Performance analysis of L ‐branch maximal ratio combining over generalised η − μ fading channels with imperfect channel estimation
2016; Institution of Engineering and Technology; Volume: 10; Issue: 10 Linguagem: Inglês
10.1049/iet-com.2015.1015
ISSN1751-8636
AutoresOsamah S. Badarneh, Fares S. Almehmadi,
Tópico(s)Wireless Communication Networks Research
ResumoIET CommunicationsVolume 10, Issue 10 p. 1175-1182 Research ArticlesFree Access Performance analysis of L-branch maximal ratio combining over generalised η − μ fading channels with imperfect channel estimation Osamah S. Badarneh, Corresponding Author Osamah S. Badarneh obadarneh@ut.edu.sa Electrical Engineering Department, University of Tabuk, 71491 Tabuk, Saudi ArabiaSearch for more papers by this authorFares S. Almehmadi, Fares S. Almehmadi Electrical Engineering Department, University of Tabuk, 71491 Tabuk, Saudi ArabiaSearch for more papers by this author Osamah S. Badarneh, Corresponding Author Osamah S. Badarneh obadarneh@ut.edu.sa Electrical Engineering Department, University of Tabuk, 71491 Tabuk, Saudi ArabiaSearch for more papers by this authorFares S. Almehmadi, Fares S. Almehmadi Electrical Engineering Department, University of Tabuk, 71491 Tabuk, Saudi ArabiaSearch for more papers by this author First published: 01 July 2016 https://doi.org/10.1049/iet-com.2015.1015Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, the authors analyse the performance of maximal-ratio combining in η − μ fading channels under imperfect channel estimation. More specifically, unified and novel analytical expressions for the average symbol error probability (SEP) and average channel capacity are obtained. The average SEP expression is valid for a wide range of coherent and non-coherent modulation schemes. The system performance is analysed and discussed through some representative numerical examples. Furthermore, to validate the correctness of the authors' derivations, the numerical results are compared with Monte-Carlo simulation results. Both results are in excellent agreement over a wide range of average signal-to-noise ratio and different values of the fading parameters. 1 Introduction There are many performance metrics that can be used to measure the quality of service provided by wireless communication systems. The average symbol error probability (SEP) [or average bit error probability (BEP)] and the average channel capacity are two important metrics. During the past years, there has been several research studies on evaluating the average SEP/BEP and the average channel capacity over various types of fading channels [1-12]. There are two approaches to evaluate the average SEP/BEP, namely, probability density function (PDF) approach and the moment-generating function (MGF) approach [13]. The PDF-based approach relies on finding the PDF of the total instantaneous signal-to-noise ratio (SNR) for a given communication system. If the PDF distribution of the total instantaneous SNR is found, then it can lead to a closed-form expressions for the average SEP/BEP. Note that, finding the PDF of the total instantaneous SNR in a simple form is typically feasible if the paths are independent and identically distributed (i.i.d.). However, finding the PDF of the total instantaneous SNR is more difficult if the paths are independent but not necessarily identically distributed (i.n.i.d.). In the latter case, the MGF-based approach is adopted. However, the MGF-based approach, first, requires to find the MGF for the PDF of the instantaneous SNR, and then, perform integral operation to obtain a desired expression for the average SEP/BEP. The average BEP of binary coherent signalling, pulse-amplitude modulation (PAM), and quadrature-amplitude modulation (QAM) coherent signalling over extended generalised-K fading channels subject to additive white generalised Gaussian noise are derived and evaluated in [1, 2]. Khatalin and Fonseka [3] investigated the average channel capacity in Rice and Hoyt fading channels with maximal-ratio combining (MRC) diversity and selection combining diversity. In [5], the average SEP of general rectangular QAM signal for single-input multiple-output system over i.n.i.d. Nakagami-q (Hoyt) fading channels. In [5], the authors provided an expression of the average channel capacity for arbitrary values of η and μ in terms of infinite series and Meijer's G-function. However, for a given accuracy, the number of terms in the series depends on the values of η and μ. A closed-form expression for average channel capacity, which is only valid for integer values of μ, is derived in [6]. In [7], Ermolova obtained closed-form expressions for the average symbol error rates of rectangular QAM modulation, based on MGF approach, in terms of Appell's and Lauricella's hypergeometric functions over the generalised η − μ and κ − μ distributions. The performance of MRC over η − μ fading channels has been reported in [8-10, 14, 15]. Milisic et al. [8] analysed the performance of L-branch MRC for non-coherent modulation over i.i.d. η − μ fading channels. The average SEP of various digital modulation schemes, such as M-ary phase shift keying (M-PSK), M-ary differential phase shift keying, and rectangular QAM, with MRC diversity over L i.n.i.d. η − μ fading channels is derived and evaluated in [9, 10]. The average SEP of arbitrary M-QAM constellations in MRC schemes over non-identical correlated channels is analysed in [14]. Dixit and Sahu [15] derived approximate expressions for the average SEP over L-branch MRC for QAM scheme over i.i.d η − μ and κ − μ fading channels. In all works [7-10], the authors derived the average SEP and average BEP using the MGF approach where the average SEP and average BEP are expressed in terms of Appell's Lauricella's hypergeometric functions. Note that, Appell's Lauricella's hypergeometric functions are evaluated numerically using their series or integral representation. In this paper, using the PDF approach, we derive exact closed-form expressions for the average SEP and the average channel capacity in MRC scheme over i.i.d. η − μ fading channels under imperfect channel estimation. Different from existing works [7-10, 14, 15], the average SEP/BEP for a wide range of coherent modulation schemes is obtained in a very simple and efficient way, using the PDF-based approach, when compared with the existing works. The derived expression for the average SEP is novel and in closed form and is expressed in terms of the well-known Fox's/Meijer's functions of two variables (bivariate Fox's H-function/Meijer's G-function). Note that, the bivariate Meijer's G-function/Fox's H-function can be accurately and efficiently computed using the algorithms presented in [16, 17], respectively. In addition, they are easier to compute compared with Appel's and Lauricella's hypergeometric functions. The derived expression is valid for any arbitrary values of η and μ. Furthermore, a closed-form expression for the average channel capacity is also obtained. In addition, the average channel capacity over Nakagami-q (Hoyt) fading channels is obtained in closed form through our results as a special case, where it is only available in the literature in terms of an infinite series. The rest of this paper is organised as follows. In Section 2, we define the channel models under consideration. In Section 3, we present our mathematical analysis. We then present some numerical results in Section 4. Finally, Section 5 concludes this paper. 2 System and channel models 2.1 System model We consider transmitting a signal s(t), with a unity variance, over slow and flat fading channels with an η − μ distribution. The received signal at the ℓ th path, (where ℓ = 1, 2, …, L), over a symbol duration Ts second can be written as (1)where αℓ = rℓexp (−jϕℓ). rℓ and ϕℓ are random variables (RVs) that represent the envelop and the phase of the η − μ fading channel, respectively. nℓ(t) represents the complex additive white Gaussian noise (AWGN) with zero mean and double-sided power spectral density of N0/2. Es denotes the transmitted energy. In practical system, the channel αℓ is not known at the receiver, and a channel estimation algorithm is used to obtain an estimate channel . The channel estimation error is defined as [18, 19] (2)The channel estimation error is assumed to be complex Gaussian random noise with . We assume that and αℓ are jointly ergodic and stationary processes. 2.2 Channel model The η − μ distribution describes the small-scale fluctuations in a non-line-of-sight environment and is originally proposed in two formats, namely, format I and format II [20]. In contrast to format 1, format 2 assumes that the in-phase and quadrature components of the fading channel within each cluster are correlated with each other and they have identical powers. The PDF of the instantaneous SNR γ of the η − μ distribution is expressed as (3)where is the gamma function, denotes the average SNR, Iv(·) is the modified Bessel function of the first kind with order v, and μ > 0 represents the number of multi-path in each cluster. The parameters h and H are functions of the fading parameter η, where . The parameter η is represented in two different formats: in format 1, it represents the power ratio between the in-phase and quadrature scattered waves of the fading signal within each cluster. In this case, h = (1 + η)2/4η and H = 1 − η2/4η. In format 2, the parameter η, where η ∈ (−1, 1), represents the correlation coefficient between the in-phase and quadrature phase components within each cluster with h = 1/1 − η2 and H = η/1 − η2. 2.3 Performance analysis 2.4 Average SEP analysis 2.4.1 Coherent detection For a large variety of coherent modulation schemes, a unified conditional symbol error rate can be expressed in terms of Gaussian Q-function [13], that is (4)where the value of the parameters ac and bc depend on the modulation schemes (refer to Table 1), is the Q-function, and γℓ is given by (5)with (6) Table 1. Parameters αc and bc for different modulation schemes Modulation scheme ac bc BPSK 1 BFSK 1 1 4-QAM, QPSK 2 1 rectangular M-QAM non-rectangular M-QAM 4 M-PSK 2 M-PAM The proof of (5) is provided in Appendix 1. According to [20], the sum of L i.i.d. squared η − μ RVs with parameters η, μ, and is also an η − μ RV with different parameters η, Lμ, and . As such, one can easily obtain the PDF of the SNR at the output of an MRC receiver as (7)where and is given in (6). The proof of (7) is provided in Appendix 2. The average SER can be obtained by performing the required statistical average in (4), that is (8)Plugging (7) into (8) yields (9)However, in order obtain a closed-form expression for Pr(e) in (8), we express the PDF in (7) in terms of the well-known Fox's H-function as in Lemma 1. Lemma 1.The PDF of the η − μ fading channels, in (7), can be expressed in terms of Fox's H-function as (10)where is the Fox's H-function. Proof.By multiplying the PDF in (7) by the quantity , the PDF becomes (11)Now, by using the representation of e−xIv(x) in terms of Meijer's G-function [21, Eq. (8.4.22.3)], the PDF of the η − μ can be expressed as (12) Finally, using the relation between the Meijer's G-function and the Fox's H-function [21, Eq. (8.3.2.21)], the expression in (10) follows. Theorem 1.For arbitrary values of η and μ, the average SEP for different coherent modulation schemes can be expressed as in (19). Proof.In order to obtain the desired result in (19), we represent the Gaussian Q-function in terms of Fox's H-function as follows: using the relation between the complementary error function erfc(x) and the Gaussian Q-function, , and with the help of [21, 8.4.14.2] and [21, Eq. (8.3.2.21)], the Gaussian Q-function in (8) can be represented in terms of Fox's H-function as (13) Now, substituting (10) and (13) into (8) yields (18). Finally, with the help of [22, Eq. (2.6.2)] the integral in (18) is solved in a closed form in terms of the bivariate Fox's H-function as in (19), where is the generalised (bivariate) Fox's H-function which is defined in terms of a double Mellin–Barnes-type integral as in (21). To the best of the author's knowledge (19) is novel and new. Note that using [22, Eq. (2.3.1)], the average SEP in (19) can be expressed in terms of bivariate Meijer's G-function as in (20). The bivariate Fox's H-function in (21) converges if the following conditions are satisfied [22] (14) (15) (16) (17)However, it is straightforward to show that the parameters of the Fox's H-function in (19) and (35) satisfy these sufficient conditions, and therefore the Fox's H-function converges (18) (19) (20)where (21)and (22) (23) (24) 2.4.2 Non-coherent detection For non-coherent modulation schemes such as DBPSK and M-ary non-coherent frequency shift keying, the average SEP can be calculated as (25)where (26)The non-negative parameters an and bn depend on the type of the modulation scheme (refer to Table 2). Table 2. Parameters an and bn for non-coherent modulation schemes Modulation scheme an bn BFSK DBPSK 1 M-FSK Theorem 2.For arbitrary values of η and μ, the average SEP for non-coherent modulation schemes can be expressed as in (28). Proof.Substituting (7) and (26) into (25) yields (27)With the help of the direct Laplace transform [23, Eq. (3.15.1.3)], and after some mathematical manipulation, the average SEP for non-coherent detection can be obtained in a closed form as (28) 2.4.3 Asymptotic analysis An asymptotic expression, in case of coherent and non-coherent detections, for the average SEP is obtained in what follows. According to [24], the asymptotic average SEP can be derived based on the behaviour of the PDF of the instantaneous SNR γ around the origin. As such and by using Taylor's series, the fλ(γ) given in (7) can be rewritten as (29)where stands for higher order terms. Case 1: For the coherent modulation, substituting (29) into (8), solving the integral, making use of the identity [25, Eq. (8.335.1)] and after some straightforward mathematical manipulations, the asymptotic SEP can be obtained as (30) Case 2: For the non-coherent modulation, substituting (29) into (25) and then solving the integral, an expression for the asymptotic SEP can be written as (31) 2.5 Average channel capacity analysis The average channel capacity can be obtained by averaging the capacity of an AWGN channel over the instantaneous received SNR γ [26], that is (32)where W represents the bandwidth of the fading channel and fλ(γ) is given in (7). To obtain a closed-form expression for in (32), we, first, represent the logarithm function ln(1 + γ) in terms of Fox's H-function using [21, Eq. (8.4.6.5)] and [21, Eq. (8.3.2.21)], respectively, that is (33)Now, plugging (10) and (33) into (32), the average channel capacity over the η − μ fading channel can be given as in (34). Finally, a closed-form expression for the average channel capacity over the η − μ fading channel is obtained in (35) after using [22, Eq. (2.6.2)]. To the best of the author's knowledge, (35) is novel and new. It is worth mentioning here that, setting η = q2 (result in h = [1 + q2]2/4q2 and H = [1 − q4]/4q2) and μ = 1/2 in (35), a closed-form expression of the average channel capacity of the well-known Nakagami-q (Hoyt) fading channel is obtained as in (37). However, there is no closed form for this channel has been reported in the literature (34) (35) (36) (37) (38) 3 Analytical and simulation results To verify the accuracy of the proposed mathematical analysis, the average SEP for coherent BPSK and M-QAM modulation schemes, non-coherent DBPSK, and the normalised average channel capacity , are numerically evaluated using the derived expressions in (20), (28), and (36), respectively. Furthermore, the analytical results are then compared with Monte-Carlo simulations results. An accurate implementation of the bivariate Meijer's G-function is given in [16]. Fig. 1 depicts the average BEP for coherent BPSK modulation over η − μ fading channels. Fig. 1 is plotted for L = 1 (no diversity) and for perfect channel estimation (i.e. ). This figure is provided here for proving the correctness of our derivations since it represents the results that have been previously reported in [27, Fig. 1]. It is seen that our results exactly match those obtained in [27, Fig. 1] and are with excellent agreement with Monte-Carlo simulations, which validates our mathematical analysis. Fig. 1Open in figure viewerPowerPoint Average BEP for coherent BPSK modulation in η − μ fading channels. L = 1, Fig. 2 plots the average BEP for BPSK modulation for η = 0.05, μ = 1, L = 1,2,3, and under imperfect channel estimation. As expected, the BEP improves as the number of branches increases. Analytical results demonstrate close match to simulation results for wide range of SNR values. Fig. 2Open in figure viewerPowerPoint Average BEP for coherent BPSK modulation with η = 0.05, μ = 1. Fig. 3 plots the average SEP of M-QAM modulation schemes, for M = 4, 8, 16, 32. The fading parameters are η = 0.8 and μ = 1.5. It can be seen that the analytical results match the Monte-Carlo results. This figure is plotted to validate our analysis and confirm that our analysis is unified and, hence, it can be applied to study the performance of other coherent modulation schemes. Fig. 3Open in figure viewerPowerPoint Average SEP of M-QAM over η − μ fading channels with η = 0.8 and μ = 1.5. L = 1, In Fig. 4, the average SEP of the non-coherent M-ary frequency shift keying (M-FSK) modulation is depicted. The figure shows the effect of channel estimation error and MRC on the system performance. As anticipated, the systems degrade when imperfect channel estimation is considered at the receiver. The figures also show great matching between analytical and simulation results which validates our analysis. Fig. 4Open in figure viewerPowerPoint Average SEP of non-coherent 8-FSK in η − μ fading channels. η = 0.3, μ = 2.5, an = 3.5, and bn = 0.5 Fig. 5 plots the normalised average channel capacity (i.e. spectral efficiency) as a function of the average SNR per symbol using (35). The analytical results are in excellent agreement with the Mote-Carlo simulations, which validates our derivations. Furthermore, it should be pointed out that Fig. 5 has been previously reported in [5, Fig. 2]. It is seen that the results in [5, Fig. 2] match our results, which validate our findings too. Fig. 5Open in figure viewerPowerPoint Average channel capacity in η − μ fading channels. L = 1, Fig. 6 depicts the effect of the imperfect channel estimation and the effect of the number of branches L on the spectral efficiency. One can note that has a great impact on the spectral efficiency. For example, the value of the spectral efficiency for L = 3 and is ∼60% larger compared with the corresponding case of L = 3 and . In addition, the figure shows that as the number of branches L increases, the spectral efficiency increases especially at high values of average SNR. For example, the value of the spectral efficiency for L = 3 is ∼140% larger compared with the corresponding case of L = 1. Fig. 6Open in figure viewerPowerPoint Average channel capacity with η = 0.05, μ = 0.75 Figs. 7 and 8 represent the average BEP (for coherent BPSK modulation) and the spectral efficiency, respectively, with no diversity (i.e. L = 1) and under perfect channel estimation (i.e. ) over Nakagami-q (so-called Hoyt) fading channels. The two figures are generated from our results as a special case of the η − μ fading channels, where η = q2 and μ = 0.5. Both figures demonstrate that the system performance improves as q increases. Fig. 7Open in figure viewerPowerPoint Average BEP for BPSK over Nakagami-q (Hoyt) fading channel. L = 1, Fig. 8Open in figure viewerPowerPoint Average channel capacity over Nakagami-q (Hoyt) fading channel. L = 1, 4 Conclusion This paper derives novel and generic closed-form expressions for the average SEP and the average channel capacity over i.i.d. η − μ fading channel with MRC and under imperfect channel estimation. These expressions are derived in terms of Fox's H-function/Meijer's G-function of two variables. Analytical results are sustained through Monte-Carlo simulation results and close match is reported for wide range of SNR and for different system parameters. 6 Appendices 6.1 Appendix 1: Proof of (5) In MRC, the received signal of a branch ℓ can be given as [28] (39)where (·)* denotes the complex conjugate operator. Using (1), (2), and (40), the combined signal can be decomposed into the signal and noise parts, that is (40)Now using (40) and after some mathematical manipulations, then the average SNR – the ratio between the average power of the instantaneous signal part and that of the noise part – for the MRC can be written as in (5). Thus, the proof is completed. 6.2 Appendix 2: Proof of (7) The envelope R of the η − μ distribution is given as (41)where Xi and Yi are mutually independent Gaussian processes with equal mean and different variance, i.e. and , and n is the number of clusters of multi-path. The total power of the ith cluster is given as (42)The PDF of γi in (42) can be easily found by following the standard procedure [20], therefore (43)where I0(·) is the modified Bessel function of the first kind and order zero, is the scattered-wave power ratio between the in-phase and quadrature components of each cluster of multi-path. The Laplace transform of the PDF in (44) can be found using [23, 3.15.1.3], that is (44)where denotes the Laplace operator and s is the Laplace variable. Knowing that γi (where i = 1, 2, …, n) are independent RVs, the PDF of with the aid of (44) can be given as (45)Now, for L-branch MRC (with equally likely transmitted symbols) the total SNR per symbol at the output of the MRC can be written as [13] (46)where γj is instantaneous SNR in jth branch of L-branch MRC receiver. Using (41) and (42), (46) can be rewritten as (47)For independent RVs of γ, the Laplace transform of the PDF of the RV γ, with the help of (44), can be obtained as (48)Finally, the PDF of γ can be found by using the inverse Laplace transform [29, Eq. (29.3.60)], hence, the proof is completed. References 1Soury H.Yilmaz F.Alouini M.: 'Average bit error probability of binary coherent signaling over generalized fading channels subject to additive generalized Gaussian noise', IEEE Commun. Lett., 2012, 16, (6), pp. 785– 788 (doi: 10.1109/LCOMM.2012.040912.112612) 2Soury H.Yilmaz F.Alouini M.: 'Error rates of M-PAM and M-QAM in generalized fading and generalized Gaussian noise environments', IEEE Commun. Lett., 2013, 17, (10), pp. 1932– 1935 (doi: 10.1109/LCOMM.2013.081913.131409) 3Khatalin S.Fonseka J.P.: 'On the channel capacity in Rician and Hoyt fading environments with MRC diversity', IEEE Trans. Veh. Technol., 2006, 55, (1), pp. 137– 141 (doi: 10.1109/TVT.2005.861205) 4Lei X.Fan P.Chen H.: ' SEP of general rectangular QAM signal with MRC diversity over Nakagami-q (Hoyt) fading channels'. IET Second Int. Conf. on Wireless, Mobile and Multimedia Networks (ICWMMN 2008), 2008 5da Costa D.B.Yacoub M.D.: 'Average channel capacity for generalized fading scenarios', IEEE Commun. Lett., 2007, 11, (12), pp. 949– 951 (doi: 10.1109/LCOMM.2007.071323) 6Peppas K.P.: 'Capacity of η − μ fading channels under different adaptive transmission techniques', IET Commun., 2010, 4, (5), pp. 532– 539 (doi: 10.1049/iet-com.2009.0409) 7Ermolova N.Y.: 'Useful integrals for performance evaluation of communication systems in generalised η − μ and κ − μ fading channels', IET Commun., 2009, 3, (2), pp. 303– 308 (doi: 10.1049/iet-com:20080189) 8Milisic M.Hamza M.Behlilovic N. et al.: ' Symbol error probability analysis of L-branch maximal-ratio combiner for generalized η − μ fading'. IEEE 69th Vehicular Technology Conf., 2009, VTC Spring 2009, 2009, pp. 1– 5 9Peppas K.Lazarakis F.Alexandridis A. et al.: 'Error performance of digital modulation schemes with MRC diversity reception over η − μ fading channels', IEEE Trans. Wirel. Commun., 2009, 8, (10), pp. 4974– 4980 (doi: 10.1109/TWC.2009.081687) 10Yu H.Wei G.Ji F. et al.: 'On the error probability of cross-QAM with MRC reception over generalized η − μ fading channels', IEEE Trans. Veh. Technol., 2011, 60, (6), pp. 2631– 2643 (doi: 10.1109/TVT.2011.2154347) 11Badarneh O.S.: 'Error rate analysis of M-ary phase shift keying in α − η − μ fading channels subject to additive Laplacian noise', IEEE Commun. Lett., 2015, 19, (7), pp. 1253– 1256 (doi: 10.1109/LCOMM.2015.2423277) 12Badarneh O.S.Kadoch M.Atawi I.E.: ' On the average bit error rate and average channel capacity over generalized fading channels'. IEEE Int. Conf. on Ubiquitous Wireless Broadband (ICUWB), 2015, pp. 1– 5 13Simon M.K.Alouini M.: ' Digital communication over fading channels' ( Wiley, New York, 2005, 2nd edn.) 14Asghari V.da Costa D.B.Äıssa S.: 'Symbol error probability of rectangular QAM in MRC systems with correlated η − μ fading channels', IEEE Trans. Veh. Technol., 2010, 59, (3), pp. 1497– 1503 (doi: 10.1109/TVT.2009.2037638) 15Dixit D.Sahu P.R.: 'Performance of L-branch MRC receiver in η − μ and κ − μ fading channels for QAM signals', IEEE Wirel. Commun. Lett., 2012, 1, (4), pp. 316– 319 (doi: 10.1109/WCL.2012.042512.120240) 16Ansari I.S.Al-Ahmadi S.Yilmaz F. et al.: 'A new formula for the BER of binary modulations with dual-branch selection over generalized-K', IEEE Trans. Commun., 2011, 59, (10), pp. 2654– 2658 (doi: 10.1109/TCOMM.2011.063011.100303A) 17Peppas K.P.: 'A new formula for the average bit error probability of dual-hop amplify-and-forward relaying systems over generalized shadowed fading channels', IEEE Wirel. Commun. Lett., 2012, 1, (2), pp. 85– 88 (doi: 10.1109/WCL.2012.012712.110092) 18Badarneh O.S.Mesleh R.: 'Performance analysis of space modulation techniques over α − μ and κ − μ fading channels with imperfect channel estimation', Trans. Emerg. Telecommun. Technol., 2015, pp. 1– 13, doi: 10.1002/ett.2940 19Mesleh R.Badarneh O.S.Younis A. et al.: 'Performance analysis of spatial modulation and space-shift keying with imperfect channel estimation over generalized η − μ fading channels', IEEE Trans. Veh. Technol., 2015, 64, (1), pp. 88– 96 (doi: 10.1109/TVT.2014.2321059) 20Yacoub M.D.: 'The κ − μ distribution and the η − μ distribution', IEEE Antennas Propag. Mag., 2007, 49, (1), pp. 68– 81 (doi: 10.1109/MAP.2007.370983) 21Prudnikov A.P.Brychkov Y.A.Marichev O.I.: ' Integrals, and series: more special functions' ( Gordon & Breach Science Publishers, New York, 1990), vol. 3 22Mathai A.Saxena R.: ' The H-function with applications in statistics and other disciplines' (Wiley Eastern, New Delhi, Halsted Press, New York, 1978) 23Prudnikov A.P.Brychkov Y.A.Marichev O.I.: ' Integrals, and series: direct Laplace transforms' ( Gordon & Breach Science Publishers, New York, 1992), vol. 4 24Wang Z.Giannakis G.B.: 'A simple and general parameterization quantifying performance in fading channels', IEEE Trans. Commun., 2003, 51, (8), pp. 1389– 1398 (doi: 10.1109/TCOMM.2003.815053) 25Gradshteyn I.S.Ryzhik I.M.: ' Table of integrals, series, and products' ( Academic Press, California, 2007, 7th edn.) 26Lee W.: 'Estimate of channel capacity in Rayleigh fading environment', IEEE Trans. Veh. Technol., 1990, 39, (3), pp. 187– 189 (doi: 10.1109/25.130999) 27da Costa D.B.Yacoub M.D.: 'Moment generating functions of generalized fading distributions and applications', IEEE Commun. Lett., 2008, 12, (2), pp. 112– 114 (doi: 10.1109/LCOMM.2008.071619) 28Cho Y.S.Kim J.Yang W.Y. et al.: ' MIMO-OFDM wireless communications with MATLAB' ( John Wiley & Sons (Asia) Pte Ltd, 2010, 1st edn.) 29Abramnowitz M.Stegun I.A.: ' Handbook of mathematical functions' ( US Department of Commerce, National Bureau of Standards, Washington, DC, 1972, 1st edn.) Citing Literature Volume10, Issue10July 2016Pages 1175-1182 FiguresReferencesRelatedInformation
Referência(s)