Analytic Results for Efficient Computation of theNuttall−Q and Incomplete Toronto Functions

Paschalis C. Sofotasios1, Khuong Ho-Van2, Tuan Dang Anh2 and Hung Dinh Quoc2

1School of Electronic and Electrical Engineering, University of Leeds, LS2 9JT, Leeds, United Kingdom.e-mail: p.sofotasios@leeds.ac.uk

2Department of Telecommunications Engineering, HoChiMinh City University of Technology, HoChiMinh City, Vietnam.e-mail: khuong.hovan@yahoo.ca, datuan@hcmut.edu.vn, hungqd76@yahoo.com

Abstract—This work is devoted to the derivation of novelanalytic results for special functions which are particularlyuseful in wireless communication theory. Capitalizing on recentlyreported series representations for the Nuttall Q−function andthe incomplete Toronto function, we derive closed-form upperbounds for the corresponding truncation error of these seriesas well as closed-form upper bounds that under certain casesbecome accurate approximations. The derived expressions aretight and their algebraic representation is rather convenient tohandle analytically and numerically. Given that the Nuttall−Qand incomplete Toronto functions are not built-in in popularmathematical software packages, the proposed results are par-ticularly useful in computing these functions when employed inapplications relating to natural sciences and engineering, such aswireless communication over fading channels.

I. INTRODUCTION

It is widely known that special functions constitute invalu-able mathematical tools in most fields of natural sciences andengineering. In the area of telecommunications, their utiliza-tion often allows the derivation of elegant closed-form repre-sentations for important performance measures such as errorprobability, channel capacity and higher-order statistics (HOS).The computational realization of such expressions is typicallystraightforward since most special functions are built-in inpopular mathematical software packages such as MAPLE,MATLAB and MATHEMATICA. Among others, the MarcumQ−function, Qm(a, b), the Nuttall Q−function, Qm,n(a, b)and the incomplete Toronto function (ITF), TB(m,n, r) wereproposed several decades ago and have been involved inanalytic solutions of various scenarios in information and com-munication theory. More specifically, Marcum Q−functionwas firstly proposed by Marcum in [1], [2] but it becamewidely known in digital communications, over fading or non-fading media, by the contributions in [3]–[8]. Its traditionaland generalized form are denoted as Q1(a, b) and Qm(a, b),respectively, and their properties and identities were presentedin [3]. Respective upper and lower bounds were reported in[8]–[11] while an exponential integral representation was pro-posed in [11]. In addition, useful closed-form representationfor the Qm(a, b) function for the special case that the orderm is an positive odd multiple of 0.5, i.e. m± 0.5 ∈ N, werederived in [9], [12].

Likewise, the Nuttall Q−function is a special function thatemerges from the generalization of the Marcum Q−function[3]. Its definition is given by a semi-infinite integral represen-tation and it can be expressed in terms of Qm(a, b) and themodified Bessel function of the first kind, In(x), for the specialcase that the sum m+n is an odd integer [13]. Establishment ofmonotonicity criteria for Qm,n(a, b) along with the derivationof tight lower and upper bounds and a closed-form expressionfor the case that m±0.5 ∈ N and n±0.5 ∈ N were reported in[9]. In the same context, the incomplete Toronto function is asimilar special function which was proposed by Hatley in [14].It is a generalization of the Toronto function, T (m,n, r), andincludes the Qm(a, b) function as a special case. Its definitionis also given in non-infinite integral form while alternativerepresentations include two infinite series [15]. Its applicationhas been used in studies relating to statistics, signal detectionand estimation, radar systems and error probability analysis[16]–[18].

Nevertheless, in spite of the evident importance of theQm,n(a, b) and TB(m,n, r) functions, they are not adequatelytabulated while they are not included as standard built-infunctions in popular software packages. Motivated by this, theauthors in [9], [21]–[24] derived explicit expressions for theQm,n(a, b) and TB(m,n, r) functions. Specifically, a closed-form expression for Qm,n(a, b) was derived in [9] for thespecific case that m ± 0.5 ∈ N and n ± 0.5 ∈ N. A similarexpression for the TB(m,nr) was subsequently derived in[21], [22]. Furthermore, simple series representations for bothQm,n(a, b) and TB(m,n, r) functions were derived in [22]–[24]. These series are particularly useful since their algebraicform is relatively simple, which render the functions conve-nient to handle analytically. However, their form is infinite andthus the determination of adequate truncation needs to be takeninto consideration for ensuring acceptable levels of accuracyfor any given application.

Motivated by this, this work is devoted to the derivationof novel upper bounds for the truncation error of the abovefunctions. The derived representations are expressed in closed-form and have a relatively simple algebraic form. In addition,simple upper bounds are derived in closed-form, which areshown to be quite tight and in certain cases they become

The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

978-1-4799-1089-2/13/$31.00 ©2013 IEEE 420

remarkably accurate closed-form approximations.

II. NOVEL RESULTS FOR THE NUTTALL Q-FUNCTION

A. Definition, Basic Properties and Existing Expressions

The Nuttall Q-function is defined by the following semi-infinite integral representation [3, eq. (86)],

Qm,n(a, b) �

∫ ∞

b

xme−x2+a

2

2 In(ax)dx, (1)

and constitutes a generalization of the Marcum Q-function,

Qm(a, b) �1

am−1

∫ ∞

b

xme−x2+a

2

2 Im−1(ax)dx. (2)

The normalized Nuttall Q−function is given byQm,n(a, b) � Qm,n(a, b)/a

n, which for n = 0 yieldsQ1,0(a, b) = Q1,0(a, b) = Q1(a, b) = Q(a, b). For the specialcase that n = m−1 it follows that Qm,m−1(a, b) = Qm(a, b)and Qm,m−1(a, b) = am−1Qm(a, b). Furthermore, when mand n are integers, the following recursion equation wasreported in [13, eq. (3)],

Qm,n(a, b) =bm−1In(ab)

ea2+b2

2

+ aQm−1,n+1(a, b)

+ (m+ n− 1)Qm−2,n(a, b).

(3)

A finite series representation expressed in terms of theQ1(a, b), In(x) functions, the gamma function, Γ(x), andthe generalized kth order Laguerre polynomial, L(n)

k (x), wasproposed in [13, eq. (8)]. However, this expression is restrictedto the case that m+ n is an odd positive integer.

As already mentioned in Section I, a simple expressions forthe Qm,n(a, b) function were reported recently by [9], [21],[24]. This expression is given by,

Qm,n(a, b) �p∑

l=0

a2l e−a2

2 Γ(p+ l)p1−2l Γ(

m+n+2l+12 , b2

2

)l!Γ(n+ l + 1)2

n−m+2l+1

2 Γ(p− l + 1),

(4)where Γ(a) �

∫∞0

ta−1exp(−t) dt and Γ(a, x) �∫∞x

ta−1exp(−t) dt denote the gamma and upper incom-plete gamma functions, respectively. As, p → ∞, the termsΓ(p + l)p1−2l/Γ(p − l + 1) vanish and (4) reduces to thefollowing exact infinite series representation,

Qm,n(a, b) =∞∑l=0

a2l e−a2

2 Γ(

m+n+2l+12 , b

2

2

)l!Γ(n+ l + 1)2

n−m+2l+1

2

, (5)

For the special case that m,n ∈ Z+, the Γ(a, x) function can

be expressed in terms of the finite series representation in [19,eq. (8.352.4)]. Hence, after necessary variable transformation(4) can be also expressed as,

Qm,n(a, b) �p∑

l=0

L∑k=0

A a2lb2kΓ(p+ l)p1−2lΓ(m+n+12 + l)

l!k!Γ(n+ l+ 1)Γ(p− l + 1)2l+k,

(6)where,

L =m+ n− 1

2+ l, (7)

andA = an2

m−n−1

2 e−a2+b

2

2 . (8)

From (6) we can also obtain straightforwardly,

Qm,n(a, b) =

∞∑l=0

L∑k=0

A a2lb2kΓ(m+n+12 + l)

l!k!Γ(n+ l + 1)2l+k, (9)

It is noted here that the above representations have a con-venient algebraic representation and are not restricted sincethey hold for all values of the involved parameters. However,an efficient way that will determine effectively after howmany terms the above series can be truncated without lossof accuracy, is undoubtedly necessary.

B. A Closed-Form Upper Bound for the Truncation Error

The accuracy of (4) is proportional to the value of p andthe series converges quickly. However, deriving a convenientclosed-form expression that is capable of determining theinvolved truncation error analytically is particularly advanta-geous.

Lemma 1. For m,n, a ∈ R and b ∈ R+, the following closed-

form upper bound for the truncation error holds,

εt ≤�n�0.5−1∑

k=0

(−1)�n�0.5Γ(2�n�0.5 − k − 1)Ik�m�0.5,�n�0.5(a, b)k!Γ(�n�0.5 − k)(2a)−k

√π2�n�0.5−

12 a2�n�0.5−1

−p∑

l=0

a2l e−a2

2 Γ(p+ l)p1−2l Γ(

m+n+2l+12 , b2

2

)l!Γ(n+ l + 1)2

n−m+2l+1

2 Γ(p− l + 1), (10)

where,

Ikm,n(a, b) =

m−n+k∑l=0

(m− n+ k

l

)(−1)k2 l−1

2

an+l−m−k×

{(−1)m−n−l−1Γ

(l + 1

2,(b+ a)2

2

)+ Γ

(l + 1

2

)

− [sgn(b− a)]l+1γ

(l+ 1

2,(b− a)2

2

)},

(11)

with γ(a, x) �∫ x

0 ta−1exp(−t) dt denoting the lower incom-plete gamma function.

Proof: The truncation error of (4) is expressed as,

εt =∞∑

l=p+1

a2l e−a2

2 Γ(p+ l)p1−2l Γ(

m+n+2l+12 , b2

2

)l!Γ(n+ l + 1)2

n−m+2l+1

2 Γ(p− l + 1), (12)

which can be equivalently expressed as

εt =

∞∑l=0

a2l e−a2

2 Γ(p+ l)p1−2l Γ(

m+n+2l+12 , b2

2

)l!Γ(n+ l + 1)2

n−m+2l+1

2 Γ(p− l+ 1)︸ ︷︷ ︸I1

−p∑

l=0

a2l e−a2

2 Γ(p+ l)p1−2l Γ(

m+n+2l+12 , b2

2

)l!Γ(n+ l + 1)2

n−m+2l+1

2 Γ(p− l + 1).

(13)

The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

421

As already mentioned, since the series in I1 tends to infinity,the terms Γ(p+ l)p1−2l/Γ(p− l + 1) vanish, yielding

I1 = Qm,n(a, b) =

∞∑l=0

a2l e−a2

2 Γ(

m+n+2l+12 , b2

2

)l!Γ(n+ l + 1)2

n−m+2l+1

2

. (14)

It is recalled that the normalized Nuttall Q-function can beupper bounded by [9, eq. (19)], namely,

Qm,n(a, b) ≤ Q�m�0.5,�n�0.5(a, b), (15)

where �n�0.5 � �n− 0.5�+0.5, with �.� denoting the integerceiling function. To this effect, by substituting (15) into (14)and then into (13), one obtains,

εt ≤ Q�m�0.5,�n�0.5 (a, b)

−p∑

l=0

a2l e−a2

2 Γ(p+ l)p1−2l Γ(

m+n+2l+12 , b

2

2

)l!Γ(n+ l + 1)2

n−m+2l+1

2 Γ(p− l + 1).

(16)

Importantly, the upper bound in (15) can be expressed inclosed-form according to the expression in [9, Corollary 1].Therefore, after substitution in (16) equation (10) is deducedthus, completing the proof1.

Remark 1. By following the same methodology as in Lemma1, a similar upper bound can be deduced for the truncationerror of the infinite series of Qm,n(a, b) in (9).

C. A Tight Upper Bound and Approximation

Besides the expressions for the special cases that m+ n isan odd positive integer and m ± 0.5 ∈ N, n ± 0.5 ∈ N, nosimple representations exist for the Nuttall Q-function.

Proposition 1. For a, b,m, n ∈ R+ and for the special cases

that b → 0 | a,m, n ≥ 32b, the following closed-form upper

bound for the normalized Nuttall Q-function is valid,

Qm,n(a, b) ≤Γ(m+n+1

2

)1F1

(m+n+1

2 , n+ 1, a2

2

)n! 2

n−m+1

2 ea2

2

. (17)

where 1F1(a, b, x) denotes the Kummer confluent hypergeo-metric function [19], [20].

Proof: According to Remark 2, equation (4) reduces tothe exact infinite series as p → ∞. By also recalling that∫∞0 f(x)dx ≥ ∫∞

af(x)dx when a ∈ R

+, it follows thatΓ(a, x) ≤ Γ(x). As a result, the Qm,n(a, b) function can bestraightforwardly upper bounded as follows:

Qm,n(a, b) ≤∞∑l=0

a2l e−a2

2 Γ(m+n+2l+1

2

)l!Γ(n+ l + 1)2

n−m+2l+1

2︸ ︷︷ ︸I2

. (18)

By recalling that the Pochhammer symbol is defined as (a)n �

Γ(a+n)/Γ(a) and expressing each gamma function as Γ(x+

1By setting n = m−1 in (4), (6), and (10), similar expressions are deducedfor the Marcum Q-function, Qm(a, b).

l) = (x)lΓ(x), one obtains,

I2 =Γ(m+n+1

2

)e−a

2

2

n!2n−m+1

2

∞∑l=0

(m+n+1

2

)la2l

l! (n+ 1)l2l. (19)

The above infinite series can be expressed in terms of theKummer’s confluent hypergeometric function, 1F1(a, b, x) �∑∞

i=0(a)i(b)i

xi

i! . Hence, by performing the necessary change ofvariables and substituting (19) into (18), equation (17) isdeduced and thus the proof is completed.

1 1.5 2 2.5 3 3.50

1

2

3

4

5

6

m

Qm

,n (a

, b)

Qm,0.5

(2.00,1.00)

eq. (4) Q

m,2.0 (2.00,1.00)

eq. (4) Q

m,2.75 (2.00,1.00)

eq. (4) Q

m,0.25 (1.25,2.75)

eq. (4) Q

m,1.0 (1.25,2.75)

eq. (4) Q

m,2.5 (1.25,2.75)

eq. (4)

Fig. 1. Qm,n(a, b) in (4).

Remark 2. When a,m, n ≥ 52 b, the upper bound in (17) be-

comes an accurate closed-form approximation for Qm,n(a, b),namely,

Qm,n(a, b) �Γ(m+n+1

2

)1F1

(m+n+1

2 , n+ 1, a2

2

)n! 2

n−m+1

2 ea2

2

. (20)

The Nuttall Q−function is neither tabulated nor built-in inpopular mathematical software packages like MAPLE, MAT-LAB and MATHEMATICA. Hence, the derived expressionsare rather useful both analytically and numerically2.

The behaviour of (4) is illustrated in Fig. 1 along withresults obtained from numerical integrations. The series wastruncated after 20 terms and one can notice the excellentagreement between analytical and numerical results. This isalso verified through the level of the involved absolute relativeerror εr �| Qm,n(a, b) − Q̃m,n(a, b) | /Qm,n(a, b) whichis less than εr < 10−11. Similarly, the behaviour of (17) isdepicted in Fig. 2. Clearly, it upper bounds the Qm,n(a, b)tightly while it becomes an accurate approximation for highervalues of a.

2Similar expressions for the Qm,n(a, b) function can be straightforwardlydeduced with the aid of the identity Qm,n(a, b) = Qm,n(a, b)/an i.e. bymultiplying equations (4), (6), (10) and (18) with an.

The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

422

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

a

Qm

,n (a

, b)

Q1.7,1.1

( a , 0.6 )

eq. (16) Q

1.4,1.4 ( a , 0.6)

eq. (16) Q

1.1,1.7 ( a , 0.6)

eq. (16)

Fig. 2. Qm,n(a, b) in (17).

III. NOVEL ANALYTIC RESULTS FOR THE INCOMPLETE

TORONTO FUNCTION

A. Definition and Basic Properties

The ITF has been also useful in telecommunications and isdefined as,

TB(m,n, r) � 2rn−m+1e−r2∫ B

0

tm−ne−t2In(2rt)dt. (21)

When B = ∞, the TB(m,n, r) function reduces to theT (m,n, r) function, while for the specific case n = (m−1)/2it is expressed in terms of the Marcum Q-function, namely,

TB

(m,

m− 1

2, r

)= 1−Qm+1

2

(r√2, B

√2). (22)

Alternative representations to the TB(a, b, r) function includetwo series which are infinite and no study has been reportedon their convergence and truncation [15]. In addition, thefollowing polynomial representation was proposed in [22],

TB(m,n, r) �p∑

k=0

Γ(p+ k)r2(n+k)−m+1γ(m+12 + k,B2

)k!p2k−1Γ(p− k + 1)Γ(n+ k + 1)er2

.

(23)which as p→∞ it reduces to,

TB(m,n, r) =

∞∑k=0

r2(n+k)−m+1γ(m+12 + k,B2

)k!Γ(n+ k + 1)er2

. (24)

B. A Closed-Form Upper Bound for the Truncation Error

A tight upper bound for the truncation error of (24) can bederived in closed-form.

Lemma 2. For m,n, r ∈ R, B ∈ R+ and m > n the

following closed-form inequality holds,

εt ≤�n�0.5−

12∑

k=0

L∑l=0

r−(2k+l)(�n�0.5 + k − 1

2

)! (L − k)!

k! l!(�n�0.5 − k − 1

2

)!(L− k − l)!

×{γ[l+12 , (B + r)2

](−1)�m�−l 22k+1

+γ[l+12 , (B − r)2

](−1)k 22k+1

}

−p∑

k=0

Γ(p+ k)r2(n+k)−m+1γ(m+12 + k,B2

)k!p2k−1Γ(p− k + 1)Γ(n+ k + 1)er2

.

(25)

Proof: Since the corresponding truncation error is ex-pressed as εt =

∑∞p+1 f(x) =

∑∞l=0 f(x) −

∑p

l f(x) andgiven that (24) reduces to an exact infinite series as p → ∞,it follows that,

εt =

∞∑k=0

r2(n+k)−m+1γ(m+12 + k,B2

)k!Γ(n+ k + 1)er2︸ ︷︷ ︸

I3

−p∑

k=0

Γ(p+ k)r2(n+k)−m+1γ(m+12 + k,B2

)k!p2k−1Γ(p− k + 1)Γ(n+ k + 1)er2

.

(26)

Given that I3 = TB(m,n, r), the εt can be upper bounded asfollows:

εt ≤ TB(�m�, �n�0.5, r)

−p∑

k=0

Γ(p+ k)r2(n+k)−m+1γ(m+12 + k,B2

)k!p2k−1Γ(p− k + 1)Γ(n+ k + 1)er2

.(27)

The TB(�m�, �n�0.5, r) function can be expressed in closed-form according to the closed-form expression in [22]. Hence,by substituting in (27) equation (25) is obtained, which com-pletes the proof.

Remark 3. By omitting the terms Γ(p+k)p1−2k/Γ(p−k+1)in (25), a closed-form upper bound can be deduced for thetruncation error of the exact infinite series in Remark 2.

C. A Tight Closed-form Upper Bound and Approximation

Capitalizing on the algebraic form of the TB(m,n, r) func-tion, a simple closed-form upper bound is derived which incertain cases becomes an accurate approximation.

Proposition 2. For m,n, r ∈ R, B ∈ R+ and m,n, r ≤ B

2 ,the following inequality holds,

TB(m,n, r) ≤ Γ(m+12

)1F1

(m+12 , n+ 1, r2

)rm−2n−1Γ(n+ 1)er2

. (28)

Proof: It is recalled that the γ(a, x) function canbe upper bounded by the Γ(a) function since Γ(a) =∫∞0 ta−1exp(−t)dt = ∫ x=∞

0 ta−1exp(−t)dt = γ(a, x = ∞).Therefore, the TB(m,n, r) function can be upper bounded as,

TB(m,n, r) ≤∞∑k=0

r2(n+k)−m+1Γ(m+12 + k

)k!Γ(n+ k + 1)er2

, (29)

The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

423

which with the aid of the identity Γ(a, n) = (a)nΓ(a) can beexpressed as,

TB(m,n, r) ≤ r2n−m+1Γ(m+12

)Γ(n+ 1) er2

∞∑k=0

(m+12

)k

(n+ 1)k

r2k

k!. (30)

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

T B(m

, n, r

)

T

4.0 (1.0, 0.5, r)

eq. (21) T

4.0 (1.0, 1.0, r)

eq. (28) T

4.0 (2.0,1.5,r)

eq. (21) T

4.0 (2.0,2.0,r)

eq. (28) T

4.0 (3.0,2.5,r)

eq. (21) T

4.0 (3.0,3.0,r)

eq. (28)

Fig. 3. TB(m,n, r) in (23)

Importantly, the above series can be expressed in closed-formin terms of the Kummer’s confluent hypergeometric function.Thus, equation (28) is deduced and the proof is completed.

Remark 4. The proof can be also completed by assumingB → ∞ in (21) and utilizing [19, eq. (8.406.3)] and [19,eq. (6.631.1)]. The incomplete Toronto function reduces thento the Toronto function, as the two functions are related bythe identity TB=∞(m,n, r) = T (m,n, r). Furthermore (28)becomes an accurate approximation when m,n, r ≤ 2B,namely,

TB(m,n, r) � Γ(m+12

)1F1

(m+12 , n+ 1, r2

)rm−2n−1Γ(n+ 1)er2

. (31)

Equation (24) is depicted in Fig. 3 along with results fromcorresponding numerical integrations.The agreement betweenanalytical and numerical results is excellent and the relativeerror for (24) is εr < 10−4 for truncation after 20 terms. In thesame context (28) is depicted in Fig. 4 along with numericalresults for three different scenarios. The involved relative erroris proportional to the value of r and is εr < 10−6 when r < 1.

To the best of the authors’ knowledge, the offered resultshave not been reported in the open technical literature andcan be particularly useful in emerging wireless technologiessuch as cognitive radio, cooperative communications, MIMOsystems, digital communications over fading channels, ulta-sound and free-space-optical communications, among others[21]–[43], and the references therein.

IV. CONCLUSION

Novel upper bounds were derived for the truncation error ofrepresentations for the Nuttall Q-function and the incomplete

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

T B(m

, n, r

)

T4.0

(1.0, 0.5, r)

eq. (33) T

4.0 (1.6, 1.6, r)

eq. (33) T

4.0 (2.0,2.5,r)

eq. (33)

Fig. 4. TB(m, n, r) in (28)

Toronto function. These expressions are given in closed-formand have a tractable algebraic form. Since the Nuttall−Qand incomplete toronto functions are not included as built-in functions in popular mathematical software packages, theoffered results are useful in computing these functions ef-ficiently. As a result, the accurate computation of criticalperformance measures in digital communications that involvethese functions becomes more feasible.

ACKNOWLEDGEMENTS

This research is funded by Vietnam National UniversityHoChiMinh City (VNU-HCM) under grant number C2013-20-09.

REFERENCES

[1] J. I. Marcum, “A statistical theory of target detection by pulsed radar:Mathematical appendix”, RAND Corp., Santa Monica, Research mem-orandum, CA, 1948.

[2] J. I. Marcum, “Table of Q-functions, U.S. Air Force Project RAND Res.Memo. M-339, ASTIA document AD 116545”, RAND Corp., SantaMonica, CA, 1950.

[3] A. H. Nuttall, “Some integrals involving the Q-Function”, Naval under-water systems center, New London Lab, New London, CT, 1972.

[4] J. G. Proakis, “Digital Communications”, 3rd ed. McGraw-Hill, 1995.[5] C. W. Helstrom, “Computing the generalized Marcum Q -function”,

Trans. on Inf. Theory, vol. 38, 4, pp. 1422-1428, Jul. 1992.[6] D. A. Shnidman, “The calculation of the probability of detection and

the generalized Marcum Q -function”, IEEE Trans. on Inf. Theory, vol.35, Issue 2, pp. 389-400, Mar. 1989.

[7] M.K. Simon, “A new twist on the Marcum Q-function and its applica-tion”, IEEE Comm. Lett., vol. 2, no. 2, pp. 39-41, Feb. 1998.

[8] M.K. Simon, M.S. Alouini, “Exponential-Type Bounds on the Gener-alized Marcum Q-Function with Application to Error Probability overFading Channels”, IEEE Trans. Commun., vol. 48, no. 3, pp. 359-366,Mar. 2000.

[9] V. M. Kapinas, S. K. Mihos, and G. K. Karagiannidis, “On themonotonicity of the generalized Marcum and Nuttall Q-functions”, IEEETrans. Inf. Theory, vol. 55, no. 8, pp. 3701-3710, Aug. 2009.

[10] A. Baricz, Y. Sun, “New Bounds for the Generalized Marcum Q-Function”, IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3091-3100, July2008.

[11] A. Annamalai, C. Tellambura, J. Matyjas, “A new twist on the gener-alized Marcum Q-function QM (a, b) with fractional-order M and itsapplications”, in 6th CCVC, 2009, pp. 1-5, 2009.

The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

424

[12] R. Li and P. Y. Kam, “Computing and bounding the generalized MarcumQ-function via a geometric approach”, in ISIT 2006, Seattle, USA, pp.1090-1094, July 2006.

[13] M.K. Simon, “The Nuttall Q-function-its relation to the Marcum Q-function and its application in digital communication performanceevaluation”, IEEE Trans. Commun., vol. 50, no. 11, pp. 1712-1715, Nov.2002.

[14] A. H. Heatley, “A short table of the Toronto functions, Trans. Roy. Soc.(Canada), vol. 37, sec. III. 1943.

[15] H. Sagon, “Numerical calculation of the incomplete Toronto function”,Proceedings of the IEEE, vol. 54, Issue 8, pp. 1095 - 1095, Aug. 1966.

[16] R. A. Fisher, “The general sampling distribution of the multiple corre-lation coefficient”, Proc. Roy. Soc. (London), Dec. 1928.

[17] J. I. Marcum, “A statistical theory of target detection by pulsed radar”,IRE Trans. on Inf. Theory, vol. IT-6, pp. 59-267, April 1960.

[18] P. Swerling, “Probability of detection for fluctuating targets”, IRE Trans.on Inf. Theory, vol. IT-6, pp. 269 - 308, April 1960.

[19] I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series, andProducts”, in 7th ed. Academic, New York, 2007.

[20] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical FunctionsWith Formulas, Graphs, and Mathematical Tables”, Dover, New York,1974.

[21] P. C. Sofotasios, “On Special Functions and Composite StatisticalDistributions and Their Applications in Digital Communications overFading Channels”, Ph.D. Dissertation, University of Leeds, UK, 2010.

[22] P. C. Sofotasios, S. Freear, “Novel Results for the Incomplete TorontoFunction and Incomplete Lipschitz-Hankel Integrals”, IEEE Interna-tional Microwave and Optoelectronics Conference (IMOC ’11), Natal,Brazil, Oct. 2011.

[23] P. C. Sofotasios, S. Freear, “Novel Expressions for the Marcum andOne Dimensional Q−Functions”, Seventh International Symposium onWireless Communication Systems (7th ISWCS ’10), York, UK, Sep.2010.

[24] P. C. Sofotasios, S. Freear, ”A Novel Representation for the NuttallQ−Function”, IEEE International Conference in Wireless InformationTechnology and Systems (ICWITS ’10), Honolulu, HI, USA, Aug. 2010.

[25] P. C. Sofotasios, E. Rebeiz, L. Zhang, T. A. Tsiftsis, D. Cabric and S.Freear, “Energy Detection-Based Spectrum Sensing over κ−μ and κ−μExtreme Fading Channels”, IEEE Trans. Veh. Techn., vol. 63, no 3, pp.1031−1040, Mar. 2013.

[26] K. Ho-Van, P. C. Sofotasios, “Bit Error Rate of Underlay Multi-hopCognitive Networks in the Presence of Multipath Fading”, in IEEEInternational Conference on Ubiquitous and Future Networks (ICUFN’13), pp. 620 - 624, Da Nang, Vietnam, July 2013.

[27] K. Ho-Van, P. C. Sofotasios, “Outage Behaviour of Cooperative Under-lay Cognitive Networks with Inaccurate Channel Estimation”, in IEEEInternational Conference on Ubiquitous and Future Networks (ICUFN’13), pp. 501-505, Da Nang, Vietnam, July 2013.

[28] K. Ho-Van, P. C. Sofotasios, “Exact BER Analysis of Underlay Decode-and-Forward Multi-hop Cognitive Networks with Estimation Errors”,IET Communications, To appear.

[29] F. R. V Guimaraes, D. B. da Costa, T. A. Tsiftsis, C. C. Cavalcante,and G. K. Karagiannidis, “Multi-User and Multi-Relay Cognitive RadioNetworks Under Spectrum Sharing Constraints,” IEEE Transactions onVehicular Technology, accepted for publication.

[30] K. Ho-Van, P. C. Sofotasios, S. V. Que, T. D. Anh, T. P. Quang,L. P. Hong, “Analytic Performance Evaluation of Underlay RelayCognitive Networks with Channel Estimation Errors”, Accepted forpublication in IEEE International Conference on Advanced Technologiesfor Communications (ATC ’13), HoChiMinh City, Vietnam, Oct. 2013.

[31] K. Ho-Van, P. C. Sofotasios, S. Freear, “Underlay Cooperative CognitiveNetworks with Imperfect Nakagami-m Fading Channel Information andStrict Transmit Power Constraint: Interference Statistics and OutageProbability Analysis”, IEEE/KICS Journal of Communications andNetworks, To appear.

[32] G. C. Alexandropoulos, P. C. Sofotasios, K. Ho-Van, S. Freear, “Symbolerror probability of DF relay selection over arbitrary Nakagami-m fadingchannels”, Journal of Engineering (Invited Paper), 2013, Accepted forpublication.

[33] M. K. Fikadu, P. C. Sofotasios, Q. Cui, M. Valkama, “Energy-OptimizedCooperative Relay Network over Nakagami−m Fading Channels”,Accepted for publication in IEEE International Conference on Wirelessand Mobile Computing, Networking and Communications (WiMob ’13),Oct. 2013.

[34] P. C. Sofotasios, T. A. Tsiftsis, K. Ho-Van, S. Freear, L. R. Wilhelmsson,M. Valkama, “The κ−μ/Inverse-Gaussian Composite Statistical Dis-tribution in RF and FSO Wireless Channels”, Accepted for publicationin IEEE Vehicular Technology Conference (VTC ’13 - Fall), Las Vegas,USA, Sep. 2013.

[35] A. Gokceoglu, Y. Zou, M. Valkama, P. C. Sofotasios, P. Mathecken,D. Cabric, “Mutual Information Analysis of OFDM Radio Link underPhase Noise, IQ Imbalance and Frequency-Selective Fading Channel”,IEEE Transactions on Wireless Communications, vol.12, no.6, pp.3048- 3059, June 2013.

[36] P. C. Sofotasios, T. A. Tsiftsis, M. Ghogho, L. R. Wilhelmsson and M.Valkama, “The η−μ/Inverse-Gaussian Distribution: A Novel PhysicalMultipath/Shadowing Fading”, in IEEE International Conference onCommunications (ICC’ 13), Budapest, Hungary, June 2013.

[37] G. C. Alexandropoulos, A. Papadogiannis, P. C. Sofotasios, “A Com-parative Study of Relaying Schemes with Decode-and-Forward overNakagami−m Fading Channels”, Journal of Computer Networks andCommunications, (Invited Paper), vol. 2011, Article ID 560528, 14pages, Dec. 2011.

[38] P. C. Sofotasios, S. Freear, “The α−κ−μ/gamma Composite Distribu-tion: A Generalized Non-Linear Multipath/Shadowing Fading Model”,IEEE INDICON ’11, Hyderabad, India, Dec. 2011.

[39] P. C. Sofotasios, S. Freear, “The η−μ/gamma and the λ−μ/gammaMultipath/Shadowing Distributions”, Australasian TelecommunicationNetworks And Applications Conference (ATNAC ’11), Melbourne, Aus-tralia, Nov. 2011.

[40] S. Harput, P. C. Sofotasios, S. Freear, “A Novel Composite StatisticalModel For Ultrasound Applications”, IEEE International UltrasonicsSymposium (IUS ’11), pp. 1387 - 1390, Orlando, FL, USA, Oct. 2011.

[41] P. C. Sofotasios, S. Freear, “The κ−μ/gamma Extreme CompositeDistribution: A Physical Composite Fading Model”, IEEE WirelessCommunications and Networking Conference (WCNC ’11), pp. 1398- 1401, Cancun, Mexico, Mar. 2011.

[42] P. C. Sofotasios, S. Freear, “The κ−μ/gamma Composite FadingModel”, IEEE International Conference in Wireless Information Tech-nology and Systems (ICWITS ’10), Honolulu, HI, USA, Aug. 2010.

[43] P. C. Sofotasios, S. Freear, “The η−μ/gamma Composite FadingModel”, IEEE International Conference in Wireless Information Tech-nology and Systems (ICWITS ’10), Honolulu, HI, USA, Aug. 2010.

The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

425