On the Monotonicity of the Generalized Marcum and Nuttall Q-Functions

PUBLISHED IN IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 55, NO. 8, A UGUST 2009 1 On the Monoton icity of the Genera lized Marcum and Nuttall Q -Functions † V asilios M. Kapinas , Member , IEEE, Sotirios K. Mihos, Student Membe r , IEEE, and George K. Karagiannidis, Senior Member , IEEE Abstract —Monotonicity criteria are established f or the gener - alized M arcum Q -fun ction, Q M ( α, β ) , th e standard Nuttall Q - function, Q M, N ( α, β ) , and the normalized Nut tall Q -function, Q M, N ( α, β ) , with respect to their real order indices M , N . Besides, closed-fo rm expressions are derived fo r the computation of the standard and normalized Nuttall Q -f unctions for the case when M , N are odd multiples of 0 . 5 and M ≥ N . By exploiting these results, novel u pper and lower bound s for Q M, N ( α, β ) and Q M, N ( α, β ) are pro posed. Furthermore, specific tight upper and lower bounds for Q M ( α, β ) , pr eviously reported in the literature, are extend ed for real values of M . The offered theoretical results can be efficiently appl ied in the stu dy of digital communications ov er fadin g channels, in the information- theoretic analysis o f multip le-input mu ltiple-outpu t systems and in the d escription of stochastic processes in probability theory , among others. Index T erms —Closed-for m expressions, generalized Marcum Q -function, lower and up per bounds, monotonicity , normalized Nuttall Q -function, standard Nuttall Q -functi on. I . I N T RO D U C T I O N A. The Nuttall Q-Fu nctions A N extended version of the (standard) Mar cum Q - function , Q ( α, β ) = R ∞ β xe − x 2 + α 2 2 I 0 ( αx ) dx , wh ere α, β ≥ 0 , or iginally appear ed in [1, Appendix, eq . ( 16)], defines th e standard 1 Nuttall Q -f unction [ 2, eq . (8 6)], given by the in tegral re presentation Q M ,N ( α, β ) = Z ∞ β x M e − x 2 + α 2 2 I N ( αx ) dx (1) where the or der indices are gen erally reals with values M ≥ 0 and N > − 1 , I N is the N th ord er m odified Bessel fu nction of the first kind [3, eq . (9 .6.3)] an d α, β are real pa rameters with α > 0 , β ≥ 0 . It is worth mentioning he re, that the negativ e values o f N , d efined ab ove, have n ot b een o f in terest in any practical applications so far . Howe ver , the extension of th e Nuttall Q -fun ction to negative values of N has been introdu ced here in order to facilitate more effectively the Manuscript recei v ed December 16, 2007; revise d January 10, 2009. This work was s upported in part by the Satellite Communications Network of Excelle nce (SatNex) project (IST -507052) and its Phase-II, SatNe x-II (IST - 27393), fund ed by the Europe an Commission (EC) un der its FP6 pr ogram. The materia l in this paper was presente d in part at the Internationa l Symposium on Wi reless Pervasi v e Computing 2008, Santorin i, Greece, May 2008. The authors are with the Electrical and Computer Engineering Department, Aristotle Univ ersity of T hessaloni ki, 5412 4 Thessaloniki, Greece (e-mail: kapinas@a uth.gr; sotmihos@gmail.com; geokarag@aut h.gr). † This work is dedicate d to the memory of Marvin K. Simon. 1 W e adopt the term “standard” for the Nuttall Q -function in order to av oid ambiguit y with its normaliz ed version to be introduced late r . relation o f this function to the mo re comm on generalized Marcum Q -fu nction, as will be shown in th e sequel. An alternative version of Q M ,N ( α, β ) is the n ormalized Nutta ll Q-function , Q M ,N ( α, β ) , which constitutes a norm alization of the form er with r espect to the par ameter α , defined simply b y the relatio n Q M ,N ( α, β ) , Q M ,N ( α, β ) α N . (2) T ypical application s inv olving the standard and normalized Nuttall Q -fun ctions inclu de: (a) the err or proba bility per- forman ce of non coheren t digital co mmunicatio n over Nak- agami fadin g chann els with interferenc e [ 4], (b) the outage probab ility o f wireless communica tion systems where the Nakagami/Rician faded desired sign als a re subject to indepen - dent and id entically distributed (i.i.d.) Rician/Nakag ami faded interferer s, respectively , under the assumptions of min imum interferen ce and signal power co nstraints [4]–[ 7], (c) the per- forman ce ana lysis and capacity statistics of uncod ed m ultiple- input m ultiple-ou tput (M IMO) systems ope rating over Rician fading channels [8]–[1 0], and (d) the extraction of the req uired log-likelihood r atio for th e de coding of differential phase- shift keying (DPSK) sig nals emp loying turb o or low-density parity - check (LDPC) cod es [11]. Since b oth typ es of th e Nu ttall Q -fu nction are not con sid- ered to be tabulated functio ns, their com putation in volved in the a foremen tioned app lications was han dled co nsidering the two distinct cases o f M + N being either odd or ev en, in order to e xpre ss th em in terms of more common functions. The possibility of doing such w hen M + N is odd was sug gested in [2], requ iring p articular combin ation of the two recursive relations [ 2, eq s. ( 87), (8 8)]. Howev er, the explicit solution was derived o nly in [4, eq. (13) ] e ntirely in terms of the Marcum Q -function a nd a finite weighted sum o f modified Bessel f unctions of the first k ind. Having all the above in mind, along with the fact that the calcu lation o f Q ( α, β ) itself requir es numer ical integration, th e issue of the efficient computatio n of (1) and (2) still rema ins open . B. The Generalized Ma r cum Q-Fu nction The gene ralized M arcum Q -fun ction [ 12] of p ositiv e re al order M , is defined b y the integral [1 3, eq. (1)] Q M ( α, β ) , 1 α M − 1 Z ∞ β x M e − x 2 + α 2 2 I M − 1 ( αx ) dx (3) 2 PUBLISHED IN IEEE TRANSACTIONS O N INFORMA TION THE OR Y , VOL. 55, NO. 8, AUGUST 2009 where α , β a re no n-negative real param eters 2 . For M = 1 , it reduce s to the popu lar standard (or fir st-order) Marcum Q - function , Q 1 ( α, β ) (or Q ( α, β ) ), while for gener al M it is related to the norm alized Nuttall Q -functio n acc ording to [1 4, eq. (4 .105) ] Q M ( α, β ) = Q M ,M − 1 ( α, β ) , α > 0 . (4) An id entical fun ction to the gen eralized Marcum Q is the probab ility of detection 3 [1, eq. (4 9)], which has a long history in ra dar comm unication s and particularly in the study o f target detection by pu lsed radar with single or multiple ob servations [1], [16]–[18]. Addition ally , Q M ( α, β ) is strongly associated with: (a) the error prob ability per formanc e of nonco herent and differentially coh erent modulatio ns over genera lized fadin g channels [14], [ 19]–[23], ( b) the sign al e nergy detection of a pr imary user over a m ultipath ch annel [24], [25], and fina lly (c) the inf ormation -theoretic stud y of MIMO sy stems [26]. Aside f rom th ese application s, the g eneralized Marcum Q - function pr esents a variety of interesting prob abilistic interpre- tations. Most ind icativ ely , for in teger M , it is the co mplemen - tary cumulative distribution fun ction (CCDF) o f a n oncentral chi-square ( χ 2 ) ran dom variable with 2 M degrees of freed om (DOF) [27, eq. (2 .45)]. This r elationship was extended in [28] to work for the c ase of odd DOF as well, throu gh a generalizatio n o f the non central χ 2 CCDF . Similar relatio ns can b e foun d in th e literatu re inv olving the g eneralized Rician [29, (2.1– 145)], the g eneralized Ray leigh [30, pp. 1] ( for α = 0 ) and the biv ariate Rayleigh [31, Append ix A], [32] (for M = 1 ) CCDF’ s. Fina lly , in a recent work [33], a new association has been der i ved b etween the generalized Mar cum Q -functio n an d a probab ilistic comp arison of two independ ent Poisson ran dom variables. More than th irty algorith ms have b een proposed in the literature for the nu merical compu tation of the stand ard an d generalized Marc um Q -fun ctions, amo ng them p ower series expansions [ 34]–[36], appro ximations an d asymptotic expres- sions [37]– [40], and N eumann series expansions [41]–[43]. Howe ver , the ab ove represen tations m ay not always pr ovide sufficient in formatio n about the relative po sition of the ap- proxim ated value with respect to the exact one, which in som e applications is high ly desired. In [44], the genera lized Marcum Q -functio n o f integer or der M has b een expre ssed as a single integral with fin ite limits, which is com putationally m ore desirable rela ti vely to other methods sug gested previously . Nev ertheless, the in tegral cann ot be comp uted analy tically and approp riate numerical in tegration technique s h av e to be applied, thereby intro ducing an approx imation error in its computatio n. In [45], a n exact representa tion for Q M ( α, β ) , when M is an odd m ultiple of 0 . 5 , was given as a finite sum of tabulated functio ns, assuming th at β 2 > α 2 + 2 M . This result was recently en hanced in [46] to a single expression that r emains accurate over all range s of the par ameters α, β , while in [47] the same expression was bo unded by particular 2 For α = 0 the right hand side of (3) can be easily shown to satisfy the limitin g value of [14, eq. (4. 71)], reproduc ed in (30). 3 For M incoherently integrat ed signals, the two funct ions are simply related by Q M ( α, β ) = P M ( α 2 2 M , β 2 2 ) , as induced by [15, eq. (7)]. utilization of novel Gaussian Q -fun ction inequa lities. Finally , in [48], a n equivalent expression to [46, eq. (1 1)] was derived, adopting a completely different (a nalytical) appro ach fro m the latter . Close insp ection of the issues men tioned above, render the existence of upp er and lower bo unds a matter o f essen tial importan ce in the compu tation o f (3). Sev eral types of bou nds for the standard [49]–[51] and generalized [37], [46], [48], [52]–[54] Marcum Q -function s have b een suggested so far . Howe ver , all the afor ementioned works consider just integer values of M , wh ich is generally tr ue wh en this p arameter represents the nu mber of indepen dent samples of a squ are- law detector ou tput. Nevertheless, in m any applicatio ns, this requirem ent does n ot ho ld. According to [14, Sec. 4.4 .2], it would be desirable to o btain alter nativ e represen tations for Q M ( α, β ) r egardless of whether M is in teger or no t. For instance, in [20], the fadin g para meter of the Nak agami- m distribution is r estricted to integer values in the lack o f a clo sed-form expression for the g eneralized Mar cum Q - function o f real ord er . Addition ally , in [2 5], [55]–[ 57], the order M of the generalized Marcu m Q -fun ction, inv olved in the en ergy d etection in various r adiometer a nd cognitive radio applications, is expressed as the produ ct of the integration time and the re ceiv er b andwidth , thus implying that in ge neral M is a non-in teger quan tity . Furth ermore, a probabilistic interpretatio n o f Q M − µ ( α, β ) , whe re M ∈ N 4 and µ = 0 . 5 , is g iv en in [58], whe re it is related to main probab ilistic characteristics of 2( M − µ ) r andom variables, while in [59]– [61], non central χ 2 random variables with fractio nal DOF are studied. Finally , in [ 62], the integran d of ( 3) has be en proved to be a p robab ility density fu nction (PDF) fo r α ≥ 0 and M > 0 , a resu lt that also has been u tilized in [48]. C. Con trib ution As describ ed in Sub section I -A, a closed- form exp ression for the compu tation o f the standard and normalized Nuttall Q -functio ns is av ailable in the literature on ly for th e case of odd M + N , with th e add itional restrictio n of in tegers M , N . In Su bsection I I-A, we derive a novel clo sed-form expression for the computation o f Q M ,N ( α, β ) and Q M ,N ( α, β ) when M , N are odd mu ltiples o f 0 . 5 and M ≥ N , being valid for all ra nges of the p arameters α, β . Besides, in Subsection II-B, we pr oceed w ith th e estab- lishment of appr opriate m onoton icity criteria, revealing the behavior of both func tions with th e sum M + N . Specifically , we demon strate that the standard Nuttall Q -func tion is strictly increasing with respect to M + N wh en M ≥ N + 1 , under the constraints of α ≥ 1 and β > 0 . For the norm alized Nuttall Q - function , a similar mon otonicity statement is proved w ithout the nece ssity of reduc ing th e ra nge of α . An alternative app roach, sufficient enou gh to facilitate th e problem of e valuating the Nuttall Q -functions, is the derivation of tight bou nds. Nevertheless, to th e best of the autho rs’ 4 Throughout the manuscript, we adopt N and N 0 notati ons for the represent ation of the positi ve and the non-negat iv e integer set, respecti v ely . In the same way , R + include s the positiv e and R + 0 the non-ne gati ve reals. KAPIN AS et al. : ON THE MONOT ONICITY OF THE GE NERALIZE D MARC UM AND NUTT ALL Q -FUNCTIONS 3 knowledge, su ch bou nds have not been reported in the litera- ture so far . Subsection II-B is completed with th e exploitation of the previous results in order to d erive n ovel upper and lo wer bound s fo r Q M ,N ( α, β ) and Q M ,N ( α, β ) when M ≥ N + 1 and β > 0 , with the extra requir ement of α ≥ 1 for the forme r . Additionally , in Su bsection I-B, the need for comp uting the generalized Marcu m Q -fun ction, Q M ( α, β ) , of re al or der M was hig hlighted, since it is a case of freq uent oc currence in various application s. Howe ver , a th oroug h liter ature search for studies co ncerning arbitrary values of M , revealed on ly [46] fo r the closed-f orm compu tation o f Q M ( α, β ) of half- odd integer or der, and the accepted pape r [63], wh ere bo unds for Q M ( α, β ) wer e introdu ced for the case when M is no t necessarily an integer . These con siderations mo ti vated u s to generalize the scop e of M in [46, eq. (16)] Q M − 0 . 5 ( α, β ) < Q M ( α, β ) < Q M +0 . 5 ( α, β ) , M ∈ N (5) as described in Sectio n I II, by providing a monoton icity formalizatio n for the gen eralized Mar cum Q -functio n, name ly that Q M ( α, β ) is strictly increasing with respect to its ord er M > 0 f or α ≥ 0 an d β > 0 . This interesting statem ent was also recently p resented in [62], usin g a different appr oach. As a consequ ence, novel upp er and lower b ound s for Q M ( α, β ) of po siti ve real or der are derived. W e finalize the paper with some con cluding remar ks, given in Section IV. I I . M O N OT O N I C I T Y O F T H E N U T TA L L Q - F U N C T I O N S A. Novel Closed-F orm Repr esentation s So far , clo sed-form expression for either typ e of the Nuttall Q -functio n is not available in the literatu re. In this section, we d eriv e such a representatio n for the case when M , N are odd m ultiples of 0 . 5 and M ≥ N , throu gh th e theo rem and corollary established below . Before proceed ing further with the correspo nding proof s, som e de finitions o f essential fu nctions and n otations used, would be very co n venient. Hereafter, Γ , γ and Γ ( · , · ) will denote the Euler gam ma [3, eq . (6 .1.1)] , the lower inco mplete gam ma [ 3, eq. (6 .5.2)] and the upper inco mplete gamm a [3, eq. ( 6.5.3) ] function s, respectively , defin ed by the in tegrals Γ( z ) = Z ∞ 0 t z − 1 e − t dt, γ ( z , x ) = Z x 0 t z − 1 e − t dt Γ ( z , x ) = Γ( z ) − γ ( z , x ) , z ∈ R + , x ∈ R . Notations n ! , ( m ) n and  m n  imply the factorial [3, eq. (6.1.6)] , the rising factorial ( Pochhamm er’ s symbo l) [3, eq. (6 .1.22) ] and the bin omial co efficient [3, eq. ( 24.1.1 C)] , respectively , defined by n ! = Q n k =1 k fo r n ∈ N ; = 1 fo r n = 0 , ( m ) n = ( m + n − 1)! ( m − 1)! for m ∈ N , n ∈ N 0 and  m n  = m ! n !( m − n )! for m, n ∈ N 0 , m ≥ n . Finally , sg n( z ) = z / | z | f or z 6 = 0 ; = 0 for z = 0 , stands for the sign um fu nction. Theor em 1 (Closed -form for th e stan dar d Nuttall Q): The standard Nuttall Q -func tion, Q M ,N ( α, β ) , when m = M + 0 . 5 ∈ N , n = N + 0 . 5 ∈ N and M ≥ N , can be ev aluated for α > 0 , β ≥ 0 by the following closed-f orm expression: Q M ,N ( α, β ) = ( − 1) n (2 α ) − n + 1 2 √ π × n − 1 X k =0 ( n − k ) n − 1 (2 α ) k k ! I k m,n ( α, β ) where th e term I k m,n ( α, β ) is given by I k m,n ( α, β ) = ( − 1 ) k +1 m − n + k X l =0  m − n + k l  2 l − 1 2 α m − n + k − l ×  ( − 1) m − n − l − 1 Γ  l + 1 2 , ( β + α ) 2 2  − (sgn( β − α )) l +1 γ  l + 1 2 , ( β − α ) 2 2  +Γ  l + 1 2  . (6) Pr oof: Given th at n = N + 0 . 5 ∈ N , the mo dified Bessel function of the first kin d, I N , can be expr essed by the finite sum [6 4, eq. (8 .467) ], which after some manip ulations can b e written as I N ( z ) = ( − 1) n (2 z ) − n + 1 2 √ π e z n − 1 X k =0 ( n − k ) n − 1 (2 z ) k k ! ×  1 − ( − 1 ) k e 2 z  , n = N + 1 2 ∈ N , z ∈ R . (7) Therefo re, using ( 1) and ( 7), the stand ard Nuttall Q -fu nction satisfies Q M ,N ( α, β ) = ( − 1) n (2 α ) − n + 1 2 √ π n − 1 X k =0 ( n − k ) n − 1 (2 α ) k k ! ×  Z ∞ β x m − n + k e − ( x + α ) 2 2 dx − ( − 1) k Z ∞ β x m − n + k e − ( x − α ) 2 2 dx  . (8) The calcu lation o f th e in tegral difference in (8) can be effec- ti vely facilitated b y the following definition I k L ( α, β ) = Z ∞ β x L e − ( x + α ) 2 2 dx − ( − 1) k Z ∞ β x L e − ( x − α ) 2 2 dx (9) where L = m − n + k . Since we examin e the case when M ≥ N or eq uiv alently m ≥ n , it follows that in the above expression the e xponen t L is a non -negative integer . Therefore, using [ 65, eq. (1.3.3 .18)], (9) o btains the form I k L ( α, β ) = L X l =0  L l  α L − l  ( − 1) L − l Z ∞ β + α x l e − x 2 2 dx − ( − 1) k Z ∞ β − α x l e − x 2 2 dx  . (10) The two integrals in volved in (10) can be co nsidered as special cases of the mor e g eneral o ne I l b = Z ∞ b x l e − x 2 2 dx, b ∈ R 4 PUBLISHED IN IEEE TRANSACTIONS O N INFORMA TION THE OR Y , VOL. 55, NO. 8, AUGUST 2009 which fo r the case of n on-negative values of b can b e calcu- lated fro m [64, eq. (3.3 81.3)] as I l b = 2 l − 1 2 Γ  l + 1 2 , b 2 2  , b ≥ 0 (11) while for negative values o f b , [ 64, eq s. (3.38 1.1), (3 .381.4 )] can be co mbined to yield I l b = 2 l − 1 2  Γ  l + 1 2  + ( − 1) l γ  l + 1 2 , b 2 2  , b < 0 . (12) Therefo re, a single expre ssion for the integral I l b for any re al value of b can b e derived, by merging (11) an d (12) with the help o f [64, eq. (8.35 6.3)], in ord er to satisfy I l b = 2 l − 1 2  Γ  l + 1 2  − [sgn( b )] l +1 γ  l + 1 2 , b 2 2  . Thus, ( 9) is equivalent to I k L ( α, β ) = L X l =0  L l  α L − l  ( − 1) L − l I l β + α − ( − 1) k I l β − α  = ( − 1) k +1 L X l =0  L l  2 l − 1 2 α L − l  Γ  l + 1 2  + ( − 1) L − l − k − 1 Γ  l + 1 2 , ( β + α ) 2 2  − [sgn( β − α )] l +1 γ  l + 1 2 , ( β − α ) 2 2  which, a fter the substitutio n L = m − n + k , yields (6), thu s completing th e pro of. Cor o llary 1 (Closed- form for the normalized Nuttall Q): The norm alized Nuttall Q -function , Q M ,N ( α, β ) , whe n m = M + 0 . 5 ∈ N , n = N + 0 . 5 ∈ N and M ≥ N , can be ev aluated for α > 0 , β ≥ 0 by the following closed-f orm expression: Q M ,N ( α, β ) = ( − 1) n 2 − n + 1 2 √ π α 2 n − 1 × n − 1 X k =0 ( n − k ) n − 1 (2 α ) k k ! I k m,n ( α, β ) where th e term I k m,n ( α, β ) is given by (6). Pr oof: The proof follo ws immediately from ( 2) and Theorem 1. B. Lower an d Upp er Boun ds In this section, n ovel lower and up per b ounds for the normalized and stand ard Nuttall Q -fun ctions ar e p roposed. Lemma 1: Th e fun ction G s ( r , x ) , defined by G s ( r , x ) , Γ( r + s, x ) Γ( r ) , r , x ∈ R + (13) is strictly incr easing with respect to r fo r all s ∈ R + 0 . Pr oof: By multiplyin g both the nu merator a nd d enomin a- tor of (13) by the upp er inc omplete gamma function , Γ( r , x ) , we o btain G s ( r , x ) = Γ( r + s, x ) Γ( r , x ) G 0 ( r , x ) where fro m (13) o ne can observe that the term G 0 ( r , x ) is the compleme nt of the regular ized lower inco mplete gam ma fu nc- tion P ( r , x ) with respect to unity , defined in [3, eq . (6 .5.1)] by P ( r , x ) = γ ( r,x ) Γ( r ) for all r > 0 and x ∈ R . Fortunately , P ( r, x ) for r , x > 0 is equ al to the cum ulativ e distribution functio n (CDF) of the standard gam ma distrib ution Gamma ( r , 1) , which is strictly decr easing with respect to the shape para meter r . Additionally , th is importan t result has also been proved ana- lytically in [ 66, eq. (59) ], thus implying that G 0 ( r , x ) is strictly increasing with respect to r > 0 for all x > 0 . Fur thermore , in [67], it has b een de monstrated tha t the functio n R ( p, q, x ) =  Γ( p, x ) Γ( q , x )  1 p − q , p > q > 0 , x > 0 (14) is increasing with respect to q . By substituting p = r + s and q = r into (14), we r ealize that the r atio Γ( r + s, x ) / Γ( r , x ) is inc reasing with respe ct to r for s > 0 , while it remain s constant f or the trivial case o f s = 0 . Theref ore, it increa ses with r > 0 f or all x > 0 , s ≥ 0 , an d the pro of is complete. The o utcome of Lemma 1 will be u tilized fo r the estab- lishment o f the next theo rem, conc erning the mo notonicity proper ty of the nor malized Nu ttall Q -f unction. Theor em 2 (Monoto nicity of the no rmalized Nuttall Q): The normalized Nuttall Q -function , Q M ,N ( α, β ) , wher e M > 0 , N > − 1 an d α, β > 0 , is strictly increasing with respect to the sum M + N , u nder the requirem ent o f constant difference M − N ≥ 1 . Pr oof: Co mbining (1), ( 2) and using th e series rep resen- tation of the mod ified Bessel functio n of the first kind in ter ms of the g amma functio n [6 4, eq. (8.4 45)], we ob tain Q M ,N ( α, β ) = e − α 2 2 ∞ X k =0 α 2 k k !Γ( k + N + 1)2 2 k + N × Z ∞ β x 2 k + M + N e − x 2 2 dx (15) where we have interc hanged the ord er of integratio n an d sum- mation, since all in tegrand quantities of the norm alized Nuttall Q -functio n are Riemann integrable on [ β , ∞ ) . Additionally , the integral in (15) is the case of ( 11), thus yielding Z ∞ β x 2 k + M + N e − x 2 2 dx = 2 k + M + N − 1 2 × Γ  k + M + N + 1 2 , β 2 2  . Therefo re, (1 5) reads Q M ,N ( α, β ) = e − α 2 2 ∞ X k =0 α 2 k 2 k + N − M +1 2 k ! Γ  k + M + N +1 2 , β 2 2  Γ( k + N + 1) . (16) Introd ucing the variables v = M + N and c = M − N an d taking the p artial deriv ativ e of both sides of (16) with respect to v , we can easily obtain ∂ ∂ v Q v + c 2 , v − c 2 ( α, β ) = e − α 2 2 ∞ X k =0 α 2 k 2 k + 3 − c 2 k ! × ∂ ∂ u ( v ) G c − 1 2  u ( v ) , β 2 2  (17) KAPIN AS et al. : ON THE MONOT ONICITY OF THE GE NERALIZE D MARC UM AND NUTT ALL Q -FUNCTIONS 5 where the function u ( v ) = k + 1 + v − c 2 has b een emp loyed for no tational convenience. W e no te h ere th at, app lying the W eierstrass M-test [6 8], the series in (17) can be p roved to conv erge unifo rmly , th us enabling o ne to inter change the ord er of differentiation a nd summation . Hence, recalling Lemma 1 and the r equiremen t of α > 0 , that follows from the d efinition of the n ormalized Nuttall Q - function , th e pr oof is com plete. In Figs. 1(a) and 1(b), the normalize d Nuttall Q -fu nction has been plotted versus the sum M + N for several values of α, β , consid ering M − N = 1 a nd M − N = 2 , respectively . Howe ver , we n ote h ere that Theorem 2 implies no n-integer differences M − N as well. For the interpr etation o f the next pro position we defin e th e pair o f half-integer r ounding o perators ⌊ x ⌋ 0 . 5 and ⌈ x ⌉ 0 . 5 that map a real x to its ne arest left an d right h alf-odd integer , respectively , acco rding to the relations 5 ⌊ x ⌋ 0 . 5 = ⌊ x − 0 . 5 ⌋ + 0 . 5 ⌈ x ⌉ 0 . 5 = ⌈ x + 0 . 5 ⌉ − 0 . 5 (18) where ⌊ x ⌋ and ⌈ x ⌉ den ote the integer floor and ceiling function s. Ad ditionally , we recall that if δ x ∈ [0 , 1) is the fractional pa rt of x , then ⌊ x ⌋ = x − δ x . Cor o llary 2 (Bo unds on the normalized Nuttall Q): Th e following ineq ualities can ser ve as lo wer and upp er bo unds on the norm alized Nu ttall Q -fu nction, Q M ,N ( α, β ) , where α, β > 0 and M , N > 0 . 5 , f or the case when M ≥ N + 1 and δ M = δ N (i.e. M − N ∈ N ): Q M ,N ( α, β ) ≥ Q ⌊ M ⌋ 0 . 5 , ⌊ N ⌋ 0 . 5 ( α, β ) Q M ,N ( α, β ) ≤ Q ⌈ M ⌉ 0 . 5 , ⌈ N ⌉ 0 . 5 ( α, β ) . (19) with th e equalities above being valid o nly for th e case of h alf- odd integer values o f M , N . Pr oof: The pro of follows im mediately fro m Theo rem 2. For the calculation o f th e boun ds in ( 19), the qu antities Q ⌊ M ⌋ 0 . 5 , ⌊ N ⌋ 0 . 5 ( α, β ) and Q ⌈ M ⌉ 0 . 5 , ⌈ N ⌉ 0 . 5 ( α, β ) can be ev alu- ated exactly by utilizing the results of Corollary 1. Moreover, for the c ase of M , N ∈ N , the prop osed bo unds obtain th e simplified fo rm Q M ,N ( α, β ) > Q M − 0 . 5 ,N − 0 . 5 ( α, β ) Q M ,N ( α, β ) < Q M +0 . 5 ,N +0 . 5 ( α, β ) . (20) In Figs. 2(a) and 2(b ), th e normalized Nuttall Q - function along with its lo wer and upper bounds are depicted versus β for several values of α and M , resp ectiv ely , wh ile the parameter N is restricted accord ing to the relation N = M − c with c ∈ N taking values c = 2 in Fig. 2 (a) an d c = 1 , 2 , 3 in Fig. 2(b ). I t is evident, th at the bound s prop osed in ( 19) are very tight, especially the upper o ne fo r δ M (= δ N ) < 0 . 5 and the lower one for δ M (= δ N ) > 0 . 5 , the latter be ing the case illustrated in Fig. 2(b). In order to ob tain lower and uppe r bo unds for th e stan dard Nuttall Q -f unction, a similar proce dure can be carr ied out. The 5 The defining equatio ns of (18 ) can be easily verified to be val id in any arbitra ry segment [ n, n + 1) , where n ∈ Z . P S f r a g r e p l a c e m e n t s M + N Q M ,N ( α, β ) M + N Q M , N ( α , β ) α = 7 . 5 , β = 6 . 5 α = 5 . 5 , β = 5 . 5 α = 0 . 5 , β = 3 . 5 α = 3 . 5 , β = 1 . 5 α = 2 . 0 , β = 2 . 0 α = 1 . 5 , β = 3 . 5 4 6 8 10 12 14 16 18 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 4 6 8 1 0 1 2 1 4 1 6 1 8 0 1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 (a) Q M ,N ( α, β ) versus M + N for M − N = 1 . P S f r a g r e p l a c e m e n t s M + N Q M , N ( α , β ) M + N Q M ,N ( α, β ) α = 7 . 5 , β = 6 . 5 α = 5 . 5 , β = 5 . 5 α = 0 . 5 , β = 3 . 5 α = 3 . 5 , β = 1 . 5 α = 2 . 0 , β = 2 . 0 α = 1 . 5 , β = 3 . 5 4 6 8 1 0 1 2 1 4 1 6 1 8 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 4 6 8 10 12 14 16 18 0 1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 (b) Q M ,N ( α, β ) versu s M + N for M − N = 2 . Fig. 1. Monotoni city of Q M ,N ( α, β ) with respect to the sum M + N for se veral real val ues of α, β . next theor em will be pa rticularly usefu l fo r the fulfillme nt o f such a der i vation. Theor em 3 (Monoto nicity of the stan dar d Nuttall Q): The standard Nu ttall Q -function , Q M ,N ( α, β ) , where M > 0 , N > − 1 and α ≥ 1 , β > 0 , is strictly increasing with respect to the sum M + N , u nder the requirem ent o f constant difference M − N ≥ 1 . Pr oof: In Theorem 2, it has been proved th at ∂ ∂ v Q v + c 2 , v − c 2 ( α, β ) > 0 (21) where we have sub stituted v = M + N and c = M − N . From (2), (21) and after using the quotient r ule for partial differentiation, we obtain ∂ ∂ v Q v + c 2 , v − c 2 ( α, β ) > ln α 2 Q v + c 2 , v − c 2 ( α, β ) . 6 PUBLISHED IN IEEE TRANSACTIONS O N INFORMA TION THE OR Y , VOL. 55, NO. 8, AUGUST 2009 P S f r a g r e p l a c e m e n t s β Q 5 , 3 ( α, β ) β Q M , 2 . 7 ( 3 . 5 , β ) M = 5 . 0 , N = 3 . 0 M = 4 . 5 , N = 2 . 5 M = 5 . 5 , N = 3 . 5 α = 0 . 5 α = 4 α = 6 . 5 M = 3 . 7 M = 4 . 7 M = 5 . 7 0 2 4 6 8 10 12 0 1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 7 . 0 8 . 0 3 4 5 6 7 8 1 0 − 2 1 0 − 1 1 0 0 1 0 1 1 0 2 (a) Q 5 , 3 ( α, β ) versu s β for se veral v alues of α . P S f r a g r e p l a c e m e n t s β Q 5 , 3 ( α , β ) β Q M , 2 . 7 (3 . 5 , β ) M = 5 . 0 , N = 3 . 0 M = 4 . 5 , N = 2 . 5 M = 5 . 5 , N = 3 . 5 α = 0 . 5 α = 4 α = 6 . 5 M = 3 . 7 M = 4 . 7 M = 5 . 7 0 2 4 6 8 1 0 1 2 0 1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 7 . 0 8 . 0 3 4 5 6 7 8 10 − 2 10 − 1 10 0 10 1 10 2 (b) Q M , 2 . 7 (3 . 5 , β ) versus β for seve ral values of M . Fig. 2. Bounds of Q M ,N ( α, β ) for M − N ∈ N and sev eral valu es of α, β . Since th e Nuttall Q - function is strictly p ositiv e, then for α ≥ 1 it fo llows that ∂ ∂ v Q v + c 2 , v − c 2 ( α, β ) > 0 and th e proo f is complete. Cor o llary 3 (Bo unds on the standard Nuttall Q): The fol- lowing in equalities can ser ve as lower and uppe r boun ds o n the standard Nuttall Q - function , Q M ,N ( α, β ) , wher e α ≥ 1 , β > 0 and M , N > 0 . 5 , for the case when M ≥ N + 1 and δ M = δ N (i.e. M − N ∈ N ): Q M ,N ( α, β ) ≥ Q ⌊ M ⌋ 0 . 5 , ⌊ N ⌋ 0 . 5 ( α, β ) Q M ,N ( α, β ) ≤ Q ⌈ M ⌉ 0 . 5 , ⌈ N ⌉ 0 . 5 ( α, β ) . (22) with th e equalities above being valid o nly for th e case of h alf- odd integer values o f M , N . Pr oof: The p roof follows im mediately fro m Theo rem 3. Similarly to th e ca se of the norm alized Nuttall Q , in the calculation of the bounds from (22), the q uantities Q ⌊ M ⌋ 0 . 5 , ⌊ N ⌋ 0 . 5 ( α, β ) and Q ⌈ M ⌉ 0 . 5 , ⌈ N ⌉ 0 . 5 ( α, β ) ca n be ev alu- ated exactly fr om Theorem 1. Finally , for M , N ∈ N , th e standard Nuttall Q -fu nction can b e simp ly bou nded by Q M ,N ( α, β ) > Q M − 0 . 5 ,N − 0 . 5 ( α, β ) Q M ,N ( α, β ) < Q M +0 . 5 ,N +0 . 5 ( α, β ) (23) which constitutes th e cou nterpart of (20) for the standard Nuttall Q -fu nction. I I I . M O N O T O N I C I T Y A N D B O U N D S F O R T H E G E N E R A L I Z E D M A R C U M Q - F U N C T I O N Recently , Li and Kam in [4 6, eq. (11 )], following a geo - metric appr oach, presented a novel closed-for m formu la for the evaluation of Q M ( α, β ) , fo r th e case w hen M is an od d multiple o f 0 . 5 and α > 0 , β ≥ 0 , given b y Q M ( α, β ) = 1 2 erfc  β + α √ 2  + 1 2 erfc  β − α √ 2  + 1 α √ 2 π M − 1 . 5 X k =0 β 2 k 2 k k X q =0 ( − 1) q (2 q )! ( k − q )! q ! × 2 q X i =0 1 ( αβ ) 2 q − i i !  ( − 1) i e − ( β − α ) 2 2 − e − ( β + α ) 2 2  (24) where er fc ( z ) = (2 / √ π ) R ∞ z e − t 2 dt is the compleme ntary error fun ction [3, eq. (7.1.2 )]. Th is representatio n in volves only e lementary function s and is convenient for evaluation both numer ically and analytically . For the trivial case when α = 0 , e xact values of the ge neralized Mar cum Q -functio n can be obtaine d from [46, eq. ( 12)] Q M (0 , β ) = erfc  β √ 2  + e − β 2 2 √ 2 π M − 1 . 5 X k =0 β 2 k +1 2 k − 1 × k X q =0 ( − 1) q ( k − q )! q !(2 q + 1) . (25) Follo wing an algebraic app roach, an alternative more co mpact closed-for m expression, equivalent to ( 24), can be derived, considerin g the next steps. Particularly , in [58, eq. (10) ] it has been proved that the gene ralized Marc um Q -fu nction o f order m − µ , with m positive integer and 0 ≤ µ < 1 can be written in terms of the gen eralized Marcu m Q -functio n of order 1 − µ a s Q m − µ ( α, β ) = e − α 2 + β 2 2 m − 1 X n =1  β α  n − µ I n − µ ( αβ ) + Q 1 − µ ( α, β ) , α 6 = 0 . By substituting µ = 0 . 5 in the ab ove eq uation and no ting that for this case the m odified Bessel fu nction of the first k ind c an KAPIN AS et al. : ON THE MONOT ONICITY OF THE GE NERALIZE D MARC UM AND NUTT ALL Q -FUNCTIONS 7 be rep laced by (7), w e o btain Q m − 0 . 5 ( α, β ) = α r 2 π e − ( α + β ) 2 2 m − 1 X n =1 ( − 2 α 2 ) − n × n − 1 X k =0 ( n − k ) n − 1 k ! (2 αβ ) k  1 − ( − 1 ) k e 2 αβ  + Q 0 . 5 ( α, β ) , m ∈ N (26) where once again ( m ) n denotes the Poch hammer’ s symbo l and the term Q 0 . 5 ( α, β ) can be deriv ed fro m the definition of the gener alized Marcum Q -function in (3), by usin g [3, eq. (1 0.2.14 )] as follows Q 0 . 5 ( α, β ) = r 2 π Z ∞ β e − x 2 + α 2 2 cosh ( ax ) dx. The ab ove integral can be compu ted in closed -form as Q 0 . 5 ( α, β ) = 1 2 erfc  β + α √ 2  + 1 2 erfc  β − α √ 2  = Q ( β + α ) + Q ( β − α ) (27) where Q d enotes the Gaussian Q -fun ction (o r Gaussian probab ility integral) [3, eq . (26. 2.3)], defined by Q ( z ) = (1 / √ 2 π ) R ∞ z e − t 2 / 2 dt . Using (26) and (2 7), the g eneralized Marcum Q -fu nction of half-odd integer order can be com puted for all α > 0 , β ≥ 0 fro m the expression Q M ( α, β ) = α r 2 π e − ( α + β ) 2 2 M − 0 . 5 X n =1 ( − 2 α 2 ) − n × n − 1 X k =0 ( n − k ) n − 1 k ! (2 αβ ) k  1 − ( − 1 ) k e 2 αβ  + Q ( β + α ) + Q ( β − α ) , M + 0 . 5 ∈ N . (28) W e note her e that a similar result to (28) has been r ecently reported in the literature [48, eq. ( 16)]. In o rder to examine the special case when α = 0 , we first notice that f rom (4) a nd (16) an alternative expression—eq uiv alent to [3, eq. (26.4. 25)]—fo r the g eneralized Marcum Q - function can be derived, written as Q M ( α, β ) = e − α 2 2 ∞ X k =0 α 2 k 2 k k ! Γ  k + M , β 2 2  Γ( k + M ) , α > 0 , β ≥ 0 (29) which for integer M falls in to the series e xpan sion [35, eq. (4)] . Since Q M ( α, β ) is a continuo us fu nction of α for all β ≥ 0 and M > 0 , the a bove equation can be extend ed to be asympto tically valid for th e case when α = 0 a s well, with the correspo nding limitin g value giv en b y Q M (0 , β ) = Γ  M , β 2 2  Γ( M ) . ( 30) This last r esult also appears in [14, eq. (4.71 )], wh ere Q M (0 , β ) h as b een der i ved directly from (3) by apply ing the small argu ment f orm o f the modified Bessel fun ction. It has been proved in [46] th at (24), ( 25) alon g with (5) can define tight upper and lower boun ds f or the gener alized Marcum Q -function o f integer or der . It seem s apparent, tha t in or der to derive bo unds for Q M ( α, β ) of re al ord er M , a strict inequality , in volving the who le range o f M , has to b e established. Su ch a gener alization co ncept can be f ormalized throug h the fo llowing theorem . Theor em 4 (Monoto nicity of the generalized Mar cum Q): The generalized Ma rcum Q -fun ction, Q M ( α, β ) , is strictly increasing with respec t to its real or der M > 0 for a ll α ≥ 0 , β > 0 . Pr oof: Con cerning the case when α = 0 , we notice that (30) can be rewritten as Q M (0 , β ) = 1 − P  M , β 2 2  . Howe ver , in [66, eq. ( 59)] the regular ized lower incom plete gamma function P ( r, x ) has been pr oved to decr ease mo no- tonically with respect to r > 0 for all x > 0 . Addition ally , for α > 0 , (4) imp lies that the no rmalized Nuttall Q - function with N = M − 1 falls into the g eneralized Marcu m Q - function of o rder M . Nevertheless, accord ing to Theorem 2, Q M ,M − 1 ( α, β ) is strictly increa sing with respect to 2 M − 1 for M > 0 , an d the pro of is c omplete. The resu lt of Th eorem 4 has also recently demonstra ted by Sun and Baricz in [ 62], where two to tally different proof s were given. The first one com bines the ser ies for m of the gen - eralized Mar cum Q -fun ction pr esented in (29), (30) togeth er with the fact that the regula rized u pper incomplete gamm a function Q ( r , x ) = 1 − P ( r, x ) is strictly increasing w ith respect to r > 0 fo r each x > 0 , orig inally stated by T ricomi in [66]. A slightly different analytical proof to this can also be found in [60, Th. 1]. The seco nd proof exploits th e intere sting relationship b etween the genera lized Mar cum Q -f unction and the reliab ility fu nction (or CCDF) R o f a χ 2 random variable with 2 M DOF and no ncentrality parameter α , namely th e fact that if β ∼ χ 2 2 M ,α then R ( β ) = Q M ( √ α, √ β ) . The interested reader is ref erred to [62, Th. 3 .1] for more inform ation. Recalling the relation between the n ormalized Nuttall and the g eneralized Mar cum Q -fun ctions, that is (4), Fig. 1(a) verifies graph ically the results of The orem 4, since it actually depicts Q M ( α, β ) versus the te rm 2 M − 1 . Cor o llary 4 (Bo unds on the generalized Ma r cum Q): The following inequalities can serve as lower and uppe r bo unds on the gen eralized M arcum Q -fu nction Q M ( α, β ) of real order M > 0 . 5 fo r all α ≥ 0 , β > 0 . Q ⌊ M ⌋ 0 . 5 ( α, β ) ≤ Q M ( α, β ) ≤ Q ⌈ M ⌉ 0 . 5 ( α, β ) . (31) with th e equalities above being valid only for th e case of h alf- odd integer values o f M . Pr oof: The p roof follows im mediately fro m Theo rem 4. In Corollary 4, the quan tities Q ⌊ M ⌋ 0 . 5 ( α, β ) and Q ⌈ M ⌉ 0 . 5 ( α, β ) can be evaluated exactly either from (24), (2 5) or (27), (30), while for M ∈ N (31) red uces to Q M − 0 . 5 ( α, β ) < Q M ( α, β ) < Q M +0 . 5 ( α, β ) which com es as a compleme nt to the inequa lities o f (20) and (23). This la st r esult was originally demo nstrated in [46, eq. (1 6)], where the autho rs fo llowing a geometr ic app roach 8 PUBLISHED IN IEEE TRANSACTIONS O N INFORMA TION THE OR Y , VOL. 55, NO. 8, AUGUST 2009 P S f r a g r e p l a c e m e n t s β Q 4 ( α, β ) β Q M ( 2 . 5 , β ) M = 4 . 0 M = 3 . 5 M = 4 . 5 α = 0 . 5 α = 3 . 5 α = 5 . 5 M = 2 . 7 M = 8 . 3 1 2 3 4 5 6 7 8 9 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 1 2 3 4 5 6 7 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 (a) Q 4 ( α, β ) versu s β for se veral val ues of α . P S f r a g r e p l a c e m e n t s β Q 4 ( α , β ) β Q M (2 . 5 , β ) M = 4 . 0 M = 3 . 5 M = 4 . 5 α = 0 . 5 α = 3 . 5 α = 5 . 5 M = 2 . 7 M = 8 . 3 1 2 3 4 5 6 7 8 9 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 1 2 3 4 5 6 7 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 (b) Q M (2 . 5 , β ) versus β for seve ral va lues of M . Fig. 3. Bounds of Q M ( α, β ) for sev eral real value s of α, β and M . (In (a) bounds proposed by Li and Kam, [46]). propo sed tigh t lo wer and up per bound s fo r the generalized Marcum Q -functio n o f in teger order M , whic h h av e been proved to o utperfo rm oth er existing ones. This can be easily verified from Fig. 3(a), where Q 4 ( α, β ) has been plotted versus β for several values of α . Ther efore, for the ca se of rea l M , one can expect even further en hancemen t in the strictness of e ither the lower boun d ( for δ M > 0 . 5 ) or the upper one (for δ M < 0 . 5 ) . This is clearly dep icted in Fig. 3(b ), where the curves Q 2 . 5 (2 . 5 , β ) and Q 8 . 5 (2 . 5 , β ) constitute very tig ht lower and u pper boun ds of Q 2 . 7 (2 . 5 , β ) and Q 8 . 3 (2 . 5 , β ) , respectively , fo r all range of β . I V . C O N C L U S I O N Applicable mono tonicity criteria wer e established for the normalized and standard Nuttall and the g eneralized Mar cum Q -functio ns. Specifically , it was proved that the two Nuttall Q - function s are strictly in creasing with respect to the real sum M + N for the case when M ≥ N + 1 , while the gener alized Marcum Q - function incr eases mon otonically with respect to its real order M . Ad ditionally , novel closed -form expressions for both types of the Nuttall Q -fun ction were given for the case when M , N ar e odd mu ltiples o f 0 . 5 and M ≥ N . Regarding the ge neralized Mar cum Q - function of half-o dd integer or der , an altern ativ e mo re comp act closed- form expression, eq uiv- alent to the alr eady existing one, was derived. By exploiting these re sults, novel lower and upper bou nds were p roposed for the Nuttall Q -fu nctions when M ≥ N + 1 , w hile the recen tly propo sed b ounds fo r the gen eralized Marcu m Q -fun ction of integer M , were approp riately utilized in ord er to extend their validity over real values of M . R E F E R E N C E S [1] J. I. 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He recei ved the diploma degree in electric al and computer engine ering from Aristotle Univ ersity of Thessaloniki, Greece, in 2000. Since 2005, he has been working tow ard the Ph.D. degre e in telecommunic ations engineering. His current research interests include wireless communicati on theory and digita l communicatio ns ov er fading channe ls, givin g special focus to space- time block coding techniques. Sotirios K . Mihos was born in Thessaloniki, Greece, in April 1984. He is an undergraduat e student at the Aristotle Univ ersity of Thessaloniki, Greece, where he is working to ward the diploma degree in electric al and computer enginee ring. His research interests span a wide range of subject areas includi ng computer science, electronic s and automatic control, with a special focus on their relatio nship to pure mathematic s. George K. Karagiannidis (M’97–SM’04) was born in Pithagorio n, Samos Island, Greece. He recei ved the Univ ersity and Ph.D. degree s in electric al enginee ring from the Univ ersity of Patras, Patras, G reece , in 1987 and 1999, respect iv ely . From 2000 to 2004, he was a Senior Researcher at the Institute for Space Applicati ons and Remote Sensing, National Observ atory of Athens, Greece. In June 2004, he joined Aristotle Uni versit y of Thessaloniki, Thessa- loniki, Greece, where he is currently an Assistant Professor in the Electrica l and Computer Engineering Department . His current research interests include wireless communication theory , digital communicati ons ove r fadi ng channels, coopera ti ve di versity systems, cognit iv e radio, satellite communications, and wireless optical communications. He is the author or coauthor of more than 80 technical papers publishe d in scientific journals and presented at internation al conference s. He is also a coauthor of two chapter s in books and a coauthor of the G reek edition of a book on mobile communicati ons. He serves on the editorial board of the E U R A S I P J O U R N A L O N W I R E L E S S C O M M U N I C ATI O N S A N D N E T W O R K - I N G . Dr . Karagiannidi s has been a member of T echni cal Program Committees for sev eral IEEE conference s. He is a member of the editorial boards of the I E E E T R A N S A C T I O N S O N C O M M U N I C AT I O N S and the I E E E C O M M U N I C A - T I O N S L E T T E R S . He is co-recipi ent of the Best Pape r A ward of the Wi reless Communicat ions Symposium (WCS) in IEEE Internation al Conference on Communicat ions (ICC’ 07), Glasgow , U.K., June 2007. He is a full member of Sigma Xi.