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Sum of Non-Identical Independent Squared í µí¼‚-í µí¼‡ Variates and Applications in the Performance Analysis of DS-CDMA Systems

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Sum of Non-Identical Independent Squared í µí¼‚-í µí¼‡ Variates and Applications in the Performance Analysis of DS-CDMA Systems
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  2718 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 2010 Sum of Non-Identical Independent Squared   -   Variates andApplications in the Performance Analysis of DS-CDMA Systems Kostas P. Peppas, Fotis Lazarakis, Theodore Zervos, Antonis Alexandridis, and Kostas Dangakis  Abstract —In this paper, new expressions for the probabilitydensity function (pdf) of the sum of non-identical independentsquared   -   random variables are derived. Based on ournewly derived results, we provide a pdf-based approach forthe performance analysis of 1-D and 2-D RAKE receivers inthe context of an asynchronous direct sequence code divisionmultiple access system operating over   -   fading channels. Forthe considered system, useful performance metrics such as theoutage probability, the channel capacity and the average bit errorprobability are investigated. Extensive numerical and computersimulation results are presented that illustrate the proposedmathematical analysis.  Index Terms —  -   fading, probability density function, RAKEreceiver, outage probability, channel capacity, average bit errorprobability. I. I NTRODUCTION T HE statistical distribution of the sum of the squaredenvelopes of faded signals occurs extensively in sev-eral wireless communications applications. Such applicationsinclude the performance analysis of maximal-ratio combin-ing (MRC) and post-detection equal-gain combining (EGC)diversity techniques as well as co-channel interference of cellular mobile radio. Representative past examples can befound in [1]–[3]. In [1], the authors made use of a resultproposed back to 1985 by Moschopoulos [4] to addressthe performance of MRC and post-detection EGC diversityreceivers as well as the performance of cellular systems overNakagami-   fading channels. In [2], the authors proposed anintegral representation for the pdf of the sum of independentgamma variates with arbitrary fading parameters while in [3]an alternative closed-form expression was presented.Recently, the so-called   -   fading distribution that includesas special cases the Nakagami-   and the Hoyt distribution,has been proposed as a more  ß exible model for practicalfading radio channels [5]. The   -   distribution  Þ ts well toexperimental data and can accurately approximate the sumof independent non-identical Hoyt envelopes having arbitrarymean powers and arbitrary fading degrees [6]. Moreover,it hasgained interest in the  Þ eld of performance evaluation of digitalcommunications over fading channels [7]–[12]. To the bestof the authors’ knowledge, however, there are no publishedresults concerning the distribution of the sum of the squaredenvelopes of    -   faded signals. Manuscript received December 23, 2009; revised April 12, 2010; acceptedJune 3, 2010. The associate editor coordinating the review of this paper andapproving it for publication was X.-G. Xia.The authors are with the Laboratory of Wireless Communications, In-stitute of Informatics and Telecommunications, National Center for Sci-enti Þ c Research–“Demokritos,” Patriarhou Grigoriou and Neapoleos, AgiaParaskevi, 15310, Athens, Greece (e-mail:  󰁻 kpeppas,  ß az, tzervos, aalex,kdang 󰁽 @iit.demokritos.gr).Digital Object Identi Þ er 10.1109/TWC.2010.071410.091839 In this paper we provide new expressions for the pdf of the sum of independent non-identically distributed (i.n.i.d)squared   -   random variables. The pdf is given in threedifferent formats: An in Þ nite series representation, an integralrepresentation as well as an accurate closed form expression.Based on these formulas, we evaluate the outage probability(OP), and the Shannon capacity and the average bit errorrate probability (ABEP) of 1-D and 2-D RAKE  1 receiversoperating in a   -   fading environment with arbitrary fadingparameters along the diversity branches.The remainder of this paper is structured as follows: SectionII presents the derivation of analytical expressions for the pdf of the sum of i.n.i.d squared   -   random variables. SectionIII applies these results to derive formulas for the OP, ABEP,and Shannon capacity of 1-D and 2-D RAKE receivers overi.n.i.d   -   diversity paths. Numerical and computer simulationresults are given in Section IV while Section V concludes thepaper.II. S UM OF  S QUARED I . N . I . D   -   VARIATES Let us consider     ≥  󰀱  independent squared   -   randomvariables    ℓ ,  ℓ  󰀽 󰀱 , 󰀲 ,     each with pdf      ℓ 󰀨   󰀩 󰀽 󰀲 √   ℓ ℓ 󰀫 󰀱󰀲 ℎ ℓ 󰀨  ℓ 󰀩    ℓ  󰀱󰀲 ℓ    ℓ  󰀱󰀲 ℓ    ℓ 󰀫 󰀱󰀲 ℓ 󰁥󰁸󰁰 􀀨  󰀲  ℓ   ℓ ℎ ℓ   ℓ 􀀩 󰃗    ℓ  󰀱󰀲 􀀨 󰀲  ℓ   ℓ   ℓ   ℓ 􀀩 , (1)where  󰀨  󰀩  is the Gamma function [13, eq. 8.310],      󰀨  󰀩  is themodi Þ ed Bessel function of the  Þ rst kind and arbitrary order    [13, eq.(8.406.1)],      󰀽   ⟨    ⟩ is the average SNR with   ⟨⟩ denoting expectation. The pdf of     ℓ  may be expressed in twoformats. In Format 1, the parameter  󰀰  󰀼  ℓ  󰀼  ∞  denotesthe ratio between the in-phase and quadrature components of the fading signal within each cluster, which are assumed tobe independent from each other and to have different powers.In Format 2,   󰀱  󰀼  ℓ  󰀼  󰀱  denotes the correlation betweenthe powers of the in-phase and quadrature scattered waves ineach multi-path cluster. In both formats, the parameter   ℓ  󰀾  󰀰 denotes the number of multi-path clusters. Also, in Format1,  ℎ ℓ  󰀽 󰀨󰀲 󰀫    󰀱 ℓ  󰀫   ℓ 󰀩 / 󰀴  and    ℓ  󰀽 󰀨   󰀱 ℓ    ℓ 󰀩 / 󰀴  whereasin Format 2,  ℎ ℓ  󰀽 󰀱 / 󰀨󰀱   󰀲 ℓ 󰀩  and    ℓ  󰀽   ℓ / 󰀨󰀱   󰀲 ℓ 󰀩 . The  -   fading distribution comprises both Hoyt  󰀨  ℓ  󰀽 󰀰 . 󰀵󰀩  andNakagami-   󰀨  ℓ  →  󰀰 , ℓ  → ∞ , ℓ  → 󰂱 󰀱󰀩 . We de Þ ne thesum of      independent random variables    ℓ  as    ≜   ∑ ℓ 󰀽󰀱   ℓ .  (2) 1 The receiver that uses both antenna and multipath diversity is known asa two-dimensional (2-D) RAKE receiver1536-1276/10$25.00 c ⃝  2010 IEEE  IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 2010 2719 The moment generating function (MGF) of     , de Þ ned as ℳ    󰀨  󰀩 󰀽   ⟨ 󰁥󰁸󰁰󰀨   Υ󰀩 ⟩ , with the help of [7, eq. (6)] maybe expressed as [9, eq. (2)], ℳ    󰀨  󰀩 󰀽   ∏ ℓ 󰀽󰀱 󰀨󰀱 󰀫  ℓ 󰀩   ℓ 󰀨󰀱 󰀫  ℓ 󰀩   ℓ ,  (3)where   ℓ  󰀽    ℓ 󰀲  ℓ 󰀨 ℎ ℓ    ℓ 󰀩  and   ℓ  󰀽    ℓ 󰀲  ℓ 󰀨 ℎ ℓ 󰀫   ℓ 󰀩 . The pdf of     may be obtained by taking the inverse Laplace transform of  ℳ    󰀨  󰀩 , i.e.       󰀨   󰀩 󰀽    󰀱 󰁻ℳ    󰀨  󰀩󰀻  󰀻   󰁽  where    󰀱 󰁻 󰀻  󰀻  󰁽 denotes inverse Laplace trasform. It is also noted that thepdf of     independent and identically distributed (i.i.d.)   -  channels is an   -   distribution with parameters   ,    and    [5].  A. In  Þ nite series representation of the pdf of the sum of independent    -   variates We may observe that in (3), each factor of the form  󰀨󰀱 󰀫  ℓ 󰀩   ℓ is the MGF of a gamma-distributed random variablewith parameters   ℓ  and   ℓ . Similarly, each factor of the form 󰀨󰀱 󰀫   ℓ 󰀩   ℓ is the MGF of a gamma-distributed randomvariable with parameters   ℓ  and   ℓ . Hence, the pdf of thesum of      independent squared   -   variates may be obtainedas the pdf of the sum of   󰀲    independent gamma variates withsuitably de Þ ned parameters. Using [1, eq. (2)], the pdf of     may be expressed as      󰀨   󰀩 󰀽   ∏  󰀽󰀱 󰀨     󰀩    ∞ ∑  󰀽󰀰      󰀲 ∑  ℓ 󰀽󰀱  ℓ 󰀫   󰀱         󰀨   󰀫 󰀲 󲈑  ℓ 󰀽󰀱  ℓ 󰀩 ,  (4)where      󰀽 󰁭󰁩󰁮 󰁻  ℓ , ℓ 󰁽  and the coef  Þ cients      may berecursively obtained as    󰀫󰀱  󰀽 󰀱   󰀫 󰀱  󰀫󰀱 ∑  󰀽󰀱 ⎡⎣   ∑  󰀽󰀱   􀀨 󰀱        􀀩  󰀫   ∑  󰀽󰀱   􀀨 󰀱        􀀩  ⎤⎦    󰀫󰀱   ,   󰀽 󰀰 , 󰀱 , 󰀲 ,..., (5)with    󰀰  󰀽 󰀱 .  B. Integral representation of the pdf of the sum of independent   -   variates An alternative expression for the pdf of      may be obtainedby using the Gil-Pelaez result [14] to obtain the inverseLaplace transform of (3). The cdf of      may be obtained as      󰀨   󰀩 󰀽    󰀱 􀁻 ℳ    󰀨  󰀩   󰀻  󰀻   􀁽 󰀽 󰀱󰀲    󰀱  ∫   ∞ 󰀰 ℑ󰁻ℳ    󰀨   󰀩    󰁽  , (6)where    󰀽 √  󰀱 , and  ℑ󰁻󰁽  is the imaginary part. By ex-pressing  ℳ    󰀨   󰀩  in polar form and substituting in (6), thecumulative distribution function (cdf) of      may be expressedas       󰀨   󰀩 󰀽 󰀱󰀲 −  󰀱  ∫   ∞ 󰀰 󰁳󰁩󰁮 󰁻∑  ℓ 󰀽󰀱   ℓ 󰁛󰁡󰁲󰁣󰁴󰁡󰁮󰀨  ℓ  󰀩 󰀫 󰁡󰁲󰁣󰁴󰁡󰁮󰀨  ℓ  󰀩󰁝 −   󰁽  ∏  ℓ 󰀽󰀱 󰁛󰀨󰀱 󰀫   󰀲  󰀲 ℓ 󰀩 󰀨󰀱 󰀫   󰀲  󰀲 ℓ 󰀩󰁝 ℓ 󰀲 󰀬 (7) The corresponding pdf may be obtained by taking the deriva-tive of (7) with respect to    , yielding       󰀨   󰀩 󰀽 ∫   ∞ 󰀰 󰁣󰁯󰁳 󰁻∑  ℓ 󰀽󰀱   ℓ 󰁛󰁡󰁲󰁣󰁴󰁡󰁮󰀨  ℓ  󰀩 󰀫 󰁡󰁲󰁣󰁴󰁡󰁮󰀨  ℓ  󰀩󰁝 −   󰁽  ∏  ℓ 󰀽󰀱 󰁛󰀨󰀱 󰀫   󰀲  󰀲 ℓ 󰀩 󰀨󰀱 󰀫   󰀲  󰀲 ℓ 󰀩󰁝 ℓ 󰀲 󰀮 (8) Both integrals can be numerically evaluated in an ef  Þ cientway, for example by using the Gauss-Legendre quadrature rule[15, eq. (25.4.29)] over (8) or (7) [2]. C. A Closed-Form expression of the pdf of the sum of inde- pendent    -   variates A closed-form expression of the pdf of      may be obtainedby making use of the following Laplace transform pair [16,p. 259]  󰁻    󰀱 Φ 󰀨  󰀩󰀲  󰀨  󰀱 ,...  󰀻  󰀻  󰀱 ,...,   󰀩󰀻  󰀻  󰁽 󰀽󰀨  󰀩   ∏  󰀽󰀱 󰀨 󰀱      󰀩    ,  󰀾  󰀰 ,  (9)where  Φ 󰀨  󰀩󰀲  󰀨  󰀩  is the con ß uent Lauricella hypergeometricfunction de Þ ned in [16, eq. 10, p. 62] Φ 󰀨  󰀩󰀲  󰀨  󰀱 ,...  󰀻  󰀻  󰀱 ,...  󰀩 󰀽 ∞ ∑  󰀱 , 󰀲 ,  ,  󰀽󰀰 󰀨  󰀱 󰀩  󰀱  ... 󰀨   󰀩   󰀨  󰀩  󰀱 󰀫 ... 󰀫     󰀱 󰀱  󰀱 !  ...       !  ,  (10)with  󰀨  󰀩   󰀽 󰀨   󰀫   󰀩 / 󰀨  󰀩  being the Pochhammer symbol.By comparing (9) and (3), the pdf of      may be obtained as      󰀨   󰀩 󰀽 󲈏  ℓ 󰀽󰀱 󰀨  ℓ  ℓ 󰀩   ℓ   󰀲 ∑   󰀽󰀱   ℓ  󰀱  󰀨 󰀲 󲈑  ℓ 󰀽󰀱  ℓ 󰀩  Φ 󰀨󰀲   󰀩󰀲  󰀨  󰀱 , 󰀱 ,   ,   ,   , 󰀲   ∑  󰀽󰀱  ℓ ,     󰀱 ,     󰀱   ,       ,       ⎞⎠ . (11)It is noted that the both the integral and in Þ nite seriesrepresentations for the pdf of      are much more convenient foraccurate and ef  Þ cient numerical evaluation than this accurateclosed form, especially for large values of     , i.e.   󰀾  󰀶 .[3]. However, accurate results may be obtained by expressingthe series in (10) as multiple integrals [17] that can be easilyevaluated numerically [3].III. A PPLICATIONS TO THE  P ERFORMANCE  A NALYSIS OF DS-CDMA S YSTEMS  A. System and Channel Model We consider a general asynchronous DS-CDMA systemoperating over an   -   fading channel and assume that there  2720 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 2010 are     simultaneously transmitting users. Three rectangularwaveforms with unit height and durations     ,      and    ,respectively are employed. The  Þ rst, for the cell-speci Þ c PNrandomization code sequence that is common to all channels.The second, for the user-speci Þ c orthogonal Walsh sequencesthat separate the user channels and the third for the user datawaveform. The processing gain of the system is de Þ ned as     󰀽  /   . For simplicity, it is assumed that      󰀽     ,the length of the orthogonal Walsh sequences is equal to    , whereas that of the PN sequences is signi Þ cantly higher.The RAKE receiver employs      antennas. The low-passequivalent impulse response of the bandpass channel from thetransmitter to the   th receive antenna for a given user, maybe described by tapped delay line model with    paths. Theinstantaneous power of the   th path,    󰀲 ,  follows a squared  -   distribution with parameters   , ,   ,  and  Ω ,  󰀽   󰁛   󰀲 , 󰁝 ,   󰀽 󰀰 , 󰀱 ,...  󰀱 . An exponentially decaying power pro Þ leis assumed, i.e.  Ω ,  󰀽 Ω , 󰀰  󰁥󰁸󰁰󰀨    󰀩  with     being the decayfactor. In addition, the total time average channel gain per an-tenna for each user is normalized to one, i.e. 󲈑   󰀱  󰀽󰀰  Ω ,  󰀽 󰀱 .Without loss of generality, we assume identical power pro Þ leamong receive antennas, i.e.  Ω ,  󰀽 Ω   for    󰀽 󰀱 ,...,   .Under the standard Gaussian assumption that invokes thecentral limit theorem to approximate the sum of the multiple-access interference signals as an additive white Gaussianprocess additional to the background Gaussian noise process,the instantaneous Signal-to-Interference-plus-Noise Ratio atthe output of the 2-D RAKE receiver for the desired useris given by [3, eq. (24)]    󰀽   󰀰    ∑  󰀽󰀱    󰀱 ∑  󰀽󰀰 󰀨   , 󰀩 󰀲 󰀽    ∑  󰀽󰀱    󰀱 ∑  󰀽󰀰   , ,  (12)where   󰀰  󰀽 󰁻 󰀲󰀨    󰀱󰀩󰀳     󰀫  󰀱  Ω 󰀰     󰀫   󰀰    󰁽  󰀱 with     ≤    beingthe number of paths combined by the RAKE receiver,     the received signal energy per bit per antenna and   󰀰  theAWGN one-sided power spectral density. We may observe thatthe random variables    ,  follow a squared   -   distributionwith parameters   , ,   ,  and    ,  󰀽   󰀰 Ω ,  󰀽   󰀰 Ω  ,    󰀽 󰀱 ,...,   ,  󰀽 󰀰 ,...,   󰀱 . Using the normalizationcondition with for  Ω , ,  Ω   may be expressed as  Ω   󰀽  −  󰀨󰀱   −  󰀩󰀱   −   .  B. Outage Probability The outage probability is de Þ ned as the probability that theinstantaneous SINR at the output of the 2-D RAKE receiver,   , falls below a speci Þ ed threshold    󰁴󰁨 , i.e.     󰀨   󰁴󰁨 󰀩 󰀽  󰀨  󰀼   󰁴󰁨 󰀩 󰀽      󰀨   󰁴󰁨 󰀩 . Using (4) and with the help of [13,eq. (8.356.3)], an in Þ nite series representation of the outageprobability may be obtained as    󰀨   󰁴󰁨 󰀩 󰀽        ∏  󰀽󰀱    󰀱 ∏  󰀽󰀰 󰀨  ,  , 󰀩   󰀬 󰃗 ∞ ∑  󰀽󰀰    ⎡⎣ 󰀱   󰀨   󰀫 ,    󰁴󰁨    󰀩 󰀨   󰀫    󰀩 ⎤⎦ , (13)where     ≜  󰀲 󲈑     󰀽󰀱 󲈑    󰀱  󰀽󰀰   , . Also,      󰀽󰁭󰁩󰁮 󰁻  ,  , 󰁽 ,  󰀨  ,  󰀩  is the incomplete gamma function[13, eq. 8.350.2] and the coef  Þ cients      are given by (5).Moreover, using (7), an integral representation of the outageprobability may readily be obtained as    󰀨   󰁴󰁨 󰀩 󰀽 󰀱󰀲    󰀱  ∫   ∞ 󰀰 󰁳󰁩󰁮󰁛   󰀨  󰀩    󰁴󰁨 󰁝   󰀨  󰀩  ,  (14)where   󰀨  󰀩 ≜    ∑  󰀽󰀱    󰀱 ∑  󰀽󰀰  , 󰁛󰁡󰁲󰁣󰁴󰁡󰁮󰀨  ,  󰀩 󰀫 󰁡󰁲󰁣󰁴󰁡󰁮󰀨  ,  󰀩󰁝 , (15a)   󰀨  󰀩 ≜     ∏  󰀽󰀱    󰀱 ∏  󰀽󰀰 󰁛  󰀱 󰀫  󰀲  󰀲 ,  󰀱 󰀫  󰀲  󰀲 ,  󰁝 󰀬 󰀲 ,  (15b)and   ,  󰀽   󰀰 Ω  󰀲  󰀬 󰀨 ℎ 󰀬    󰀬 󰀩  and   ,  󰀽   󰀰 Ω  󰀲  󰀬 󰀨 ℎ 󰀬 󰀫   󰀬 󰀩 Finally, by integrating (11) term-by-term, a closed form ex-pression for     󰀨   󰁴󰁨 󰀩  may be obtained as    󰀨   󰁴󰁨 󰀩 󰀽 󲈏     󰀽󰀱 󲈏    󰀱  󰀽󰀰  󰀨  ,  , 󰀩   󰀬     󰁴󰁨 󰀨󰀱 󰀫   󰀩 Φ 󰀨󰀲      󰀩󰀲  󰀨  󰀱 , 󰀰 , 󰀱 , 󰀰 ,   ,    ,   󰀱 ,    ,   󰀱 , 󰀱 󰀫 ,     󰁴󰁨  󰀱 , 󰀰 ,     󰁴󰁨  󰀱 , 󰀰   ,     󰁴󰁨     ,   󰀱 ,     󰁴󰁨     ,   󰀱 􀀩 . (16) C. Channel Capacity For fading channels, the ergodic channel capacity charac-terizes the long-term achievable rate averaged over the fadingdistribution and depends on the amount of available channelstate information (CSI) at the receiver and transmitter [18].Two adaptive transmission schemes are considered: Optimalrate adaptation with constant transmit power (ORA) and opti-mal simultaneous power and rate adaptation (OPRA). Underthe ORA scheme that requires only receiver CSI, the capacityis known to be given by [18] ⟨   ⟩   󰀽 󰀱󰁬󰁮󰀲 ∫   ∞ 󰀰     󰀨   󰀩󰁬󰁮󰀨󰀱 󰀫   󰀩    (17)In order to obtain an analytical expression of   ⟨   ⟩   for theconsidered DS-CDMA system, we  Þ rst make use of the in Þ niteseries representations of the pdf of      given by (4). Then,by expressing the exponential and the logarithm in terms of Meijer-G functions [19, Eq.(8.4.6.5)], [19, Eq. (8.4.6.2)] andapplying the result given in [19, Eq. (2.24.1.1)], the followingexpression for the capacity may be obtained: ⟨   ⟩   󰀽      󰁬󰁮󰀲    ∏  󰀽󰀱    󰀱 ∏  󰀽󰀰 󰀨  ,  , 󰀩   󰀬 ∞ ∑  󰀽󰀰     󰀱 , 󰀳󰀳 , 󰀲 󐁛     󰀱 , 󰀱 , 󰀱      󰀱 , 󰀰 󐁝 󰀨   󰀫   󰀩  , (18)where   , ,  󰁛  󰁝  is the Meijer-G function [13, Eq. (9.301)]. Forthe OPRA scheme, the capacity is known to be given by [18,Eq. (7)] ⟨   ⟩   󰀽 ∫   ∞   󰀰 󰁬󰁯󰁧 󰀲 􀀨     󰀰 􀀩      󰀨   󰀩 ,  (19)  IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 2010 2721 where    󰀰  is the cutoff SNR below which transmission issuspended. By substituting (4) to (19), expressing the loga-rithm and the exponential in terms of Meijer-G functions [19,Eq.(8.4.6.5)], [19, Eq. (8.4.3.1)] and with the help of [19, Eq.(2.24.1.1)], ⟨   ⟩   may be obtained as ⟨   ⟩   󰀽      󰁬󰁮󰀲    ∏  󰀽󰀱    󰀱 ∏  󰀽󰀰 󰀨  ,  , 󰀩   󰀬 ∞ ∑  󰀽󰀰     󰀰 , 󰀳󰀳 , 󰀲 󰁛      󰀰  󰀱 , 󰀱 , 󰀱      󰀰 , 󰀰 󰁝 󰀨   󰀫   󰀩  . (20)  D. Average Bit Error Probability The conditional bit error probability     󰀨   󰀩  in an AWGNchannel may be expressed in uni Þ ed form as     󰀨   󰀩 󰀽  󰀨 ,  󰀩󰀲󰀨  󰀩 where    and    are parameters that depend on the speci Þ cmodulation scheme. For example,    󰀽 󰀱  for binary phaseshift keying (BPSK) and  󰀱 / 󰀲  for binary frequency shift keying(BFSK). Also,    󰀽 󰀱  for non-coherent BFSK and binarydifferential PSK (BDPSK) and  󰀱 / 󰀲  for coherent BFSK/BPSK.The average bit error probability (ABEP) for the consideredsystem may be obtained by averaging     󰀨   󰀩  over the pdf of     i.e.,     󰀽 ∫   ∞ 󰀰    󰀨   󰀩     󰀨   󰀩 .  (21)Using (4) in conjunction with (21) and with the help of [13,eq. 6.455] the ABEP may be obtained as     󰀽        󰀫  󰀲󰀨  󰀩    ∏  󰀽󰀱    󰀱 ∏  󰀽󰀰 󰀨  ,  , 󰀩   󰀬 󰃗 ∞ ∑  󰀽󰀰󰀲   󰀱 􀀨 󰀱 ,  󰀫    󰀫  󰀻   󰀫    󰀫 󰀱󰀻 󰀱󰀱 󰀫    􀀩 󰃗    󰀨   󰀫    󰀫  󰀩󰀨󰀱 󰀫    󰀩  󰀫   󰀫  󰀨   󰀫    󰀫 󰀱󰀩 ,  (22)where  󰀲   󰀱 󰀨  󰀩  is the Gauss hypergeometric function [19, eq.(7.2.1.1)]. Also, by substituting (8) to (21), the ABEP isexpressed as a two-fold integral. This expression may besimpli Þ ed by performing integration by parts and after somealgebraic manipulations as follows     󰀽 󰀱󰀲  ∫   ∞ 󰀰 􀁻 󰁣󰁯󰁳󰁛   󰀨  󰀩󰁝   󰁳󰁩󰁮󰀨  󰁡󰁲󰁣󰁴󰁡󰁮󰀨 / 󰀩󰀩󰀨  󰀲 󰀫  󰀲 󰀩 󰀯 󰀲  󰀫 󰁳󰁩󰁮󰁛   󰀨  󰀩󰁝 􀁛 󰀱     󰁣󰁯󰁳󰀨  󰁡󰁲󰁣󰁴󰁡󰁮󰀨 / 󰀩󰀩󰀨  󰀲 󰀫  󰀲 󰀩 󰀯 󰀲 􀁝􀁽    󰀨  󰀩 . (23)This integral can be ef  Þ ciently evaluated by means of theGauss-Legendre quadrature integration rule. Finally, an alter-native ABEP expression may be obtained by substituting (11)to (21). By integrating the corresponding in Þ nite series term-by-term and with the help of [15, eq. (6.5.37)], the ABEP maybe obtained in closed form as     󰀽󰀨    󰀫  󰀩 󲈏     󰀽󰀱 󲈏    󰀱  󰀽󰀰  󰀨  ,  , 󰀩   󰀬 󰀲    󰀨  󰀩󰀨    󰀫 󰀱󰀩    󰀨󰀲      󰀩  󰀨    󰀫 , 󰀱 , 󰀰 , 󰀱 , 󰀰 ,   ,    ,   󰀱 ,    ,   󰀱 󰀻    󰀫 󰀱󰀻   󰀱  󰀱 , 󰀰 ,   󰀱  󰀱 , 󰀰   ,   󰀱     ,   󰀱 ,   󰀱     ,   󰀱 􀀩 , (24)where    󰀨  󰀩   󰀨 , 󰀱 ,   ,  󰀻  󰀻  󰀱 ,   ,  󰀩  is the Lauricellamultiple hypergeometric function of     variables de Þ ned as[19, Eq. (7.2.4.57)], [19, Eq. (7.2.4.15)]:   󰀨  󰀩   󰀨 , 󰀱 ,   ,  󰀻  󰀻  󰀱 ,   ,  󰀩 󰀽󰀨  󰀩󰀨    󰀩󰀨  󰀩 ∫   󰀱󰀰    󰀱 󰀨󰀱   󰀩     󰀱  ∏  󰀽󰀱 󰀨󰀱     󰀩     󰀽 ∞ ∑  󰀱 , 󰀲 ,  ,  󰀽󰀰 󰀨  󰀩    󰀨  󰀩     ∏  󰀽󰀱 󰀨   󰀩   󰀨    󰀫 󰀱󰀩      ,  ∣   ∣ 󰀼  󰀱 . (25)where      󰀽 󲈑  󰀽󰀱   . The integral in (25) exists for  ℜ 󰁻    󰁽  󰀾  󰀰  and  ℜ 󰁻  󰁽  󰀾  󰀰  where  ℜ 󰁻󰁽  denotes the real part.By making use of [3, eq. (49)], it becomes obvious that theconvergence conditions for the integral and the in Þ nite seriesrepresentations of the Lauricella functions are satis Þ ed. It isnoted that the Lauricella function can be ef  Þ ciently evaluatednumerically by means of its integral representation.IV. N UMERICAL AND  C OMPUTER  S IMULATION  R ESULTS In this section, the performance of 1-D and 2-D RAKEreceivers in the context of an asynchronous uplink directsequence-CDMA system operating in i.n.i.d   -   fading chan-nels is investigated. A  Þ xed processing gain     󰀽 󰀱󰀲󰀸  isconsidered. Also, a fading channel with    󰀽     󰀽 󰀴  tapsis assumed for all test cases of interest. Finally, for all theconsidered scenarios, Format 1 is assumed.With the aid of eq. (14) and eq. (13), in Fig. 1, theoutage probability      of the considered system is illustratedas a function of      / 󰀰 , for     󰀽 󰀲󰀵  users,      󰀽 󰀱 , 󰀲 ,   ℎ  󰀽 󰀰 dB and for various values of the decay factor   . The fading parameters are    󰀽 󰁛󰀰 . 󰀱 , 󰀰 . 󰀲 , 󰀰 . 󰀳 , 󰀰 . 󰀴󰁝  and   󰀽 󰁛󰀰 . 󰀵 , 󰀱 , 󰀱 . 󰀴 , 󰀱 . 󰀶󰁝 . It can be seen that      improves withan increase of      / 󰀰  and/or a decrease of     . Furthermore, as itcan be observed, the incorporation of MRC diversity receptionschemes signi Þ cantly enhances the outage probability of theconsidered system. Moreover, as expected, eq. (14) and eq.(13) provide identical results for all the considered test cases.Fig. 2 depicts the channel capacity for the ORA scheme asa function of      / 󰀰  with and without MRC diversity for   󰀽 󰁛󰀰 . 󰀴 , 󰀰 . 󰀵 , 󰀰 . 󰀶 , 󰀰 . 󰀷󰁝 ,    󰀽 󰁛󰀰 . 󰀷󰀵 , 󰀱 , 󰀱 . 󰀵 , 󰀲󰁝 ,     󰀽 󰀰 . 󰀵  and fordifferent values of     . As it can be observed, the capacityimproves with an increase of      / 󰀰  and/or a decrease of     .For the same test case, Fig. 3 plots the achievable ABEPperformance for BPSK as a function of      / 󰀰  when      󰀽 󰀱 and      󰀽 󰀲  and as it is evident ABEP improves as     / 󰀰 increases and/or     decreases. In order to evaluate ABEP, (22),(23) and (24) are used and as expected, identical results areprovided for all the test cases of interest. In Figs 1, 2 and 3,
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