding regions is not exactly 1, so there is still a small gap on the logarithm scale. When s → ∞, the exact integral region in (4) becomes very large, and so do the bounding regions in (15) and (16). Therefore, the integral of the PDF in the bounding regions approaches 1, which makes the bounds tight at the right tail. 2 Conventionally, the powers of the fading amplitudes are normalized, and such assumption would impose a constraint on the means and the variances [ ] K ∑ 2 = 1. of the accompanying Gaussian RVs as E |hl |2 = µ2k,l + σk,l k=1

3

R

F (r) using Gauss-Legendre R

Bounds on F R (r), N=5

5

Bounds on F (r), N=10 R

10

VII. N UMERICAL R ESULTS

2.5

F (r) using the trapezoidal method

6

Outage happens when γM RC falls below a predetermined threshold γth , thus the outage probability of MRC can be expressed as M RC Pout (γth ) = Pr (γM RC < γth ) ( L K ) ) (√ ∑∑ (90) γth γth 2 = Pr = F√γM RC Gk,l < Pt Pt l=1 k=1 √ where γM RC follows the generalized Beckmann distribution. Given the transmit power Pt , the covariance matrix Rg and the mean vector µg of the Gk,l ’s, we can obtain the outage probability bounds according to (47). Since γPtht → 0 when Pt → ∞, we can also obtain the outage probability of MRC in large SNR regions according to (70). In addition, we can use numerical integration approaches based on (64) to estimate the outage probability of MRC because in this case, numerical evaluation using the bounds could be time-consuming when the branch number L is larger than 6 and all of the eigenvalues of Rg are different.

2

7

0.26

-3

0.5

1

Bounds on F R (r), N=20

0.27 1.5

2

2.5

3

r

CDF bounds of the Beckmann distribution and the CDF using Fig. 4. numerical integral estimation approaches. The parameters of the Beckmann distribution are µX = µY = 1, σX = 1 and σY = 0.5. The panel width for the numerical trapezoidal integration is 0.001. The number of points for Gauss-Legendre quadrature is 5.

The Gauss-Legendre quadrature works well when r is small, but shows larger discrepancy than the other approaches for large r values. This is because the integrand in (3) is poorly approximated by polynomials when r is large3 . Note that it is also possible to generalize the Gauss-Legendre quadrature to higher dimensions [28, Chap. 5]. The time costs for the Gauss-Legendre quadrature, the bounds, and the trapezoidal integration are, respectively, 0.5 × 10−4 s, 1.0 × 10−4 s (N = 5), and 5.0 × 10−4 s. Fig. 5 illustrates the relative differences between the upper and lower bounds, i.e. (FR,U B (r) − FR,LB (r))/FR (r) where FR,U B (r) is the upper bound in (20) and FR,LB (r) is the lower bound in (23). It can be observed that the relative difference between the upper and the lower bounds is a decreasing function of N , thus the bounds become tighter when N grows. Furthermore, the decreasing rate of the relative difference becomes slower for large N , which indicates that increasing N cannot provide significant accuracy improvement when N is already large (N > 50). Fig. 6 compares the three CDF estimation methods based on the bounds in (47), the CF-based integration in (64) and the asymptotic expression in (70). It can be observed that the Monte Carlo simulation becomes unreliable when s is less than 0.04, and this is because the smaller the probability is, the larger number of samples are required to estimate the probability. In contrast, the asymptotic expression in (70) is reliable when s is small, but it cannot provide accurate estimation when s > 3.16, and this is because the PDF in the integral region of (66) varies across a large range, and using the PDF value at the origin fcM (0) to approximate fcM (cM ) leads to a large error. Numerical integration based on the CF is inaccurate in both CDF tails. For the left tail where s is small, the CF-based integration is not accurate because the truncation error becomes larger than the exact CDF value; for the right 3 Increasing the order of Gauss-Legendre quadrature cannot ensure the approximate CDF approach 1 for large s because the integrand involves exponential functions and erf (·).



0.2 r=0.2 r=0.4 r=0.6 r=0.8

0.18

10 2

0.14

10 0

0.12 0.1 0.08

10

-2

FS(s)

(F R,UB (r) - F R,LB (r))/F R(r)

0.16

0.06 0.04

10 -4

0.02 0 50

100

150

200

250

Monte Carlo Simulation Asymptotic CDF CDF Bounds in (47) Numerical Integration in [-60,60] Numerical Integration in [-200,200] Numerical Integration in [-1000,1000]

10 -6

300

N

Fig. 5. Difference between the upper and lower bounds as a function of N . The parameters of the Beckmann distribution are µX = µY = 1, σX = 1 and σY = 0.5. TABLE I C OMPARISON B ETWEEN T HE CDF E STIMATION A PPROACHES Estimation Approaches Monte Carlo simulation CF integration in [−60, 60] CF integration in [−200, 200] CF integration in [−103 , 103 ] Asymptotic expression CDF bounds

Time Cost (sec.) 3.10 0.22 0.68 3.30 4.90 × 10−4 0.36

Range of Application s ∈ [0.04, +∞] s ∈ [0.13, 25.12] s ∈ [0.06, 25.12] s ∈ [0.03, 25.12] s ∈ [0, 3.16] s ∈ [0, +∞)

tail where s becomes large, the factor e−jts in the integrand of (64) has rapid oscillations, which makes it difficult to approximate using numerical integration. By enlarging the numerical integral region from [−60, 60] to [−103 , 103 ], it can be observed that the CF-based integration method becomes more accurate, but the improvement in accuracy is limited. Considering that the computation time4 for numerical integration (3.30 sec.) is longer than that of the Monte Carlo simulation (3.10 sec.), it is not preferable to use the CF-based integration method to estimate the CDF tails. The bounding technique in (47) is the best method that can provide accurate CDF estimation in all integral regions, and its computation time is acceptable (0.36 sec.). We comment that the asymptotic estimation technique and the bounding technique can be jointly used. For example, the bounds can be used to predict when the asymptotic CDF converges to a certain range of accuracy. Table I compares the time complexity and the ranges of application of the four CDF estimation methods for ready reference. In Fig. 7, we compare the bounds in (47) and (57) when some of the eigenvalues of Rg are identical. It can be observed that the bounds in (57) are tighter than the bounds in (47) for a fixed N value. This is because the bounds in (47) are based on M −1 times of the bounding processes while the bounds in

10

-2

10

-1

10

0

10

1

10

2

s

Fig. 6. Comparison between the CDF estimation approaches based on the bounds in (47), the CF integration in (64) and the asymptotic expression in (70). The number of degrees of freedom is M = 3, and the mean vector of the accompanying Gaussian RVs is µg = [1; 1.5; 2], and the covariance matrix is Rg = [1.1, 0.2, 0; 0.2, 0.8, 0.3; 0, 0.3, 1]. The interval of the numerical integration is 0.0081. The number of the samples for the Monte Carlo simulation is 107 . The value of N for the bounds is 8.

2

10 0

N=2

10

-2

FS(s)

10

-1

N=2

N=8

10

Monte Carlo Simulation CDF Bounds in (47) CDF Bounds in (57)

-3

1

1.5

2

2.5

3

3.5

4

s

Fig. 7. Comparison between the bounds in (47) and (57). The number of degrees of freedom is M = 4. The mean vector is µg = [1; 1.5; 2; 0]. The covariance matrix is Rg = [0.7, 0, 0.1, 0; 0, 0.7, 0, 0.1; 0.1, 0, 0.7, 0; 0, 0.1, 0, 0.7], which has eigenvalues 0.8, 0.8, 0.6 and 0.6.

4 The computation time refers to the time cost to achieve all of the points of a specific estimation method in a figure.



Numerical integration The approximation in [29] Upper bound, N=5 Lower bound, N=5 Upper bound, N=30 Lower bound, N=30 Upper bound, N=120 Lower bound, N=120

Outage Probability

Outage Probability

10 -2

10 -4 Zoom in

10

Numerical integration Upper bound, N=5 Lower bound, N=5 Upper bound, N=20 Lower bound, N=20 Upper bound, N=50 Lower bound, N=50

10 -1

10 -2

2 10 -3 1.5

×10 -4

Zoom in

1

-6

10

-7

10 -8 25

10 -4

0.5

52

33

33.5

34

30

53

35

35

numerical integration is

−

2 wz

eq

2

ln

γth /1000. A 0 P t hl

45

50

55

P t (dBm)

Pt (dBm)

Fig. 8. Outage probability of an FSO system with pointing error. The parameters are σX = 0.2m, σY = 0.3m, µX = 0m, µY = −0.1m, wze = 1.37m, γth = 1mW √ (0dBm), A0 = 0.8, hl = 0.01. The step of the

54 40

Fig. The µY c = √

9. Outage probability of a UWOC system with pointing error. parameters are σX = 0.1 m, σY = 0.15 m, µX = 0.05 m, = 0.1 m, dl = 5 m, γth = 10 mW (10 dBm), D = 0.1 m, 0.1/m and σ0 (dl ) = 0.2 m. The step of the numerical integration is ( ) 2 (d ) 8σ0 l −2σ02 (dl ) ln exp (cdl ) γth /1000. D2 P t

(57) are based on K −1 times of the bounding processes. Thus the bounds in (57) suffer less bounding error. The time cost for the Monte Carlo simulation is 2.32 secs while calculating the bounds costs 1.67 secs using (47), and 0.12 sec using (57) with N = 8. For N = 2, calculating the bounds costs 0.04 sec using (47) and 0.03 sec using (57). Therefore, we can conclude that numerical evaluation of the bounds in (57) is more efficient. Furthermore, it can also be observed that the time improvement by using (57) is more significant when N is large. Fig. 8 shows the exact outage probability and its bounds versus the transmit power in dBm of a pointing-error-limited FSO system. The exact outage probability is given in (79) and can be evaluated according to (3), and the bounds are given in (80). The approximation in [29, eq. (10)] is also added for comparison, which suggested to use the Rayleigh distribution to approximate the Beckmann The interval of the √ distribution. / wz2eq γth numerical integration is − 2 ln A0 Pt hl 1000, which is based on (3). It can be observed that the outage probability bounds get tighter as N increases. The tightness of the bounds significantly increases when N increases from 5 to 30. However, when N increases from 30 to 120, the tightness of the bounds does not significantly increase. This indicates that the value of N does not need to be too large. It can also be observed from Fig. 8 that using the Rayleigh distribution to approximate the Beckmann distribution in modeling the pointing error might result in discrepancy in large SNR region. Fig. 9 presents the outage probability and its bounds for a UWOC system as functions of the transmit power in dBm. Eq. (84) shows the exact outage probability, and its value is using numerical integration with interval size √ estimated wz2eq − 2 ln A0γPtht hl /1000, and the bounds are given in (85). We assume a scenario with pure sea water (c = 0.1/m) and

the link distance is dl = 5 m. It can be observed again that the bounds get tighter as N increases from 5 to 50. The difference between the upper and lower bounds is less than 2 dBm at Pt = 45 dBm. Fig. 8 and Fig. 9 are examples of using the CDF bounds to estimate the complementary CDF of the Beckmann distribution. It can be observed that the differences of the bounds slowly grow as Pt increases. However, increasing the N value can significantly decrease the difference, and this only results in a linear increase in the time cost. Fig. 10 compares the estimated outage probabilities of 2branch MRC estimated using Monte Carlo simulation, CFbased numerical integration, asymptotic expression and the bounds in (47). The covariance matrix of the accompanying Gaussian RVs Rg is assumed to be identical for the compared systems, while their mean vector µg is varied for comparison. The asymptotic expression in (70) fails to provide accurate estimation in low SNR region, which agrees with the result in Fig. 6 that the asymptotic CDF expression is not reliable for small s. When µg = [0.2, 0.5, 0.3, 0.6]T , the asymptotic outage probability converges to the exact outage probability at Pt = 5 dB, but when µg = [1.0; 1.4; 1; 1.6]T , the asymptotic outage probability converges at Pt = 30 dB. This indicates that the converging speed of the asymptotic expressions is highly related to the statistics of channels, thus the convergence could be unpredictable at a particular SNR. In contrast, CF-based numerical integration can provide accurate outage probability estimation in low to medium SNR regions, but the estimation becomes unreliable for large SNR regions (Pt > 30 dB). In Fig. 10, the Monte Carlo simulation employs 107 samples, and it provides good estimation for outage probabilities less than 10−5 . However, the approach becomes unreliable when the product of the exact outage probability and the number of



Outage Probability of MRC

10

Monte Carlo Simulation Asymptotic CDF Outage Probability Bounds by (47) Numerical Integration in [-1200,1200]

0

µG = [0.2, 0.5, 0.3, 0.6]T 10 -5

µG = [1.0, 1.4, 1.0, 1.6]T 10

-10

-10

0

10

20

30

40

P (dB) t

Fig. 10. Comparison between different estimation approaches for the MRC outage probability over two correlated Beckmann fading channels with γth = 1 W. The covariance variance of the accompanying Gaussian RVs, i.e. g = [G1,1 , G1,2 , G2,1 , G2,2 ], is Rg = [1.2, 0.8, 0.6, 0.0; 0.8, 1.1, 0.8, 0.6; 0.6, 0.8, 1.0, 0.8; 0.0, 0.6, 0.8, 1.0]. The mean vector µg is varied for comparison.

samples is less than 102 . From Fig. 10, it can be observed that the bounding technique in (47) is the most reliable compared the other three approaches. Its range of application is not restricted to a certain SNR region. The time cost for the four approaches is exemplified as follows: Monte Carlo (2.65 sec.), CF-based numerical integration (2.63 sec.), asymptotic expression (2 × 10−4 sec.) and the bounds (2.24 sec.). VIII. C ONCLUSION Closed-form upper and lower bounds for the CDF of the Beckmann distribution were derived, where the bounds can be made arbitrarily tight by adjusting an integer-valued parameter. The bounding technique was generalized to higher dimensions, leading to closed-form bounds of the generalized Beckmann distribution. A CF-based single-fold integral expression and an asymptotic expression for the CDF were also derived. These CDF estimation approaches were compared and used to evaluate the outage probabilities of FSO, UWOC and MRC over arbitrarily correlated generalized Beckmann fading channels. R EFERENCES [1] J. Cheng and T. Berger, “Performance analysis for maximal-ratio combining in correlated generalized Rician fading,” in Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers, vol. 2, Nov. 2003, pp. 1672–1675. [2] N. Youssef, C.-X. Wang, M. Patzold, I. Jaafar, and S. Tabbane, “On the statistical properties of generalized Rice multipath fading channels,” in IEEE Veh. Technol. Conf. (VTC), vol. 1, May 2004, pp. 162–165. [3] J. G. Proakis and M. Salehi, Digital Communications, 5th ed. New York: McGraw-Hill, 2008. [4] F. Yang, J. Cheng, and T. A. Tsiftsis, “Free-space optical communication with nonzero boresight pointing errors,” IEEE Trans. Commun., vol. 62, no. 2, pp. 713–725, Feb. 2014.

[5] Y. Dong, S. Tang, and X. Zhang, “Effect of random sea surface on downlink underwater wireless optical communications,” IEEE Commun. Lett., vol. 17, no. 11, pp. 2164–2167, Nov. 2013. [6] Y. Xie and Y. Fang, “A general statistical channel model for mobile satellite systems,” IEEE Trans. Veh. Technol., vol. 49, no. 3, pp. 744– 752, May 2000. [7] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves From Rough Surfaces. Norwood, MA: Artech, 1987. [8] W. Braun and U. Dersch, “A physical mobile radio channel model,” IEEE Trans. Veh. Technol., vol. 40, no. 2, pp. 472–482, May 1991. [9] J. F. Paris, “Advances in the statistical characterization of fading: From 2005 to present,” International Journal of Antennas and Propagation, vol. 2014, pp. 1–5, July 2014. [10] V. A. Aalo, G. P. Efthymoglou, and C. Chayawan, “On the envelope and phase distributions for correlated Gaussian quadratures,” IEEE Commun. Lett., vol. 11, no. 12, pp. 985–987, Dec. 2007. [11] P. Dharmawansa, N. Rajatheva, and C. Tellambura, “Envelope and phase distribution of two correlated Gaussian variables,” IEEE Trans. Commun., vol. 57, no. 4, pp. 915–921, Apr. 2009. [12] W. Dahech, N. Hajri, N. Youssef, T. Kawabata, and M. Pätzold, “Outage statistics for Beckmann fading channels in non-isotropic scattering environments,” in Asia-Pacific Conference on Communications (APCC), Oct. 2015, pp. 164–168. [13] N. C. Beaulieu and X. Jiandong, “A novel fading model for channels with multiple dominant specular components,” IEEE Wireless Commun. Lett., vol. 4, no. 1, pp. 54–57, Feb. 2015. [14] J. P. P. Martin, J. M. R. Jerez, and F. J. Lopez-Martinez, “Generalized MGF of Beckmann fading with applications to wireless communications performance analysis,” to be published in IEEE Trans. Commun., 2017. [15] G. Ropokis, A. Rontogiannis, P. Mathiopoulos, and K. Berberidis, “An exact performance analysis of MRC/OSTBC over generalized fading channels,” IEEE Trans. Commun., vol. 58, no. 9, pp. 2486–2492, Sept. 2010. [16] B. Zhu, J. Cheng, N. Al-Dhahir, and L. Wu, “Asymptotic analysis and tight performance bounds of diversity receptions over Beckmann fading channels with arbitrary correlation,” IEEE Trans. Commun., vol. 64, no. 5, pp. 2220–2234, May 2016. [17] B. Zhu, Z. Zeng, and J. Cheng, “Arbitrarily tight bounds on cumulative distribution function of Beckmann distribution,” in International Conference on Computing, Networking and Communications (ICNC), Jan. 2017, pp. 41–45. [18] T. Song, Q. Wang, M.-W. Wu, and P.-Y. Kam, “Performance of laser inter-satellite links with dynamic beam waist adjustment,” Opt. Express, vol. 24, no. 11, pp. 11 950–11 960, May 2016. [19] R. Li, P. Y. Kam, and H. Fu, “New representations and bounds for the generalized Marcum Q-function via a geometric approach, and an application,” IEEE Trans. on Commun., vol. 58, no. 1, pp. 157–169, Jan. 2010. [20] P. Y. Kam and R. Li, “Computing and bounding the first-order Marcum Q-function: A geometric approach,” IEEE Trans. on Commun., vol. 56, no. 7, pp. 1101–1110, Jul. 2008. [21] M. D. Yacoub, “The κ-µ distribution and the η-µ distribution,” IEEE Antennas Propag. Mag., vol. 49, no. 1, pp. 68–81, Feb. 2007. [22] L. Zhang, C. Liu, J. Zhang, F. Wu, and W. Yang, “Performance of relay networks in fading environments with dominant specular components,” China Commun., vol. 13, no. 12, pp. 69–78, Dec. 2016. [23] B. Zhu, F. Yang, J. Cheng, and L. Wu, “Performance bounds for diversity receptions over arbitrarily correlated Nakagami-m fading channels,” IEEE Trans. Wireless Commun., vol. 15, no. 1, pp. 699–713, Jan. 2016. [24] N. C. Beaulieu and Q. Xie, “An optimal lognormal approximation to lognormal sum distributions,” IEEE Trans. Veh. Technol., vol. 53, no. 2, pp. 479–489, Mar. 2004. [25] A. A. Farid and S. Hranilovic, “Outage capacity optimization for freespace optical links with pointing errors,” Journal of Lightwave Tech., vol. 25, no. 7, pp. 1702–1710, July 2007. [26] Y. Dong, S. Tang, and X. Zhang, “Effect of random sea surface on downlink underwater wireless optical communications,” IEEE Commun. Lett., vol. 17, no. 11, pp. 2164–2167, Nov. 2013. [27] B. M. Cochenour, L. J. Mullen, and A. E. Laux, “Characterization of the beam-spread function for underwater wireless optical communications links,” IEEE J. Ocean. Eng., vol. 33, no. 4, pp. 513–521, Oct. 2008. [28] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration. Academic Press, 1984. [29] R. Boluda-Ruiz, A. Garc´ıa-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Novel approximation of misalignment fading modeled by Beckmann distribution on free-space optical links,” Opt. Express, vol. 24, no. 20, pp. 22 635–22 649, Oct. 2016.


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