A Novel Unified Expression for the Capacity and Bit Error Probability of Wireless Communication Systems over Generalized Fading Channels

IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 1 A No v el Unified Exp ression for the Capacity and Bit Error Prob ability of W irele ss Communication Systems ov e r Generalize d F ading Channels Ferkan Y ilmaz and Mohamed-Slim Alouini Electrical Engineering Progra m, Di vis ion of Physical Sciences and Engineering, King Abdullah Uni v ersity of S cience and T echnology (KA UST), Thuwal, Mekkah Province, Saudi Arabia. Email(s): { ferkan.yilmaz, slim.alouini } @kaust.edu.sa Abstract Analysis of the average binary error probabilities (ABEP) and av erage capacity (A C) of wireless commu nica- tions systems over gener alized fading channels have been considered separately in the past. This paper in troduces a novel mom ent generating function (MGF)-based un ified e xpr ession for the AB EP and A C of single and multiple link c ommun ication with maximal ratio com bining. In add ition, this paper pro poses the hy per-Fox’ s H fading model as a unified fading distribution of a majority of the well-kn own ge neralized fading mo dels. As such , we offer a generic u nified perf ormance expression that can b e easily calculated and that is applicable to a wide variety of fading scen arios. T he m athematical formalism is illustrated with some selected num erical examp les that validate the correctness of our n e wly d eriv ed results. Index T erms Unified perform ance expression, average bit erro r rate, average cap acity , max imal ratio combining, hyper-Fox’ s H fading ch annels, generalized Gamm a fading, composite fading channels, extend ed gene ralized-K fading , momen t generating function. I . I N T R O D U C T I O N The a verage b inary error probabi lities (ABEP) and ave rage capacity (A C) are important performance metrics of wi reless communication sys tems op erating ov er fading channels. As such, consid erable efforts IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 2 hav e been dev oted so far t o develop analytical to ols/framew orks to e valuate these t wo performance metrics [1, and the references therein]. Howe ver , and to the best authors’ knowledge, these tools/ frame works were deve loped separately and the computation of these two performance metrics was viewed as two independent problems. For instance, based on Craig’ s representation of the compl ementary error function , a uni fied moment generating functi on (MGF)-based approach was deve loped to com pute the ABEP of a wid e variety of modulation t echniques over generalized fading [1, and t he references therein]. More recently , o ther MGF-based approaches [2]– [4] were also prop osed for the capacity calculation of wi reless channels sub ject to generalized fading. In contrast, this paper presents a nove l MGF-based unified e xpr ession for the exact ev aluation of both the ABEP and A C of s ingle an multiple li nks generalized fa ded channels. The paper introduces also the hyper-F ox’ s H distribution as a versatile fading model including a variety of well-known models as special cases. W ith these two un ifying frameworks at hand, we propose new generic expressions for the ABEP and A C wi th and wi thout div ersity reception. W e also present som e selected n umerical examples to validate our newly deriv ed results. The remainder of this paper is organized as follo ws. In Section II, a unified performance measure analysis of dive rsity recei vers over additive wh ite Gaussian noise (A WGN) channels is int roduced and some key results are present ed. In Section III, n e w results for single li nk and multiple link reception are presented and applied t o the newly proposed unifying hyper-F ox’ s H fading model. Finally , the main results are summarized and s ome conclus ions are drawn in the last section. I I . U N I FI E D C O N D I T I O NA L P E R F O R M A N C E E X P R E S S I O N A compact form for the conditional bit error probabi lity (BEP) P B E P ( γ end ) for a certain value of instantaneous SNR γ end for d iff erent binary modu lations was propos ed b y W ojnar in [5, Eq. (13)] as P B E P ( γ end ) = Γ ( b, aγ end ) 2Γ ( b ) , a, b ∈  1 , 1 2  , (1) where a depends on the type of modu lation ( 1 2 for orthogon al frequency shift keying (FSK), 1 for antipodal phase shift keying (PSK)), b depends on the typ e of detection ( 1 2 for coherent, 1 for non-coherent), and Γ ( · , · ) denot es the com plementary incom plete Gamm a functi on [6, E q. (6.5.3 )]. In the foll owing t heorem, we i ntroduce an alternative representation of (1) using the incomplete beta function. Theor em 1 (Uni fied BEP Expressi on Us ing t he Incomplete Beta Function) . An alternati ve r epr esentat ion IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 3 of th e compa ct form of the conditi onal bit err or pr ob ability P B E P ( γ end ) r epr esented in (1) is given by P B E P ( γ end ) = 1 2 − exp ( − i π b ) 2Γ ( b ) lim d →∞ d b B  − a d γ end ; b, 1 − d  , (2) wher e the parameters a and b depend on t he p articular form of the modu lation and detection as mentioned above (see T able I), and i = √ − 1 is t he imaginary unit with the pr operty i 2 = − 1 , an d B ( z ; a, b ) = R z 0 u a − 1 (1 − u ) b − 1 du i s the incomplete beta funct ion [6, Eq. (6.6.1)] . Pr oo f: See Appendix A. In addi tion to the BEP p erformance measure, there exists another important performance measure commonly u sed in the literature, which is known as the conditional capacity . Explicitl y , the conditi onal capacity i s a m easure of how much error-free information can be transmitt ed and recei ved through the channel. T he condition al no rmalized channel capacity P C ( γ end ) i n n ats/s/Hz for a certain value of instantaneous SNR γ end at the out put of the recei ver i s well-known to be gi ven by P C ( γ end ) = log (1 + a γ end ) , nats/s/Hz , (3) where a ∈ R + represents the transmiss ion power , and log ( · ) i s the natural logarithm (i.e., the logarithm to the base e ) [6, Eq. (4.1.1)]. W e introduce in the following t heorem a ne w alternativ e incom plete beta function-based representati on of (3). Theor em 2 (Capacity Expression Us ing the Incomplete Beta Functio n) . An alternative r epr esentation of the conditional capacity P C ( γ end ) given in (3) is given by P C ( γ end ) = − B ( − aγ end ; 1 , 0) . (4) Pr oo f: See Appendix B . T o the best of autho rs’ knowledge, bo th (2) and (4) are not av ailable in t he literature. M ore importantly and usin g these two incomplete beta representations of the BEP and channel capacity measures, one can readily give, as shown in the following corollary , a unified performance expression whose s pecial cases include t he BEP and channel capacity . Corollary 1 (Unified Performance Expression Us ing Incomplete Beta Function) . A compact and unified form of the conditi onal p erformance measure P U P ( γ end ) (which i nclude both the conditio nal BEP and IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 4 the conditional channel capacity) is given by P U P ( γ end ) = α + β exp ( − i π b ) d b 2Γ( b ) B  − a d γ end ; b, 1 − d  . (5) which reduces to (1) f or α = 1 , β = − 1 a nd d → ∞ , a nd which r educes to (3) for α = 0 , β = 2 , a = 1 , b = 1 and d = 1 . Pr oo f: Based on Theorem 1 and Theorem 2, the proof is obvious. In the l imit as d → ∞ , (5) mi ght produce approximate results due to numerical comp utation limi ts of st andard mathematical soft ware packages. In this cont ext, usin g b oth [7, Eq. (7.2 .2/13)] and [7, Eq. (7.3.1/28)], the u nified expression given by Corollary 1 can be represented wi thout limit as sho wn i n the following corollary . Corollary 2 (Un ified Performance Ex pression Using t he Hypergeometric Function) . The unified perfor- mance measur e P U P ( γ end ) can als o be r epresented as P U P ( γ end ) = 1 − n 2 ( 1 + ( − 1) n ( aγ end ) b Γ( b + 1) n F 1 h Λ ( n ) b ; b + 1; − aγ end i ) , (6) which r educes to (1) for n = 1 , and which r edu ces to (3) for a = 1 , b = 1 and n = 2 . Mor eover , in (6) , the coef ficient set Λ ( m ) a is defined as Λ ( n ) a ≡ n times z }| { a, a, . . . , a, (7) and p F q [ · ; · ; · ] denotes the generalized hyper geometric functi on [7, Eq. (7.2.3 /1)]. Pr oo f: Using both [7, Eq. (7.2.2/13)] and [7, Eq. (7. 3.1/28)], t he proof is o bvious. Note that, in addit ion to the hypergeometric function representation given by Corollary 2, the u nified expression can also be giv en i n terms o f other special functions such as the MacRobert’ s E function and the Meijer’ s G functi on, as shown in the following corollaries. These expressions are useful since they will facilitate in Section III the uni fied analysi s of BEP and capacity in generalized fading en v ironments. Corollary 3 (Unified Performance Expression Using t he M acRobert’ s E Function) . The unified perf or- mance measur e P U P ( γ end ) can als o be written as P U P ( γ end ) = 1 − n 2 ( 1 − ( − 1) n ( aγ end ) b Γ( b ) E  Λ ( n ) b ; b + 1; 1 aγ end  ) (8) wher e E [ · ; · ; · ] i s the MacRobert ’ s E function [8, Sec. (9.4 )] , [7, Sec. (2.2 3)] . Moreo ver , (8) r educes to IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 5 (1) f or n = 1 , an d it r educes t o (3) for a = 1 , b = 1 and n = 2 . Pr oo f: Using the generalized hyp er geometric functi on representation of the MacRobert’ s E function [7, Sec. (2.23)], i.e., substitutin g the following equality E [ a 1 , a 2 , . . . , a r ; b 1 , b 2 , . . . , b q ; az ] = Q p r =1 Γ( a r ) Q q r =1 Γ( b r ) p F q  a 1 , a 2 , . . . , a r ; b 1 , b 2 , . . . , b q ; − 1 az  (9) into (6) results in (8), which proves Coroll ary 3. Corollary 4 (Unified Performance Expression Us ing t he Meijer’ s G Funct ion) . The unified performance measur e P U P ( γ end ) has this additiona l Meijer’ s G function-based r epr esent ation given by P U P ( γ end ) = 1 − n 2 ( 1 − ( − 1) n Γ( b ) G 1 ,n n, 2 " aγ end      Λ ( n ) 1 b, 0 #) , (10) wher e G m,n p,q [ · ] is the Meijer’ s G f unction [7, Eq. (8.3.22)] . Furthermore , (10) r educes to (1) for n = 1 , and it als o r educes to (3) for a = 1 , b = 1 an d n = 2 . Pr oo f: Using the relation b etween M acRobert’ s E and Meijer’ s G function [7, Eq. (8.4.51 /8)], we can writ e E  Λ ( n ) b ; b + 1; 1 aγ end  = G 1 ,n n, 2 " aγ end      Λ ( n ) 1 − b 0 , − b # . (11) Then, substitutin g (11) into (6) results in (10), which proves Corollary 4. It i s worth to mention that t he Meijer’ s G function is an s pecial function defined by t he Mellin- Barnes type integral which contains products and quo tients o f the Eul er gamma function s, and it is as s uch considered as a generalization of hypergeometric functions and oth er special functions such as exponential, bessel, logarith m, sine / cosine i ntegral functions. Therefore, the PDF of some well-known fading dis tributions can be represented i n terms of M eijer’ s G function . In this context, referring to [7, Eq. (2.24.1/1)], the representation in Coroll ary 4 is more useful than the representatio ns gi ven in Corollaries 1, 2 and 3 from numerical s implicity and computation start-points. In the fol lowing secti on, we u se our new representations in the analysis o f the BEP and capacity measures in generalized fading en v ironments. I I I . U N I FI E D P E R F O R M A N C E E X P R E S S I O N OV E R F A D I N G C H A N N E L S Over th e past four decades, the ABEP and t he A C measures hav e been considered as two different problems, and many different solutio ns ranging from bounds to approxi mations, integral expressions, and IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 6 closed-form formulas have been presented for t hese two differe nt measures for a variety of m odulation schemes, diversity com bining techniqu es, and fading distributions. At this p oint, we would also like to highlight again that th ese two different performance measures can be compactl y combin ed as mentio ned in the previous sections and can also be considered as a singl e problem. As such, for a certain nonnegative distribution o f the instantaneous SNR γ end at the output of the recei ver (i.e., γ end is distributed over (0 , ∞ ) according to p γ end ( γ ) which is the probability density fun ction (PDF) of γ end ), the avera ge uni fied performance (A UP) expression is completely represented as P AU P = E [ P U P ( γ end )] = Z ∞ 0 P U P ( γ ) p γ end ( γ ) dγ , (12) where E [ · ] denotes the expectation operator , and as proposed in Section II, P U P ( γ ) is simply t he conditional un ified p erformance expression. A. Singl e Link Reception W e consi der an optim um receiv er em ploying binary modulation and operatin g in a slow non-selective generalized fading en vironment corrupted by A WGN noise. In such a case, γ end = γ ℓ is th e instantaneous SNR at the ou tput of t he recei ver , and it is dis tributed over ( 0 , ∞ ) according t o the PDF p γ ℓ ( γ ) . Referring to (12), the A UP expression can thus written as P AU P = 1 − n 2 ( 1 − ( − 1) n Γ( b ) Z ∞ 0 G 1 ,n n, 2 " aγ      Λ ( n ) 1 b, 0 # p γ ℓ ( γ ) dγ ) . (13) The PDF of the distribution of the fading is a nonn egati ve function (i.e., p γ ℓ ( γ ) ≥ 0 for γ ≥ 0 ) and a nonnegativ e function can be expressed in terms o f Meijer’ s G or Fox’ s H functi on using integral transforms theory . In oth er words, the PDF of a variety of stati stical en velope di stributions, such as Rayleig h, Rician, exponential, Nakagam i-m, W eibull, generalized Nakagami- m , lo gnormal, K-dist ribution, generalized-K, etc., can be expressed in terms of either Meijer’ s G or Fox’ s H function [9]. As a consequence of that, (13) can be readily reduced t o a clos ed-form solut ion by exploiting [7, Eq. (2.24.1/1 )]. In thi s context, the Fox’ s H channel fading model is i ntroduced in the following definitio n. Definition 1 (Hyper-F ox’ s H F ading Channel) . Let γ ℓ be a hyper-F o x’ s H fadi ng di stribution r epr esenting the i nstantaneous SNR γ ℓ at the output of the single-link r eceiver , which fol lows a pr obabili ty law in IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 7 general such that its hyper-F ox’ s H PDF is gi ven by p γ ℓ ( γ ) = K ℓ X n ℓ =1 η n ℓ H M n ℓ ,N n ℓ P n ℓ ,Q n ℓ   c n ℓ γ       ( ϑ n ℓ j , θ n ℓ j ) P n ℓ j =1 ( ϕ n ℓ j , φ n ℓ j ) Q n ℓ j =1   , γ > 0 , (14) with the conditi ons max j =1 , 2 ,..,M n ℓ  − ϕ n ℓ j φ n ℓ j  < S n ℓ < min j =1 , 2 ,..,N n ℓ  1 − ϑ n ℓ j θ n ℓ j  , n ℓ ∈ { 1 , 2 , . . . , K ℓ } , (15) such that S = S 1 ∩ S 2 ∩ . . . ∩ S K ℓ 6 = ∅ , wher e bot h { ( ϕ n ℓ j , φ n ℓ j ) Q n ℓ j =1 } and { ( ϑ n ℓ j , θ n ℓ j ) P n ℓ j =1 } coefficient sets ar e such parameters that the y support the conditio ns given by [7, Section 8.3.1] , and wher e η n ℓ ∈ R and c n ℓ ∈ R + ar e such two parameters that (14) certai nly supp orts R ∞ 0 p γ ℓ ( γ ) dγ = 1 . In (14) , H m,n p,q [ · ] denotes t he F ox’ s H funct ion [7, Eq. (8.3.1/1)] 1 , 2 . It i s worth to n otice t hat the PDF of most non-negative d istributions can b e comp actly expressed or accurately approximated in th e form of (14). For example, as s een in the first colu mn of T ables II,III,V and V , the PDFs commonly us ed in the literature for modeli ng of instantaneous SNR dis tribution γ end are listed in the form of hyper -Fox’ s fa ding dist ribution. F r om this perspective, wher e hyper-F ox’ s H distribution pr ovides an unified framework on mo deling of f ading distribution, the unified and generalized r esul t r e gar ding the A UP expr essio n of s ingle-link r eception can b e obtained by m eans of substitu ting (14) into (13) and using [11, Theorem 2.9] wi th [7, Eq. (8.3.2/21)] such that the A UP expression over hyper-F o x’ s H fading channel is gi ven by P AU P = 1 − n 2    1 − ( − 1) n Γ( b ) K ℓ X n =1 η n ℓ c n ℓ H M n ℓ + n,N n ℓ +1 P n ℓ +2 ,Q n ℓ + n   c n ℓ a       (1 − b, 1) , ( ϑ n ℓ j + θ n ℓ j , θ n ℓ j ) P n ℓ j =1 , ( 1 , 1) Λ ( n ) ((0 , 1)) , ( ϕ n ℓ j + φ n ℓ j , φ n ℓ j ) Q n ℓ j =1      (16) with t he conditi on P ∩  S 1 ∩ S 2 ∩ . . . ∩ S K ℓ  6 = ∅ , where 0 < P < b and the con vergence region set { S n ℓ } is defined in (15). Let u s consider some special cases of the h yper -Fox’ s H fading model in order t o check analytical simplicit y , accuracy and correctness of the A UP expre ssion giv en by (13). Example 1 (Unified Performance M easure in Generalized Nakagami- m (GNM ) Fading Channels) . In GNM fading channels, th e distribution of the instantaneous SNR γ ℓ follows a generalized Gam ma PDF 1 For more information about the Fox’ s H function, the r eaders are referred to [ 10], [11] 2 Using [7, Eq. (8.3.22)], the Fox’ s H function can be represented in terms of the Meijer’ s G function [7, Eq. (8.2.1)] which is a built-in function in the most popular mathematical software packages such as MA THEMA TICA® . IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 8 giv en by p γ ℓ ( γ ) = ξ ℓ Γ ( m ℓ )  β ℓ ¯ γ ℓ  ξ ℓ m ℓ γ ξ ℓ m ℓ − 1 e −  β ℓ ¯ γ ℓ  ξ ℓ γ ξ ℓ (17) for 0 ≥ γ < ∞ , where the parameters m ℓ ≥ 1 / 2 , ξ ℓ > 0 and ¯ γ ℓ > 0 are t he fading figure, t he s haping parameter and the local mean power of the ℓ th GNM d istribution, and β ℓ = Γ ( m ℓ + 1 / ξ ℓ ) / Γ ( m ℓ ) . It may be useful to notice that the special or limiti ng cases of the GNM distri bution are we ll-known in the lit erature as t he Rayleigh ( m ℓ = 1 , ξ ℓ = 1) , exponential ( m ℓ = 1 , ξ ℓ = 1 / 2) , Half-Normal ( m ℓ = 1 / 2 , ξ ℓ = 1) , Nakagami- m ( ξ ℓ = 1) , Gamma ( ξ ℓ = 1 / 2) , W eibull ( m ℓ = 1) , lognormal ( m ℓ → ∞ , ξ ℓ → 0) , and A WGN ( m ℓ → ∞ , ξ ℓ = 1) . Referring to Definition 1 in o rder to obt ain the A UP expression P AU P for generalized Gam ma fading channels by m eans of usi ng (16), one can readily represents the PDF p γ ℓ ( γ ) of the generalized Gamma distribution in terms of the hyper-F ox’ s H distribution by means of using [11, Eq. (2.9.4 )], i. e., us ing the Fox’ s H representation given in the first row of T able V p γ ℓ ( γ ) = β ℓ ¯ γ ℓ Γ ( m ℓ ) H 1 , 0 0 , 1  β ℓ ¯ γ ℓ γ     ( m ℓ − 1 /ξ ℓ , 1 /ξ ℓ )  (18) where means that the coefficients are abs ent. Next, expressing (18) in terms of (14) with K = 1 , and then subs tituting (18) into (16) (i.e., by m eans of mapping the parameters o f two PDF mod el (14 ) and (18), and then utilizing (16)), one can readily obtain the A UP expression for GNM fading channels as P AU P = 1 − n 2 ( 1 − ( − 1) n Γ( b )Γ( m ℓ ) H n +1 , 1 2 ,n +1 " β ℓ a ¯ γ ℓ      (1 − b, 1) , (1 , 1) Λ ( n ) (0 , 1) , ( m ℓ , 1 /ξ ℓ ) #) (19) Here, it might be useful to notice that, in addition to this A UP expression for GN M fading channels, T able I offers simpli fied expressions for the A UP of a va riety of commonly used fading channels in order to facilitate for the readers the use of our uni fied BEP and A C results. In o rder to check the v alidity and completeness of (19), s ubstituti ng n = 1 result s in the ABEP of signal transmissio n over GNM fading channels as expected, t hat is P AB E P = 1 2  1 − 1 Γ( b )Γ( m ℓ ) H 2 , 1 2 , 2  β ℓ a ¯ γ ℓ     (1 − b, 1 ) , ( 1 , 1) (0 , 1) , ( m ℓ , 1 /ξ ℓ )  . (20) After performi ng some algebraic manip ulations either by means of using [11, Property 2 .11] or by recognizing that t he cum ulative distribution function (CDF) of a Fox’ s H distribution can be written IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 9 in two differe nt forms given in [12, Eq. (4.17)] and [12, Eq. (4.19 )], (20) can be simpl ified to P AB E P = 1 2Γ( b )Γ( m ℓ ) H 1 , 2 2 , 2  β ℓ a ¯ γ ℓ     (1 , 1) , (1 − b, 1) ( m ℓ , 1 /ξ ℓ ) , (0 , 1)  . (21) in perfect agreement wi th [13, Eq. (10)]. Note that i n the special case of Nakagami- m fading channels, substitut ing ξ ℓ = 1 , it is straight forward to show that (20) reduces to [5, Eq. (17)] by means of s ome algebraic mani pulations using [7, E q.(8.4.49/13)] and [7, Eq.(7. 3.1/28)] togeth er . In addi tion, for Rayleigh fading channel (i.e., m ℓ = ξ ℓ = 1 ), (20) simplifies to P AB E P = 1 2  1 −  a ¯ γ ℓ 1+ a ¯ γ ℓ  b  which agrees with the four resul ts given in [1] for BFSK ( a = 1 / 2 and b = 1 / 2 ), BPSK ( a = 1 and b = 1 / 2 ), non-coherent BFSK (NC-BFSK) ( a = 1 / 2 and b = 1 ) and differe ntially encoded BPSK (BDPSK) ( a = 1 and b = 1 ). In additi on to t hese expected ABEP consequences of (19), the AC of the GNM fading channels can be also easily obtai ned by setting b = 1 and n = 2 in (19), yielding P AC = 1 Γ( m ℓ ) H 3 , 1 2 , 3  β ℓ a ¯ γ ℓ     (0 , 1) , (1 , 1) (0 , 1) , (0 , 1) , ( m ℓ , 1 /ξ ℓ )  , (22) where a denotes the t ransmitted power . Setti ng ξ ℓ = 1 and then using [7, Eq.(8.3.2 /21)], (22) further reduces to th e well-kn own result P AC = G 3 , 1 2 , 3 h m ℓ a ¯ γ ℓ    0 , 1 0 , 0 ,m ℓ i / Γ( m ℓ ) which is the A C of the Nakagami- m fading channels [14, Eq. (3)]. Furth ermore, us ing [11, Eq. (2.1.7)] and [7, Eq. (8.4.11.3)], then recalling the relation between the first order E n integral E 1 ( x ) and exponential i ntegral E i ( x ) such as E 1 ( x ) = − E i ( − x ) , for Rayleigh fading channel ( m ℓ = ξ ℓ = 1 ), (22) sim plifies to [1, Eq. (15.26)] as expected. In additi on to the revie w above regar ding the unified performance m easures in GNM fading chann els, let us consider the shadowing id entified as a main cause no t only for reducing energy but also causing performance loss and in stability at the recei ver . Specifically , shado wing is the effe ct t hat t he recei ved signal power , i.e., the m ean of ins tantaneous SNR fluctuates due t o objects ob structing the propagatio n path between transm itter and receiver . Additi onally , in the architecture o f next-generation wireless comm u- nication systems, a spectrum well above 6 0 GHz frequency will be used, causing greater sus ceptibility to shadowing. In th is context, the modeling of shadowing plays an i mportant role in the context of designin g systems and ev alu ating the corresponding performance. The m ost well-known shadowing di stribution in the literature is the L ognormal d istribution [1, and the reference s therein]. Example 2 (Unified Performance Measure i n L ognormal Fading Channels) . The PDF of lognormal distribution can be well approxim ated in the form o f (16) as shown in the second row of T able IV IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 10 such that p γ ℓ ( γ ) = 1 √ π K X k =1 w k ω k H 0 , 0 0 , 0  γ ω k     (23) where µ ℓ (dB) and σ ℓ (dB) are the mean and the standard deviation of γ ℓ , and where ω k is defined as ω k = 10 ( √ 2 σ ℓ u k + µ ℓ ) / 10 such that for k ∈ { 1 , 2 , . . . , K } , { w k } and { u k } are the weight factors and the zeros (abscissas) of the K -o rder Hermit e polynomial [6, T able 25.10]. M apping the parameters and coefficients between (23) and (14 ), one can readily give the unified performance measure over l ognormal fading channels by means of ut ilizing (16) and then usi ng [7, Eq.(8.3.2/2 1)] as P AU P = 1 − n 2 ( 1 − ( − 1) n √ π Γ( b ) K X n =1 w k G n, 1 2 ,n " 1 a ω k      1 − b, 1 Λ ( n ) (0) #) . (24) As expected, setting n = 1 in (24) results in P AB E P = 1 2 √ π K P n =1 w k Γ( b, aω k ) / Γ( b ) . M oreover , setti ng b = 1 and n = 2 , (24) simplifies i nto P AB E P = 1 2 √ π K P n =1 w k log (1 + aω k ) as expected. In additi on to t he cont ext of shadowing, composite fading chann els (include both s hadowing and fading) plays an im portant role in desi gning and modeli ng wireless com munication s ystems. T o the best o f our knowledge, another general compo site fading model is the extended generalized-K (EGK) fading. In the following example, unified performance measure P AU P is analyzed over EGK fa ding channels [15 ]. Example 3 (Unified Performance Measure in Extend ed Generalized-K (EGK) Fading Channels) . The distribution of the i nstantaneous SNR in the EGK fading channel follows th e PDF gi ven b y [15, Eq. (3)], [16], that is, p γ ℓ ( γ ) = ξ ℓ Γ( m ℓ )Γ( m sℓ )  β ℓ β sℓ Ω sℓ  ξ ℓ m ℓ γ ξ ℓ m ℓ − 1 Γ m sℓ − m ℓ ξ ℓ ξ sℓ , 0 ,  β ℓ β sℓ Ω sℓ  ξ ℓ γ ξ ℓ , ξ ℓ ξ sℓ ! (25) defined over 0 ≤ γ < ∞ , where the p arameters m ℓ (0 . 5 ≤ m ℓ < ∞ ) and ξ ℓ (0 ≤ ξ ℓ < ∞ ) represent t he fading figure (div ersity sev erity / order) and t he fading shaping factor , respectively , whil e m sℓ (0 . 5 ≤ m sℓ < ∞ ) and ξ sℓ (0 ≤ ξ sℓ < ∞ ) represent t he shadowing sev erity and the shadow- ing shapi ng factor (inho mogeneity), respectively . In addit ion, the parameters β ℓ and β sℓ are defined as β ℓ = Γ ( m ℓ + 1 / ξ ℓ ) / Γ ( m ℓ ) and β sℓ = Γ ( m sℓ + 1 / ξ sℓ ) / Γ ( m sℓ ) , respectively . In addition , Γ ( · , · , · , · ) is the extended i ncomplete Gamma function defined as Γ ( α, x, b, β ) = R ∞ x r α − 1 exp  − r − br − β  dr , where α, β , b ∈ C and x ∈ R + [17, Eq. (6.2 )]. The Fox’ s H representation of (25) is give n by at the s econd row IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 11 of T able V, that i s p γ ℓ ( γ ) = β ℓ β sℓ Γ( m ℓ )Γ( m sℓ ) ¯ γ sℓ H 2 , 0 0 , 2  β ℓ β sℓ ¯ γ sℓ γ     ( m ℓ − 1 /ξ ℓ , 1 /ξ ℓ ) , ( m sℓ − 1 /ξ sℓ , 1 /ξ sℓ )  . (26) Mapping the parameters and coefficients of (26) to t hose of the hy per -Fox’ s H fading model defined in Definition 1 with K = 1 , the unified performance m easure P AU P can be readily expressed as P AU P = 1 − n 2 ( 1 − ( − 1) n Γ( b )Γ( m ℓ )Γ( m sℓ ) H n +2 , 1 2 ,n +2 " β ℓ β sℓ a ¯ γ sℓ      (1 − b, 1) , (1 , 1) Λ ( n ) ((0 , 1)) , ( m ℓ , 1 ξ ℓ ) , ( m sℓ , 1 ξ sℓ ) #) . (27) Note that (27 ) reduces to the ABEP in EGK fading channels [16, Eq . (34 )] by m eans of s etting n = 1 and using the same steps in t he d eri vation of (21). Moreover , for n = 2 and b = 1 , (27) simpl ifies into the A C in EGK fading channels [16, E q. (42)] as expected. In o rder to check the analytical accuracy and correctness, some numerical and sim ulation results regarding t he ABEP and A C performances o f sing le lin k reception over fading channels are depicted in Fig. 1, Fig. 2 and Fig. 3, and they show that these num erical and simulation results are i n p erfect agreement. B. Multi ple Link Reception W e consider an L -branch maximal ratio combiner (MRC) diver sity system employing binary modulation and operating in a slow non-sel ectiv e mu tual independent and no t-necessarily identically dist ributed generalized fading en vironment corrupted by A WGN nois e. Th e instantaneou s SNR γ end at the output of the MRC receiv er is considered as the sum of t he inst antaneous SNRs of the branches, that is, γ end = L X ℓ =1 γ ℓ , (28) where L denotes the num ber of branches, and where for ℓ ∈ { 1 , 2 , 3 , . . . , L } , γ ℓ is the i nstantaneous SNR the ℓ th branch is subjected to. The A UP expression P AU P of the L -branch MRC combiner can be obtained by av eraging the (instantaneous ) u nified performance measure P U P ( γ end ) given by Corollary 4 over the PDF of γ end = P L ℓ =1 γ ℓ as s hown in (12). Due to seve ral reasons (e.g., insufficient ant enna spacing or coupling among RF layers), correlation may exist among diversity branches of th e L -branch MRC. With IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 12 that, t he AC using (12 ) i n volves an L -fold integral give n by P AU P = 1 − n 2 ( 1 − ( − 1) n Γ( b ) Z ∞ 0 Z ∞ 0 . . . Z ∞ 0 | {z } L -fold G 1 ,n n, 2 " a L X ℓ =1 r ℓ      Λ ( n ) 1 b, 0 # × p γ 1 ,γ 2 ,...,γ L ( r 1 , r 2 , . . . , r L ) dr 1 dr 2 . . .d r L ) , (29) where p γ 1 ,γ 2 ,...,γ L ( r 1 , r 2 , . . . , r L ) is the joint m ultiv ariate PDF of th e in stantaneous SNRs { γ ℓ } L ℓ =1 . In (29), the L -fold i ntegration i s tedious and canno t be partioned into the product of one dimensional i ntegrals e ven if the instantaneous SNRs { γ ℓ } L ℓ =1 are assum ed mutually independent. Additionall y , it is clear that the numerical ev aluation of (29 ) is complex requiri ng a long time t o comput e the desired resul t as the number of branches L increases. Fortunately , after performing some algebraic m anipulations, the A UP expression can be readily obtained in t erms of a single int egral expression using an MGF-based approach [1] as presented in the foll owing theorem. Theor em 3 (A verage Unified Expression of the L -branch Di versity Combiners over Correlated Not-Nec- essarily Identicall y Distributed Fading Channels) . The exact A UP of L -branch diversity combiner over mutually not-necessarily independent nor i dentically d istributed f ading c hannels is given by P AU P = 1 − n 2 ( 1 + ( − 1) n Γ( b ) Z ∞ 0 G 1 ,n n +1 , 2 " a s      Λ ( n ) 1 , 1 b, 0 #  ∂ ∂ s M γ 1 ,γ 2 ,...,γ L ( s )  ds ) , (30) wher e the parameter s a ∈ R + , b ∈ (0 , 1 ] , n ∈ { 1 , 2 } ar e selected accor ding to desir ed performance measur e, and wher e M γ 1 ,γ 2 ,...,γ L ( s ) ≡ E [exp( − s P ℓ γ ℓ )] is the jo int MGF of the corr elated instant aneous SNRs γ 1 , γ 2 , . . . , γ L of th e branches. Pr oo f: See Appendix C . It is worth accentuating that, in order to find the A UP expression of the diversity combiner , the MGF- based technique proposed i n Theorem 3 eliminates the necessity of finding the PDF o f the inst antaneous SNR γ end = P L ℓ =1 γ ℓ through t he in verse Laplace transform (IL T) of the joi nt MGF M γ 1 ,γ 2 ,...,γ L ( s ) ≡ E [exp( − s P ℓ γ ℓ )] . Shortly , Theorem 3 suggests that one can readily o btain the A UP expression usi ng the joint MGF M γ 1 ,γ 2 ,...,γ L ( s ) . Addi tionally , t he in tegral in (30) can be accurately est imated by employing the Gauss-Chebyshe v Quadrature (GCQ) formula [6, Eq.(25.4.39)], i.e., P AU P = 1 − n 2 ( 1 + ( − 1) n Γ( b ) N X n =1 w n G 1 ,n n +1 , 2 " a s n      Λ ( n ) 1 , 1 b, 0 #  ∂ ∂ s M γ 1 ,γ 2 ,...,γ L ( s )     s = s n  ) , (31) IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 13 which con ver ges rapidly and steadily , requiring only few terms for an accurate result. In (31), the coef ficients w n and s n are defined as w n = π 2 sin  2 n − 1 2 N π  4 N cos 2  π 4 cos  2 n − 1 2 N π  + π 4  and s n = ta n  π 4 cos  2 n − 1 2 N π  + π 4  , (32) respectiv ely , where t he t runcation index N could be chosen m ore th an N = 30 to obtain a high level of accurac y . Despit e the fact that the novel t echnique represented by Theorem 3 is easy to use, and referring to b oth (30 ) and (31), let us consider i ts special cases in order to check its analytical simplicity and accurac y . Special Cas e 1 (A verage Bit Error Probabilities o f L -Branch Div ersity Combi ner) . As m entioned before, the A BEP P AB E P = 1 2 E [Γ( b, aγ end ) / Γ( b )] of the L -branch diversity combiner can be readily obtain ed through setting n = 1 in (30), s uch that P U P | n =1 results i n P AB E P = 1 2 + 1 2Γ( b ) Z ∞ 0 G 1 , 1 2 , 2  a s     1 , 1 b, 0  ∂ ∂ s M γ 1 ,γ 2 ,...,γ L ( s )  ds. (33) T o the best of the aut hors’ knowledge, the result giv en by (34) i s a new result which readily sim plifies into the performance of BFSK ( a = 1 / 2 and b = 1 / 2 ), BPSK ( a = 1 and b = 1 / 2 ), non-coherent BFSK ( a = 1 / 2 and b = 1 ) and BDPSK ( a = 1 and b = 1 ). Special Case 2 (A verage Capacity of L -Branch Diversity Combiner) . The A C P AC = E [log (1 + aγ end )] of the L -branch d iv ersity combiner can be readily obtained through settin g b = 1 and n = 2 in (30), such that P U P | n =2 ,b =1 results i n P AC = − Z ∞ 0 G 1 , 2 3 , 2  a s     1 , 1 , 1 1 , 0  ∂ ∂ s M γ 1 ,γ 2 ,...,γ L ( s )  ds, (34) where i t may be useful to notice that G 1 , 2 3 , 2 h a s    1 , 1 , 1 1 , 0 i = G 0 , 2 2 , 1  a s   1 , 1 0  by means of employing [7, Eq. (8.2.2/ 9)]. Then, usi ng both [7, Eq. (8.2.2/14)] and G 0 , 2 2 , 1  a s   1 , 1 0  = − Ei  − s a  [7, Eq. (8.4.11/1)], where Ei ( · ) is t he exponential integral function [8, Eq. (8.211/1 )], (34) simplifies to P AC = Z ∞ 0 Ei ( − s/a )  ∂ ∂ s M γ 1 ,γ 2 ,...,γ L ( s )  ds, (35) which is a well-known result given by Di Renzo et al . in [3, Eq. (7)]. Note that the spatial correlatio n between all fading amplitu des can be determined from the phys ical parameters of the m odel, which includes antenna spacing, antenna arrangement, angl e spread, and angle IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 14 of arri val. In the case of there does not exist any correlation between all fading am plitudes γ 1 , γ 2 , . . . , γ L for t he b ranches of the di versity combiner , the A UP expression i s gi ven in the following corollary . Corollary 5 (A verage Unified Expression of the L -branch Diversity Combiners ov er Mutually Independent Non-Identically Distributed Fading Channels) . The exact A UP expr ession of L -branch MRC over mut ually independent and non-identi cally distributed fading channels is g iven b y P AU P = 1 − n 2      1 + ( − 1) n Γ( b ) Z ∞ 0 G 1 ,n n +1 , 2 " a s      Λ ( n ) 1 , 1 b, 0 # L X ℓ =1  ∂ ∂ s M γ ℓ ( s )  L Y k =1 k 6 = ℓ M γ k ( s ) ds      , (36) wher e, for ℓ ∈ { 1 , 2 , . . . , L } , M γ ℓ ( s ) ≡ E [ exp ( − sγ ℓ )] is the MGF of the in stantaneous SNR γ ℓ that the ℓ th branch is subjected to. Pr oo f: When there is no correlation b etween all inst antaneous SNRs γ 1 , γ 2 , . . . , γ L of the branches, one can readily write M γ 1 ,γ 2 ,...,γ L ( s ) = Q L ℓ =1 M γ ℓ ( s ) whos e deriva tion wit h respect to s is ∂ ∂ s M γ 1 ,γ 2 ,...,γ L ( s ) = L X ℓ =1  ∂ ∂ s M γ ℓ ( s )  L Y k =1 k 6 = ℓ M γ k ( s ) . (37) Finally , subs tituting (37) into (30) results in (36), which proves Corollary 5. As menti oned before and nicely s hown in T ables II , III, IV and V, t he PDF of seve ral non-negative distributions can be compactly expressed or accurately approxim ated i n the form of (14). Using either [12, Eq. (2.12)] or [18, Eq. (2.10)], th e M GF of the hyper -Fox’ s H fading channel is give n by M γ ℓ ( s ) = K ℓ X n ℓ =1 η n ℓ c n ℓ H M n ℓ ,N n ℓ +1 P n ℓ +1 ,Q n ℓ   c n ℓ s       (1 , 1) , ( ϑ n ℓ j + θ n ℓ j , θ n ℓ j ) P n ℓ j =1 ( ϕ n ℓ j + φ n ℓ j , φ n ℓ j ) Q n ℓ j =1   (38) with t he con ver g ence region ℜ { s } ≥ 0 . Then, with the aid of [7, Eq. (8. 3.2/15)], t he deriv ativ e o f (38) can be readily deriv ed as ∂ ∂ s M γ ℓ ( s ) = − K ℓ X n ℓ =1 η n ℓ c n ℓ s H M n ℓ ,N n ℓ +1 P n ℓ +1 ,Q n ℓ   c n ℓ s       (0 , 1) , ( ϑ n ℓ j + θ n ℓ j , θ n ℓ j ) P n ℓ j =1 ( ϕ n ℓ j + φ n ℓ j , φ n ℓ j ) Q n ℓ j =1   (39) Finally , subst ituting both (38) and (39) int o (36), the A UP expre ssion of t he L -branch diversity combiner IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 15 performing over hyper-F ox’ s H fading chann el can be readil y obtained in the form of Corollary 5 as P AU P = 1 − n 2 ( 1 − ( − 1) n Γ( b ) Z ∞ 0 G 1 ,n n +1 , 2 " a s      Λ ( n ) 1 , 1 b, 0 # × L X ℓ =1 K ℓ X n ℓ =1 η n ℓ c n ℓ s H M n ℓ ,N n ℓ +1 P n ℓ +1 ,Q n ℓ   c n ℓ s       (0 , 1) , ( ϑ n ℓ j + θ n ℓ j , θ n ℓ j ) P n ℓ j =1 ( ϕ n ℓ j + φ n ℓ j , φ n ℓ j ) Q n ℓ j =1   × L Y k =1 k 6 = ℓ K k X n k =1 η n k c n k H M n k ,N n k +1 P n k +1 ,Q n k   c n k s       (1 , 1) , ( ϑ n k j + θ n k j , θ n k j ) P n k j =1 ( ϕ n k j + φ n k j , φ n k j ) Q n k j =1   ds  . (40) Let u s cons ider some special cases of t he hyp er -Fox’ s H fading model, i.e., the special cases of (40). Note that t he MGFs of some comm only us ed fading dist ributions (such as one-sided Gaussian, expo- nential, Gamma, W eibull, hyper -Gamma, Nakagami q (Hoyt), Nakagami n (Rice) , Maxwell, lognormal, K-distribution, generalized-K, generalized Gamma, extended g eneralized Gamm a and F ox’ s H) and t heir deriv atives are giv en in details in T ables II,III,IV and V. Using the MGF of generalized Gam ma distrib ution and its deriv ative given in the first row in T a ble V, and then subst ituting them into (40), the A UP expression of the L -branch combiner over GNM fading channels is giv en by P AU P = 1 − n 2 ( 1 − ( − 1) n Γ( b ) " L Y k =1 1 Γ ( m k ) # Z ∞ 0 1 s G 1 ,n n +1 , 2 " a s      Λ ( n ) 1 , 1 b, 0 # × L X ℓ =1 H 1 , 1 1 , 1 " β ℓ ¯ γ ℓ s      (0 , 1) ( m ℓ , 1 ξ ℓ ) # L Y k =1 k 6 = ℓ H 1 , 1 1 , 1 " β k ¯ γ k s      (1 , 1) ( m k , 1 ξ k ) # ds ) . (41) Note that in the special case of singl e link, i.e., L = 1 , (41) not surprising ly s implifies in to (19 ) by means of some alg ebraic m anipulations u sing [7, E q.(2.25.1/1)] and [7, Eq.(8.3.2/2 1)] together . Some algebraic manipulation s subst ituting t he shaping parameters ξ 1 = ξ 2 = · · · = ξ N = 1 into (41) and then utilizin g [7, Eqs. (8 .3.2/7), (8.3.2/21 ) and then (8.4.2/5)] result in the unified expression of the L -branch diversity combiner performing over m utually independent and not necessarily ident ically distributed Nakagami- m fading channels, that i s, P AU P = 1 − n 2 ( 1 − ( − 1) n Γ( b ) Z ∞ 0 G 1 ,n n +1 , 2 " a s      Λ ( n ) 1 , 1 b, 0 # L P ℓ =1 ¯ γ ℓ 1 + ¯ γ ℓ s m ℓ L Q ℓ =1  1 + ¯ γ ℓ s m ℓ  m ℓ ds ) . (42) For id entically d istributed Nakagami- m fa ding channels (i.e., m 1 = m 2 = . . . = m L = m and ¯ γ 1 = ¯ γ 2 = . . . = ¯ γ L = ¯ γ ), app arently as a result of both empl oying Γ ( mL + 1)  1 + ¯ γ s m  − mL − 1 = G 1 , 1 1 , 1  ¯ γ s m   − mL 0  IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 16 [7, Eq.(8.4.2/5)] and then usi ng the integral equality of two Meij er’ s G functions [7, Eq.(2.24.1/1)], (42) simplifies to P AU P = 1 − n 2  1 − ( − 1) n Γ( b )Γ( mL ) G n +1 , 1 2 ,n +1  m a ¯ γ     1 − b, 1 Λ ( n ) 0 , mL  (43) which is the sp ecial case of (19) with ξ ℓ = 1 , m ℓ = m , ¯ γ ℓ = ¯ γ and L = 1 as it is expected. In additio n, for n = 1 , foll owing the same steps in the deriv at ion of (21) from (20), the uni fied expression given by (42) readily reduces to the ABEP of the L -branch div ersity combin er ov er identi cal N akagami- m fa ding channels can be readily o btained as P AB E P = 1 2Γ( b )Γ( mL ) G 1 , 2 2 , 2  m a ¯ γ     1 , 1 − b 0 , mL  . (44) For n = 2 and b = 1 , (43) simpli fies to the well-known result [4, Eq. (33)], that is, P AC = 1 Γ( mL ) G 3 , 1 2 , 3  m a ¯ γ     0 , 1 0 , 0 , mL  , (45) where a denotes the t ransmitted power . As an illustration of the mathematical formalism pre sented above, some numerical and simulation results regarding the ABEP and A C performance of multiple link reception over fa ding channels are depicted in Fig. 4, Fig. 5 and Fig. 6, and these figures show that these analytical and simul ation results are in perfect agreement. I V . C O N C L U S I O N In this paper , we presented a unified performance expression com bining the ABEP and A C of wireless communication syst ems over generalized fading channels. M ore precisely , this p aper introduces an MGF- based unified expression for t he AB EP and A C of sin gle and multiple l ink communication with an L -branch MRC combinin g. In addi tion, the hyper-F ox’ s H fading m odel is proposed as a un ified fading d istribution for a m ajority of the well-known generalized fading models in order to provide more general and more generic result s wh ich can be readily simpli fied to som e pu blished result s for some well-known fading distributions. W e explicitly offer a g eneric unified performance expression t hat can be easily calculated and that is applicable to a wide v ariety of fading scenarios. Finally , as an i llustration o f t he mathemati cal formalism, some simulations hav e been carr ied out for dif ferent scenarios of fading en vi ronment, and numerical and simu lation results were sh own to be in perfect agreement. IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 17 A C K N OW L E D G M E N T S This work is supported by King Abd ullah University of Science and T echnology (KA UST). A P P E N D I X A U N I FI E D B E P E X P R E S S I O N U S I N G T H E I N C O M P L E T E B E T A F U N C T I O N Note that, substitu ting the lo wer incom plete gamma γ ( a, z ) = R z 0 e − u u a − 1 du [6, Eq. (6.5.2)] and using the relation between the lower and upper i ncomplete Gamma functi ons (i.e., γ ( a, z ) = Γ ( a ) − Γ ( a, z ) ) [6, Eq. (6.5.2)], the condi tional bit error probability P B E R ( γ end ) can be written as P B E P ( γ end ) = 1 2 − γ ( b, aγ end ) 2Γ ( b ) . (A.1) Substitutin g an alternati ve representation of the l owe r in complete g amma function γ ( a, z ) = b − 1 ( az ) b 1 F 1 [ a ; a + 1; − z ] [7, Eq. (7.11.3/1)] in (A.1) and then usi ng the well-known lim it representation 1 F 1 [ − z ; a ; a + 1 ] = lim d →∞ 2 F 1 [ d, a ; a + 1; − z /d ] [7, E q. (7.2.2/ 13)], we can writ e P B E P ( γ end ) = 1 2 − ( aγ end ) b 2Γ ( b + 1) lim d →∞ 2 F 1 h d, b ; b + 1 ; − a d γ end i , (A.2) where 1 F 1 [ · ; · ; · ] and 2 F 1 [ · ; · , · ; · ] are the Kummer confluent and Gauss hypergeometric functions, re- spectiv ely . Finally , s ubstitutin g the Gauss hypergeometric representation o f the incomplete beta functi on, i.e., 2 F 1 [ z ; d, b ; b + 1] = bz − b B ( z ; b, 1 − d ) [7, Eq. (7.3.1/28)] into (A.2) results in (2), which pro ves Theorem 1. A P P E N D I X B C A P A C I T Y E X P R E S S I O N U S I N G T H E I N C O M P L E T E B E TA F U N C T I O N Using [7, Eq. (7.3.3/7)], i .e., 2 F 1 h m n , 1; m n + 1; − z i = − m n z − m/n n − 1 X k =0 e − i(2 k +1) m n π log  1 − z m/n e i(2 k + 1) m n π  , (B.1) one can readily show by s etting m = n = 1 in (B.1) that the conditio nal capacity P C ( γ end ) can also be represented as P C ( γ end ) = γ end 2 F 1 [1 , 1; 2; − γ end ] . (B.2) Substitutin g the G auss hyper geom etric representation of th e incomp lete beta function, i .e., 2 F 1 [ z ; a, b ; b + 1] = bz − b B ( z ; b, 1 − a ) [7, Eq. (7.3.1/28)] into (B.2) results in (4), which proves Theorem 2. IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 18 A P P E N D I X C A V E R A G E U N I FI E D E X P R E S S I O N O F T H E L - B R A N C H D I V E R S I T Y C O M B I N E R S OV E R C O R R E L A T E D N O T - N E C E S S A R I L Y I D E N T I C A L L Y D I S T R I B U T E D F A D I N G C H A N N E L S W e utilize the MacRobert’ s E function representation of the UP expression (i.e., see Corollary 3) instead of the other representations. More specifically , usin g the well-kn own the i ntegral formula of the MacRobert’ s E function [19, Eq. (6)], [8, Eq . (7. 814/2)], given by Z ∞ 0 exp ( − u ) u α p +1 − 1 E h { α i } p i =1 ; { ρ j } q j =1 ; z u i du = E  { α i } p +1 i =1 ; { ρ j } q j =1 ; z  (C.1) where ℜ { α p +1 } > 0 and z ∈ R + , and performing som e alg ebraic manipulati ons, one can readily obtain an alternati ve representation of (C.1) as ( aγ end ) b E  Λ ( n ) b ; b + 1; 1 aγ end  = Z ∞ 0 u b − 1 e − u aγ end E  Λ n − 1 b ; b + 1; 1 u  du. (C.2) Accordingly , sub stituting (C.2) and the tot al i nstantaneous SNR γ end = P L ℓ =1 γ ℓ into (8), it is straightfor- ward to show t hat the A UP expression P AU P = E [ P U P ( γ end )] for an L -branch diversity com biner can be explicitly given by P AU P = 1 − n 2 ( 1 − ( − 1) n Γ( b ) Z ∞ 0 u b − 1 E " exp − u a P L ℓ =1 γ ℓ !# E  Λ ( n − 1) b ; b + 1; 1 u  du ) . (C.3) Note that, E h exp( − s/ P L ℓ =1 γ ℓ ) i with s = u/a can be considered as th e MGF of the reciprocal di stribution of γ end = P L ℓ =1 γ ℓ . In order to p roceed further , we present the following lem ma. Lemma 1 (MGF of Reciprocal Distribution [20, Theorem 1 ]) . Given any no nne gati ve continuou s RV R distributed over (0 , ∞ ) with the PDF p R ( r ) and the MGF M R ( s ) for ℜ{ s } ≥ 0 . Then, t he MGF of its r ecipr ocal distribution e R (i.e., e R ≡ 1 / R ) is given by M e R ( s ) = − Z ∞ 0 J 0  2 √ su   ∂ ∂ u M R ( u )  du (C.4) with the con ver gence r e gion ℜ{ s } ≥ 0 , wher e J 0 ( · ) is the zer oth-or der Bessel function of th e first kind defined in [6, Eq. (6.1 9.7)] . IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 19 Accordingly , applying Lemma 1 on th e expectation part E h exp( − ( u/a ) / P L ℓ =1 γ ℓ ) i of (C.3) results as P AU P = 1 − n 2 ( 1 + ( − 1) n Γ( b ) Z ∞ 0 Z ∞ 0 u b − 1 J 0  2 r up a  × E  Λ ( n − 1) b ; b + 1; 1 u  du " ∂ ∂ p exp − p L X ℓ =1 γ ℓ !# dp ) . (C.5) Note that using the Hankel transform of the MacRobert’ s E functi on [8, E q. (7.823/1)], and then applying [21, Eq. (1)], the inner integral of (C.5) can be easily obtained i n closed-form as Z ∞ 0 u b − 1 J 0  2 r up a  E  Λ ( n − 1) b ; b + 1; 1 u  du = G 1 ,n n +1 , 2 " a p      Λ ( n ) 1 , 1 b, 0 # . (C.6) Finally , replacing th e i nner integral in (C.5) with the Hankel trans form of MacRobert’ s E function gi ven by (C.6) and using the definition of joi nt MGF (i.e., M γ 1 ,γ 2 ,...,γ L ( p ) ≡ E [exp( − p P ℓ γ ℓ )] ), (C.5) simpl ifies to (30), which prov es Theorem 3. R E F E R E N C E S [1] M. K. Simon and M.-S. Alouini, Di gital Communication over F ading Channels , 2nd ed. John Wile y & Sons, Inc., 2005. [2] V . Bhaskar, “Capacity of MRC on correlated Rician fading channels, ” International Journal of Electr onics and Communications (A EU) , vol. 56, no. 5, pp. 708–711, May . 2008. [3] M. Di Renzo, F . Graziosi, and F . Santucci, “Channel capacity ov er generalized fading channels: A novel MGF-based approach for performance analysis and design of wireless communication systems, ” IEEE T ransa ctions on V ehicular T echnolog y , vol. 59, no. 1, pp. 127–14 9, Jan. 2010. [4] F . Y ilmaz and M.-S. Alouini, “ A unified MGF-based capacity analysis of div ersity combiners over fading channels, ” submitted to IEEE T ransactions on Communications , Dec. 2010, av ailable at htt p://arxi v .org/abs/1012 .2596. [5] A. W ojnar, “Unkno wn bounds on performance in nakagami channels, ” vol. 34, no. 1, pp. 22–24, Jan. 1986. [6] M. Abramowitz and I. A. S tegun , Handbook of Mathematical Functions with F ormulas, Graphs, and Mathematical T ables , 9th ed. Ne w Y ork: Dov er P ublications, 1972. [7] A. P . Prudniko v, Y . A. Brychko v, and O. I. Mariche v, Inte gral and Series: V olume 3, Mor e Special Functions . CRC Press Inc., 1990. [8] I. S . Gradshteyn and I. M. Ryzhik, T able of Integr als, Series, and Produc ts , 5th ed. San Diego, CA: Academic Press, 1994. [9] K. Y ao, M. K. Simon, and E. Biglieri, “Unified theory on wireless communication fading statistics based on SIRP , ” in Pr oceedings of IEEE 5th W orkshop on Signal Pro cessing Advances in W ir eless C ommunications (SP A WC 2004) , July 2004, pp. 135–139. [10] A. M. Mathai, R. K. S axen a, and H. J. Haubold, The H-F unction: Theory and Applications , 1st ed. Dordrecht, Heidelberg, London, Ne w Y ork: Springer Science, 2009. [11] A. Kil bas and M. Saigo, H-T ransforms: Theory and Applications . Boca Raton, F L: CRC P ress LLC, 2004. [12] J. I. D. Cook, “The H-F unction and Probability Density F unctions of C ertain A lgebraic Combinations of Independent Random V ariables with H-Function Probability Distribution, ” Ph.D. dissertation, The Univ ersity of T exas, Austin, TX, May 1981. IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 20 [13] V . Aalo, T . Piboongungon, and C.-D. Iskande r, “Bit-error rate of binary digital modulation schemes in generalized Gamma fading channels, ” IE EE Communications Letters , vol. 9, no. 2, pp. 139–141, Feb . 2005. [14] N. C. Sagias, G. S. T ombras, and G. K. Karagiannidis, “Ne w results for the shannon channel capacity in generalized fading channels, ” IEEE Communications Letters , vol. 9, no. 2, pp. 97–99, Feb. 2005. [15] F . Y ilmaz and M.-S. Alouini, “ A new simple model for composite fading channels: Second order statistics and channel capacity , ” in 7th International Symposium on W ir eless Communication Systems (ISWCS 2010) , Sep. 2010, pp. 676–680. [16] ——, “Extended generalized-K (EGK): A ne w simple and general mod el for composite fading channels, ” Submited to IEEE T ransactions on Communications , Dec. 2010, available at http:// arxi v .org/abs/101 2.2598. [17] M. A. Chaudhry and S . M. Zubair, On a Class of I ncomplete Gamma Functions with Applications . Boca Raton-London-Ne w Y ork-W ashington, D. C.: Chapman & Hall/CRC, 2002. [18] B. D. Carter and M. D. Springer, “The distribution of products, quotients and powers of independent H-function variates, ” SIAM Jo urnal on Applied Mathematics , vol. 33, no. 4, pp. 542–558, Dec. 1977. [19] F . Ragab, “The inv erse L aplace transform of an expone ntial function, ” Communications on Pur e and Applied Mathematics , vol. 11, pp. 115–127, 1958. [20] F . Y ilmaz, O. Kucur, and M.-S. Alouini, “ A unified framewo rk for the statistical characterization of the SNR of amplify-and-forward multihop channels, ” in IEEE 17th International Confer ence on T elecommunications (ICT 2010) , Apr . 2010, pp. 324–330. [21] T . M. MacRobert, “Integrals in volving E -functions, ” Mathematische Zeitschrift , vol. 69, no. 1, pp. 234–236, F eb . 1958. [22] S. Ekin, F . Y ilmaz, H. Celebi, K . A. Qaraqe, M.-S. A louini, and E. Serpedin, “ Achiev able capacity of a spectrum sharing system over hyper fading channels, ” in Pro ceedings of IEEE Global T elecommunications Confer ence (GL OBECOM 2009) , Nov . 2009, pp. 1–6. [23] P . M. Shankar, “Error rates in generalized shadowed fading channels, ” W ireless P ersonal Commun. , vol. 28, pp. 233–238, Feb. 2004. [24] E. W . Stacy, “ A generalization of the G amma distr ibution, ” The Annals of Math. Statistics , vol. 3, no. 33, pp. 1187–11 92, Sep. 1962. IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 21 T ABLE I P A R A M E T E R S a , b , n A N D d F O R T H E B E P A N D C A PAC I T Y P E R F O R M A N C E M E A S U R E S Perf ormance Measure a b n d BEP of orthogonal coherent BFSK [1, Eq. (8.43)], P BE P ( γ end ) = Q  √ γ end  , where Q ( · ) is the Gaussian Q-function having a one-to-one mapping with the complementa ry error function erfc ( · ) , i.e., Q ( z ) = 1 2 erfc( z/ √ 2) . 1 2 1 2 1 ∞ BEP of orthogonal noncoheren t BFSK [1, Eq. (8.69)], P BE P ( γ end ) = 1 2 exp  − γ end 2  . 1 2 1 1 ∞ BEP of antipoda l coherent BPSK [1, E q. (8.19)], P BE P ( γ end ) = Q p 2 γ end  . 1 1 2 1 ∞ BEP of antipoda l diff erentia lly coheren t BPSK (DPSK) [1, Eq. (8.85)], P BE P ( γ end ) = 1 2 exp ( − γ end ) . 1 1 1 ∞ BEP of correlat ed coherent binary signaling [1, Chapter 8, footnote 6], P BE P ( γ end ) = Q p 2 aγ end  . a ∈ [0 , 1] 1 2 1 ∞ Shannon Capacit y [1, Eq. (15.22)], P C ( γ end ) = log (1 + aγ end ) , where a is the transmitt ed power and log ( · ) is the natural logarithm (i.e., the logarithm to the base e ) [6, Eq. (4.1.1)]. a ∈ R + 1 2 1 IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 22 T ABLE II M G F S O F S O M E W E L L - K N OW N F A D I N G C H A N N E L M O D E L S Instantaneous SNR Distribution, i.e. , p γ ℓ ( γ ) MGF M γ ℓ ( s ) a nd its derivative ∂ ∂ s M γ ℓ ( s ) One-Sided Ga ussian [1, Sec. 2.2 .1.4] p γ ℓ ( γ ) = s 2 π ¯ γ ℓ exp  − γ 2 2 ¯ γ ℓ  , = 1 2 √ π ¯ γ ℓ H 1 , 0 0 , 1  γ 2 ¯ γ ℓ     ( − 1 2 , 1 )  , where ¯ γ ℓ is the av erage po wer (i.e., ¯ γ ℓ ≥ 0 ). M γ ℓ ( s ) = 1 √ π H 1 , 1 1 , 1  1 2 ¯ γ ℓ s     (1 , 1 ) ( 1 2 , 1 )  = 1 √ π G 1 , 1 1 , 1  1 2 ¯ γ ℓ s     1 1 2  = 1 p 1 + 2 ¯ γ ℓ s , ∂ ∂ s M γ ℓ ( s ) = 2 ¯ γ ℓ √ π H 2 , 1 2 , 2  1 2 ¯ γ ℓ s     (2 , 1 ) , (1 , 1) ( 3 2 , 1 ) , (2 , 1)  = 2 ¯ γ ℓ √ π G 2 , 1 2 , 2  1 2 ¯ γ ℓ s     2 , 1 3 2 , 2  = − ¯ γ ℓ (1 + 2 ¯ γ ℓ s ) 3 / 2 , where G m,n p,q [ · ] and H m,n p,q [ · ] represent the Meijer’ s G function [7, Eq. (8.2.1/1)] and Fox’ s H function [7 , Eq. (8.3.1/1)], respect iv ely . Note that one-sided Gaussian fading coincides with the worst-case fadin g or equival ently , the largest amount of fading (AoF) for all Gaussian-ba sed fading distrib utions. Exponential [1, Eq. (2.7)] p γ ℓ ( γ ) = 1 ¯ γ ℓ exp  − γ ¯ γ ℓ  = 1 ¯ γ ℓ H 1 , 0 0 , 1  γ ¯ γ ℓ     (0 , 1 )  , where ¯ γ ℓ is the av erage po wer (i.e., ¯ γ ℓ ≥ 0 ). M γ ℓ ( s ) = H 1 , 1 1 , 1  1 s ¯ γ ℓ     (1 , 1 ) (1 , 1 )  = G 1 , 1 1 , 1  1 s ¯ γ ℓ     1 1  = 1 1 + ¯ γ ℓ s , ∂ ∂ s M γ ℓ ( s ) = ¯ γ ℓ H 2 , 1 2 , 2  1 s ¯ γ ℓ     (2 , 2 ) , (1 , 1) (2 , 1 ) , (2 , 1)  = ¯ γ ℓ G 2 , 1 2 , 2  1 s ¯ γ ℓ     2 , 1 2 , 2  = − ¯ γ ℓ (1 + s ¯ γ ℓ ) 2 , Gamma [1, Eq. (2.21) ] p γ ℓ ( γ ) = 1 Γ( m ℓ )  m ℓ ¯ γ ℓ  m ℓ γ m ℓ − 1 exp  − m ℓ γ ¯ γ ℓ  , = m ℓ Γ( m ℓ ) ¯ γ ℓ H 1 , 0 0 , 1  m ℓ ¯ γ ℓ γ     ( m ℓ − 1 , 1)  , where ¯ γ ℓ is the average po wer , and where m ℓ ( 0 . 5 ≤ m ℓ ) denotes the fading figure. Moreover , Γ( · ) is the Gamma functio n [8, Sec. 8.31]. M γ ℓ ( s ) = 1 Γ( m ℓ ) H 1 , 1 1 , 1  m ℓ ¯ γ ℓ s     (1 , 1 ) ( m ℓ , 1 )  = G 1 , 1 1 , 1 h m ℓ ¯ γ ℓ s    1 m ℓ i Γ( m ℓ ) =  1 + ¯ γ ℓ m ℓ s  − m ℓ , ∂ ∂ s M γ ℓ ( s ) = 1 Γ( m ℓ + 1) H 2 , 1 2 , 2  m ℓ ¯ γ ℓ s     (2 , 1 ) , (1 , 1) ( m ℓ + 1 , 1) , (2 , 1)  = G 2 , 1 2 , 2 h m ℓ ¯ γ ℓ s    2 , 1 m ℓ +1 , 2 i Γ( m ℓ + 1) = − ¯ γ ℓ  1 + ¯ γ ℓ m ℓ s  − m ℓ − 1 , Note that the Nakagami- m distrib ution spans via the m parameter the widest range of amount of fadi ng (AoF) among all the multipath distrib utions [1]. As such, Nakagami- q (Hoyt) and Nakagami- n (Rice) can also be closely approximat ed by Nakagami- m distribut ion [1, Eq. (2. 25)], [1, Eq. (2.26)]. W eibull [1, Eq. (2.27) ] p γ ℓ ( r ) = ξ ℓ  ω ℓ ¯ γ ℓ  ξ ℓ r ξ ℓ − 1 exp  −  ω ℓ ¯ γ ℓ  ξ ℓ r ξ ℓ  , = ω ℓ ¯ γ ℓ H 1 , 0 0 , 1  ω ℓ ¯ γ ℓ γ     (1 − 1 /ξ ℓ , 1 /ξ ℓ )  , where ω ℓ = Γ(1 + 1 /ξ ℓ ) and where ξ ℓ ( 0 < ξ ℓ ) denotes the fadi ng shaping factor . Moreov er , ¯ γ ℓ is the av erage po wer . M γ ℓ ( s ) = H 1 , 1 1 , 1 " ω ℓ Ω sℓ s      (1 , 1 ) (1 , 1 ξ ℓ ) # = s 4 k l (2 π ) 2 k +2 l − 2 G 2 l, 2 k 2 k, 2 l   ω 2 k ℓ (2 k ) 2 k s 2 k Ω 2 k sℓ (2 l ) 2 l       − Ξ ( − 2 k ) (2 k ) Ξ (1) (2 l )   , ∂ ∂ s M γ ℓ ( s ) = Ω sℓ ω ℓ H 2 , 1 2 , 2 " ω ℓ Ω sℓ s      (2 , 1 ) , (1 , 1) (1 + 1 ξ ℓ , 1 ξ ℓ ) , (2 , 1) # = √ 16 k 3 l q (2 π ) 2 k +2 l − 2 s G 2 l +1 , 2 k 2 k +1 , 2 l +1   ω 2 k ℓ (2 k ) 2 k s 2 k Ω 2 k sℓ (2 l ) 2 pl       − Ξ ( − 2 k ) (2 k ) , 0 Ξ (1) (2 l ) , 1   , where the Meijer’ s G representa tions are giv en for the rational va lue of the fadi ng s haping fac tor ξ sℓ (that is, we let ξ sℓ = k/l , where k , and l are arbi trary positiv e integer s.) through the mediu m of algebraic manipulat ions utilizi ng [7, Eq. (8.3.2.22)]. In addi tion, the coef ficient Ξ ( x ) ( n ) of the Meijer’ s G function is a set of coeffic ients such that it is defined as Ξ ( x ) ( n ) ≡ x n , x +1 n , . . . , x + n − 1 n with x ∈ C and n ∈ N . IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 23 T ABLE III M G F S O F S O M E W E L L - K N OW N F A D I N G C H A N N E L M O D E L S Instantaneous SNR Distribution, i.e. , p γ ℓ ( γ ) MGF M γ ℓ ( s ) a nd its derivative ∂ ∂ s M γ ℓ ( s ) Hyper-Gamma [22, Eq. (3 )] p γ ℓ ( γ ) = K X k =1 ξ ℓk Γ ( m ℓk )  m ℓk ¯ γ ℓk  m ℓk γ m ℓk − 1 e − m ℓk ¯ γ ℓk γ , = K X k =1 ξ ℓk m ℓk Γ( m ℓk ) ¯ γ ℓk H 1 , 0 0 , 1  m ℓk ¯ γ ℓk γ     ( m ℓk − 1 , 1)  , where m ℓk ( 0 . 5 ≤ m ℓk ) is the fading figure, ¯ γ ℓk ( 0 < ¯ γ ℓk ) is the a verage po wer , and ξ ℓk ( 0 < ξ ℓk ) is the accru ing factor , of the k th fading en vironment. M γ ℓ ( s ) = K X k =1 ξ ℓk H 1 , 1 1 , 1 h m ℓk s ¯ γ ℓk    (1 , 1) ( m ℓk , 1) i Γ( m ℓk ) = K X k =1 ξ ℓk G 1 , 1 1 , 1 h m ℓk s ¯ γ ℓk    1 m ℓk i Γ( m ℓk ) = K X k =1 ξ ℓk  1 + ¯ γ ℓk m ℓk s  − m ℓk , ∂ ∂ s M γ ℓ ( s ) = K X k =1 ξ ℓk H 2 , 1 2 , 2 h m ℓk s ¯ γ ℓk    (1 , 1) , (0 , 1) ( m ℓk , 1) , (1 , 1) i s Γ( m ℓk ) = K X k =1 ξ ℓk G 2 , 1 2 , 2 h m ℓk s ¯ γ ℓk    1 , 0 m ℓk , 1 i s Γ( m ℓk ) = − K X k =1 ξ ℓk ¯ γ ℓk  1 + ¯ γ ℓk m ℓk s  − m ℓk − 1 , where Γ( · ) is the Gamma function [6, Eq. (6.1.1)]. In addition, It may be useful to notice that the s um of the accruing probabilit ies ξ ℓk , k ∈ { 1 , 2 , . . . , K } of K possible fading en vironments is unit such that P K k =1 ξ ℓk = 1 . Po wer of Nakagami- q (Hoy t) [1, Eq . (2. 11)] p γ ℓ ( γ ) = 1 + q 2 ℓ 2 q ℓ ¯ γ ℓ e − (1 + q 2 ℓ ) 2 4 q 2 ℓ ¯ γ ℓ γ I 0  1 − q 4 ℓ 4 q 2 ℓ ¯ γ ℓ γ  , = lim K →∞ K X k =0 Φ k m k H 1 , 0 0 , 1 h m k Ω k γ    ( m k − 1 , 1) i Γ( m k )Ω k , where q ℓ ( 0 < q ℓ < 1 ) is the Nakagami-q fading parameter and ¯ γ ℓ ( 0 < ¯ γ ℓ ) is the averag e power . In addition, I 0 ( · ) is the zerot h order modified Bessel funct ion of the first kind [6, Eq. (9. 6.20)]. M γ ℓ ( s ) = lim K →∞ K X k =0 Φ k Γ( m k ) H 1 , 1 1 , 1  m k s Ω k     (1 , 1 ) ( m k , 1 )  =  1 + 2 ¯ γ ℓ s + (2 ¯ γ ℓ s ) 2 q 2 ℓ (1 + q 2 ℓ ) 2  − 1 2 , ∂ ∂ s M γ ℓ ( s ) = lim K →∞ K X k =0 Φ k s Γ( m k ) H 2 , 1 2 , 2  m k s Ω k     (1 , 1 ) , (0 , 1) ( m k , 1 ) , (1 , 1)  = − ¯ γ ℓ  1 + 4 q 2 ℓ ¯ γ ℓ s (1+ q 2 ℓ ) 2   1 + 2 ¯ γ ℓ s + (2 ¯ γ ℓ s ) 2 q 2 ℓ (1+ q 2 ℓ ) 2  3 2 , where m k , Ω k are defined as m k = 2 k + 1 and Ω k = 4(2 k + 1) q 2 ℓ ¯ γ ℓ / (1 + q 2 ℓ ) 2 , respecti vely . In additi on, the weighting coeffici ents { Φ k } are giv en by Φ k = 2 q ℓ √ π (1+ q 2 ℓ ) Γ( k + 1 2 ) Γ( k +1)  1 − q 2 ℓ 1+ q 2 ℓ  2 k for all k ∈ N . It may be useful to notice that the series expression of the MGF for the Nakagami- q (Hoyt) and its deri vati ve are con verging very fast such that 10 summation terms is generall y enough. Po wer of Nakagami- n (Rice) [1, Eq . (2.1 6)] p γ ℓ ( γ ) = (1 + n 2 ℓ ) e − n 2 ℓ ¯ γ ℓ e − (1+ n 2 ℓ ) ¯ γ ℓ γ I 0 2 n ℓ s 1 + n 2 ℓ ¯ γ ℓ γ ! , = lim K →∞ K X k =0 Ψ k m k H 1 , 0 0 , 1 h m k Ω k γ    ( m k − 1 , 1) i Γ( m k )Ω k , where n ℓ ( 0 < n ℓ ) and ¯ γ ℓ ( 0 < ¯ γ ℓ ) are the LOS figure and av erage po wer , respecti vel y . M γ ℓ ( s ) = lim K →∞ K X k =0 Ψ k Γ( m k ) H 1 , 1 1 , 1  m k s Ω k     (1 , 1 ) ( m k , 1 )  = 1 + n 2 ℓ (1 + n 2 ℓ ) + ¯ γ ℓ s exp  − n 2 ℓ ¯ γ ℓ s (1 + n 2 ℓ ) + ¯ γ ℓ s  , ∂ ∂ s M γ ℓ ( s ) = lim K →∞ K X k =0 Ψ k s Γ( m k ) H 2 , 1 2 , 2  m k s Ω k     (1 , 1 ) , (0 , 1) ( m k , 1 ) , (1 , 1)  = − ¯ γ ℓ 1 + ¯ γ ℓ s (1+ n 2 ℓ ) 2  1 + ¯ γ ℓ s 1+ n 2 ℓ  3 exp  − n 2 ℓ ¯ γ ℓ s (1 + n 2 ℓ ) + ¯ γ ℓ s  , where m k and Ω k are defined as m k = k + 1 and Ω k = ( k + 1) ¯ γ ℓ 1+ n 2 ℓ , respecti vel y . In additio n, the weighti ng coef ficients Ψ k are gi ven by Ψ k = n 2 k ℓ exp  − n 2 ℓ  / Γ( k + 1) . It may be useful to notice that the line-of-sight (LOS) figure i.e. n ℓ is relate d to the Rician K ℓ fac tor by K ℓ = n 2 ℓ which corresponds to the ratio of the power of the L OS (specular) component to the avera ge po wer of the scattere d component . IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 24 T ABLE IV M G F S O F S O M E W E L L - K N OW N F A D I N G C H A N N E L M O D E L S Instantaneous SNR Distribution, i.e. , p γ ℓ ( γ ) MGF M γ ℓ ( s ) a nd its derivative ∂ ∂ s M γ ℓ ( s ) Maxwell [1, Eq. ( 2.53)] p γ ℓ ( γ ) = s 27 γ 2 π ¯ γ 3 ℓ e − 3 2 ¯ γ ℓ γ = 3 H 1 , 0 0 , 1 h 3 γ 2 ¯ γ ℓ    ( 1 2 , 1) i √ π ¯ γ ℓ , where ¯ γ ℓ is the av erage po wer . M γ ℓ ( s ) = 2 √ π H 1 , 1 1 , 1  3 2 ¯ γ ℓ s     (1 , 1 ) ( 3 2 , 1 )  = 2 √ π G 1 , 1 1 , 1  3 2 ¯ γ ℓ s     1 3 2  = 3 √ 3 (3 + 2 ¯ γ ℓ s ) 3 / 2 , ∂ ∂ s M γ ℓ ( s ) = 2 √ π s H 2 , 1 2 , 2  3 2 ¯ γ ℓ s     (1 , 1 ) , (0 , 1) ( 3 2 , 1 ) , (1 , 1)  = − 9 √ 3 ¯ γ ℓ (3 + 2 ¯ γ ℓ s ) 5 / 2 , Lognormal [ 1, Eq. (2.53 )] p γ ℓ ( γ ) = ξ √ 2 π σ ℓ γ e − (10 log 10 ( γ ) − µ ℓ ) 2 2 σ 2 ℓ , = 1 √ π K X k =1 w k ω k H 0 , 0 0 , 0  γ ω k     , where µ ℓ (dB) and σ ℓ (dB) are the mean and the standard de viati on of γ ℓ . M γ ℓ ( s ) = 1 √ π K X k =1 w k H 1 , 0 0 , 1  ω k s     (0 , 1 )  = 1 √ π K X k =1 w k exp ( − ω k s ) , ∂ ∂ s M γ ℓ ( s ) = − 1 √ π K X k =1 w k ω k H 1 , 0 0 , 1  ω k s     (0 , 1 )  = − 1 √ π K X k =1 w k ω k exp ( − ω k s ) , where ω k is defined as ω k = 10 ( √ 2 σ ℓ u k + µ ℓ ) / 10 such that for k ∈ { 1 , 2 , . . . , K } , { w k } and { u k } are the weight fac tors and the zeros (abscissas) of the K -order Hermite polynomial [6, T able 25.10]. Po wer of K-Distribution [ 1, Eq. (2.15 )] p γ ℓ ( γ ) = 2  m sℓ ¯ γ sℓ  m sℓ +1 2 γ m sℓ − 1 2 Γ( m sℓ ) K m sℓ − 1  2 r m sℓ γ ¯ γ sℓ  , = m sℓ H 2 , 0 0 , 2 h m sℓ ¯ γ sℓ s    (0 , 1) , ( m sℓ − 1 , 1) i Γ( m sℓ ) ¯ γ sℓ , M γ ℓ ( s ) = 1 Γ( m sℓ ) H 2 , 1 1 , 2  m sℓ ¯ γ sℓ s     (1 , 1 ) (1 , 1 ) , ( m sℓ , 1 )  = 1 Γ( m sℓ ) G 2 , 1 1 , 2  m sℓ ¯ γ sℓ s     1 1 , m sℓ  , ∂ ∂ s M γ ℓ ( s ) = 1 Γ( m sℓ ) , s H 3 , 1 2 , 3  m sℓ ¯ γ sℓ s     (1 , 1 ) , (0 , 1) (1 , 1 ) , (1 , 1) , ( m sℓ , 1 )  = 1 Γ( m sℓ ) , s G 3 , 1 2 , 3  m sℓ ¯ γ sℓ s     1 , 0 1 , 1 , m sℓ  , where m sℓ ( 1 2 ≤ m sℓ ) denotes the shadowi ng sev erity , and ¯ γ sℓ ( 0 < ¯ γ sℓ ) represents the avera ge power . In additio n, K n ( · ) is the n th order modified Bessel function of the second kind [6, Eq. (9.6.24)]. Po wer of Generalized-K [2 3, Eq. (5 )] p γ ℓ ( γ ) = 2  m sℓ m ℓ ¯ γ sℓ  φ ℓ 2 γ φ ℓ 2 − 1 Γ( m sℓ ) K ψ ℓ  2 r m sℓ m ℓ γ ¯ γ sℓ  , = m ℓ m sℓ H 2 , 0 0 , 2 h m ℓ m sℓ ¯ γ sℓ γ    ( m ℓ − 1 , 1) , ( m sℓ − 1 , 1) i Γ( m ℓ )Γ( m sℓ ) ¯ γ sℓ , M γ ℓ ( s ) = 1 Γ( m ℓ )Γ( m sℓ ) H 2 , 1 1 , 2  m sℓ m ℓ ¯ γ sℓ s     (1 , 1 ) ( m ℓ , 1 ) , ( m sℓ , 1 )  = 1 Γ( m ℓ )Γ( m sℓ ) G 2 , 1 1 , 2  m sℓ m ℓ ¯ γ sℓ s     1 m ℓ , m sℓ  , ∂ ∂ s M γ ℓ ( s ) = 1 Γ( m ℓ )Γ( m sℓ ) s H 3 , 1 2 , 3  m sℓ m ℓ ¯ γ sℓ s     (1 , 1 ) , (0 , 1) ( m ℓ , 1 ) , ( m sℓ , 1 ) , (1 , 1)  = G 3 , 1 2 , 3 h m sℓ m ℓ ¯ γ sℓ s    1 , 0 m ℓ ,m sℓ , 1 i Γ( m ℓ )Γ( m sℓ ) s , where φ ℓ = m sℓ + m ℓ and ψ ℓ = m sℓ − m ℓ , and where the paramet ers m ℓ ( 0 . 5 ≤ m ℓ ) and m sℓ ( 0 . 5 ≤ m sℓ ) represent the fading figure (di versi ty sev erity / order) and the shado wing se verit y , respecti vel y . ¯ γ sℓ ( 0 < ¯ γ sℓ ) represents the ave rage power . IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 25 T ABLE V M G F S O F S O M E W E L L - K N OW N F A D I N G C H A N N E L M O D E L S Instantaneous SNR Distribution, i.e. , p γ ℓ ( γ ) MGF M γ ℓ ( s ) a nd its derivative ∂ ∂ s M γ ℓ ( s ) Generalized Ga mma [24] p γ ℓ ( γ ) = ξ ℓ  β ℓ ¯ γ ℓ  m ℓ ξ ℓ γ m ℓ ξ ℓ − 1 Γ( m ℓ ) exp −  β ℓ ¯ γ ℓ γ  ξ ℓ ! , = β ℓ Γ( m ℓ ) ¯ γ ℓ H 1 , 0 0 , 1 " β ℓ ¯ γ ℓ γ      ( m ℓ − 1 ξ ℓ , 1 ξ ℓ ) # , M γ ℓ ( s ) = H 1 , 1 1 , 1  β ℓ ¯ γ ℓ s     (1 , 1) ( m ℓ , 1 ξ ℓ )  Γ( m ℓ ) = 2 π l m ℓ k p (2 π ) k + l k l Γ( m ℓ ) G l,k k,l   k k l l  β ℓ ¯ γ ℓ s  k       Ξ ( k ) (1) Ξ ( l ) ( m ℓ )   , ∂ ∂ s M γ ℓ ( s ) = H 2 , 1 2 , 2  β ℓ ¯ γ ℓ s     (1 , 1) , (0 , 1) ( m ℓ , 1 ξ ℓ ) , (1 , 1)  Γ( m ℓ ) s = 2 π l m ℓ k 2 p (2 π ) k + l k l Γ( m ℓ ) s G l +1 ,k k +1 ,l +1   k k l l  β ℓ ¯ γ ℓ s  k       Ξ ( k ) (1) , 0 Ξ ( l ) ( m ℓ ) , 1   , where the parameters m ℓ (0 . 5 ≤ m ℓ < ∞ ) and ξ ℓ (0 ≤ ξ ℓ < ∞ ) represent the fading figure (div ersity seve rity / order) and the fadi ng shaping facto r , respec ti vely , while ¯ γ ℓ (0 ≤ ¯ γ ℓ < ∞ ) is the avera ge power . In additi on, the parameter β ℓ is defined as β ℓ = Γ  m ℓ + 1 /ξ ℓ  / Γ  m ℓ  , and referring the coef ficien ts of the Meijer’ s G function, Ξ ( x ) ( n ) is a s et of coef ficients such that it is defined as Ξ ( x ) ( n ) ≡ x n , x +1 n , . . . , x + n − 1 n with x ∈ C and n ∈ N . Extended Genera lized Gamma [ 16] p γ ℓ ( γ ) = ξ ℓ  β ℓ β sℓ ¯ γ sℓ  ξ ℓ m ℓ Γ( m ℓ )Γ( m sℓ ) γ ξ ℓ m ℓ − 1 × Γ m sℓ − m ℓ ξ ℓ ξ sℓ , 0 ,  β ℓ β sℓ ¯ γ sℓ  ξ ℓ γ ξ ℓ , ξ ℓ ξ sℓ ! , = H 2 , 0 0 , 2  β ℓ β sℓ ¯ γ sℓ γ     ( m ℓ − 1 ξ ℓ , 1 ξ ℓ ) , ( m sℓ − 1 ξ sℓ , 1 ξ sℓ )  Γ( m ℓ )Γ( m sℓ ) ¯ γ sℓ β ℓ β sℓ , M γ ℓ ( s ) = H 2 , 1 1 , 2  β ℓ β sℓ ¯ γ s,ℓ s     (1 , 1) ( m ℓ , 1 ξ ℓ ) , ( m sℓ , 1 ξ sℓ )  Γ( m ℓ )Γ( m sℓ ) = Φ ℓ G kl s + lk s , kk s kk s , kl s + lk s "  β ℓ β sℓ Ψ ℓ ¯ γ sℓ s  kk s      Ξ ( k s k ) (1) Ξ ( lk s ) ( m ℓ ) , Ξ ( l s k ) ( m sℓ ) # Γ( m ℓ )Γ( m sℓ ) , ∂ ∂ s M γ ℓ ( s ) = H 2 , 1 1 , 2  β ℓ β sℓ ¯ γ s,ℓ s     (1 , 1) ( m ℓ , 1 ξ ℓ ) , ( m sℓ , 1 ξ sℓ )  Γ( m ℓ )Γ( m sℓ ) = Φ ℓ k k s G kl s + lk s +1 , kk s kk s +1 , kl s + lk s +1 "  β ℓ β sℓ Ψ ℓ ¯ γ sℓ s  kk s      Ξ ( k s k ) (1) , 0 Ξ ( lk s ) ( m ℓ ) , Ξ ( l s k ) ( m sℓ ) , 1 # Γ( m ℓ )Γ( m sℓ ) s , where Φ ℓ = 2 π ( l k s ) m ℓ ( l s k ) m sℓ / p (2 π ) lk s + l s k + k k s − 1 ll s and Ψ ℓ = ( lk s ) 1 ξ ℓ ( l s k ) 1 ξ sℓ / ( kk s ) , and where the paramete rs m ℓ (0 . 5 ≤ m ℓ < ∞ ) and ξ ℓ (0 ≤ ξ ℓ < ∞ ) represent the fading figure (div ersity sev erity / order) and the fadi ng shaping factor , respecti vely , while m sℓ (0 . 5 ≤ m sℓ < ∞ ) and ξ sℓ (0 ≤ ξ sℓ < ∞ ) represent the shadowin g sev erity and the shadowi ng shaping factor (inhomogeneity) , respect iv ely . In addition, the paramete rs β ℓ and β sℓ are defined as β ℓ = Γ  m ℓ + 1 /ξ ℓ  / Γ  m ℓ  and β sℓ = Γ  m sℓ + 1 /ξ sℓ  / Γ  m sℓ  , respect iv ely , where Γ ( · ) is the Gamm a func tion [6, E q. (6.5.3)]. In addit ion, Γ ( · , · , · , · ) is the extended incompl ete Gamma function defined as Γ ( α, x, b, β ) = R ∞ x r α − 1 exp  − r − br − β  dr , where α, β , b ∈ C and x ∈ R + [17, E q. (6.2)] . Referring the coef ficients of the Meijer’ s G function, Ξ ( x ) ( n ) is a set of coef ficient s such that it is defined as Ξ ( x ) ( n ) ≡ x n , x +1 n , . . . , x + n − 1 n with x ∈ C and n ∈ N . Fox’ s H distrib ution [1 8, Eq . (3.1 )], [9] p γ ℓ ( γ ) = K ℓ H m,n p,q  G ℓ γ     ( a i , α i ) i =1 , 2 ,...,p ( b j , β j ) j =1 , 2 ,...,q  where K ℓ ∈ R and G ℓ ∈ R are such two numbers that R ∞ 0 p γ ℓ ( γ ) dγ = 1 . M γ ℓ ( s ) = K ℓ G ℓ H m,n +1 p +1 ,q  G ℓ s     (1 , 1 ) , ( a i + α i , α i ) i =1 , 2 ,...,p ( b j + β j , β j ) j =1 , 2 ,...,q  , ∂ ∂ s M γ ℓ ( s ) = − K ℓ G ℓ s H m,n +1 p +1 ,q  G ℓ s     (0 , 1 ) , ( a i + α i , α i ) i =1 , 2 ,...,p ( b j + β j , β j ) j =1 , 2 ,...,q  , where max i ∈{ 1 , 2 ,..,m } {− b i /β i } < min i ∈{ 1 , 2 ,..,n } { (1 − a i ) /α i } where a i , b i ∈ R and α i , β i ∈ R + . IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 26 5 10 15 20 25 30 35 40 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Numerical (BDPSK) Numerical (BPSK) Simulation P S f r a g r e p l a c e m e n t s SNR [dB] ABEP [error/sn] (a) (b) (c) (d) (e) Fig. 1. ABEP performance comparison of BPSK and BDP SK binary modulations over generalized Gamma fading channel with the parameters: (a) m ℓ = 2 , ξ ℓ = 0 . 25 , (b) m ℓ = 2 , ξ ℓ = 0 . 5 , (c) m ℓ = 2 , ξ ℓ = 0 . 75 , (d) m ℓ = 2 , ξ ℓ = 1 . 0 and (e) m ℓ = 2 , ξ ℓ = 1 . 5 . The number of samples is chosen as 10 7 in the computer-based simulations. IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 27 5 10 15 20 25 30 35 40 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Numerical (NC−BFSK) Numerical (BFSK) Simulation P S f r a g r e p l a c e m e n t s SNR [dB] ABEP [error/sn] (a) (b) (c) (d) (e) Fig. 2. ABEP performance comparison of BFSK and NC-FSK binary modulations over generalized Gamma fading channels wit h the parameters: (a) m ℓ = 2 , ξ ℓ = 0 . 25 , (b) m ℓ = 2 , ξ ℓ = 0 . 5 , (c) m ℓ = 2 , ξ ℓ = 0 . 75 , (d) m ℓ = 2 , ξ ℓ = 1 . 0 and (e) m ℓ = 2 , ξ ℓ = 1 . 5 . The number of samples is chosen as 10 7 in the computer-based simulations. IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 28 0 5 10 15 20 25 30 35 40 0 1 2 3 4 5 6 7 8 9 10 P S f r a g r e p l a c e m e n t s SNR [dB] A C [nat/sn/Hz] ( a ) , ( b ) , ( c ) , ( d ) , ( e ) Fig. 3. AC of wireless communication system over generalized Gamma fading channel with the parameters: (a) m ℓ = 2 , ξ ℓ = 0 . 25 , (b) m ℓ = 2 , ξ ℓ = 0 . 5 , (c) m ℓ = 2 , ξ ℓ = 0 . 75 , (d) m ℓ = 2 , ξ ℓ = 1 . 0 and (e) m ℓ = 2 , ξ ℓ = 1 . 5 . The number of samples is chosen as 10 6 in the computer-ba sed simulations. IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 29 5 10 15 20 25 30 35 40 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Numerical (BDPSK) Numerical (BPSK) Simulation P S f r a g r e p l a c e m e n t s SNR [dB] ABEP [error/sn] L = 1 L = 2 L = 3 L = 4 L = 5 Fig. 4. ABEP performance comparison of BP SK and BDPSK binary modulations for L -branch MRC recei ver ove r generalized Gamma fading channels with t he parameters m ℓ = 2 and ξ ℓ = 0 . 25 . The number of samples is chosen as 10 7 in the computer-based simulations. IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 30 5 10 15 20 25 30 35 40 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Numerical (NC−BFSK) Numerical (BFSK) Simulation P S f r a g r e p l a c e m e n t s SNR [dB] ABEP [error/sn] L = 1 L = 2 L = 3 L = 4 L = 5 Fig. 5. ABEP performance comparison of BFS K and NC-FSK binary modulations for L -branch MRC receiv er ove r generalized Gamma fading channels with t he parameters m ℓ = 2 and ξ ℓ = 0 . 25 . The number of samples is chosen as 10 7 in the computer-based simulations. IEEE TRANSA CTIONS ON COMMUNICA TIONS, V OL. X, NO. XX, J AN. 2011 31 0 5 10 15 20 25 30 35 40 0 1 2 3 4 5 6 7 8 9 10 Numerical Simulation P S f r a g r e p l a c e m e n t s SNR [dB] A C [nat/sn/Hz] L = 1 , 2 , 3 , 4 , 5 Fig. 6. AC of wirel ess communication systems with an L -branch MRC receiver ov er generalized Gamma fading channel wi th the parameters m ℓ = 2 , ξ ℓ = 0 . 25 . The number of samples is chosen as 10 6 in the computer-based simulations.