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Technol., vol. 60, no. 6, pp. 2482– 2494, Jul. 2011. 859 An Approach to the Second-Order Statistics of Maximum-Ratio Combining-Like Reception Over Independent Nakagami Channels Jia-Chin Lin, Senior Member, IEEE Abstract—A statistical approach facilitating performance assessment in maximum-ratio combining (MRC)-like reception over independent nonidentically distributed (INID) Nakagami-m fading channels was investigated. Using the proposed method, generalized simple closed-form second-order statistics, i.e., level-crossing rate (LCR) and average fade duration (AFD), can be readily formulated. These second-order statistics can be classified as generalized because multiple (i.e., not limited to two) identically or nonidentically distributed diversity branches can be incorporated. These closed formulations can be accurately applied in various practical environments where nonidentically distributed branches are presented, e.g., single-user single-input–multiple-output (SU-SIMO) applications, relay networks, and macrodiversity reception in wideband code-division multiple access (W-CDMA) communications. The analytic formulations derived in this work were comprehensively verified in computer simulations. The derived closed forms were also carefully verified by comparison with those obtained using the true MRC in the degenerate scenarios, i.e., those in which only two diversity branches exist or where identically distributed diversity branches can be assumed. The derived closed formulations of the second-order statistics were shown to be generalized, accurate, easily evaluated, and clearly insightful with respect to their physical interpretation although they are approximate to those of the true MRC. Index Terms—Average fade duration (AFD), diversity combining, level-crossing rate (LCR), Nakagami-fading channel. I. I NTRODUCTION Level-crossing rate (LCR) and average fade duration (AFD) are important second-order statistical quantities that have been extensively studied, particularly in the area of diversity reception. For example, Yacoub et al. proposed a novel method for evaluating LCR and AFD over a single Nakagami-fading channel [1]. A few years later, the LCR and AFD of diversity reception combined over several Nakagami-fading channels were studied [2], [3]. Although the latter studies completely derived closed-form second-order statistics regarding selection combining (SC), equal-gain combining, and maximum-ratio combining (MRC) reception, they rely on the strong assumption that Nakagami-fading channels are independent and identically distributed (IID). The LCR and AFD of the SC over independent but nonidentically distributed (INID) Nakagami-fading channels have also been previously studied [4]. An alternate method of evaluating the LCR and AFD of low-order MRC over dual-branch unbalanced channels has also been studied in an environment with two INID Nakagami-fading channels [5]. A previous study [6] examined the LCR and AFD of MRC on an INID Nakagami-fading channel for which the branches had a common fading figure, i.e., an equal maximum Doppler shift but unequal average signal powers. Recently, Rutagemwa et al. provided expressions for the LCR and AFD of MRC reception over INID dual-branch Nakagami-fading channels Manuscript received March 24, 2011; revised September 8, 2011 and November 1, 2011; accepted December 13, 2011. Date of publication December 21, 2011; date of current version February 21, 2012. This work was supported in part by the National Science Council, Taiwan, under Contract NSC 100-2221-E-008-049-MY3. The review of this paper was coordinated by Prof. X. Wang. The author is with the Department of Communication Engineering, National Central University, Jhongli 320, Taiwan (e-mail: jiachin@ieee.org). Digital Object Identifier 10.1109/TVT.2011.2180729 0018-9545/$31.00 © 2012 IEEE 860 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 2, FEBRUARY 2012 [7]. In this paper, a two-user-based cooperative diversity system over nonidentically distributed Nakagami-fading channels was investigated. Consequently, unequal Nakagami fading factors, different signal powers, and different maximal Doppler shifts were taken into consideration [7]. The LCR and AFD expressions provided in the previous study [7] cannot be considered to be closed-form second-order statistics because they involve unsolvable integration and an infinite series. To date, no closed formulations of the LCR and AFD of MRC reception over independent Nakagami-fading channels have been reported for the generalized condition in which the number of diversity branches is not limited to two and the diversity branches are not necessarily identically distributed. The primary goal of this work was to evaluate the LCR and AFD of MRC-like reception over independent Nakagami-fading channels and to derive simple closed formulations for more than two identically or nonidentically distributed Nakagami branches. Approximations of the second-order statistics in the INID scenario were suggested in a previous work [8], [9]; this method can approximate the second-order statistics by exploiting an α−µfading channel, in which an extra degree of freedom (DOF) is added to each diversity branch. In this technique [8], [9], several intermediate parameters, e.g., α, µ, and r̂, must be numerically computed in advance according to tested scenarios. The addition of a DOF in conjunction with the numerically calculated parameters inevitably effaces the physical meanings, which must be clearly presented in an effective channel model. Differing from the previous work [8], [9], the method proposed herein does not create extra DOFs, and the formulations are derived in a simple, closed-form, and exact manner. In addition, many previous studies [3]–[5], [8]–[10] have focused on the exact formulations in theory and thus omitted Monte Carlo simulation experiments. Comprehensive simulations were conducted in this work to validate the statistical analyses, to verify the accuracy of the closed formations derived herein, and to provide insight on the second-order statistics of MRC-like reception. The proposed method first unifies the variances of envelope derivatives among diversity branches to assure of the independence between the envelope and the envelope derivative obtained after diversity combining. In accordance with the independence, the joint probability density function (JPDF) of the envelope and the envelope derivative can be obtained. Accordingly, the first- and second-order statistics can be derived as those in IID scenarios. The derived closed formulations were also compared with those found in the literature. Although these formulations are approximate to those of the true MRC, they are exact, generalized with respect to diversity-branch number and diversity-branch distribution, and simple in terms of computational complexity. The closed-form first- and second-order statistics investigated in this study are highly applicable to various currently existing wireless applications with more than two nonidentically distributed diversity branches, e.g., single-user single-input–multiple-output (SUSIMO) systems, relay networks [7], or macrodiversity reception in wideband code-division-multiple-access (W-CDMA) communication systems. The Nakagami probability density function (PDF) employed to represent the envelope distribution of the lth diversity branch is given as [1] m  ml r 2 − Ωl   Γ ml , FRl (r) = ml 2 r Ωl Γ(ml )  , r > 0, ml ≥ 0.5 (2) b where Γ(a, b) = a xa−1 e−x dx is the lower incomplete Gamma function [1]. Therefore, the pdf of the received power Γl = Rl2 on the lth diversity branch can be written as  m fΓl (γ) = ml l γ ml −1 ml γ exp − m Ωl Ωl l Γ(ml )  , γ > 0. (3) In brief, it is in the form of a Gamma pdf and can be expressed as Γl ∼ Gamma(αl , βl )| αl =ml , where αl and βl represent the shape βl =Ωl /ml and scale parameters of the Gamma pdf, respectively. In L-branch MRC reception, the received signals in individual branches are appropriately amplified with matched complex gains; therefore, they are coherently combined (i.e., to form a cophased sum). As a result, MRC reception accumulates the power delivered through all branches,  and the resultant power at the MRC output can be written L as ΓMRC = l=1 Rl2 . Because the pdf of the sum of independent Gamma RVs has been previously derived [10], the pdf of the resultant envelope can be formulated accordingly. III. D ERIVATIONS OF LCR AND AFD In essence, a Gamma-distributed RV can be considered as the sum DOF. For of a set of statistically IID central χ2 RVs with a single 2m any integral value of 2ml , Γl can be rewritten as Γl = i=1l Ul,i = 2ml 2 Xl,i , where Ul,i , i = 1, 2, . . . , 2ml , are IID central χ2 RVs i=1 with one DOF, and Xl,i , i = 1, 2, . . . , 2ml , are IID Gaussian RVs with zero mean and variance σl2 = Ωl /2ml . Additionally, the time derivative Ṙl of Rl can be expressed as 2ml 1  Ṙl = Xl,i Ẋl,i . Rl (4) i=1 Given that Xl,i , l = 1, 2, . . . , L, i = 1, 2, . . . , 2ml , are Gaussian RVs, the derivatives Ẋl,i are also IID Gaussian distributed RVs with zero 2 = 2π 2 ν 2 σl2 = (πν)2 Ωl /ml , where ν mean and a variance of σẊ l denotes the maximum Doppler frequency. As a result, the conditional pdf of (Ṙl |Xl,1 , Xl,2 , . . . , Xl,2ml ) has a Gaussian distrib2 = ution with zero mean and a variance of σṘ |X ,X ,···,X l II. S IGNAL M ODELS 2ml l r2ml −1 exp fRl (r) = m Γ(ml )Ωl l where Rl is the random variable (RV) representing the envelope of the received signal on the lth diversity branch; Ωl = E{Rl2 }; ml = E2 {Rl2 }/Var{Rl2 }; E{·} and Var{·} denote the expectation and variance of their arguments, respectively; and Γ(ml ) = value ∞ m −1 −x x l e dx is the Gamma function [1]. The cumulative distrib0 ution function (CDF) of Rl can be expressed as l,1 l,2 l,2ml π 2 ν 2 (Ωl /ml ), l = 1, 2, . . . , L, because of the linear relation given by (4), where Xl,1 , Xl,2 , . . . , Xl,2ml are given. A. Independence Between RMRC and ṘMRC , r>0 ml ≥ 0.5 (1) To proceed with the derivation of the LCR and AFD, it is first necessary to evaluate the JPDF of the RV ṘMRC and the RV RMRC . By defining RMRC , Φ1 , Φ2 , . . . , ΦL−1 to be a set of L IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 2, FEBRUARY 2012 hyperspherical coordinates, an L-to-L transformation can thus be defined as ⎧ ⎪ ⎪ R1 = RMRC cos(Φ1 ) ⎪ ⎪ R2 = RMRC sin(Φ1 ) cos(Φ2 ) ⎪ ⎪ ⎨ R3 = RMRC sin(Φ1 ) sin(Φ2 ) cos(Φ3 ) (5) . .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ RL = RMRC sin(Φ1 ) sin(Φ2 ) sin(Φ3 ) sin(Φ4 ) ⎩ Therefore, the JPDF of the RV ṘMRC and the RV RMRC can be rewritten as π/2 fṘMRC ,R1 ,...,RL ··· fṘMRC ,RMRC (ṙ, r) = 0 0 × (ṙ, r1 , . . . , rL )|J|dφ1 · · · dφL−1 L−1 L−1 (6) where |J| = r sin(L−1−i) φi is the Jacobean of the aforei=1 mentioned hyperspherical transformation. Because the JPDF of the RVs ṘMRC , R1 , . . . , RL can be expressed as fṘMRC ,R1 ,...,RL (ṙ, r1 , . . . , rL) = fṘMRC ,R1 ,R2 ,...,RL(ṙ|r1 , r2 , . . . , rL)·fR1 ,R2 ,...,RL (r1 , r2 , . . . , rL) (7) the JPDF of ṘMRC and RMRC can be formulated as π/2 π/2 fṘMRC ,RMRC (ṙ, r) fṘMRC ,R1 ,...,RL (ṙ|r1 , . . . , rL ) ··· = ·  l=1 fRl (rl )|J|dφ1 dφ2 · · · dφL−1 . (8) µṘCnd. = E{ṘMRC |R1 = r1 , . . . , RL = rl } = 0 = Var{ṘMRC |R1 = r1 , . . . , RL = rl } = L  r2 l l=1 r2 σṙ2l (9) L where r2 = l=1 rl2 , because ṘMRC can be expressed as ṘMRC = L (1/RMRC ) l=1 Rl Ṙl and Ṙl , l = 1, 2, . . . , L are independent of one another. In fact, the RV (ṘMRC |R1 = r1 , . . . , RL = rL ) is a Gaussian RV because its form is a linear combination of Gaussian RVs Ṙl , l = 1, 2, . . . , L. It can be seen from (9) that even the second-order moment of the Gaussian RV (ṘMRC |R1 = r1 , R2 = r2 , . . . , RL = rL ) exhibits a dependence of ṘMRC on R1 , . . . , RL . The JPDF of the RV ṘMRC and the RV RMRC shown in (8) can be further reformulated as fṘMRC ,RMRC (ṙ, r) = · ⎧ L ⎪ ⎨  ⎪ ⎩ l=1 ⎩  1 ⎡ exp 2 2πσṘ Cnd. ⎡  ⎢ ⎢ fRl (rl ) ⎣rL−1 ⎣  B. Preunification As mentioned in (9), the RV (ṘMRC |R1 = r1 , R2 = r2 , . . . , RL = L 2 rL ) has zero mean and the variance σṘ = l=1 rl2 /r2 σṙ2l . If σṙ2l Cnd. 2 can be unified for l = 1, 2, . . . , L, σṘ can thus be reformulated as Cnd. follows: Cnd. For any distribution of the RV (ṘMRC |R1 = r1 , R2 = r2 , . . . , RL = rL ), its mean and variance can be found as ⎧ ⎨ 2 means that ṘMRC may depend on RMRC = This expression of σṘ Cnd. r. In fact, in the following derivations of the second-order statistics, the JPDF of the RV ṘMRC and the RV RMRC is inevitably required. However, no method found from the aforementioned derivations or in any previously published study can directly provide the JPDF of the RV ṘMRC and the RV RMRC . From an original study [11] to the more recent study [7], the independence between the envelope and the envelope derivative is still an empirical assumption with no statistical proof. The independence between the RV ṘMRC and the RV RMRC therefore becomes the most critical hurdle preventing the formation of any closed-form second-order statistics because only the marginal pdfs of the RV ṘMRC and the RV RMRC can be individually obtained. The JPDF of the RV ṘMRC and the RV RMRC can then be obtained as the product of the marginal pdf of the RV ṘMRC and that of the RV RMRC with the prerequisite proof of independence between these two RVs. 2 σṘ L Cnd.  π/2 6 · · · 2m)(π/2) and 0 sin2m+1 xdx = (2 · 4 · 6 · · · (2m))/(1 · 3 · 5 · · · (2m + 1)), m = 1, 2, · · ·. Although the terms in the former and latter braces in (10) look like the marginal pdfs of ṘMRC and RMRC , respectively, the independence between the RV Ṙ MRC and the RV L 2 = l=1 (rl2 /r2 )σṙ2l . RMRC has not yet been proven because σṘ 0 0 2 σṘ  π/2 where 0 sinL−1−i φi dφi can be replaced by means of the follow π/2 ing relations [13]: 0 sin2m xdx = (1 · 3 · 5 · · · (2m − 1))/(2 · 4 · Cnd. · · · sin (ΦL−2 ) sin (ΦL−1 ) . π/2 861 π 2 L−1 i=1 0  − ṙ2 2 2σṘ Cnd. ⎫ ⎬ ⎭ ⎤⎤⎫ ⎪ ⎬ ⎥⎥ sinL−1−i φi dφi ⎦⎦ ⎪ ⎭ = σṙ21 L l=1 r2 rl2 = σṙ21 . (11) Thus, the first brace in (10) becomes irrelevant with R1 = r1 , R2 = r2 , . . . , RL = rL . As a result, the RV ṘMRC can be assured to be independent of R1 , R2 , . . . , RL , given that the variances of Ṙl , l = 1, 2, . . . , L, are unified as a common value. This method is, so far, the only way to further proceed with the derivations. To arrive at a simple closed formulation, appropriate real-valued gains cl are applied to the individual diversity branches. Finally, the signalenvelope squares are then summed across the L diversity branches, so that the envelope of the MRC  with preunification (MRC/PU) can be L c R2 , where cl = (ml /Ωl )βC is expressed as RMRC/PU = l=1 l l the real-valued weighting coefficient used to unify the nonidentically distributed Gamma RVs to yield the common parameter βC , i.e., R̄l2 = cl Rl2 ∼ Gamma(αl , βl )| αl =ml . The preunification method, in fact, βl =βC balances all scale parameters to a common value. Using this method, the common scale parameter βC must be carefully determined, so that the MRC/PU reception can approach a true MRC reception as the L nonidentically distributed branches are combined. To associate the nonidentically-distributed-branch case with the identically-distributedbranch case and to associate the MRC/PU with the true MRC, the weighting coefficients cl , l = 1, 2, . . . , L adopted in the MRC/PU proposed herein can be set to cl = (ml /Ωl )βC , l = 1, 2, . . . , L, L where βC = L( l=1 ml /Ωl )−1 . Based on the aforementioned branch-balancing method, the pdf of the resultant envelope of MRC/PU reception is thus reduced to (10) fRMRC/PU (r) =  2 2r2(mT −1) exp − βrC mT βC Γ(mT )  , r>0 (12) 862 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 2, FEBRUARY 2012 L where mT = l=1 ml . Therefore, the pdf of the received power obtained using the MRC/PU can be written as fΓMRC/PU (γ) =  γ γ mT −1 exp − mT βC βC Γ(mT )  , γ > 0. (13) Because the diversity branches are assumed to be independent and they have the same scale parameter βC , the pdf of the RV ṘMRC/PU can be written as fṘMRC/PU ,R1 ,R2 ,...,RL (ṙ|r1 , r2 , . . . , rL ) =  1 exp 2 2πσṘ Cnd.PU  − ṙ2 2 2σṘ Cnd.PU  (14) where the variance of the RV (ṘMRC/PU |R1 = r1 , R2 = r2 , . . . , 2 = Var{ṘMRC/PU |R1 = r1 , RL = rL ) can be rewritten as σṘ Cnd.PU 2 2 R2 = r2 , . . . , RL = rL } = π ν βC . The aforementioned equations show that ṘMRC/PU is independent of R1 , R2 , . . . , RL . Therefore, the JPDF of RMRC/PU and ṘMRC/PU can be written as fṘMRC/PU ,RMRC/PU (ṙ, r) = fṘMRC/PU (ṙ)fRMRC/PU (r). Therefore, the LCR and AFD on the MRC/PU reception over the L INID Nakagami-fading branches can be expressed as √ NR = mT βC Γ(mT )  Γ mT , TR = √  2 2πβC νR(2mT −1) exp − βRC R2 βC  mT βC  2 2πβC νR(2mT −1) exp − βRC  . (15) (16) C. Special Case: IID In the special case where all diversity branches are assumed to be IID, i.e., ml = m̄ and Ωl = Ω̄, l = 1, 2, . . . , L, the weighting factors all degenerate to unity, i.e., cl = 1, l = 1, 2, . . . , L. The MRC/PU reception considered here is, of course, equivalent to the true MRC reception. The pdf of RMRC/PU is thus reformulated as fRMRC/PU (r) = 2m̄Lm̄ Γ(Lm̄)Ω̄  r √ Ω̄ 2(Lm̄−1)  exp − m̄r2 Ω̄  , Cnd.PU Tρ = Γ(Lm̄, m̄ρ2 ) exp(m̄ρ2 ) √ 2πν(m̄ρ2 )Lm̄−0.5 IV. N UMERICAL R ESULTS AND C OMPUTER S IMULATIONS To validate the proposed approach and demonstrate the effectiveness and accuracy of the closed-form derivations of the first- and secondorder statistics studied herein, several Monte Carlo simulations were conducted. In the following experiments, the Nakagami-fading simulator proposed in the previous study [14] was employed. The carrier frequency was set to be fc = 2.0 GHz, and the mobile speed was set to be either v = 240 or v = 3 km/h. The LCRs were normalized by the maximum Doppler frequency ν, the AFDs were normalized by 1/ν, and the envelopes of either the MRC/PU reception or the true MRC reception were normalized by the square root of the total power, i.e., √ R/ ΩT , where ΩT denotes the total power of the L diversity branches to enable a fair comparison. A. INID Diversity Branches r2 , . . . , RL = rL } = π 2 ν 2 (Ω̄/m̄). Based on the aforementioned independence property between RMRC/PU and ṘMRC/PU , the LCR and AFD for the MRC/PU reception over IID Nakagami-fading branches can be formulated as √ 2πν (m̄ρ2 )Lm̄−0.5 exp(−m̄ρ2 ) Γ(Lm̄) √ where ρ = R/ Ω̄. It is obvious that the closed-form statistics derived in (18) are equivalent to those obtained using the true MRC on IID diversity branches in the previous studies [2], [3]. r > 0 (17) and the corresponding cdf is given as FRMRC/PU (r) = Γ(Lm̄, m̄r2 /Ω̄)/Γ(Lm̄). Because the diversity branches are assumed to be IID, the variance of (ṘMRC/PU |R1 = r1 , R2 = r2 , . . . , RL = 2 = Var{ṘMRC/PU |R1 = r1 , R2 = rL ) can be modified as σṘ Nρ = Fig. 1. PDFs of the resultant powers obtained using either the true MRC or the MRC/PU. Case I: (m1 , Ω1 ) = (1/2, 1/8), (m2 , Ω2 ) = (1, 1/8), (m3 , Ω3 ) = (2, 2/8), and (m4 , Ω4 ) = (4, 4/8). Case II: (m1 , Ω1 ) = (1/2, 1/16), (m2 , Ω2 ) = (1/2, 2/16), (m3 , Ω3 ) = (1, 1/16), (m4 , Ω4 ) = (1, 2/16), (m5 , Ω5 ) = (2, 2/16), (m6 , Ω6 ) = (3, 2/16), (m7 , Ω7 ) = (4, 2/16), and (m8 , Ω8 ) = (5, 4/16). (18) Fig. 1 shows the pdfs of the resultant powers obtained using either the true MRC reception or the MRC/PU reception derived in this paper in an environment where Nakagami-fading branches are considered to be INID. There are four diversity branches in Case I: (m1 , Ω1 ) = (1/2, 2/16), (m2 , Ω2 ) = (1, 2/16), (m3 , Ω3 ) = (2, 4/16), and (m4 , Ω4 ) = (4, 8/16). There are eight diversity branches in Case II: (m1 , Ω1 ) = (1/2, 1/16), (m2 , Ω2 ) = (1/2, 2/16), (m3 , Ω3 ) = (1, 1/16), (m4 , Ω4 ) = (1, 2/16), (m5 , Ω5 ) = (2, 2/16), (m6 , Ω6 ) = (3, 2/16), (m7 , Ω7 ) = (4, 2/16), and (m8 , Ω8 ) = (5, 4/16). The following can be observed: the simulation results for the pdfs of the resultant powers agreed with the analytic results derived in the previous study [10] when the true MRC was performed, the simulation results for the pdfs of the resultant powers matched the analytic results derived in (13) when the MRC/PU was performed, and both the simulation and analytic results for the pdfs of the resultant IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 2, FEBRUARY 2012 Fig. 2. Normalized LCRs obtained using either the true MRC or the MRC/PU, when mobile speed is 240 km/h. Case I: (m1 , Ω1 ) = (1/2, 1/8), (m2 , Ω2 ) = (1, 1/8), (m3 , Ω3 ) = (2, 2/8), and (m4 , Ω4 ) = (4, 4/8). Case II: (m1 , Ω1 ) = (1/2, 1/16), (m2 , Ω2 ) = (1/2, 2/16), (m3 , Ω3 ) = (1, 1/16), (m4 , Ω4 ) = (1, 2/16), (m5 , Ω5 ) = (2, 2/16), (m6 , Ω6 ) = (3, 2/16), (m7 , Ω7 ) = (4, 2/16), and (m8 , Ω8 ) = (5, 4/16). 863 Fig. 4. Normalized LCRs obtained using either the true MRC or the MRC/PU, when the mobile speed is 3 km/h. Case III: (m1 , Ω1 ) = (2, 4/8), (m2 , Ω2 ) = (1, 2/8), (m3 , Ω3 ) = (1/2, 1/8), and (m4 , Ω4 ) = (1/2, 1/8). Case IV: (m1 , Ω1 ) = (2, 4/16), (m2 , Ω2 ) = (2, 4/16), (m3 , Ω3 ) = (1, 2/16), (m4 , Ω4 ) = (1, 2/16), (m5 , Ω5 ) = (1, 1/16), (m6 , Ω6 ) = (1, 1/16), (m7 , Ω7 ) = (1/2, 1/16), and (m8 , Ω8 ) = (1/2, 1/16). Fig. 3. Normalized AFDs obtained using either the true MRC or the MRC/PU, when the mobile speed is 240 km/h. Case I: (m1 , Ω1 ) = (1/2, 1/8), (m2 , Ω2 ) = (1, 1/8), (m3 , Ω3 ) = (2, 2/8), and (m4 , Ω4 ) = (4, 4/8). Case II: (m1 , Ω1 ) = (1/2, 1/16), (m2 , Ω2 ) = (1/2, 2/16), (m3 , Ω3 ) = (1, 1/16), (m4 , Ω4 ) = (1, 2/16), (m5 , Ω5 ) = (2, 2/16), (m6 , Ω6 ) = (3, 2/16), (m7 , Ω7 ) = (4, 2/16), and (m8 , Ω8 ) = (5, 4/16). Fig. 5. Normalized AFDs obtained using either the true MRC or the MRC/PU, when the mobile speed is 3 km/h. Case III: (m1 , Ω1 ) = (2, 4/8), (m2 , Ω2 ) = (1, 2/8), (m3 , Ω3 ) = (1/2, 1/8), and (m4 , Ω4 ) = (1/2, 1/8). Case IV: (m1 , Ω1 ) = (2, 4/16), (m2 , Ω2 ) = (2, 4/16), (m4 , Ω4 ) = (1, 2/16), (m5 , Ω5 ) = (1, 1/16), (m3 , Ω3 ) = (1, 2/16), (m6 , Ω6 ) = (1, 1/16), (m7 , Ω7 ) = (1/2, 1/16), and (m8 , Ω8 ) = (1/2, 1/16). powers obtained using the MRC/PU were close to the analytic results derived in the previous study [10]. Figs. 2 and 3 show the normalized LCRs and the normalized AFDs, respectively, obtained using the true MRC reception or the MRC/PU reception in the same environment as in the aforementioned simulation. Figs. 4 and 5 show the normalized LCRs and the normalized AFDs, respectively, obtained using the true MRC reception or the MRC/PU reception when the mobile speed is set to 3 km/h. There are four diversity branches in Case III: (m1 , Ω1 ) = (2, 4/8), (m2 , Ω2 ) = (1, 2/8), (m3 , Ω3 ) = (1/2, 1/8), and (m4 , Ω4 ) = (1/2, 1/8). There are eight diversity branches in Case IV: (m1 , Ω1 ) = (2, 4/16), (m2 , Ω2 ) = (2, 4/16), (m3 , Ω3 ) = (1, 2/16), (m4 , Ω4 ) = (1, 2/16), (m5 , Ω5 ) = (1, 1/16), (m6 , Ω6 ) = (1, 1/16), (m7 , Ω7 ) = (1/2, 1/16), and (m8 , Ω8 ) = (1/2, 1/16). It can be observed from these figures that the simulation results for the LCR and AFD obtained using the MRC/PU reception approached the analytic results derived in (15) and (16) and that the simulation and analytic results for the LCR and AFD obtained using the MRC/PU reception were close to those obtained using the true MRC reception. The aforementioned observations demonstrate that the PU did not significantly change the first- and second-order statistics of the MRC/PU reception from those of the true MRC reception and that the 864 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 2, FEBRUARY 2012 derived in (15) and (16), and that both the simulation and analytic results for the LCR and AFD obtained using the MRC/PU approached those obtained via the numerical method derived in the previous study [7] for the true MRC reception. In addition, it can be observed from Fig. 6 that the normalized LCR curves of the MRC/PU have a roughly common peak. After some manipulations, the common peak of the normalized LCR curves can be lo cated at the normalized level ρ, i.e., ρ= L(βB /βT )(1 − 1/(2mT )), L where βB = βC /L = ( l=1 ml /Ωl )−1 and βT = ΩT /mT . Four observations should be noted in Figs. 6 and 7. 1) The peak of a normalized LCR curve approaches the level ρ = 3.01 dB as the value of mT is increased. 2) The off-peak portions of the normalized LCR curve more rapidly decrease as the value of mT is increased. 3) Crossovers occur on the normalized AFD curves, appearing at roughly . ρ = 3.01 dB. 4) As ρ increases beyond ρ, the normalized AFDs with larger values of mT more rapidly increase. C. Remarks Fig. 6. Normalized LCRs. Case A: (m1 , Ω1 ) = (1/2, 1/4), (m2 , Ω2 ) = (1, 3/4); Case B: (m1 , Ω1 ) = (1, 1/4), (m2 , Ω2 ) = (2, 3/4); Case C: (m1 , Ω1 ) = (2, 1/4), (m2 , Ω2 ) = (4, 3/4); and Case D: (m1 , Ω1 ) = (1, 1/2), (m2 , Ω2 ) = (2, 1/2). The accuracy of the MRC/PU L should be remarked here. It can be easily proven that (1/L) l=1 cl = 1 because cl = (ml /Ωl )βC , L l = 1, 2, . . . , L and βC = L( l=1 ml /Ωl )−1 . This implies that the preunification has unity power gain, and therefore, the MRC/PU attempts to keep its output mean and variance unchanged from the mean and variance obtained by the true MRC. Since the secondorder statistics of the MRC reception are taken into investigation, the second moments of the true MRC and the MRC/PU are compared here. On the one hand, the second moment of the  true MRC reception L L can be obtained as E{ΓMRC } = l=1 αl βl = l=1 Ωl . It can be seen that mT · βmin ≤ E{ΓMRC } ≤ mT · βmax βmin = minl {βl ; l = 1, 2, . . . , L} βmax = maxl {βl ; l = 1, 2, . . . , L}. (19) On the other hand, the second moment of the MRC/PU reception can be derived as E{ΓMRC/PU } = mT βC . βC is the harmonic mean of β1 , β2 , . . . , βL , and it can be proven that [15] βmin ≤ βC = Fig. 7. Normalized AFDs. Case A: (m1 , Ω1 ) = (1/2, 1/4), (m2 , Ω2 ) = (1, 3/4); Case B: (m1 , Ω1 ) = (1, 1/4), (m2 , Ω2 ) = (2, 3/4); Case C: (m1 , Ω1 ) = (2, 1/4), (m2 , Ω2 ) = (4, 3/4); and Case D: (m1 , Ω1 ) = (1, 1/2), (m2 , Ω2 ) = (2, 1/2). PU step can facilitate the formulation of closed-form LCR and AFD equations, whereas closed-form statistics for the true MRC reception are very difficult, if not impossible, to derive. B. Unbalanced Dual Branches Figs. 6 and 7 show the normalized LCRs and the normalized AFDs, respectively, obtained using either the true MRC or the MRC/PU in an environment where dual diversity branches are considered to be INID Nakagami fading. In this experiment, Case A is for (m1 , Ω1 ) = (1/2, 1/4), (m2 , Ω2 ) = (1, 3/4); Case B is for (m1 , Ω1 ) = (1, 1/4), (m2 , Ω2 ) = (2, 3/4); Case C is for (m1 , Ω1 ) = (2, 1/4), (m2 , Ω2 ) = (4, 3/4); and Case D is for (m1 , Ω1 ) = (1, 1/2), (m2 , Ω2 ) = (2, 1/2). It can be seen from these figures that the simulation results for the LCR and AFD obtained using the MRC/PU were close to the analytic results 1 β1 + 1 β2 L + ··· + 1 βl ≤ βmax . (20) Therefore, it can be easily obtained that mT · βmin ≤ E{ΓMRC/PU } = mT · βC ≤ mT · βmax . (21) The two equalities in the preceding equation stand when Ω1 /m1 = Ω2 /m2 = · · · = ΩL /mL , which is not necessary to correspond to the IID scenario. When βl = Ωl /ml is a constant for any l = 1, 2, . . . , L, E{ΓMRC/PU } = E{ΓMRC }. The difference between the aforementioned second moments can be found as ! ! 0 ≤ !E{ΓMRC/PU }−E{ΓMRC }! ≤ mT (βmax − βmin ). (22) Although the difference (mT βmax − mT βmin ) confines the inaccuracy of the MRC/PU, the inconsistency among βl , ∀l = 1, 2, . . . , L, inevitably results in a drift (or a bias) of the LCR and AFD curves from those obtained using the true MRC. This may be observed from the aforementioned simulations. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 2, FEBRUARY 2012 V. C ONCLUSION Novel analytic expressions for the LCR and AFD in MRC-like reception over INID Nakagami-fading branches have been thoroughly studied. Using the proposed approach, generalized closed-form expressions can be derived for any number of diversity branches and for IID or INID Nakagami-fading branches. Furthermore, the analytic solutions derived in this paper have been carefully validated by comprehensive simulation. The closed-form statistics that have been proposed here not only feature high accuracy and easy calculation but also provide a reliable benchmark for evaluating the effectiveness of diversity-reception systems. R EFERENCES [1] M. D. Yacoub, J. E. Bautista, and L. G. D. R. Guedes, “On higher order statistics of the Nakagami-m distribution,” IEEE Trans. Veh. Technol., vol. 48, no. 3, pp. 790–794, May 1999. [2] M. D. Yacoub, C. R. C. M. da Silva, and J. E. Vargas Bautista, “Secondorder statistics for diversity-combining techniques in Nakagami-fading channels,” IEEE Trans. Veh. Technol., vol. 50, no. 6, pp. 1464–1470, Nov. 2001. [3] C. D. Iskander and P. T. Mathiopoulos, “Analytical level crossing rates and average fade durations for diversity techniques in Nakagamifading channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1301–1309, Aug. 2002. [4] X. Dong and N. C. Beaulieu, “Average level crossing rate and average fade duration of selection diversity,” IEEE Commun. Lett., vol. 5, no. 10, pp. 396–398, Oct. 2001. [5] X. Dong and N. C. Beaulieu, “Average level crossing rate and average fade duration of low-order maximal ratio diversity with unbalanced channels,” IEEE Commun. Lett., vol. 6, no. 4, pp. 135–137, Apr. 2002. [6] D. Li and V. K. Prabhu, “Second order statistics for maximal-ratio combining in unbalanced Nakagami channels,” in Proc. IEEE GLOBECOM, 2004, pp. 3399–3403. [7] H. Rutagemwa, V. Mahinthan, J. W. Mark, and X. Shen, “Second order statistics of non-identical Nakagami fading channels with maximalratio combining,” in Proc. IEEE GLOBECOM, Nov. 27–Dec. 1, 2006, pp. 1–5. [8] D. B. D. Costa, J. C. S. S. Filho, M. D. Yacoub, and G. Fraidenraich, “Crossing rates and fade durations for diversity-combining schemes over α–µ fading channels,” IEEE Trans. Wireless Commun., vol. 6, no. 12, pp. 4263–4267, Dec. 2007. [9] D. B. D. Costa, M. D. Yacoub, and J. C. S. S. Filho, “An improved closedform approximation to the sum of arbitrary Nakagami-m variates,” IEEE Trans. Veh. Technol., vol. 57, no. 6, pp. 3854–3858, Nov. 2008. [10] G. K. Karagiannidis, K. C. Sagias, and T. A. Tsiftsis, “Closed-form statistics for the sum of squared Nakagami-m variates and its applications,” IEEE Trans. Commun., vol. 54, no. 8, pp. 1353–1359, Aug. 2006. [11] M. Nakagami, “The m-distribution a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, Ed. Elmsford, NY: Pergamon, 1960. [12] W. C. Y. Lee, “Statistical analysis of the level crossings and duration of fades of the signal from an energy density mobile radio antenna,” Bell Syst. Tech. J., vol. 46, pp. 417–448, Feb. 1967. [13] M. R. Spiegel, Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1968. [14] T. M. Wu, “Generation of Nakagami-m fading channels,” in Proc. IEEE Veh. Technol. Conf., May 2006, vol. 6, pp. 2787–2792. [15] Y.-L. Chou, Statistical Analysis. New York: Holt Int., 1969. 865 Transmit Antenna Selection OFDM Systems With Transceiver I/Q Imbalance Behrouz Maham, Member, IEEE, and Olav Tirkkonen, Senior Member, IEEE Abstract—One of the serious imperfections affecting orthogonal frequency-division multiplexing (OFDM) systems is transceiver in-phase/ quadrature-phase (I/Q) imbalance. In this paper, the effect of I/Q imbalances on the transmit antenna selection (TAS) OFDM system is studied. The optimum antenna selection in presence of transmit and receive I/Q imbalances are proposed. We derive closed-form expressions for ergodic capacity of the system with transmit and receive I/Q imbalances. Moreover, some tight and simple lower and upper bounds for the ergodic capacity of the TAS OFDM system are derived. The simplicity of the proposed formula gives insight into the performance and optimization of the system. Finally, the analytical results are confirmed by simulations. Index Terms—Ergodic capacity, in-phase/quadrature-phase (I/Q) imbalance, multiple-antenna processing, orthogonal frequency-division multiplexing (OFDM). I. I NTRODUCTION Multiple-antenna–multiple-output (MIMO) orthogonal frequencydivision multiplexing (OFDM) systems, involving clever processing in both spatial and frequency domains, have been proposed for WiFi, WiMax, and fourth-generation cellular systems, as well as the IEEE 802.16 standard for wireless Internet access [1]. In particular, transmit antenna selection (TAS) is a low-complexity MIMO approach that selects a single transmit antenna to maximize the signal-to-noise ratio (SNR) at the receiver [2], [3]. It was shown in [4] and [5] that TAS preserves the same diversity order as conventional MIMO that utilizes all the transmit antennas. This diversity order was found to be a product of the number of antennas at the source and the destination. A low-cost implementation of such physical layers is desirable in view of mass deployment but challenging due to impairments associated with the analog components. A major source of analog impairments in highspeed wireless communications systems is the in-phase/quadraturephase (I/Q) imbalance [6], [7]. The I/Q imbalance is the mismatch between I and Q balances due to the analog imperfection and introduced both in the up- and downconversion at the transceivers. In general, it is difficult to efficiently and entirely eliminate such imbalances in the analog domain due to power consumption, size, and cost of the devices. Therefore, efficient compensation techniques in the digital baseband domain are needed for the transceivers [8], [9]. The analysis of capacity of single-antenna OFDM systems with transceiver I/Q imbalance is studied in [10]. In this paper, we find a closed-form Manuscript received May 28, 2011; revised October 2, 2011; accepted November 14, 2011. Date of publication December 5, 2011; date of current version February 21, 2012. This work was supported in part by the Research Council of Norway through Project 176773/S10 and in part by the Finnish Funding Agency for Technology and Innovation through the project entitled “DIRTY-RF: Advanced Techniques for RF Impairment Mitigation in Future Wireless Radio Systems.” The review of this paper was coordinated by Prof. W. Choi. B. Maham is with the School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran 14395-515, Iran (e-mail: bmaham@ut.ac.ir). O. Tirkkonen is with the Department of Communications and Networking, Aalto University, 00076 Aalto, Finland, and Nokia Research Center, 00180 Helsinki, Finland (e-mail: olav.tirkkonen@tkk.fi). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2011.2178081 0018-9545/$31.00 © 2012 IEEE