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Signal Processing

Signal Processing 94 (2014) 691–702

0165-16http://d

n CorrE-m

npurjc@

journal homepage: www.elsevier.com/locate/sigpro

Copulas for statistical signal processing (Part I): Extensionsand generalization

Xuexing Zeng c, Jinchang Ren a,n, Zheng Wang b, Stephen Marshall a,Tariq Durrani a

a Centre for Excellence in Signal and Image Processing (CeSIP), Dept. of Electronic and Electrical Engineering, University of Strathclyde, 204George Street, Glasgow, United Kingdomb School of Computer Software, Tianjin University, Tianjin 300072, Chinac School of Information Engineering, China University of Geosciences, Beijing, China

a r t i c l e i n f o

Article history:Received 21 January 2013Received in revised form16 May 2013Accepted 7 July 2013Available online 19 July 2013

Keywords:CopulasStatistical signal processingParameter estimationArbitrary marginal distributions

84/$ - see front matter & 2013 Elsevier B.V.x.doi.org/10.1016/j.sigpro.2013.07.009

esponding author. Tel.: +44 141 5482384.ail addresses: Jinchang.Ren@strath.ac.uk,yahoo.com (J. Ren).

a b s t r a c t

Existing works on multivariate distributions mainly focus on limited distribution func-tions and require that the associated marginal distributions belong to the same family.Although this simplifies problems, it may fail to deal with practical cases when themarginal distributions are arbitrary. To this end, copula function is employed since itprovides a flexible way in decoupling the marginal distributions and dependencestructure for random variables. Among different copula functions, most researches focuson Gaussian, Student's t and Archimedean copulas for simplicity. In this paper, to extendbivariate copula families, we have constructed new bivariate copulas for exponential,Weibull and Rician distributions. We have proved that the three copula functions ofexponential, Rayleigh and Weibull distributions are equivalent, constrained by only oneparameter, thus greatly facilitating practical applications of them. We have also provedthat the copula function of log-normal distribution is equivalent to the Gaussian copula.Moreover, we have derived the Rician copula with two parameters. In addition, themodified Bessel function or incomplete Gamma function with double integrals in thecopula functions are simplified by single integral or infinite series for computationalefficiency. Associated copula density functions for exponential, Rayleigh, Weibull, log-normal, Nakagami-m and Rician distributions are also derived.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

In general, the marginal distribution functions can bederived through straightforward integration from a knownmultivariate joint distribution function. However, conver-sely, a unique joint distribution is not readily availablefrom the marginal distributions. Although many multi-variate distribution functions have been defined, in mostcases the marginal distributions are assumed to be within

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the same family such as multivariate Gaussian distribution.In practical situations, this assumption does not alwayshold true, as practically the marginal distributions can bearbitrary and may belong to different families of prob-ability distributions. For example, the probability distribu-tions of pixel intensities for infrared image and visibleimages of the same objective are usually different. Conse-quently, some popular multivariate distributions such asthe multivariate Gaussian distribution may not be the idealchoice in modeling real problems since it can only capturethe linear dependence between random variables.

Copula functions, however, make it possible to capturemore sophisticated non-linear dependencies between ran-dom variable [1,2]. Copulas have provided great potential

X. Zeng, et al. / Signal Processing 94 (2014) 691–702692

in statistics and probability theory and have been successfullyapplied in many applications covering finance [3], signalprocessing [4–17], and communications [18,19]. Let FXY(x,y) be a joint cumulative distribution function (cdf)with two marginal cdfs FX(x) and FY(y) of two randomvariables x and y, respectively. According to the Sklar'stheorem [1], there exists a copula function C(.) for any xand y such that

FXY ðx; yÞ ¼ CðFXðxÞ; FY ðyÞÞ ð1Þ

If FX(x) and FY(y) are continuous, the copula function C(.) isunique [1 (Eq. (2.3.1))]. A bivariate copula is a bivariatecumulative distribution function defined on the unit cubewith two uniformmarginal distributions on the interval [0, 1].Let u¼FX(x) and v¼FY(y), the copula function C (u, v) isdefined as [1]

Cðu; vÞ ¼ FXY ðx; yÞ ¼ FXY ½FX�1ðuÞ; FY�1ðvÞ� ð2Þ

Note that FX() and FY() are increasing but may not bestrictly increasing, F�1

X () andF�1Y () are quasi-inverse func-

tions of FX() and FY(), respectively. If FX() and FY() arestrictly increasing, the case certainly becomes the ordinaryinverse. Detailed definition of Quasi-inverse function isgiven in [1 (Section 2.3)]. Due to the non-uniqueness of thedefinition of the associated inverse of the cdf for the caseof discrete-valued random variables, the correspondingcopula also becomes non-unique. As a result, in theremainder of this paper only continuous valued randomvariables are focused.

A bivariate copula has the following properties[1 (Section 2.2)]:

(1)

For any

u; v∈½0; 1�; Cðu;0Þ ¼ Cð0; vÞ ¼ 0; Cðu;1Þ ¼ u and Cð1; vÞ ¼ v

ð3Þ

(2)

For any

u1; u2; v1; v2½0; 1�; and u1≤u2; v1≤v2; Cð:Þ satisfyCðu2; v2Þ�Cðu2; v1Þ≥Cðu1; v2Þ�Cðu1; v1Þ ð4Þ

Eq. (3) means that the joint probability is zero if anymarginal probability is 0, and if one marginal probability is

1, the joint probability is the same as the probability ofanother marginal probability. Eq. (4) is called 2-increasingproperty of copula functions [1 (Section 2.9)].

A significant advantage of the copula is that copulafunctions represent dependence structure without usingmarginal distributions. Actually, its variables u and v areboth uniformly distributed, thus it is straight forwardto transform u and v into the marginal variables x and yby the inverse marginal cdf, respectively [1 (Section 2.9)].This characteristic implies that the marginal distribu-tions can be arbitrary when a suitable dependence struc-ture (copula) is given. Moreover, let f XY ðx; yÞ represent ajoint probability density function (pdf) for random vari-ables x and y, and cðu; vÞ represent copula density function,according to the definition of copula density function, the

joint pdf can be obtained as [3 (Section 4.5)]

cðu; vÞ ¼ ∂2Cðu; vÞ∂u∂v

¼ ∂2C½FXðxÞ; FY ðyÞ�∂FXðxÞ∂FY ðyÞ

¼ f XY ðx; yÞf XðxÞf Y ðyÞ

ð5Þ

where f XðxÞ and f Y ðyÞ represent the marginal pdfs of x andy, respectively. This means that the joint pdf can beexpressed as the product of a copula density functionand arbitrary marginal pdfs if a suitable copula is adopted.

The copulas that are most frequently applied areGaussian copula [1 (Eq. (2.3.6))], Student's t copula [3(Section 4.8.2)] and four Archimedean copula [1 (Ch 4)]including Clayton [3 (Section 4.8.4)], Frank [3 (Section4.8.4)], Gumbel [2 (Section 4.8.4)] and Ali–Mikhail–Haqcopulas [1 (Section 3.3.2)]. In Refs. [8–10], Gaussian copulais considered for supervised classification of syntheticaperture radar (SAR) images, validating biometric authen-tication and modeling stochastic dependence in large-scale integration of wind power respectively. Gaussianand Student's t copulas are used in [11,12] for unsuper-vised classification of radar signals and fusion of correlatedsensor decisions. In Ref. [13], four Archimedean copulas,Clayton, Frank, Gumbel and Ali–Mikhali–Haq, are appliedfor change detection in remote sensing applications. Clay-ton, Frank, Gumbel, Gaussian and Student's t copulas areused [14,15] for location estimation of random signalsource and hypothesis testing using heterogeneous data,respectively. In Ref. [16], the four Archimedean copulasalong with Gaussian and Student's t copulas are employedfor supervised SAR image classification. In Ref. [17], Ray-leigh copula is used for change detection from SAR images.

Note that a copula function represents a specificdependence structure, thus choosing a suitable copula isalways crucial for practical applications. Obviously, exist-ing copulas used above are inappropriate to deal with realand complex cases, such as exponential, Rayleigh, Weibull,Nakagami-m and Rician distributions. As a result, there is astrong need to explore new copula families, which hasmotivated our work in this paper to derive and constructnew copulas from existing bivariate distributions includingexponential, Rayleigh, Weibull, log-normal, Nakagami-mand Rician as these have been widely used in modelingsignal processing, communication and other stochasticproblems [20–25]. Note that copulas of Rayleigh, log-normal and Nakagami-m distributions have been dis-cussed in Refs. [4–6]. The Rayleigh copula function hasbeen firstly presented with 3 parameters [4], and thenumber of parameter was then reduced to 1 [5], whereits applications can be found in Ref. [17] for changedetection in SAR images. In this paper, a new form ofRayleigh copula defined by a single parameter is derived.The log-normal copula was firstly given in Ref. [5], wherein this paper, the log-normal copula is proved to beequivalent to Gaussian copula [1 (Eq. (2.3.6))]. The detailstatus about the current studies on bivariate copulas issummarized in Table 1.

As can be seen, there are many N/A cells in the tablewhich indicates new areas to be explored. Basically, wewill focus on these unexplored parts, which inevitablyform the novelty and main contributions of the paper asmentioned below. Firstly, new copulas families are pro-posed for exponential, Weibull and Rician distributions as

Table 1Summary of existing research and our works in this paper.

Copulas Copula function Copula density function

Existing Our work Existing Our work

Exponential N/A Eqs. (10) and (14) N/A Eq. (28)Rayleigh [4,5] Deriving new forms in Eqs. (10) and (14) N/A Eq. (28)Weibull N/A Eqs. (10) and (14) N/A Eq. (28)Log-normal [5] To prove log-normal copula equals to Gaussian copula. See Eq. (13) N/A Eq. (27)Nakagami-m [6] Using results in [6], see Eq. (19) N/A Eq. (30)Rician N/A Eqs. (22), (23) and (25) N/A Eq. (32)

Note that it is proven in this paper that bivariate exponential, Rayleigh and Weibull copulas are equivalent.N/A in cells represents unexplored areas.

X. Zeng, et al. / Signal Processing 94 (2014) 691–702 693

well as a new expression for Rayleigh copula function.Secondly, the relationships among these copulas arerevealed to discover the equivalence of log-normal toGaussian and exponential to Rayleigh and Weibull. Thirdly,the copula density functions for all the six copula functionsdiscussed above are derived.

2. Bivariate copula functions

Several methods for constructing a copula functionhave been summarized in [2 (Section 1.14)]. Here, we willadopt the inversion method to derive some new copulafunctions [2 (Section 1.14.5)].

2.1. Bivariate gaussian copula

The Gaussian copula is perhaps the most popularcopula in applications [2]. The bivariate Gaussian copulais defined by two standard Gaussian distributions, andwritten as [1 (Section 2.3)], [3 (Section 4.8.1)]

Cðu; vÞ ¼Ψ ðΦ�1ðuÞ;Φ�1ðvÞÞ

¼ 1

2πffiffiffiffiffiffiffiffiffiffiffiffi1�ρ2

p Z Φ�1ðuÞ

�1

Z Φ�1ðvÞ

�1exp � s2�2ρst þ t2

2ð1�ρ2Þ

� �dsdt

ð6Þwhere ρ∈½�1;1� is the only Gaussian copula parameter,Φ(.) denotes the cdf of standard univariate Gaussiandistribution and Ψ (.) denotes the standard bivariate Gaus-sian distribution function. The standard Gaussian cdf canbe written as ΦðzÞ ¼ 1=

ffiffiffiffiffiffi2π

p R z�1 expð�t2=2Þdt andΦ�1

ðuÞ ¼ffiffiffi2

perf�1ð2u�1Þ. Here erf(.) function is called error

function, which is defined as erf ðxÞ ¼ 2=ffiffiffiπ

p R x0 expð�t2Þdt.

Actually, according the theorem 2.4.3 in Ref. [1], we canprove that the Gaussian copula can be defined by twogeneralized (non-standard) Gaussian distributions as fol-lows. The theorem 2.4.3 in Ref. [1] can be described as: Let Xand Y be continuous random variables with copula CXY. If αXand βY are strictly increasing on RanX and RanY, respec-tively, then CαXβY ¼ CXY . Thus CXY is invariant under strictlyincreasing transformations of X and Y [1 (Section 2.4)].

Let FX and FY denote the standard Gaussian distributionfunctions of X and Y. Transforming X and Y to X′ andY′respectively as follows: αX ¼ XsX þ μX and βY ¼ YsY þ μYrespectively, where μX and μY are expected values, sX andsY are standard derivations of X′and Y′respectively, and

then the standard Gaussian distribution becomes general-ized Gaussian distribution. Note that αX and βY are strictlyincreasing on RanX and RanY respectively, and thereforewe can determine that the copula defined by two general-ized normal distribution is consistent with the copuladefined by two standard normal distribution.

2.2. Bivariate log-normal copula

Log-normal distributions are usually used to model theslow fading channel for communication systems [24]. Let

two log-normal pdfs of x and y be f XðxÞ ¼ 1=xffiffiffiffiffiffi2π

psX

� �expð�1=2ðlogðxÞ�μX=sXÞ2Þ and f Y ðyÞ ¼ 1=y

ffiffiffiffiffiffi2π

psY

� �exp

ð�1=2ðlogðyÞ�μY=sY Þ2Þ, respectively. Note that both log(x)and log(y) have a normal distribution, where μX and μY arethe corresponding mathematical expectations and sX andsY the corresponding standard deviations of the tworandom variables of log(x) and log(y), respectively. Let FXand FY denote the standard Gaussian distribution functionsof X and Y, we can respectively transform X to X′ and Y toY′ using αX ¼ expðμX þ XsXÞ and βY ¼ expðμY þ YsY Þ, thusboth X′and Y′satisfy a log-normal distribution. Note thatαXand βYare strictly increasing on RanX and RanY, respec-tively, thus the log-normal copula is consistent with theGaussian copula. The log-normal copula, with four moreparameters A, B, m, n used, was firstly derived in Ref. [5],but the author did not point out this equivalence. This doesnot affect the final definition of log-normal copula func-tion, as we can simply transform X and Y as followsαX ¼ logðX=AÞ�mμX=msX and βY ¼ logðY=BÞ�nμY=nsY . Notethat αX and βY are strictly increasing on RanX and RanYrespectively, and thus the definition of log-normal isinvariant.

2.3. Bivariate Weibull/Rayleigh/exponential copula

The Weibull distribution is usually used for modelingchannel fading of communications [22] and wind speed inpower systems [26]. Let the two Weibull pdfs be f XðxÞ ¼βX=ΩXxβX�1expð�ðxβX=ΩXÞÞ and f Y ðyÞ ¼ βY=ΩYyβY�1 expð�ðyβY =ΩY ÞÞ,; respectively, where βX≥0, βY≥0, Ωx ¼ EðxβX Þand ΩY ¼ EðyβY Þ, and Eð:Þ denotes the operator of mathe-matical expectation. Then, the associated marginal cdfsbecome FXðxÞ ¼ 1�expð�ðxβX=ΩXÞÞ and FY ðyÞ ¼ 1�exp

X. Zeng, et al. / Signal Processing 94 (2014) 691–702694

ð�ðyβY =ΩY ÞÞ. Thus, the corresponding bivariate Weibulldensity function is obtained as [22]

f XY ðx; yÞ ¼βXβYxβX�1yβY�1

ΩXΩY ð1�ρÞ exp�1

ð1�ρÞ ðxβX

ΩXþ yβY

ΩYÞ

� �

I02xβX=2yβY=2

ffiffiffiρ

pð1�ρÞ ffiffiffiffiffiffiffiffiffiffiffiffiffi

ΩXΩYp

!ð7Þ

where 0≤ρo1 denotes the power correlation coefficient.The modified Bessel functions of the first kind with nthorder are defined as [27 Eq. (9.6.10)]

InðzÞ ¼z2

� �n∑1

k ¼ 0

ðz=2Þ2nk!Γðnþ kþ 1Þ ð8Þ

where Γð:Þis the Gamma function [27 (Eqs. 6.1.1 and 6.1.6)].Note that the inverse marginal cdfs are obtained as x¼ F�1

XðuÞ ¼ ½�ΩX lnð1�uÞ�1=βX andy¼ F�1

Y ðvÞ ¼ ½�ΩY lnð1�vÞ�1=βY .As a result, the Weibull copula can be rewritten as

Cðu; vÞ ¼Z b1

0

Z b2

0

βXβYxβX�1yβY�1

ΩXΩYρ0exp �ðz1 þ z2Þ½ �I0ð2

ffiffiffiffiffiffiffiffiffiffiffiffiρz1z2

p Þdxdy

ð9Þwhere we have b1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ΩX lnð1�uÞ

pand b2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�ΩY lnð1�vÞp

; z1 ¼ ðρ0ΩXÞ�1xβX∈½0;w1�where w1 ¼�logð1�uÞ=ρ0, and z2 ¼ ðρ0ΩY Þ�1yβY∈½0;w2� where w2 ¼�logð1�vÞ=ρ0 and ρ0 ¼ 1�ρ.

Since the associated Jacobean determinant is deter-

mined as

����� ∂x=∂z1 ∂x=∂z2∂y=∂z1 ∂y=∂z2

�����¼ðð1�ρÞ2ΩXΩY Þ=ðβXβYxβX�1yβY�1Þ.

Consequently, Eq. (9) can be rewritten as

Cðu; vÞ ¼ ρ0Z a1

0

Z a2

0expð�z1�z2ÞI0ð2

ffiffiffiffiffiffiffiffiffiffiffiffiρz1z2

p Þdz1z2 ð10Þ

where we have a1 ¼�lnð1�uÞ=ρ0 and a2 ¼�lnð1�vÞ=ρ0.Consequently, we can find that ρ is the only parameterfor Weibull copula.

Next, we will discuss the case of exponential copula.Consider as an example, a single-server queuing system, asa common assumption, the inter-arrival time and theservice time satisfy an exponential distribution. Downton'sbivariate exponential distribution is a suitable choice forthe queuing system with infinitely servers [2 (Section5.14)].

Let two marginal exponential probability density func-tions be f XðxÞ ¼ λexpð�λxÞ and f Y ðyÞ ¼ μexpð�μyÞ, wherex≥0, y≥0, λ≥0 and μ≥0. The expectations of x and y can bedecided as 1/λ and 1/μ, respectively. Thus, their twomarginal cdfs can be determined as FXðxÞ ¼ 1�expð�λxÞand FY ðyÞ ¼ 1�expð�μyÞ, respectively. Finally, the asso-ciated Downton's bivariate exponential distribution has apdf as [2 (Eq. 5.5.7)]

f XY ðx; yÞ ¼λμ

1�ρexp � λx

1�ρ� μy1�ρ

� I0

21�ρ

ffiffiffiffiffiffiffiffiffiffiffiffiρλμxy

p� ð11Þ

where ρ is the correlation coefficient of x and y satisfying0≤ρo1, and I0 (.) is the modified Bessel function of thefirst kind with 0th order. Clearly, x and y are independentonly if ρ¼0.

From Eq. (10), we have found that ρ is the onlyparameter for Weibull copula. This implies that we candefine the Weibull copula by two Weibull distributions

with arbitrary parameters βX and βY respectively. As amatter of fact, the Weibull distribution becomes exponen-tial distribution when βX ¼ 1 and βY ¼ 1, while this will notaffect the final definition of Weibull copula. Therefore, wecan determine that the exponential copula equals to theWeibull copula.

Next, we will discuss the Rayleigh copula function. TheRayleigh distribution is frequently used in communica-tion domain to model the received signal amplitudes inurban and suburban areas [23]. Let two marginal Rayleighdensity functions be f XðxÞ ¼ 2x=ΩXexpð�ðx2=ΩXÞÞ andf Y ðyÞ ¼ 2y=ΩYexpð�ðy2=ΩY ÞÞ, where x≥0; y≥0. The asso-ciated two marginal cdfs are FXðxÞ ¼ 1�expð�ðx2=ΩXÞÞand FY ðyÞ ¼ 1�expð�ðy2=ΩY ÞÞ, where Ωx ¼ Eðx2Þ and ΩY ¼Eðy2Þ, and Eð:Þdenotes the operator of mathematical expec-tation. The corresponding bivariate Rayleigh density func-tion can be derived as [25 (Eq. 6.2)]

f XY ðx; yÞ ¼4xy

ΩXΩYρ0exp

�1ρ0

x2

ΩXþ y2

ΩY

� � �I0

2xyρ0

ffiffiffiffiffiffiffiffiffiffiffiffiffiρ

ΩXΩY

r� ð12Þ

where 0≤ρo1 is the power correlation coefficient of xand y.

We have proven that ρ is the only parameter forWeibull copula in Eq. (10), and therefore we can definethe Weibull copula by two Weibull distribution witharbitrary parameters βX and βY respectively. We can simplylet βX ¼ 2 and βY ¼ 2, then the Weibull distributionbecomes Rayleigh distribution, and this will not changethe final definition of Weibull copula. Consequently, wecan determine that the Rayleigh copula equals to theWeibull copula as well.

To compute the bivariate Weibull/exponential/Rayleighcopula, the following formula can be used to reduce thedouble integral to a single integral [28]:

ð1�ρÞZ x

0

Z y

0e�ðsþtÞI0ð2

ffiffiffiffiffiffiffiρst

pÞdsdt ¼ ð1�eρy�yÞ

þ eρy�yKðρy; xÞ�eρx�xKðy; ρxÞ ð13Þwhere ρ≥0 and Kðx; yÞ ¼ e�y

R x0 e

�sI0ð2ffiffiffiffiffisy

p Þds.After algebraic manipulation, Eq. (10) can be replaced

by the following formula that only includes a singleintegral for simplicity:

Cðu; vÞ ¼ 1þ eρa2�a2

Z ρa2

0e�sI0ð2

ffiffiffiffiffiffiffia1s

p Þds�1� �

�e�a1

Z a2

0e�sI0ð2

ffiffiffiffiffiffiffiffiffiffiρa1s

p Þds ð14Þ

As Kðx; yÞ ¼ e�yR x0 e�sI0ð2

ffiffiffiffiffisy

p Þds. K(x,y) can also beexpressed by Marcum's Q function as follows [29]:

QMða; bÞ ¼Z 1

bx

xa

� �M�1e�ððx2þa2Þ=2ÞIM�1ðaxÞdx ð15Þ

Let Q1(a,b) denote the special case of Marcum's Qfunction with M¼1 and Q1ða; bÞ ¼

R1b xe�ðx2þa2=2ÞI0ðaxÞdx,

we haveZ b

0xe�ðx2þp2Þ=2I0ðaxÞdx¼ ½1�Q1ða=p; bpÞ�e�ða2=2p2Þ=p2 ð16ÞAfter algebraic manipulation, we can derive the

relationship between K(x,y) and Q1(x,y) as Kðx; yÞ

00.2

0.40.6

0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

uv

C(u,v)

u

v

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

00.2

0.40.6

0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

uv

C(u,v)

u

v

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 1. Bivariate Weibull/exponential/Rayleigh copula.

X. Zeng, et al. / Signal Processing 94 (2014) 691–702 695

¼ 1�Q1ðffiffiffiffiffiffi2y

p;ffiffiffiffiffiffi2x

pÞ. As a result, Eqs. (10) and (14) can be

rewritten as [5 (Eq. (4))]:

Cðu; vÞ ¼ 1�ð1�vÞa3�ð1�uÞð1�a3Þa3 ¼Q1ð

ffiffiffiffiffiffiffiffi2a1

p;ffiffiffiffiffiffiffiffi2a2

pÞ ð17Þ

Apparently, this has validated that Eq. (14) is a newrepresentation of bivariate Rayleigh copula that onlyrequires a single integral.

The computation results of bivariate Weibull/Rayleigh/Exponential copula with the parameter ρ¼0.3 and ρ¼0.8are shown in Fig. 1(a) and (c), respectively, and theircontour plots are shown in Fig. 1(b) and (d) respectively,which have validated the properties of the copula asdefined in Eqs. (3) and (4). In the plots of generatedcontours, we can clearly find that the isolines have thefollowing characteristics. Firstly, on each separate isoline,its curvature increases when the difference between u andv decreases, which reaches the maximum when u¼v.Secondly, the isolines have lower curvaturess when (u+v)is large but higher curvaturess when (u+v) is small.Thirdly, when ρ increases from 0 to 1, the maximumcurvature achieved in each isoline increases, though theremaining part of isolines becomes flatter than those of asmaller ρ. In an extreme case when ρ¼0, perfectly parallel

straight lines with the slope of �451 will be produced inthe contour plot. Moreover, the perpendicular lines withcorners lying on the diagonal will appear when ρ¼1.

2.4. Bivariate Nakagami-m copula

Nakagami-m distribution is frequently used to modelfading in wireless environment [23]. Considering twomarginal Nakagami-m pdfs of random variables x and yas f XðxÞ ¼ 2mmx2m�1=ΓðmÞΩm

X expð�ðmx2=ΩXÞÞ and f Y ðyÞ¼ 2mmy2m�1=ΓðmÞΩm

Y expð�ðmy2=ΩY ÞÞ, respectively, wherex40 and y40. The marginal cdfs can be determined asFXðxÞ ¼ Pðm;mx2=ΩXÞ and FY ðyÞ ¼ Pðm;my2=ΩY Þ, where P(.)is the regulated lower incomplete Gamma function beingdefined as Pða; zÞ ¼ γða; zÞ=ΓðaÞ ¼ 1

ΓðaÞR z0 t

a�1expð�tÞdt[27(Eq. (6.5.1))], and γ(.) is the lower incomplete Gammafunction defined as γða; zÞ ¼ R z

0 ta�1expð�tÞdt[27 (Eq.

(6.5.2))]. The associated joint pdf can then be derived as [6]

f XY ðx; yÞ ¼4ρmð1�ρÞ�1ðmxyÞmΓðmÞð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ρΩXΩYp Þmþ1 expð�z1�z2ÞIm�1ð2

ffiffiffiffiffiffiffiffiffiffiffiffiρz1z2

p Þ

ð18Þ

X. Zeng, et al. / Signal Processing 94 (2014) 691–702696

where 0≤ρo1 is the power correlation coefficient of x andy, ρ0 ¼ 1�ρ, z1 ¼mx2=ρ0ΩX and z2 ¼my2=ρ0ΩY . The copulafunction of the bivariate Nakagami-m distribution hasbeen defined with two parameters ρ and m as [6]

Cðu; vÞ ¼ ð1�ρÞmΓðmÞ ∑

1

k ¼ 0

ρkΓðmþ kÞ2k!

� �P mþ k;

P�1ðm;uÞ1�ρ

" #

P mþ k;P�1ðm; vÞ

1�ρ

" #ð19Þ

The computation results of bivariate Nakagami-mcopula with the parameter ρ¼0.3; m¼2, ρ¼0.8; m¼2and ρ¼0.8; m¼5 are shown in Fig. 2(a), (c) and (e)respectively, and their contour plots are shown in Fig. 2(b), (d) and (f) respectively, which have validated theproperties of the copula defined in Eqs. (3) and (4). Overall,the generated isolines have similar characteristics as thosein the results of bivariate Weibull/Rayleigh/Exponentialcopula functions: the curvature decrease when u+vincreases and reaches the extreme when u¼v; When ρincreases, the maximum curvature achieved also increases.Again if the parameter ρ is set to 0, perfectly parallelstraight lines with the slope of �451 will be shown in thecontour plot. If the parameter ρ is set to 1, the perpendi-cular lines with corners lying on the diagonal will begenerated. In addition, the contours seem insensitive tothe parameter m when its value is changed from 2 to 5.

2.5. 1.5 Bivariate Rician copula

Rician distribution is widely used to model the ampli-tude fluctuations of received signals from different multi-path fading [23]. Considering two marginal Rician pdfsof random variables x and y as f XðxÞ ¼ x=s2expð�ðx2 þa2=2s2ÞÞI0ðax=s2Þ and f Y ðyÞ ¼ y=s2expð�ðy2 þ a2=2s2ÞÞI0ðay=s2Þ, respectively, where x≥0 and y≥0, and a is thenon-centrality parameter. When a¼0, the Rician distribu-tion degrades to the Rayleigh distribution. The marginalcdfs of x and y are FXðxÞ ¼ 1�Q1ða=s; x=sÞ and FY ðyÞ¼ 1�Q1ða=s; y=sÞ, respectively [30 (Eq. (2.18))], and theassociated bivariate Rician pdf is given in [30,31] as

f XY ðx; yÞ ¼xy

ð1�ρ2Þs4 exp � x2 þ y2 þ 2ð1�ρÞa22ð1�ρ2Þs2

�

∑þ1

k ¼ 0εkIk

xyρð1�ρ2Þs2�

Ikax

ð1þ ρÞs2�

Ikay

ð1þ ρÞs2�

ð20Þwhere 0≤ρo1 is the correlation coefficient of x and y and

ε is the Neumann factor defined as εm ¼ 1 m¼ 02 m40

[30].

The associated bivariate Rican copula can be written as

Cðu; vÞ ¼Z x0

0

Z y0

0

xyð1�ρ2Þs4 exp � x2 þ y2 þ 2ð1�ρÞa2

2ð1�ρ2Þs2�

∑þ1

k ¼ 0εkIk

xyρð1�ρ2Þs2�

Ikax

ð1þ ρÞs2�

Ikay

ð1þ ρÞs2�

dxdy

ð21Þ

where x0 ¼ sQ�11 ða=s;1�uÞ and y0 ¼ sQ�1

1 ða=s;1�vÞ.Let z1 ¼ xs�1, z2 ¼ ys�1 and z¼ as�1, thus we have

0≤z1≤Q�11 ðz;1�uÞ and 0≤z2≤Q�1

1 ðz;1�vÞ. The associated

Jacobean determinant becomes

�����∂x=∂z1 ∂x=∂z2∂y=∂z1 ∂y=∂z2

�����¼ s2. As

a result, the bivariate Rician copula function in Eq. (32) canbe derived as

Cðu; vÞ ¼Z b1

0

Z b2

0

z1z21�ρ2

exp � z21 þ z22 þ 2ð1�ρÞz22ð1�ρ2Þ

�

∑þ1

k ¼ 0εkIk

ρz1z21�ρ2

� Ik

z1z1þ ρ

� Ik

z2z1þ ρ

� dz1dz2 ð22Þ

where b1 ¼Q�11 ðz;1�uÞ and b2 ¼Q�1

1 ðz;1�vÞ.Note that the Rician factor k can be defined as

K ¼ a2=2s2, and thus z¼ffiffiffiffiffiffiffi2K

p. Therefore, the bivariate

Rician copula function can also be written as

Cðu; vÞ ¼Z b1

0

Z b2

0

z1z21�ρ2

exp � z21 þ z22 þ 4ð1�ρÞK2ð1�ρ2Þ

�

∑þ1

k ¼ 0εkIk

ρz1z21�ρ2

� Ik

z1ffiffiffiffiffiffiffi2K

p

1þ ρ

!Ik

z2ffiffiffiffiffiffiffi2K

p

1þ ρ

!dz1dz2

ð23Þwhere b1 ¼Q�1

1 ðffiffiffiffiffiffiffi2K

p;1�uÞ and b2 ¼Q�1

1 ðffiffiffiffiffiffiffi2K

p;1�vÞ.

Also note that the bivariate Rician copula relies only onthe parameters z (or K) and ρ. Also note that Marcum's Qfunction QM(a,b) in Eq. (15) monotonically decreases from1 to 0 as b varies from 0 to 1, hence the bisection method[32] has been proposed for evaluating the inverse Mar-cum's function in [33]. The form of the bivariate Riciancopula is complicated, but it can be readily computed bythe numerical integral method [32]. Note that the bivariateRician distribution function can be represented by theinfinite series representation by expanding the modifiedfirst kind of Bessel function as follows [31,34]:

FXY ðx; yÞ ¼ exp � a2s�2

1þ ρ

� ∑1

k ¼ 0ð�1Þk

∑1

l;m;n ¼ 0

ð1þ ρÞ1�l0 ð1�ρÞ1þl0ρ2mþk as

� �2l02l0 l!m!n!ðlþ kÞ!ðmþ kÞ!ðnþ kÞ!

!

γðlþmþ kþ 1;ðx=sÞ22ð1�ρ2ÞÞγðnþmþ kþ 1;

ðy=sÞ22ð1�ρ2ÞÞ ð24Þ

where l0 ¼ lþ nþ k, andγ(.) is the lower incompleteGamma function defined as [27(Eq. (6.5.2))] as γða; zÞ¼ R z

0 ta�1expð�tÞdt.

Let u¼FX(x), and v¼FY(y), then x=s¼Q�11 ðz;1�uÞ and

y=s¼Q�11 ðz;1�vÞ where z¼ a=s, then we can derive the

bivariate Rician copula function C(u,v) expressed by infi-nite series from FXY(x,y) as

Cðu; vÞ ¼ exp�z2

1þ ρ

� ∑1

k ¼ 0εkð�1Þk

∑1

l;m;n ¼ 0

ð1þ ρÞ1�l0 ð1�ρÞ1þl0ρ2mþkz2l0

2l0 l!m!n!ðlþ kÞ!ðmþ kÞ!ðnþ kÞ!

γ lþmþ kþ 1;½Q�1

1 ðz;1�uÞ�22ð1�ρ2Þ

!

γðnþmþ kþ 1;½Q�1

1 ðz;1�vÞ�22ð1�ρ2Þ Þ ð25Þ

Although Eq. (25) can be directly used to computebivariate Rician copula, it suffers from the vast amount ofcomputation needed in resolving the inverse Marcum Q

X. Zeng, et al. / Signal Processing 94 (2014) 691–702 697

function. Assume we need to determine the result ofbivariate Rician copula C(u1,v1) with parameters ρ1 andz1, we need firstly calculate Q�1

1 ðz1;1�u1Þ and Q�11

ðz1;1�v1Þ for avoiding resolving the inverse Marcum Qfunction in the incomplete Gamma function of the infinitesummation as they are irrelevant to k, l, m and n. Alter-natively, Eq. (24) can be applied to compute bivariate

00.2

0.40.6

0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

uv

C(u,v)

00.2

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0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

uv

C(u,v)

00.2

0.40.6

0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

uv

C(u,v)

Fig. 2. Bivariate Naka

Rician copula as follows: assume we need to calculatethe result of bivariate Rician copula C(u1,v1) with para-meters ρ1 and z1, we can simply get x1 and y1 fromu1 ¼ 1�Q1ðz1; x1Þ andv1 ¼ 1�Q1ðz1; y1Þ. Next, we can sub-stitute x1 and y1 to Eq. (24) and also let a¼ z1 to computethe value of C(u1,v1) with parameters ρ1 and z1. In this waythe difficulty in computing the inverse Marcum Q function

0.1

0.1

0.1

0.1 0.1

0.2

0.2

0.20.2

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.8

0.9

u

v0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.10.1

0.1 0.1 0.1

2.0

0.2

0.2 0.2

0.30.3

0.3 0.3

0.4

0.40.4

5.0

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.9

u

v

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.10.1

0.1 0.1 0.1

2.0

0.2

0.2 0.2

0.30.3

0.3 0.3

0.4

0.40.4

0.5

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.9

u

v

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

gami-m copula.

00.2

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0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

uv

C(u,v)

0.1

0.1

0.1

0.1 0.10.2

0.2

0.20.2

0.3

0.30.3

0.3

0.4

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.8

0.9

u

v

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

00.2

0.40.6

0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

uv

C(u,v)

0.10.1

0.10.1 0.1

0.2

0.2

0.20.2

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.8

0.9

u

v

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

00.2

0.40.6

0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

uv

C(u,v)

0.10.1

0.1 0.1 0.1

2.0

0.2

0.2 0.2

0.30.3

0.3 0.3

0.4

0.40.4

5.0

0.50.5

0.6

0.6

0.7

0.7

0.8

0.9

u

v

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 3. Bivariate Rician copula.

X. Zeng, et al. / Signal Processing 94 (2014) 691–702698

in the infinite summation in Eq. (25) is also avoidedthough the whole process is still quite time consuming.The truncation error upper bound for the infinite series inEq. (24) is given in Ref. [34], and the truncation is relevantto u, v, ρ and z. It also has been pointed out that computingthe nested infinite summations is usually tedious even ifthey are truncated, since the calculation time and accuracy

solely depend on the available computing power [34]. Inaddition, Eqs. (22) and (23) can also be used to determinethe bivariate Rician copula using numerical integration[32]. This method is adopted in this paper, since theformer two methods using either Eqs. (25) or (24) aretoo time consuming to compute the bivariate Riciandistribution with 30 different values of u and v.

X. Zeng, et al. / Signal Processing 94 (2014) 691–702 699

The computation results of bivariate Rician copula withthe parameter ρ¼0.3; z¼0.5, ρ¼0.3; z¼2 and¼0.8; z¼2are shown in Fig. 3(a), (c) and (e) respectively, and theircontour plots are shown in Fig. 3(b), (d) and (f) respectively,which have validated the properties of the copula defined inEqs. (3) and (4). In this group of plots, we can find that thegenerated contours are insensitive to the parameter z. Onthe other hand, the isolines feature the same characteristicsas the previous two cases when u, v and ρ change.

3. Copula density functions

In practical applications, copula density functions areoften more frequently used than the copula function itself,e.g., copula density has been applied to estimate mutualinformation [7]. From the copula function, its densityfunction can be derived by definition in Eq. (26) below,where specific density functions for various copulas arediscussed in detail in this section.

cðu; vÞ ¼ ∂2Cðu; vÞ∂u∂v

¼ ∂2C½FXðxÞ; FY ðyÞ�∂FXðxÞ∂FY ðyÞ

ð26Þ

3.1. Bivariate log-normal copula

We have proved in Section (2.2) that the bivariate log-normal copula and bivariate Gaussian copula are equiva-lent. Therefore, the density functions of bivariate log-normal copula and bivariate normal copula are alsoequivalent and can be written as [7]

cðu; vÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffi1�ρ2

p exp2ρΦ�1ðuÞΦ�1ðvÞ�ρ2f½Φ�1ðuÞ�2 þ ½Φ�1ðvÞ�2g

2ð1�ρ2Þ

!

ð27Þwhere Φ(.) represents the cdf of standard Gaussiandistribution.

3.2. Bivariate exponential, Rayleigh and Weibull copulas

Since bivariate exponential, Rayleigh and Weibull copu-las have been proved to be equivalent, here only bivariate

00.2

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0

0.5

10

1

2

3

4

uv

c(u,v)

Fig. 4. Bivariate Weibull/exponent

exponential copula is considered in deriving the densityfunction. According to the Eq. (11), bivariate exponentialcopula has the form as

Cðu; vÞ ¼Z logð1�uÞ=�λ

0

Z logð1�vÞ=�μ

0

λμ

1�ρ

exp � λxþ μy1�ρ

� I0

2ffiffiffiffiffiffiffiffiffiffiffiffiρλμxy

p

1�ρ

� dxdy

where x¼�lnð1�uÞ=λ and y¼�lnð1�vÞ=μ. Note that1=FX

′ðxÞ� �¼ 1=λexpð�λxÞ� �and 1=FY

′ðyÞ� �¼ 1=μexp�

ð�μyÞÞ. After algebraic manipulation, we can derive thecopula density function as

cðu; vÞ ¼ 11�ρ

expρ½lnðu0Þ þ lnðv0Þ�

1�ρ

� I0

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρlnðu0Þlnðv0Þ

p1�ρ

!

ð28Þwhere u0 ¼ 1�u and v0 ¼ 1�v.

From Eq. (28), it also can be found again that thebivariate exponential (and also Rayleigh and Weibull)copula rely only on the parameter ρ. Plots of the bivariateexponential (also Rayleigh and Weibull) copula densityfunctions with ρ¼0.3 and ρ¼0.8 are given in Fig. 4(a) and(b) respectively. If the parameter ρ is set to 0, we have c(u,v)¼1 for arbitrary values of u and v. When the parameterρ increases from 0 to 1, higher values of c(u,v) occur whenu and v are close. On the other hand, c(u,v) decreases andapproaches to 0 when the values of u and v are dissimilar.

3.3. Bivariate Nakagami-m copula

The copula function of bivariate Nakagami-m distribu-tion has been defined in [5] as

Cðu; vÞ ¼ ð1�ρÞmΓðmÞ ∑

1

k ¼ 0

ρkΓðmþ kÞ2k!

� �P mþ k;

P�1ðm;uÞ1�ρ

" #

P mþ k;P�1ðm; vÞ

1�ρ

" #

where mx2=ΩX ¼ P�1ðm;uÞ and my2=ΩY ¼ P�1ðm; vÞ.Note that the partial derivatives of the regulated lowerincomplete Gamma function have been evaluated in

00.2

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0.81

0

0.5

10

5

10

15

uv

c(u,v)

ial/Rayleigh copula density.

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0.81

0

0.5

10

1

2

3

4

uv

c(u,v)

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0

0.5

10

5

10

15

uv

c(u,v)

00.2

0.40.6

0.81

0

0.5

10

5

10

15

uv

c(u,v)

Fig. 5. Bivariate Nakagami-m copula density.

X. Zeng, et al. / Signal Processing 94 (2014) 691–702700

[27 (Eq. (6.5.25))] as

∂Pða; zÞ∂z

¼ e�zza�1

ΓðaÞ ð29Þ

Thus we have 1=FX′ðxÞ ¼ ΓðmÞΩX=2mx½P�1ðm;uÞ�m�1exp

½P�1ðm;uÞ� and 1=FY′ðyÞ ¼ ΓðmÞΩX=2my½P�1ðm; vÞ�m�1exp

½P�1ðm; vÞ�. After algebraic manipulation, we can derive thecopula density of Nakagami-m distribution as

cðu; vÞ ¼ ΓðmÞ ð_u_vÞ�ðm�1Þ=2

ð1�ρÞρðm�1Þ=2 exp�ρð_u þ_vÞ

1�ρ

� Im�1

2ffiffiffiffiffiffiffiffiffiρ_u_v

p1�ρ

!

ð30Þ

where_u ¼ P�1ðm;uÞ,_v ¼ P�1ðm; vÞ,Γð:Þis the Gamma function[27 (Eqs. 6.1.1 and 6.1.6)] and P(.) is regulated lower incom-plete Gamma function. From Eq. (30), again it validates thatthe bivariate Nakagami-m copula rely only on the parametersρ and m.

Plots of the bivariate Nakagami-m copula density withparameter ρ¼0.3; m¼2, ρ¼0.8; m¼2 and¼0.8; m¼5 areshown in Fig. 5(a), (b) and (c) respectively. If the parameterρ is close to 0, the value of c(u,v) approaches 1 for arbitraryvalues of u and v. If the parameter ρ is close to 1, highervalues of c(u,v) occur when the values of u and v are closeto each other; c(u,v) decreases and approaches to 0 whenthe difference of u and v increases. If the parameter isfixed, for small values of u and v, c(u,v) increases when m

is increasing; whilst for high values of u and v, c(u,v)decreases when the parameter m increases.

3.4. Bivariate Rician copula

The copula function of bivariate Rician distribution hasbeen defined in Eq. (32) as

Cðu; vÞ ¼Z x0

0

Z y0

0

xyð1�ρ2Þs4 exp � x2 þ y2 þ 2ð1�ρÞa2

2ð1�ρ2Þs2�

∑þ1

k ¼ 0εkIk

xyρð1�ρ2Þs2�

Ikax

ð1þ ρÞs2�

Ikay

ð1þ ρÞs2�

dxdy

where x0 ¼ sQ�11 ða=s;1�uÞ and y0 ¼ sQ�1

1 ða=s;1�vÞ.Let z1 ¼ x=s∈½0; a1�, z2 ¼ y=s∈½0; a2� and z¼ a=s, where

a1 ¼ Q�11 ðz;1�uÞ and a2 ¼Q�1

1 ðz;1�vÞ. Note that thepartial derivative of Marcum's Q function can be repre-sented as [35]

∂Q1ða; bÞ∂b

¼�aI0ðabÞexp � a2 þ b2

2

!ð31Þ

Therefore, we have 1=FX′ðxÞ ¼ sexpf�ð1=2Þ ½Q�1

1 ðz;1�uÞ�2 þ ðzÞ2g=Q�1

1 ðz;1�uÞI0ðz; x=sÞ and 1=FY′ðyÞ ¼ sexp

f�ð1=2Þ½Q�11 ðz;1�vÞ�2 þ z2g=Q�1

1 ðz;1�vÞI0ðz; y=sÞ. Finally,we can derive the density function of bivariate Rician copula

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0.5

10.5

1

1.5

2

2.5

uv

c(u,v)

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10

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4

6

8

10

uv

c(u,v)

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0.5

10

2

4

6

8

10

12

uv

c(u,v)

Fig. 6. Bivariate Rician copula density.

X. Zeng, et al. / Signal Processing 94 (2014) 691–702 701

density function as

cðu; vÞ ¼ expð�ðρ2ða21 þ a22Þ þ 2ρðρ�1Þz2=2ð1�ρ2ÞÞÞð1�ρ2ÞI0ðz; a1ÞI0ðz; a2Þ

∑1

k ¼ 0εkIk

ρa1a21�ρ2

� Ik

a1z1þ ρ

� Ik

a2z1þ ρ

� ð32Þ

From Eq. (32), it can be found again that the bivariateRician copula rely only on the parameters ρ and z. Plots ofthe bivariate of Rician copula density with parameterρ¼0.3; z¼0.5, ρ¼0.8; z¼0.5 and¼0.8; z¼2 are shown inFig. 6(a), (b) and (c) respectively. If the parameter ρ is closeto 0, c(u,v) is close to 1 for arbitrary values of u and v. If theparameter ρ is close to 1, higher values of c(u,v) occurwhen u and v are close to each other; c(u,v) decreases andapproaches to 0 when the difference between u and v

increases. If the parameter ρ is fixed, for small values of uand v, the values of c(u,v) increases when m increases;whilst for high values of u and v, c(u,v) will decrease evenm is increasing.

4. Conclusions

Although copulas have been successfully applied formany signal/image processing and communication pro-blems, their applications are constrained due to the lack ofsuitable copulas and in-depth analysis of these functions.As a result, we focus on extension and generalization of

copulas. Firstly, copula functions associated with thebivariate exponential, Weibull and Rician distributionsare derived. Secondly, a new representation of bivariateRayleigh copula with a single parameter, using singleintegral, is also derived. Thirdly, we have proved thatlog-normal copula is equivalent to Gaussian copula thathas only one parameter, and Rayleigh copula is equivalentto exponential and Weibull copula that also has only oneparameter. The Rician copula is derived to have twoparameters. Fourthly, all the modified Bessel function orincomplete Gamma function involved double integrals inthe copula functions are represented by single integral orinfinite series. Finally, the density functions of log-normal,exponential, Rayleigh, Weibull, Nakagami-m and Riciancopulas are also derived. Since only bivariate copulas arefocused in this paper, multivariate cases will be studied inthe future for further improved applicability. Relevantdetails on random variables generation of these newcopulas and criteria for optimal copula selection as wellas simulation and case studies will be discussed in the PartII of the paper.

Acknowledgments

Partial work of this paper is funded by the two NationalScience Foundation of China (NSFC) Projects (61003201,

X. Zeng, et al. / Signal Processing 94 (2014) 691–702702

61202165) and a joint project funded by Royal Society ofEdinburgh and NSFC (61211130125).

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[3] U. Cherubini, E. Luciano, W. Vecchiato, Copula Methods in Finance,John Wiley & Sons, New York, 2004.

[4] T.S. Durrani, X. Zeng, Copulas for bivariate probability distributions,Electronics Letters 43 (4) (2007) 248–249.

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