Average Channel Capacity for Generalized Fading Scenarios
IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 12, DECEMBER 2007
949
Average Channel Capacity for Generalized Fading Scenarios Daniel Benevides da Costa, Student Member, IEEE, and Michel Daoud Yacoub, Member, IEEE
Abstract— Novel expressions for the average channel capacity (ACC) of single-branch receivers operating in generalized fading channels, namely η-µ and κ-µ, are derived. The expressions are written in terms of the well-known Meijer’s G-functions, which are easily implemented in the most popular computing softwares. In addition, it is shown that, for any given fading condition, the Nakagami-m ACC defines regions of capacity: it lowerbounds the η-µ ACC and it upperbounds the κ-µ ACC. In the same way, it lowerbounds the α-µ ACC for α < 2 and µ > m, and it upperbounds the α-µ ACC for α > 2 and µ < m.
R2 Es /N0 , where Es and N0 denote the average symbol energy and the single-sided power density, respectively, and R stands for the envelope. Then [1] ∞ CW log2 (1 + γ)fΥ (γ)dγ (1) 0
In what follows, (1) shall be derived for the η-µ and κ-µ fading scenarios. For the α-µ one, the ACC, as derived in [6], is reproduced here for convenience.
Index Terms— Average channel capacity, generalized fading channels.
A. α-µ Fading Scenario I. I NTRODUCTION
T
HE study of average channel capacity (ACC) in fading scenarios have been given special attention over the years [1]–[6]. As well known, such a metric provides an upper bound for the maximum transmission rate and was initially established for the Gaussian environment. Recently, in [3], [4], [6] the ACC for single-branch receivers operating in Hoyt (Nakagami-q), Nakagami-m, Rice, Weibull, and in Generalized Gamma fading channels has been derived. In this paper, we derive novel expressions for the average channel capacity of single-branch receivers operating in generalized fading channels, namely η-µ and κ-µ [7]. The expressions are written in terms of the well-known Meijer’s G-functions [8, Eq. 9.301], which are easily implemented in the most popular computing softwares. In addition, this Letter also shows a very interesting result, which maintains that, for any given fading condition, the Nakagami-m ACC defines regions of capacity: it lowerbounds the η-µ ACC and it upperbounds the κ-µ ACC. In the same way, capitalizing on a result available in the literature in which the Generalized Gamma ACC is derived [6], it is shown that the Nakagamim ACC lowerbounds the average channel capacity of the α-µ fading scenario [9] for α < 2 and µ > m and it upperbounds it for α > 2 and µ < m. For comparison purposes, the capacity of the additive white Gaussian noise (AWGN) channel is depicted. II. AVERAGE C HANNEL C APACITY Let W be the fading channel bandwidth and C its ACC. Define fΥ (·) as the probability density function (PDF) of the instantaneous signal-to-noise ratio (SNR) per symbol Υ = Manuscript received August 8, 2007. The associate editor coordinating the review of this letter and approving it for publication was Prof. Ioannis Psaromiligkos. This work was partly supported by FAPESP (05/59259-7). The authors are with the Wireless Technology Laboratory (WissTek), Department of Communications, School of Electrical and Computation Engineering, State University of Campinas, DECOM/FEEC/UNICAMP, PO Box 6101, 13083-852 Campinas, SP, Brazil (e-mail: {daniel, michel}@wisstek.org). Digital Object Identifier 10.1109/LCOMM.2007.071323.
The α-µ fading scenario comprises both Weibull (µ = 1) and Nakagami-m (α = 2) [9]. In such an environment, α > 0 models the non-linearity of the propagation medium, whereas µ > 0 denotes the number of multipath clusters. The ACC for an α-µ fading scenario is given as [6] 1 W √ k+αk−3 ln(2) k β αµ 2 Γ(µ)(2π) 2 αk αµ αk −αk αµ 2 β Φ , Φ ,− , 1 − k(α+1), αk 2 2 2 2 2 αk αµ × Gαk,k(α+1) αµ k k Φ(k, 0), Φ αk 2 ,− 2 ,Φ 2 ,− 2 (2) ∞ z−1 where Γ(z) = 0 t exp(−t)dt is the gamma function, β = γ¯ Γ(µ)/Γ(µ + 2/α), γ¯ = E (Υ), E(·) denotes expectation, Φ(n, ) /n, ( + 1)/n, . . . , ( + n − 1)/n, with being an arbitrary real value and n a positive integer. In the same way, αk/2 and k are positive integers, although α may assume any positive value. (Note that k is arbitrary and is chosen so that αk/2 be integer.)
C=
B. η-µ Fading Scenario The η-µ fading scenario comprises both Hoyt (µ = 0.5) and Nakagami-m (η → 0, η → ∞, η → ±1) [7]. The corresponding PDF is presented in two Formats. In Format 1, the in-phase and quadrature components of the fading signal within each cluster are assumed to be independent from each other and to have different powers, with the parameter 0 < η < ∞ given by the ratio between them. In Format 2, the inphase and quadrature components of the fading signal within each cluster are assumed to have identical powers and to be correlated with each other, with the parameter −1 < η < 1 representing the correlation coefficient between them. For both formats, the parameter µ denotes the number of multipath clusters. From [7, Eq. 27], the PDF of Υ can be obtained, after a transformation of variates, as √ 1 1 2µHγ 2 π µµ+ 2 hµ γ µ− 2 2µγh − fΥ (γ) = Iµ− 12 1 1 exp γ¯ γ¯ Γ(µ)H µ− 2 γ¯ µ+ 2 (3)
c 2007 IEEE 1089-7798/07$25.00
950
IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 12, DECEMBER 2007
where Iν [·] is the modified Bessel function of the first kind and arbitrary order ν [10, Eq. 9.6.20], h and H are functions of η and varies from one format to another [7]. More specifically, in Format 1, h = (2+η −1 +η)/4 and H = (η −1 −η)/4, whereas, in Format 2, h = 1/(1 − η 2 ) and H = η/(1 − η 2 ). Using an infinite series representation for Iν (·) [8, Eq. 8.445], and 1 replacing this and (3) in (1), an integral with integrands γ µ− 2 , ln(1 + γ), and exp(−2µhγ/¯ γ ) appears. We then express the logarithm and exponential functions in terms of the Meijer’s 1,1 G-functions [11, Eq. 11], i.e., ln(1 + γ) = G1,2 2,2 [γ|1,0 ] and γ |− exp(−2µhγ/¯ γ ) = G1,0 0 ]. Knowing that the integral 0,1 [2µhγ/¯ from the product of a power and two Meijer’s G-functions is also a Meijer’s G-function [11, Eq. 21], the ACC for an η-µ fading scenario is obtained as 2n √ ∞ H 2−2n−2µ W 2 π
C= ln(2) Γ(µ)hµ n=0 n! Γ n + µ + 12 h 0, 1 3,1 2µh (4) × G2,3 γ¯ 2n + 2µ, 0, 0
Fig. 1. Average channel capacity of a single-branch receiver undergoing α-µ fading.
C. κ-µ Fading Scenario The κ-µ fading scenario comprises both Rice (µ = 1) and Nakagami-m (κ → 0) [7]. The multipath clusters are assumed to have the scattered waves with identical powers but within each cluster a dominant component is found that presents an arbitrary power. The parameter κ is defined as the ratio between the total power of the dominant components and the total power of the scattered waves, whereas the parameter µ denotes the number of multipath clusters. From [7, Eq. 11], the PDF of Υ can be obtained, after a transformation of variates, as µ+1 µ−1 µ(1 + κ) 2 γ 2 µ(1 + κ)γ fΥ (γ) = µ−1 − µ+1 exp γ¯ κ 2 exp(µκ) γ¯ 2
κ(1 + κ)γ (5) × Iµ−1 2µ γ¯ Following the same procedure as before, the ACC for a κ-µ fading scenario is given as C=
∞
W (µκ)n 1 ln(2) exp(µκ) n=0 n! Γ(n + µ) µ(1 + κ) 0, 1 (6) × G3,1 2,3 n + µ, 0, 0 γ¯
To the best of the authors’ knowledge, (4) and (6) are new. D. Special Cases The expressions derived here are general and exact. They specialize to some particular cases already available in the literature. The ACC of the Rice scenario, given in [4, Eq. 5], can be attained from (6) by setting µ = 1. The Nakagamim ACC, given in [4, Eq. 3], can be obtained from anyone of the two generalized fading scenarios whose formulations are derived here. More specifically, it arises (i) from (4) for µ = m and η → 0 or η → ∞ in Format 1, and for µ = m and η → ±1 in Format 2; or (ii) from (6) for κ → 0. The ACC of Hoyt fading channels, given in [3], can be obtained in an
Fig. 2. Average channel capacity of a single-branch receiver undergoing η-µ fading Format 1.
exact manner from (4) by setting µ = 0.5. (Note, however, that the result of [3] does not appear in terms of the Meijer’s G-function.) E. Channel Capacity for a Fixed Nakagami m Parameter Equations [9, Eq. 16], [7, Eq. 13], and [7, Eq. 30] relate the Nakagami parameter m with those of α-µ, η-µ and κµ distributions. From these, and as shown in [7], [9], an infinite number of curves of the α-µ, η-µ and κ-µ distributions are found that present the same Nakagami-m parameter. The graphs in [7] reveal that the Nakagami-m cumulative distribution function (CDF) upperbounds the η-µ and lowerbounds the κ-µ CDFs. As for the α-µ CDF, nothing is reported in [9]. We looked for a similar behavior for the α-µ CDF and we found out a very interesting feature, as follows: the Nakagamim CDF upperbounds the α-µ CDF for α < 2 and µ > m and it lowerbounds it otherwise (α > 2 and µ < m). As shown in the next section, the corresponding ACCs present a similar behavior. In this case, however, the Nakagami-m ACC upperbounds the κ-µ ACC, whereas it lowerbounds the η-µ ACC. In the same way, the Nakagami-m ACC lowerbounds
DA COSTA and YACOUB: AVERAGE CHANNEL CAPACITY FOR GENERALIZED FADING SCENARIOS
Fig. 3. Average channel capacity of a single-branch receiver undergoing κ-µ fading.
951
1 axes, µ = 1.5 and α varies from 1 to 3. Note that as α increases, the system capacity also increases. In the set 2 axes, α = 1.5 and µ ranges from 0.8 to 3. Again, a performance improvement is observed as µ augments. Figs. 2 and 3 plot the normalized ACC for the η-µ (Format 1) and κ-µ fading scenarios, respectively, as a function of the average SNR per symbol γ¯ . Varying one parameter and keeping the other constant, a behavior similar to that of the α-µ scenario is observed. Note that the formulas for ACC in these cases are given in terms of infinite series. The number of terms in the series required for a given accuracy varies with the parameters. However, in all of the cases shown here, for an error smaller than 10−6 the number of required terms was not greater than 20. For a fixed Nakagami parameter m = 0.5, Fig. 4 plots the η-µ, κ-µ (set 1 axes), and α-µ ACC (set 2 axes). As commented before, the Nakagami-m ACC divides the fading plane in two regions, in which the lower region is related to the κ-µ fading, whereas the upper region is related to the η-µ fading. As for the α-µ fading case, the upper region is obtained for α ≤ 2 and µ ≥ m, and the lower region is obtained otherwise. R EFERENCES
Fig. 4. Average channel capacity for the α-µ, η-µ, and κ-µ fading scenarios using the same Nakagami-m parameter m (m = 0.5).
the α-µ ACC for α ≤ 2 and µ ≥ m and it upperbounds it otherwise. III. S OME N UMERICAL P LOTS In this section, some representative plots illustrate the normalized ACC, C/W , of single receivers subject to α-µ, η-µ, and κ-µ fading as a function of the average SNR per symbol γ¯ . For each case, in the the same figure, two sets of axes are used: set 1, comprising the left vertical axis and the bottom horizontal axis; set 2, comprising the right vertical axis and the top horizontal axis. Fig. 1 shows the normalized ACC for α-µ fading scenarios as a function of the average SNR per symbol γ¯ . In the set
[1] W. C. Y. Lee, “Estimate of channel capacity in Rayleigh faing environment,” IEEE Trans Veh. Technol., vol. 39, no. 3, pp. 187–189, Aug. 1990. [2] M.-S. Alouini and A. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1165–1181, July 1999. [3] J. Cheng and T. Berger, “Capacity of Nakagami-q (Hoyt) fading channels with channel side information,” in Proc. IEEE Int. Conf. Commun. Technol., vol. 2, 2003, pp. 1915–1918. [4] N. C. Sagias, G. S. Tombras, and G. K. Karagiannidis, “New results for the Shannon channel capacity in generalized fading channels,” IEEE Commun. Lett., vol. 2, no. 9, pp. 97–99, Feb. 2005. [5] S. Khatalin and J. P. Fonseka, “On the channel capacity in Rician and Hoyt fading environments with MRC diversity,” IEEE Trans. Veh. Technol., vol. 55, no. 1, pp. 137–141, Jan. 2006. [6] N. C. Sagias and P. T. Mathiopoulos, “Switched diversity receivers over generalized Gamma fading channels,” IEEE Comm. Lett., vol. 9, no. 10, pp. 871–873, Oct. 2005. [7] M. D. Yacoub, “The κ-µ distribution and the η-µ distribution,” IEEE Ant. Prop. Mag., vol. 49, no. 1, pp. 68–81, Feb. 2007. [8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York: Academic, 2000. [9] M. D. Yacoub, “The α-µ distribution: a physical fading model for the Stacy distribution,” IEEE Trans. Veh. Technol., vol. 56, no. 1, pp. 27–34, Jan. 2007. [10] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York, 1972. [11] V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in reduce systems,” Proc. Intern. Conf. on Symbolic and Algebraic Computation, pp. 212–224, 1990.
949
Average Channel Capacity for Generalized Fading Scenarios Daniel Benevides da Costa, Student Member, IEEE, and Michel Daoud Yacoub, Member, IEEE
Abstract— Novel expressions for the average channel capacity (ACC) of single-branch receivers operating in generalized fading channels, namely η-µ and κ-µ, are derived. The expressions are written in terms of the well-known Meijer’s G-functions, which are easily implemented in the most popular computing softwares. In addition, it is shown that, for any given fading condition, the Nakagami-m ACC defines regions of capacity: it lowerbounds the η-µ ACC and it upperbounds the κ-µ ACC. In the same way, it lowerbounds the α-µ ACC for α < 2 and µ > m, and it upperbounds the α-µ ACC for α > 2 and µ < m.
R2 Es /N0 , where Es and N0 denote the average symbol energy and the single-sided power density, respectively, and R stands for the envelope. Then [1] ∞ CW log2 (1 + γ)fΥ (γ)dγ (1) 0
In what follows, (1) shall be derived for the η-µ and κ-µ fading scenarios. For the α-µ one, the ACC, as derived in [6], is reproduced here for convenience.
Index Terms— Average channel capacity, generalized fading channels.
A. α-µ Fading Scenario I. I NTRODUCTION
T
HE study of average channel capacity (ACC) in fading scenarios have been given special attention over the years [1]–[6]. As well known, such a metric provides an upper bound for the maximum transmission rate and was initially established for the Gaussian environment. Recently, in [3], [4], [6] the ACC for single-branch receivers operating in Hoyt (Nakagami-q), Nakagami-m, Rice, Weibull, and in Generalized Gamma fading channels has been derived. In this paper, we derive novel expressions for the average channel capacity of single-branch receivers operating in generalized fading channels, namely η-µ and κ-µ [7]. The expressions are written in terms of the well-known Meijer’s G-functions [8, Eq. 9.301], which are easily implemented in the most popular computing softwares. In addition, this Letter also shows a very interesting result, which maintains that, for any given fading condition, the Nakagami-m ACC defines regions of capacity: it lowerbounds the η-µ ACC and it upperbounds the κ-µ ACC. In the same way, capitalizing on a result available in the literature in which the Generalized Gamma ACC is derived [6], it is shown that the Nakagamim ACC lowerbounds the average channel capacity of the α-µ fading scenario [9] for α < 2 and µ > m and it upperbounds it for α > 2 and µ < m. For comparison purposes, the capacity of the additive white Gaussian noise (AWGN) channel is depicted. II. AVERAGE C HANNEL C APACITY Let W be the fading channel bandwidth and C its ACC. Define fΥ (·) as the probability density function (PDF) of the instantaneous signal-to-noise ratio (SNR) per symbol Υ = Manuscript received August 8, 2007. The associate editor coordinating the review of this letter and approving it for publication was Prof. Ioannis Psaromiligkos. This work was partly supported by FAPESP (05/59259-7). The authors are with the Wireless Technology Laboratory (WissTek), Department of Communications, School of Electrical and Computation Engineering, State University of Campinas, DECOM/FEEC/UNICAMP, PO Box 6101, 13083-852 Campinas, SP, Brazil (e-mail: {daniel, michel}@wisstek.org). Digital Object Identifier 10.1109/LCOMM.2007.071323.
The α-µ fading scenario comprises both Weibull (µ = 1) and Nakagami-m (α = 2) [9]. In such an environment, α > 0 models the non-linearity of the propagation medium, whereas µ > 0 denotes the number of multipath clusters. The ACC for an α-µ fading scenario is given as [6] 1 W √ k+αk−3 ln(2) k β αµ 2 Γ(µ)(2π) 2 αk αµ αk −αk αµ 2 β Φ , Φ ,− , 1 − k(α+1), αk 2 2 2 2 2 αk αµ × Gαk,k(α+1) αµ k k Φ(k, 0), Φ αk 2 ,− 2 ,Φ 2 ,− 2 (2) ∞ z−1 where Γ(z) = 0 t exp(−t)dt is the gamma function, β = γ¯ Γ(µ)/Γ(µ + 2/α), γ¯ = E (Υ), E(·) denotes expectation, Φ(n, ) /n, ( + 1)/n, . . . , ( + n − 1)/n, with being an arbitrary real value and n a positive integer. In the same way, αk/2 and k are positive integers, although α may assume any positive value. (Note that k is arbitrary and is chosen so that αk/2 be integer.)
C=
B. η-µ Fading Scenario The η-µ fading scenario comprises both Hoyt (µ = 0.5) and Nakagami-m (η → 0, η → ∞, η → ±1) [7]. The corresponding PDF is presented in two Formats. In Format 1, the in-phase and quadrature components of the fading signal within each cluster are assumed to be independent from each other and to have different powers, with the parameter 0 < η < ∞ given by the ratio between them. In Format 2, the inphase and quadrature components of the fading signal within each cluster are assumed to have identical powers and to be correlated with each other, with the parameter −1 < η < 1 representing the correlation coefficient between them. For both formats, the parameter µ denotes the number of multipath clusters. From [7, Eq. 27], the PDF of Υ can be obtained, after a transformation of variates, as √ 1 1 2µHγ 2 π µµ+ 2 hµ γ µ− 2 2µγh − fΥ (γ) = Iµ− 12 1 1 exp γ¯ γ¯ Γ(µ)H µ− 2 γ¯ µ+ 2 (3)
c 2007 IEEE 1089-7798/07$25.00
950
IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 12, DECEMBER 2007
where Iν [·] is the modified Bessel function of the first kind and arbitrary order ν [10, Eq. 9.6.20], h and H are functions of η and varies from one format to another [7]. More specifically, in Format 1, h = (2+η −1 +η)/4 and H = (η −1 −η)/4, whereas, in Format 2, h = 1/(1 − η 2 ) and H = η/(1 − η 2 ). Using an infinite series representation for Iν (·) [8, Eq. 8.445], and 1 replacing this and (3) in (1), an integral with integrands γ µ− 2 , ln(1 + γ), and exp(−2µhγ/¯ γ ) appears. We then express the logarithm and exponential functions in terms of the Meijer’s 1,1 G-functions [11, Eq. 11], i.e., ln(1 + γ) = G1,2 2,2 [γ|1,0 ] and γ |− exp(−2µhγ/¯ γ ) = G1,0 0 ]. Knowing that the integral 0,1 [2µhγ/¯ from the product of a power and two Meijer’s G-functions is also a Meijer’s G-function [11, Eq. 21], the ACC for an η-µ fading scenario is obtained as 2n √ ∞ H 2−2n−2µ W 2 π
C= ln(2) Γ(µ)hµ n=0 n! Γ n + µ + 12 h 0, 1 3,1 2µh (4) × G2,3 γ¯ 2n + 2µ, 0, 0
Fig. 1. Average channel capacity of a single-branch receiver undergoing α-µ fading.
C. κ-µ Fading Scenario The κ-µ fading scenario comprises both Rice (µ = 1) and Nakagami-m (κ → 0) [7]. The multipath clusters are assumed to have the scattered waves with identical powers but within each cluster a dominant component is found that presents an arbitrary power. The parameter κ is defined as the ratio between the total power of the dominant components and the total power of the scattered waves, whereas the parameter µ denotes the number of multipath clusters. From [7, Eq. 11], the PDF of Υ can be obtained, after a transformation of variates, as µ+1 µ−1 µ(1 + κ) 2 γ 2 µ(1 + κ)γ fΥ (γ) = µ−1 − µ+1 exp γ¯ κ 2 exp(µκ) γ¯ 2
κ(1 + κ)γ (5) × Iµ−1 2µ γ¯ Following the same procedure as before, the ACC for a κ-µ fading scenario is given as C=
∞
W (µκ)n 1 ln(2) exp(µκ) n=0 n! Γ(n + µ) µ(1 + κ) 0, 1 (6) × G3,1 2,3 n + µ, 0, 0 γ¯
To the best of the authors’ knowledge, (4) and (6) are new. D. Special Cases The expressions derived here are general and exact. They specialize to some particular cases already available in the literature. The ACC of the Rice scenario, given in [4, Eq. 5], can be attained from (6) by setting µ = 1. The Nakagamim ACC, given in [4, Eq. 3], can be obtained from anyone of the two generalized fading scenarios whose formulations are derived here. More specifically, it arises (i) from (4) for µ = m and η → 0 or η → ∞ in Format 1, and for µ = m and η → ±1 in Format 2; or (ii) from (6) for κ → 0. The ACC of Hoyt fading channels, given in [3], can be obtained in an
Fig. 2. Average channel capacity of a single-branch receiver undergoing η-µ fading Format 1.
exact manner from (4) by setting µ = 0.5. (Note, however, that the result of [3] does not appear in terms of the Meijer’s G-function.) E. Channel Capacity for a Fixed Nakagami m Parameter Equations [9, Eq. 16], [7, Eq. 13], and [7, Eq. 30] relate the Nakagami parameter m with those of α-µ, η-µ and κµ distributions. From these, and as shown in [7], [9], an infinite number of curves of the α-µ, η-µ and κ-µ distributions are found that present the same Nakagami-m parameter. The graphs in [7] reveal that the Nakagami-m cumulative distribution function (CDF) upperbounds the η-µ and lowerbounds the κ-µ CDFs. As for the α-µ CDF, nothing is reported in [9]. We looked for a similar behavior for the α-µ CDF and we found out a very interesting feature, as follows: the Nakagamim CDF upperbounds the α-µ CDF for α < 2 and µ > m and it lowerbounds it otherwise (α > 2 and µ < m). As shown in the next section, the corresponding ACCs present a similar behavior. In this case, however, the Nakagami-m ACC upperbounds the κ-µ ACC, whereas it lowerbounds the η-µ ACC. In the same way, the Nakagami-m ACC lowerbounds
DA COSTA and YACOUB: AVERAGE CHANNEL CAPACITY FOR GENERALIZED FADING SCENARIOS
Fig. 3. Average channel capacity of a single-branch receiver undergoing κ-µ fading.
951
1 axes, µ = 1.5 and α varies from 1 to 3. Note that as α increases, the system capacity also increases. In the set 2 axes, α = 1.5 and µ ranges from 0.8 to 3. Again, a performance improvement is observed as µ augments. Figs. 2 and 3 plot the normalized ACC for the η-µ (Format 1) and κ-µ fading scenarios, respectively, as a function of the average SNR per symbol γ¯ . Varying one parameter and keeping the other constant, a behavior similar to that of the α-µ scenario is observed. Note that the formulas for ACC in these cases are given in terms of infinite series. The number of terms in the series required for a given accuracy varies with the parameters. However, in all of the cases shown here, for an error smaller than 10−6 the number of required terms was not greater than 20. For a fixed Nakagami parameter m = 0.5, Fig. 4 plots the η-µ, κ-µ (set 1 axes), and α-µ ACC (set 2 axes). As commented before, the Nakagami-m ACC divides the fading plane in two regions, in which the lower region is related to the κ-µ fading, whereas the upper region is related to the η-µ fading. As for the α-µ fading case, the upper region is obtained for α ≤ 2 and µ ≥ m, and the lower region is obtained otherwise. R EFERENCES
Fig. 4. Average channel capacity for the α-µ, η-µ, and κ-µ fading scenarios using the same Nakagami-m parameter m (m = 0.5).
the α-µ ACC for α ≤ 2 and µ ≥ m and it upperbounds it otherwise. III. S OME N UMERICAL P LOTS In this section, some representative plots illustrate the normalized ACC, C/W , of single receivers subject to α-µ, η-µ, and κ-µ fading as a function of the average SNR per symbol γ¯ . For each case, in the the same figure, two sets of axes are used: set 1, comprising the left vertical axis and the bottom horizontal axis; set 2, comprising the right vertical axis and the top horizontal axis. Fig. 1 shows the normalized ACC for α-µ fading scenarios as a function of the average SNR per symbol γ¯ . In the set
[1] W. C. Y. Lee, “Estimate of channel capacity in Rayleigh faing environment,” IEEE Trans Veh. Technol., vol. 39, no. 3, pp. 187–189, Aug. 1990. [2] M.-S. Alouini and A. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1165–1181, July 1999. [3] J. Cheng and T. Berger, “Capacity of Nakagami-q (Hoyt) fading channels with channel side information,” in Proc. IEEE Int. Conf. Commun. Technol., vol. 2, 2003, pp. 1915–1918. [4] N. C. Sagias, G. S. Tombras, and G. K. Karagiannidis, “New results for the Shannon channel capacity in generalized fading channels,” IEEE Commun. Lett., vol. 2, no. 9, pp. 97–99, Feb. 2005. [5] S. Khatalin and J. P. Fonseka, “On the channel capacity in Rician and Hoyt fading environments with MRC diversity,” IEEE Trans. Veh. Technol., vol. 55, no. 1, pp. 137–141, Jan. 2006. [6] N. C. Sagias and P. T. Mathiopoulos, “Switched diversity receivers over generalized Gamma fading channels,” IEEE Comm. Lett., vol. 9, no. 10, pp. 871–873, Oct. 2005. [7] M. D. Yacoub, “The κ-µ distribution and the η-µ distribution,” IEEE Ant. Prop. Mag., vol. 49, no. 1, pp. 68–81, Feb. 2007. [8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York: Academic, 2000. [9] M. D. Yacoub, “The α-µ distribution: a physical fading model for the Stacy distribution,” IEEE Trans. Veh. Technol., vol. 56, no. 1, pp. 27–34, Jan. 2007. [10] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York, 1972. [11] V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in reduce systems,” Proc. Intern. Conf. on Symbolic and Algebraic Computation, pp. 212–224, 1990.
