A Unified Performance Analysis of Wireless Communications over η – λ – μ Generalized Fading Channels

Ehab Salahat College of Engineering, Khalifa University of Science, Technology and Research, Abu Dhabi, United Arab Emirates

E-mail: ehab.salahat@ieee.org

Abstract—In this paper, we present a unified performance analysis of wireless communication over fading channels in terms of the Average Symbol Error Rates (ASER), Average Channel Capacity (ACC) and Moment Generating Function (MGF). We derive a simple and accurate exact and approximate closed-form expression for ASER, encapsulating all types of coherent digital modulations schemes, and exact expressions for the ACC and MGF, for the new fading distribution that has not been studied before, the η – λ – μ generalized fading, which accurately characterizes, thanks to the remarkable flexibility of the named parameters, the non-homogeneous nature of the propagation medium, including the λ – μ, the η – μ, the Hoyt, the Rice, the Nakagami-m, the Rayleigh, the Gamma, the Exponential, and the One-sided Gaussian fading distributions as special cases. The derived expressions are analytically-traceable and convenient to handle both numerically and analytically. Monte Carlo simulations and existing solutions for the special cases of this fading have been used to validate the derived expressions, where an excellent match is achieved.

Keywords— Symbol Error Rate; Moment Generating Function; Average Channel Capacity; Generalized Fading; Nakagami-m; η – λ – μ fading; λ – μ fading; η – μ fading.

I. INTRODUCTION ireless Communication is an active area of research in communication technology, and its applications seem limitless. Much effort is being devoted to the

problem of modeling frequency-nonselective fading and shadowing in the channel because of the growing interest in wireless systems and their applications. Previously restricted to outdoor environments, wireless systems soon have reached indoor ones, and, more recently, enclosed environments. It is well known that shadowing and multipath fading are two effects that degrade communication signals during wireless propagation. In this context, the correct characterizations of the wireless propagation environment, and in particular, its fading conditions, continues to raise interest [1-3]. A classical way of characterizing these effects has been feasible with the aid of statistical distributions. The shadowing or else the long-term signal variation is known to follow the Lognormal distribution, whereas the short-term signal variation is described by several other distributions. Statistical models such as Rayleigh, Weibull, Nakagami-q (Hoyt) and especially Nakagami-m, due to its ease of manipulation and wide range of applicability [4], have been widely considered ample in modelling multipath fading in Non-Line-Of-Sight (NLOS) communication scenarios. In the same context, the Lognormal, the Gamma and Rice distributions have been traditionally considered to be adequate for modelling the shadowing effect with regards to the characterization of

multipath fading in Line-Of-Sight (LOS) communication scenarios [4-6]. Nevertheless, as opposed to the initial applications, namely outdoor and indoor, for which the propagation conditions are reasonably well known, those for enclosed environments have yet to be better understood, where situations are encountered for which none of the preceding distributions seem to adequately fit experimental data, though one or another, may yield a moderate fitting [1]. Questions arise that concern the appropriateness of the use of the well-known small scale fading distributions, such as Nakagami-m, Weibull, and others as applied to the enclosed environments [7]. The well-known fading distributions have been derived assuming a homogeneous diffuse scattering field, resulting from randomly distributed point scatterers. With such an assumption, the central limit theorem leads to complex Gaussian processes with in-phase and quadrature Gaussian distributed variables x and y having zero means and equal standard deviations. The assumption of a homogeneous diffuse scattering field is certainly an approximation because the surfaces are spatially correlated characterizing a non-homogeneous environment [8]. Therefore, despite the undoubted usefulness of the aforementioned statistical models, their effectiveness is limited to the characterization of either shadowing or multipath fading, and fail to account concurrently for both. Based on that, the need for composite statistical distributions became significantly evident. Capitalizing on the above multipath statistical models, M. D. Yacoub derived in [9-12] four generalized fading distributions, namely, the λ – μ, the κ – μ, the η – μ and the α – μ distributions, which gained momentum in the context of studying wireless communication over fading channels. Characterization of these distributions in terms of the probability density functions (PDF) and moments is given in [9-12] and moment generating function (MGF) expressions have been derived in [13-15]. These models are distinct for the remarkable flexibility offered by their named parameters which ultimately provide a rather good fitting to experimental data of realistic and practical fading communication scenarios [16-17]. The usefulness of these models is also evident by the fact that they include as special cases the majority of the aforementioned fading distributions [9-12]. The η − μ distribution follows the pattern of the multi-cluster analysis by including 2 formats. In format 1, the in-phase and quadrature components are independent from each other and have different powers, while format 2 coincides with the λ − μ distribution, where the in-phase and quadrature components are correlated and have identical powers. Format 2 can be obtained from format 1 by a rotation of the axes [9, 13]. Analysis of the error rate performance of communication systems operating in radio channels modelled by the above

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generalized distributions has been of large research interest for the last few years (see, e.g. [13, 16-17]). Anchored on the previous generalized fading distributions, Papazafeiropoulos et al. in [18] introduced new remarkable generalized fading distribution, the η – λ – μ distribution, encompassing the previous models as special cases. To the best of our knowledge, no performance evaluation and analysis on this distribution has been reported in the literature before.

In this paper, the work is devoted to analyze the performance of wireless communication over the η – λ – μ generalized fading channels, including the λ – μ, the η – μ, the Nakagami-m, the Hoyt, the Rice, the Rayleigh, the Gamma, the Exponential, and the One-sided Gaussian fading distributions as special cases. Exact closed-form expressions for Average Symbol Error Rate (ASER), Average Channel Capacity (ACC) and Moment Generating Function (MGF) of this distribution are derived. We also derive an approximate yet very accurate ASER and ACC expressions. Noteworthy, the derived ASER expressions (exact and approximate) are valid of all coherent digital modulation schemes. All results reported in this work are new and novel.

The remaining part of this paper is structured as follows: in section II, the η – λ – μ distribution is visited. In section III, the mathematical expressions for the ASER, ACC and MGF expressions are derived. Then simulation results for different fading scenarios will be demonstrated in section IV, and will be compared with Monte Carlo simulations as well as existing results found in literature for the special cases of this distribution as a quick validity test. Finally, the paper findings are summarized in section V.

II. MATHEMATICAL DERIEVATION

A. The η – λ – μ Generalized Fading The primary objective of this paper is to derive unified and generic expressions for the ASER, ACC and MGF expressions for η – λ – μ distribution. From [18], the pdf of the received signal envelop is given in [18, eqn. (10)] by

( ) =√ ( )

( )

, (1)

where = ( )( )

, = ( )( )

, and = ( − 1) + 4 .

One of the main advantages of this distribution is that it is expressed in terms of three physical parameters, namely η, λ, and μ, which account for the unequal power of the in-phase and quadrature components of the fading signal, the correlation between these components, and the numbers of the multipath clusters, respectively [18]. The cumulative distribution function (CDF) and the nth moment expressions are given in [18, eqn. (16)] and [18, eqn. (17)]. Table I (next page) illustrates the relation between the η-λ-μ distribution and the other distributions. As indicated in the table, there exists different choices of the fading parameters that yield the same special fading scenario (e.g. Nakagami-m, Rice, Hyot, and Rayleigh).

B. Generalized Average Symbol Error Rate The ASER, and the Average Bit Error Rate (ABER) when the symbols are bits, due to fading channel can be evaluated by averaging the SER of the Additive White Gaussian Noise (AWGN) channel using the PDF of the fading envelop. Hence, the resultant expression from this averaging process usually involves the Q-function as part of the expression. The expression of the Symbol Error Rate (SER) for different coherent modulation schemes in AWGN is given by

( ) = (√ ), (2)

where Q(∙) is the Q-function defined as

( ) = ∫√

, (3)

and and are constants that depend on the digital modulation scheme (see Table II).

Hence, following the averaging process = ∫ (√ ) ( ) , (4) where ( ), after applying variable transformation to (1), is given by

( ) =√ ( )

( )

= ( ), (5) where (∙) is the modified bessel function of the first kind and arbitrary order , = , = − , = , = + 0.5, and

= √ 2 (1 − ) / 2 Γ( ) . , where we omitted the sub-indices for the sake of the compactness of the expressions. Substituting (5) back into (4) yields = ∫ (√ ) ( ) , (6) In the following, an exact and an approximate solutions to the ASER will be presented. i. Exact ASER Analysis Following from (6), and performing variable transform as = , with some manipulations, (6) can be rewritten as

= ∫ ( ) , (7)

TABLE II: VALUES OF AND FOR DIFFERNET SCHEMES

Modulation Scheme Average SER

BFSK = (√ ) 1 1 BPSK = 2 1 2

QPSK,4-QAM ≈ 2 (√ ) 2 1

M-PAM ≈ ( ) ( )

M-PSK ≈ 2 2 sin 2 2 sin

Rectangular M-QAM ≈ √

√ √

√

Non-Rectangular M-QAM ≈ 4 4

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using the Taylor series representation of ( ) given as [19] ( ) = ∑

! ( ) , (8)

then (7) can be rewritten as

= ∑ Ψ ∫ [ / ] , (9) where Ψ =

! ( ), = + 2 + and = .

Using the relations [20] ( ) = 0.5erfc

√, (10)

erfc =√ ,

, −, 10,0.5 , (11)

= ,, |

−0 , (12)

where ,, (∙) is the Meijer-G function, (9) can then be

rewritten as = ∑ Ψ ∫ ,

, −, 10, 0.5 ,

, −0 , (13)

where = /(2 ) in which the integral can be found in a simple closed-form using [21, pp. (346)], and then the final exact ASER expression is given by

= ∑ Ψ 2,22,1 Δ(1,1 − ), Δ(1,1)

Δ(1,0), Δ(1,0.5) , (14)

where Ψ = and Δ( , ℎ) = , , … , .

The expression in (14) is new and novel, and is generalized for any of the fading models presented in table I, and is valid for the study of ASER of all coherent digital modulations schemes presented in table II, involving the Meijer-G function that’s available in many software packages (e.g. Mathematica and Maple).

ii. Approximate ASER Analysis Performing change of variable to (6) as t= √ , (6) tends to be in the form

= ∫ ( ) . (15) In order to simplify the integration in (15), the Q-function has to be replaced by simpler, yet accurate, approximation from those found in literature (see e.g. [22-26]). Fig. 1 represents the relative absolute error ( ) of the Q-function approximations given in [22-26], where the notation refers to the initials of the respective authors. However, since we seek simplicity of the ASER expression, the approximation in [22], denoted as CDS in Fig. 1, and given by

Q( ) ≈ + , (16)

Fig. 1: Relative absolute error for the approximations of the

Q-Function given in [22-26].

seems a potential candidate. Noteworthy, even higher accuracy can be achieved from (16) by including more exponential terms to the approximation [22]. Replacing (16) into (15) results in

0 1 2 3 4 5 6 710-4

10-3

10-2

10-1

100

x

Rel

ativ

e A

bsol

ute

Erro

r ε(x

)

PBCSOPBCSCDSGKALYCNB (n=8)YCNB (n=12)

TABLE I: RELATION BETWEEN THE − − DISTRIBUTION AND THE OTHER DISTRIBUTIONS

# DISTRIBUTION λ μ η COMMENT REFERENCES 1 η − λ − μ 0 ≤ λ ≤ 1 0 < μ 0 ≤ η ≤ 1 − [18]

2 λ − μ 0 ≤ λ ≤ 1 μ = μ’ η → 1 η – μ format 2. [12][18] μ = 2μ’ η → 0

3 η − μ λ → 0 μ = μ’ 0 ≤ η ≤ 1 η – μ format 1. [11][15][18] λ → 1 μ = 2μ’ 0 ≤ η ≤ 1

4 Nakagami-m

λ → 0 μ = 0.5m − Follows λ–μ either setup. [11][15][18]

λ → 1 μ = m − − μ = m η → 0 Follows η–μ either setup. Generally, for intermediate

values of η and μ, = ( )

( ). − μ = 0.5m η → 1

5 Nakagami-q (Hoyt) 0 ≤ λ ≤ 1 μ = 0.5 − Follows λ – μ, with μ = 0.5, η – μ, as = , and can

be approximated from Nakagami-m by =

( ), ≤ 1.

[5][11][15][18] − μ = 1 =

6 Nakagami-n (Rice)

− − − Follows λ–μ either setup, where =( )

.

[5][11][12][18] − − − Follows η–μ , mapping formula is ( ) =( )

).

− − − Follows Nakagami-m, as = √

√

7 Rayleigh − − − Follows Nakagami-m, Nakagami-q, and Nakagami-n by setting m=1, q=1, and n=0, with corresponding parameters selected as indicated above.

[5][18]

8 One-sided Gaussian − − − Follows from Nakagami-m with m=0.5 [5][18]

Note: Nakagami-n can be also approximated from Nakagami-m as =( )

or equivalently as = √√

for m ≥ 1. Sometimes in literature, the parameter

K defines . Noteworthy, as m→ ∞, fading channel converges to the nonfading AWGN.

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= ∑ ∫ , (17)

where = {1/12,1/4} and = {1/2, 2/3}. With some manipulations to (17), the expression simplifies to = ∑ ∫ ( ) , (18)

where = + and = . Reference [27, eqn. (2.15.3.2)] shows that an integral of the form given in (18) has a closed-form solution, which gives the final ASER expression as = ∑ Λ , ; + 1; , (19)

where Λ =2

Γ( + )+

Γ( +1) , and (∙) refers to the gauss

hypergeometric function.

This new resultant expression in (19) is generalized for all for any of the fading channel presented in table I, and is valid for the study of the average bit error rates of all coherent digital modulations schemes presented in table II. This expression has also avoided any computationally-complicated functions and is very accurate as will be illustrated in section IV.

C. Moment Generating Function of the η-λ-μ Fading The MGF can be obtained using the Laplace Transform of the pdf given in (5) as ( ) = ∫ ( ) = ∫ ( ) ( ) , (20) which directly follows the integral given in [27, eqn. (2.15.3.2)], and hence the MGF is obtained in simple closed-form as

( ) =( )

( ) ( ) , ; + 1;

( ). (21)

This expression is new, novel and has not been reported in literature before. It includes all the fading scenarios presented in table I. The MGF expression in (20) can be quite useful in the performance study of mobile fading channel, for example when diversity, e.g. Maximum Ratio Combining (MRC) and Equal Gain Combining (EGC), are used.

D. Generalized Average Channel Capacity Over the last few decades, channel capacity has extensively been used as the basic tool for the analysis and design of new and more efficient techniques to improve the spectral efficiency of modern wireless communication systems and to gain insight into how to counteract the detrimental effects of multipath fading propagation. Channel capacity is defined as the highest rate at which information can be sent over the channel with arbitrarily small probability of error [28]. The average channel capacity (ACC) can be obtained by averaging the capacity of an Additive White Gaussian Noise (AWGN) channel over the distribution of the received power [29], and is defined as = ∫ log (1 + ) ( ) , (22)

where is the channel bandwidth and ( ) is given in (5). If we define ℂ= C/W to be the normalized channel capacity, then (22) can be rewritten as

ℂ = ∫ log (1 + ) ( ) , (23)

In the following, we will present an exact and an approximate expressions for the ACC. i. Exact Expression for the ACC Following from (23), transforming variable as = , and changing the base of the logarithm, (23) can be expressed as ℂ =

( ) ∫ ln(1 + ) ( ) , (24)

where = 1/ and = . Using (8), (12) and using the Meijer-G form of the natural logarithm [20], given by

ln(1 + ) = ,, |1,1

1,0 , (25) and substituting them back into (24) and collecting terms yields the integral

ℂ = ∑ δ ∫ ,, |1,1

1,0 ,, |

−0 , (26)

where δ =( ) ! ( )

and = + 2 + . This integral can be found in closed-form as given in [21, pp. (346)], and the final exact ACC expression is then

ℂ = ∑ δ 3,21,3 Δ(1,1), Δ(1,1), Δ(1,1 − )

Δ(1,1), Δ(1,0) , (27)

where δ = δ and Δ( , ℎ) = , , … , .

This ACC expression is novel and new, and has not been reported in literature before, including all the fading scenarios given in table I.

ii. Approximated Expression for the ACC Following from (23), it can be seen that the log (1 + ) makes this integral somewhat difficult. However, if this logarithm can be approximated in some way, then the integration will be straightforward. Hence, we propose the following exponential approximation

log (1 + ) = + ∑ = ∑ , (28)

where = { 9.3306, −2.6354, −4.0320, −2.3883} and ={0, 0.0372, 0.0044, 0.2739}. Those values were obtained using nonlinear curve fitting. Noteworthy, the relative absolute error is of the order 10 . Using (28) and [27, eqn. (2.15.3.2)], (23) can be rewritten as

ℂ = ∑ ∫ ( ) ( )

= ∑ Ω , ; + 1; , (29)

where Ω = 2Γ( + )

+Γ( +1)

and = + .

This new approximated expression for the ACC is very simple and accurate, and includes all fading models presented in table I. In fact the approximation of log (1 + ) can be quite useful in other applications as well.

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III. SIMULATION RESULTS The validity of the expressions of the ASER and ACC will be verified here for different values of the parameters η, λ, and μ, and will compared with the Monte Carlo simulations. For some of the special cases, the results will be also compared with other results found in literature. Note that the solid lines are the Monte Carlo simulated curves, whereas the points on the solid lines corresponds to the derived expression, where the legends were omitted for the sake clear visualization, and the focus here will be the generalized fading models presented in table I, since the special cases follow from them. Figs. 2-5 represent four case studies for the ASER of Binary Phase Shift Keying (BPSK) in the η – λ – μ (Fig. 2), η – μ (Fig. 3), λ – μ (Fig. 4) and Nakagami-m (and Rayleigh) in Fig. 5. In Fig. 2, different values of the parameters η, λ, and μ are selected over a large range of average signal-to-noise ratio (average γ) to indicate that the expression in is not constrained by average SNR limit, where an agreement can be seen with the Monte Carlo for the different scenarios presented in the figure. The second case study, presented in Fig. 3, follows the η – μ fading scenario, where the values of λ are set according to table I, (λ→0), and the values of the parameters η and μ are selected to match those found in [14, Fig. 2] for the sake of quick comparison to the solutions found in literature. From Fig. 3, it can be seen that the results of derived ASER expression fits with Monte Carlo simulation and matches the curves found in [14, Fig. 2]. As a third case study, the special case of λ – μ fading scenario is illustrated in Fig. 4, where the value of the parameter η follows from table I (η →1), and the values of λ and μ are selected as the values in [14, Fig. 3], where it should be recalled that λ - μ fading is actually η – μ: format 2. As illustrated in Fig. 4, an accurate results is achieved from the derived expression as it fits Monte Carlo simulation and those in [14, Fig. 3]. As a final test scenario for the derived ASER expression, Nakagami-m (and Rayleigh) fading is studied in Fig. 5. Following from table I, the values are set to be (λ→0, η →1, μ=0.5m, where it shows an agreement with Monte Carlo simulation (AWGN curve is included as a reference), indicating the validity of derived ASER expression for the fading scenarios presented in table I and their special cases. As for the ACC, two different general cases will be presented. Fig. 6 shows the ACC for different values of the η – λ – μ fading, whereas Fig. 7 shows the ACC for the Rayleigh, η – μ and λ – μ as special cases of this generalized fading distribution. The different cases illustrated in the figures are compared to the numerically evaluated curves. It can be seen from the figures that the exact (solid) and derived ACC (points) expressions coincide, proving the validity and the accuracy of the derived expressions. The results presented both ASER and ACC concludes the validity and the accuracy of the derived expressions

Fig. 2: ABER for different modulation schemes for η-λ-μ scenarios.

Fig. 3: BPSK ABER for different η-μ scenarios (η-μ format I), (λ→0).

Fig. 4: BPSK ABER for different λ-μ scenarios (η-μ format I), (η →1).

Fig. 5: BPSK ABER for different Nakagami-m fading parameter m (λ→0, η →1, μ=0.5m ).

0 10 20 30 40 50 60 70 8010

-20

10-15

10-10

10-5

100

Average γ

Ave

rage

BE

R

64-PSK(μ=0.65, η=0.25, λ=0.9)

32-QAM(μ=1.13, η=0.35, λ=0.21)

BPSK, QPSK, 4-QAM(μ=3.43, η=0.95, λ=0.77)

16-QAM(μ=2.25, η=0.65, λ=0.5)

0 5 10 15 20 2510

-10

10-8

10-6

10-4

10-2

100

Average γ

Ave

rage

BE

R

μ=2.5

μ=1.5

η=0.1

η=0.8

η=0.8

η=0.1

0 5 10 15 20 2510

-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Average γ

Ave

rage

BE

R

μ=1.5

μ=3.5 η=0.1η=0.8

η=0.8

η=0.1

0 10 20 30 40 50 60 70 8010-20

10-15

10-10

10-5

100

Average γ

Ave

rage

BE

R

μ=1 (m=2)

μ=2 (m=4)

AWGN

μ=1.5 (m=3)

μ=0.5 (m=1)

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Fig. 6: ACC for different scenarios of η – λ – μ fading. Fig. 7: ACC for different special cases of η – λ – μ fading.

IV. CONCLUSION

In this paper, we derived novel exact closed-form expressions for the ASER, ACC and MGF for a new generalized fading scenario, the η – λ – μ, which includes the λ – μ , the η – μ, the Hoyt, the Rice, the Nakagami-m, the Rayleigh, the Gamma, the exponential, and the One-sided Gaussian fading distributions as special cases. The derived ASER is valid for all coherent modulations. The reported results motivate researchers to explore more applications for this new fading model in the performance evaluation of such generalized channel model.

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[29] A. Goldsmith, Wireless Communications. Cambridge University Press, 2005.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

average γ

Nor

mal

ized

AC

C

Rayleigh

AWGN

η=0.1, μ=1.25

λ=0.8, μ=2.2510 11 12 13 142.5

3

3.5

4

4.5

5

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

average γ

Nor

mal

ized

AC

C

AWGN

η =0.32, λ=0.75, μ=1.5

η =0.11, λ=0.12, μ=0.23

η =0.5, λ=0.55, μ=0.51

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