REPOSITÓRIO DA PRODUÇÃO CIENTIFICA E INTELECTUAL DA UNICAMP
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DOI: 10.1109/access.2019.2907891
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DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo
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Ratio of Two Envelopes Taken From
α-µ, η-µ, and κ-µ Variates and
Some Practical Applications
CARLOS RAFAEL NOGUEIRA DA SILVA 1, NIDHI SIMMONS 2, ELVIO J. LEONARDO 3, SIMON L. COTTON 2, (Senior Member, IEEE), AND MICHEL DAOUD YACOUB 1 1Wireless Technology Laboratory, School of Electrical and Computer Engineering, State University of Campinas, Campinas 13083-970, Brazil
2Wireless Communications Laboratory, Institute of Electronics, Communications and Information Technology, Queen’s University Belfast, Belfast BT3 9DT, U.K. 3Department of Informatics, State University of Maringa, Maringa 87020-900, Brazil
Corresponding author: Carlos Rafael Nogueira Da Silva (carlosrn@decom.fee.unicamp.br)
This work was supported in part by the FAPESP under Grant 2018/16667-8, in part by the CNPq under Grant 304248/2014-2, and in part by the U.K. Engineering and Physical Sciences Research Council under Grant EP/L026074/1.
ABSTRACT In this paper, new, exact expressions for the probability density functions and the cumulative distribution functions of the ratio of random envelopes involving theα-µ, η-µ, and κ-µ fading distributions are derived. The expressions are obtained in terms of easily computable infinite series and also in terms of the multivariable Fox H-function. Some special cases of these ratios, namely, Hoyt/Hoyt,η-µ/Nakagami-m, κ-µ/Nakagami-m, κ-µ/η-µ with an integer µ for the η-µ variate, η-µ/η-µ with an integer µ for only one of the η-µ, and their reciprocals are found in novel exact closed-form expressions. In addition, simple closed-form expressions for the asymptotes of the probability density functions and cumulative distribution functions of all ratios, both for the lower and upper tails of the distributions are derived. In the same way, asymptotes for the bit error rate on a binary signaling channel are obtained in closed-form expressions. To demonstrate the practical utility of these new formulations, an application example is provided. In par-ticular, the secrecy capacity of a Gaussian wire-tap channel used for device-to-device and vehicle-to-vehicle communications is characterized using data obtained from field measurements conducted at 5.8 GHz.
INDEX TERMS α-µ distribution, κ-µ distribution, η-µ distribution, D2D, multihop systems, spectrum
sharing, spectrum sensing, V2V.
I. INTRODUCTION
In wireless systems, a number of applications make use of the ratio of random variables (RVs) representing the envelope or power of the various short term fading processes. For instance, the signal-to-interference ratio is an important per-formance metric used to assess the effect of co-channel inter-ference and is defined as the ratio of RVs. Also, in physical layer security, the probability of a positive secrecy capacity involves the ratio of RVs. Other applications can be found in wireless networks with technologies such as multihop communications, spectrum sharing and spectrum sensing to name but a few.
Among the myriad of possible fading scenarios, those involving the α-µ, η-µ, and κ-µ processes emerge as especially attractive given their generality, mathematical
The associate editor coordinating the review of this manuscript and approving it for publication was Ivan Wang-Hei Ho.
tractability, and, more importantly, practical applicability. Theα-µ distribution [1] is defined in terms of two fading parameters, namelyα and µ, related, respectively, to the non-linearity present in the wireless medium and to the num-ber of multipath clusters present in the signal propagation. It includes as special cases other important distributions, such as the Weibull and Nakagami-m. Theκ-µ distribution [2] is also written in terms of two fading parameters, namely κ and µ, the former related to the ratio between the total power of the dominant waves and the total power of the scattered waves, and the latter as before. It includes as special cases popular distributions such as the Rice and Nakagami-m. The η-µ distribution [2] is again defined in terms of two fading parameters, namelyη and µ, the former describing two different phenomena, as revisited next, and the latter as before. In Format 1,η is given by the power ratio between the quadrature and in-phase scattered waves. In Format 2, η represents the correlation coefficient between these two
quadrature components. The η-µ distribution includes as special cases Hoyt and Nakagami-m.
In this paper, a number of fundamental statistics, which will underpin the characterization of general fading channels, are derived. These are the probability density functions (PDFs) and the cumulative distribution func-tions (CDFs) for the ratio of two independent and arbitrary RVs involving theα-µ, κ-µ, and η-µ distributions.1 Exact expressions are obtained for the PDFs and CDFs in terms of the multivariable Fox H-function. Exact, reasonably sim-ple, and rapidly convergent series representations for these statistics are also provided. The motivation to use the Fox H-function comes from the fact that it is able to compactly represent a diversity of combined functions. Although not yet implemented in commercial mathematical software tools, it has gained widespread popularity for the evaluation of wireless communications performance. For instance, the sin-gle variable Fox H-function is used to describe the cascaded channel over generalized Nakagami-m fading in [4]. In [5], the Fox H-function is applied to model spherically invariant random processes, whereas in [6]–[9], expressions for the capacities and bit error probabilities for different modulation schemes over fading channels are given. The authors in [10], thoroughly analyze integral transformations involving the Fox H-function within the scope of wireless communications, whereas in [11] and [12], the multivariable Fox H-function is used to obtain expressions for the symbol error rate in the context of maximal ratio combining and equal gain combin-ing, for theα-µ channel and for the H-channel, respectively. The authors of [13] present a fading model for the vehicle-to-vehicle (V2V) channel based on Fox’s H distributions accounting for the path loss, shadowing, and multipath fading as well as communication links covering random distances. Nonetheless, the overwhelming majority of the fading models that we address in our work are not considered in [13].
One of the key challenges which presides over the use of the Fox H-function is how to evaluate it, whereby computa-tion of the multivariable Fox H-funccomputa-tion is still an open issue. To this end, the work reported in [12] provides a useful Python implementation, which, according to the authors, is precise and efficient for up to four variables. Alternatively, calculus of residues may be used to find computable series representa-tions, although the numerical computation of such series usu-ally becomes extremely inefficient as the number of variables grow beyond four. It is important to mention that, in our case,
the use of the Fox-H function has not been an end in itself but a means to arrive at mathematically tractable expres-sions, namely series expansion formulations. In our work, and more importantly, in addition to the Fox-H function, we also express the results in terms of power series (obtained by means of the use of the Fox-H function). This means that it is possible to avoid dealing with the multivariable 1In [3], these same statistics, namely PDF and CDF, are found for the
cross-product ofα-µ, η-µ, and κ-µ fading variates.
Fox H-function altogether when computing the ratios of the aforementioned RVs.
The ratio of some of these general variates have been used mainly in the context of co-channel interference and for specific scenarios. For instance, in [14] and [15], theµ parameter is assumed to be integer, whereas in [16] and [17], an outage is found but in an approximate manner. In [18], some fundamental results such as closed-form expressions are obtained in the analysis of co-channel interference for theκ-µ/κ-µ scenario. In [19], the outage is obtained using nested summations for theα-µ scenario. In [20], the author calculates the PDF of the ratio ofα-µ variates given in terms of the Fox H-function, which is a result equivalent to the one presented here. However, no expression for the CDF is offered. Also, in [21], the authors calculate the PDF and the CDF of the ratio ofα-µ RVs, but express their results either as an infinite power series or, with some reasonable restriction on the values of the parameters, as a finite power series.
The key contributions of this work may now be summa-rized as follows:
1) New, exact expressions for the probability density func-tions and the cumulative distribution funcfunc-tions of the ratio of random envelopes involving the α-µ, κ-µ andη-µ fading distributions are derived. The expres-sions are obtained in terms of the multivariable Fox H-function and also in terms of easily computable infinite series;
2) Novel closed-form expressions for many of the ratio mixtures which appear as special cases of our formu-lations are obtained. These cases include Hoyt/Hoyt, η-µ/Nakagami-m, κ-µ/Nakagami-m, κ-µ/η-µ with integerµ for the η-µ variate, η-µ/η-µ with integer µ for only one of theη-µ, and their reciprocals;
3) Novel, simple closed-form expressions for the asymp-totes of the PDFs and CDFs of all ratios are obtained, both for the lower and upper tails of the distributions; 4) Novel, simple closed-form expressions for the
asymp-totes of the bit error rate (BER) of a binary signaling channel for all ratios are obtained in a high signal-to-noise ratio (SNR) condition;
5) To demonstrate some practical applications for the results presented in this paper, we employ the ratios of the aforementioned RVs to estimate the secrecy capacity of a Gaussian wire tap channel used for device-to-device (D2D) and V2V communications using data obtained from field measurements con-ducted at 5.8 GHz.
The remainder of this paper is organized as fol-lows. SectionII reviews the statistical models considered. SectionIIIprovides exact expressions for the PDF and CDF for all combinations of ratios involvingα-µ, η-µ, and κ-µ RVs along with asymptotic expressions for the lower and upper tails. In the same way, asymptotic expressions for the BER of a binary signaling channel in high SNR condition are obtained. SectionIVdepicts some plots for the ratio PDFs.
Some practical applications are then presented in SectionV using data obtained from field measurements. Section VI provides some concluding remarks.
II. PRELIMINARIES
A. THEα-µ DISTRIBUTION
The PDF of a fading envelope R > 0 following an α-µ distribution, withα-root mean square ˆr , √αE[Rα] is given by [1] fR(r) = αrαµ−1 0(µ)Aαµexp −r α Aα (1) in which E[·] represents the expectation operator, α > 0 is associated with the non-linearities of the wireless radio channel, µ > 0 is given by µ = E2[Rα]/V[Rα] and it is related to the number of multipath clusters,0(·) is the Gamma function [22, Eq. (6.1.1)], V[·] is the variance operator and the constant A is defined as
A = rˆ
µ1/α. (2)
B. THEκ-µ DISTRIBUTION
The PDF of a fading envelope R> 0 following a κ-µ distri-bution, with rms value ˆr ,pE[R2], is written, alternatively to [2, Eq. (1)] by using the identity [22, Eq. (9.6.47)], as
fR(r) = 2 r2µ−1 eκµK2µexp −r 2 K2 0F˜1 µ; κµr2 K2 (3) in whichκ > 0 is the ratio between the total power of the dominant components and the total power of the scattered waves,µ > 0 is related to the number of multipath clusters and can be obtained in terms of the physical parameters by µ = E2[R2]/V[R2] × (1 + 2κ)/(1 + κ)2,
0F˜1(a, z) = 0F1(a, z)/ 0(a) is a particular case of the generalized
hyper-geometric function [23, Eq. (9.14.1)], and the constant K is defined as
K = √ rˆ
µ(1 + κ). (4)
C. THEη-µ DISTRIBUTION
The PDF of a fading envelope R> 0 following an η-µ distri-bution, with rms value ˆr,pE[R2], is given, alternatively to
[2, Eq. (17)] by using the identity [22, Eq. (9.6.47)] and some minor algebraic manipulations, as
fR(r) = 2hµr4µ−1 0(2µ)E4µexp −r 2h E2 0F1 µ +1 2; r4H2 4E4 (5) in which µ > 0 is related to the number of multipath clusters and is given in terms of the physical parameters by µ = E2[R2]/V[R2] × (1 + (H/h)2)/2, and the constant E is
defined as
E = √rˆ
2µ. (6)
The parameters h and H are functions ofη and are defined according to the chosen format. In Format 1, 0 < η < ∞
is the scattered-wave power ratio between in-phase and quadrature components, with h = (2 + η−1 +η)/4 and
H = (η−1−η)/4. In Format 2, −1 < η < 1 is the cor-relation coefficient between the scattered-wave in-phase and quadrature components, with h = (1 −η2)−1and H =ηh.
III. PDFS AND CDFS OF THE RATIO DISTRIBUTION
In this section, the PDF and CDF of the random variable
Z , X /Y , with X and Y being independent and arbitrarily distributed RVs following either anα-µ, κ-µ or η-µ distri-bution, are presented. While these results can be expressed in terms of the multivariate Fox H-function, unfortunately the multivariate Fox H-function does not provide any seg-regation in the number of integrations as well as making it quite difficult to infer changes in the curves by modifying the parameters. It is also recognized that while works on the software implementation of the multivariate Fox H-function are on-going, it may be a number of years before these are readily available in popular mathematical packages such as MATLAB and Mathematica. Acknowledging this, any results computed in terms of the Fox H-function are relegated to Appendix A. Instead we focus on providing computable series representations for all of the statistics of interest, which yield much greater insights into the interaction of numerator and denominator RVs.
Tables 1 and 2 show series representation for the PDF and CDF, respectively, of the ratio distribution involving the α-µ, κ-µ and η-µ fading models, in which (a)nis
Pochham-mer’s symbol [22, Eq. (6.1.22)], B(·, ·) denotes the beta func-tion [22, Eq. (6.2.2)] and the constants uab and vab with
a, b ∈ {α, κ, η}, are defined as uαα = Ax Ay , uακ =Ax Ky , uαη=Ax phy Ey , uκκ = Kx Ky, uκη =Kx phy Ey , uηη = √Ex hx phy Ey , uκα = Kx Ay, uηα = Ex Ay √ hx , uηκ = Ex Ky √ hx , and vab = z2 z2+ u2 ab . (7)
The corresponding mathematical derivation can be found in Appendix C. We maintain that all of the series derived here are convergent. The mathematical proof for this can be done by following the standard procedure presented in [24, Section 1.7 and 2.9]. It is noteworthy that in [25], the statistics for the ratio of twoα-µ variates were obtained in closed form as a sum of a hypergeometric function provided that the ratio of the involved parameters α were coprimes (i.e.,αx/αy = k/l, such that k, l ∈ N). In contrast to [25],
the series for the PDF and CDF of the ratio of twoα-µ variates provided in Tables1 and2have no restriction whatsoever. Although the convergence of these series is dependent on the conditionαx < αy this presents no restriction on their
applicability as the reciprocal distribution can be used to evaluate the required statistics. It is important to remark
TABLE 1. Series expansion for the ratio PDF.
TABLE 2. Series expansion for the ratio CDF.
that in the Appendix A the PDF and CDF of the ratio of two α-µ variates are provided in terms of the single vari-able Fox H-function again without any parameter restriction. These series yield accurate results and compute rapidly on an ordinary desktop computer.2The series involving theα-µ 2‘‘Rapidly’’ refers to the time taken for an ordinary desktop computer to
evaluate the series. In particular, for those not involving theα-µ variates, 100 terms were more than enough to arrive at the desired accuracy (10−8) with the computations taking from 0.02s to 0.2s depending on the choice of the parameters. For those involving the α-µ variates, the number of terms for the same accuracy were around 20 and the time taken varied from 0.6s to 5s. The stopping criterion for the calculations is governed by generating sufficient terms to achieve an accuracy of 10−8. In general, this is accomplished when the nth term is at the order of the required precision.
distribution are still troublesome as they involve the single variable Fox H-function and therefore are dependent on the efficiency of the Fox H-function implementation. Although a clever computational implementation of the Fox-H function can be found in [4], whenα > 2 and in the vicinity of z = 0 it may yield incorrect results. A useful implementation of the Fox H-function, customized for the needs of this paper, is shown in Appendix D. It is also worthwhile remarking that some series for the CDFs are given as double summations. While it is indeed possible to further simplify the expres-sions to obtain a single summation, exhaustive tests have shown that, surprisingly, the overall convergence speed of the double summation is faster than the single one. Of course,
by changing the order of summation of the residues, other series expansions can be found, but those given here have been proved efficient.
A. SERIES REPRESENTATIONS: RATIO OF κ-µ/α-µ, η-µ/α-µ AND η-µ/κ-µ
In the previous subsections, the PDFs and CDFs for the ratios α-µ/κ-µ, α-µ/η-µ and κ-µ/η-µ were presented. The PDF for the reciprocal distribution, i.e., ˜Z = 1/Z, may be obtained following a standard statistical procedure as
fZ˜(z) = 1 z2fZ 1 z , (8) and its CDF as FZ˜(z) = 1 − FZ 1 z . (9)
Hence, Tables1 and2can be used directly to obtain these statistics.
B. SOME SPECIAL CASES: NEW CLOSED-FORM EXPRESSIONS
For some combinations of parameters, the ratio PDF may be obtained as a closed-form expression given in terms of the hypergeometric function. For instance, by settingµx =
µy=1/2 in the η-µ distribution (thus reducing it to the Hoyt
distribution), using [26, Eq. (6.8.1.6)] along with the linear transformation given by [26, Eq. (7.3.1.3)], and after some algebraic manipulations, the PDF is given as
fZ(z) = 2phxhy(ˆrxrˆy)2(hyrˆx2+ z2hxrˆy2)z (hyrˆx4+2z2hxhyrˆx2rˆy2+ z4hxrˆy4)3/2 ×2F1 3 4, 5 4;1; 4z4Hx2Hy2(ˆrxrˆy)4 (hyrˆx4+2z2hxhyrˆx2rˆy2+ z4hxrˆy4)2 ! . (10)
Ratios involving the Nakagami-m distribution can also be obtained in terms of simpler functions. By settingαx =2 in
theα-µ distribution (thus reducing it to the Nakagami-m dis-tribution), and after some algebraic manipulations, the PDF of the ratios of Nakagami-m/κ-µ, and Nakagami-m/η-µ are given, respectively as fZ(z) = 2vµx ακ(1 − vακ)µy zeκyµyB µ x, µy ×1F1 µx+µy;µy;κyµy(1 − vακ) , and (11) fZ(z) = 2vµx αη 1 − vαη2µy zB µx, 2µy hµyy ×2F1 µx +2µy 2 , 1 +µx+2µy 2 ; 1 + 2µy 2 ; Hy2 h2 y 1 − vαη2 ! . (12)
Another interesting particular case involving theη-µ dis-tribution is found in closed-form for µ ∈ Z. By writing the 2F1(·, ·; ·; ·) function, found in the results involving
the η-µ distribution, in terms of the Legendre Q func-tion using [26, Eq. (7.3.1.102)], then using the identity [27, Eq. (07.12.03.0028.01)], and after some tedious alge-braic manipulations the results for the ratios of κ-µ/η-µ andη-µ/η-µ are obtained and given, respectively, by (13) and (14), as shown at the bottom of this page, in which Ga= Hy 1 − vaη /hy, with a ∈ {κ, η} and ξ1=2µx+µy.
Similar results are obtained in [15] in the analysis of the out-age probability overκ-µ/η-µ and η-µ/η-µ. Some important closed-form expressions for some particular cases of the ratio κ-µ/κ-µ are derived in [18]
Of course, the inverse of these ratios (κ-µ and Nakagami-m,η-µ and Nakagami-m, and η-µ and κ-µ) can be easily obtained as hinted earlier. The PDF for other important ratio distributions that are special cases of the Nakagami-m
fZ(z) = 21−µy 1 − vκηµyvµx κη zeκxµx Hy µy µy−1 X k=0 1 −µy k µy k k! (−1)k(1 − G κ)−µx−µy+k(2Gκ)−k B µx, µy 1 −µx−µyk ×1F1 −k +µx+µy;µx; κxµxvκη (1 − Gκ) + (−1) µyµ x(1 + Gκ)−µx−µy µx+µy B µy, µx+µy ×(Gκ−1) k(2G κ)−k 1 +µx+µyk 2F2 1 +µx, µx+2µy;µx, 1 + k + µx+µy; κ xµxvκη (1 + Gκ) . (13) fZ(z) = 21−µy 1 − v ηηµyv2µx ηη zhµx x Hy µy µy−1 X k=0 1 −µyk µyk k! G η−1 2Gη k 1 − G η−ξ1 B 2µx, µy (1 − ξ1)k ×2F1 ξ1 − k 2 , ξ1− k +1 2 ; 1 2+µx; Hx2v2ηη h2 x 1 − Gη 2 ! +2(−1) µyµ x 1 + Gη −ξ1 ξ1B µy, ξ1(1 + ξ1)k ×3F2 1 +µx, µx+µy, 1 2 +µx+µy; 1 + k +ξ1 2 , 2 + k +ξ1 2 ; Hx2v2ηη h2 x 1 + Gη 2 ! . (14)
TABLE 3. Left and right tail asymptotic expression for the PDF of the ratio distribution.
TABLE 4. Left and right tail asymptotic expression for the CDF of the ratio distribution.
distribution can also be derived from the above equations by making the appropriate parameter substitutions.
C. ASYMPTOTIC EXPRESSIONS 1) PDF AND CDF
To obtain closed-form expressions for the behavior of the PDF for both the lower and upper tails, the following proce-dure is performed. For the cases involving theα-µ distribu-tion, the Fox H-function is expanded in a power series using the residue theorem. By taking the poles over the0(µx+ t),
the variable z will have a positive exponent. Now, consider z small enough so that it is possible to ignore all terms in the sum other than the first one resulting in an expression for the lower tail PDF. On the other hand, by taking the residues over the other set of poles, a negative exponent on z arises. In this case, when z is high enough, all terms of the sum but the first may be ignored resulting in the upper tail PDF.
In the other scenarios, for small values of z, we may consider vab ∼= z2/u2ab and 1 − vab ∼= 1. As z is in the
vicinity of zero, summation indexes different from zero can be ignored. That is, for the lower tail asymptotic behavior, only the first term in the sum is used. Proceeding like this and after some algebraic manipulations, the lower tail PDFs presented in Table 3 are obtained. For high values of z, we consider lim
z→∞pFq(ap, bq, k(1 − vab)) = 1, vab
∼ =1 and 1 − vab ∼= u2ab/z2. Replacing these in the expressions of
Table1and performing the required summation, the upper tail
PDFs given in Table3are obtained. Of course, by integrating these expressions, the CDFs of the asymptotic behavior can also be obtained, and these are given in Table4. Note that the upper tail CDFs are indeed one, although the expressions provided in Table4can be used to achieve those for the lower tails of the reciprocal distribution by using (9).
2) BIT ERROR RATE
In a Gaussian channel, the bit error rate (BER) of binary sig-naling may be written in a compact form as [28, Eq. (8.100)]
P(γ ) =0(b, aγ )
20(b) , (15)
in which the parameters a and b are defined according to the modulation and detection schemes3and0(·, ·) is the comple-mentary Gamma function [22, Eq. (6.5.3)]. Now consider a binary signal over a ratio channel Z = h1/h2where h1, X
and h2 , Y , with X and Y as defined before. The SNR is
obtained as 0 = Es N0 h1 h2 2 . (16)
in which Es is the average energy per symbol and N0is the
noise power spectral density. The average SNR is given as ¯ γ = Es N0E[h 2 1]E[h −2 2 ]. (17)
3The authors in [28, Table 8.1] provide the range of possible values for
TABLE 5. Bit error rate at high SNR for a binary signal over the ratio channel.
After a simple variable transformation, the PDF of the SNR is given as f0(γ ) = q EZ2 2√γ ¯γ fZ s γ E Z2 ¯ γ (18)
The average BER is obtained by averaging (15) over the SNR as ¯ P = Z ∞ 0 P(γ )f0(γ ) dγ (19) The PDF of the SNR can be written as a power series using, for instance, those in Table 1 and expanding the special function therein to obtain a PDF in the form ofP
iai(γ / ¯γ)bi.
To obtain the asymptotic behavior for the BER in a high SNR regime, higher exponents in the PDF can be ignored taking only i = 0, which means that the BER behavior at high SNR levels depends only on the lower tail of the channel PDF. That is, to obtain an asymptotic equation for the BER at high SNR over a ratio channel, replace fZ(z) by the respective channel
lower tail provided in Table3. Table5gives the asymptotic behavior of the BER over a ratio channel for all possible combinations of ratios.
IV. NUMERICAL RESULTS
As an illustration, Fig.1shows plots of the PDF of the ratio of (a)κ-µ/η-µ, (b) α-µ/η-µ, and (c) κ-µ/α-µ and their asymp-totic behavior at the upper and lower tails. It is noteworthy that the slope (angular coefficient) of the lower tail depends solely on the distribution of the numerator, specifically the
parameter µ (and α if the α-µ distribution is involved), whereas the rate of decay of the upper tail is ruled by the distribution of the denominator, specifically the parameter µ (and α if the α-µ distribution is involved). This can be observed in the expressions of Tables 3 and 4 and in the illustrations of Fig.1(a), (b), (c). On the other hand, the linear coefficient (shift on the y axis) in the lower tail is more influenced by the numerator, whereas in the upper tail this is by the denominator (see Fig.1(c)). It is worth highlighting that theα-µ, κ-µ, and η-µ distributions have a finite non-zero value at z = 0 whenαµ = 1, 2µ = 1 and 4µ = 1, respectively. The same effect is encountered for the ratio distributions if the RV on the numerator also satisfies the condition for finite, non-zero value at z = 0.
In Fig.2, the BER is depicted for theκ-µ/κ-µ channel, in which solid lines are the exact solution obtained from (19) and the dashed lines are the asymptotic curves at high SNR. As expected, the simulation results coincide with analytical curves. Additionally, it can be seen from Fig. 2 that the asymptotes start to coincide with the exact curve earlier when the numerator parameterµ (or αµ if the numerator is α-µ distributed) is small.
V. SOME PRACTICAL APPLICATIONS
In this section, we demonstrate the practical utility of the new formulations derived here by analyzing the secrecy capacity of a Gaussian wire-tap channel [29], [30], for D2D and V2V communications, using data obtained from field measure-ments. The secrecy capacity for a Gaussian wire-tap channel
FIGURE 1. Plots depicting the PDF of the ratio of (a)κ-µ/η-µ, (b) α-µ /η-µ, and (c)κ-µ/α-µ with their respective lower and upper tail asymptotes.
is defined as Cs= log2 1 +γm 1 +γw , if γm> γw 0, ifγm6 γw (20)
in whichγm , X2andγw , Y2are the SNR for the main
and the wire-tap channels, respectively, and are given byγi=
Ph2i/Ni, i = (m, w), in which P and Niare the transmission
FIGURE 2. BER of the ratio ofκ-µ and η-µ RVs with µy=2,ηy=0.75, κx=2, andµx= {0.25, 1, 2.5} along with their asymptotic behavior for
high SNR.
and noise power, respectively, and hiis the channel gain. The
probability of positive secrecy capacity is given as Pr[Cs > 0] = 1 − Pr " hm hw < s Nm Nw # =1 − FZ s ¯ γwE[h2m] ¯ γmE[h2w] =FZ˜ s ¯ γmE[h2w] ¯ γwE[h2m] (21) in which: (i) FZ(·) is obtained from (24) with parameters
taken from Table8and evaluated using the exact series given in Table2in accordance with the chosen fading model for hm
and hw; (ii) FZ˜is the CDF for the reciprocal distribution given
in (9); and (iii) ¯γmand ¯γware the mean values of the SNR for
the main and wire-tap channels, respectively. A. D2D COMMUNICATIONS
The D2D measurements were conducted at 5.8 GHz within an indoor seminar room. The exact details of the measurement setup, experiments, and data analysis can be found in [31]. The measurement trial considered three persons who carried the hypothetical user equipments (UEs) A, B and E4and were positioned at points X, Y and Z respectively (see Fig.3). All three persons had the UEs positioned at their heads, and were initially stationary. The persons at positions Y and Z were then instructed to walk around randomly within a circle of radius 0.5 m from their starting positions whilst imitating a voice call. It should be noted that the channel between UEs A and B is referred to as the main channel whilst the channel between UEs A and E is referred to as the wire-tap channel.
For the analysis, the data sets were normalized to their respective local means prior to parameter estimation. To determine an appropriate window size for the extraction of the local mean signal, the raw data was visually inspected and overlaid with the local mean signal for differing win-dow sizes. In this case, a smoothing winwin-dow of 500 sam-ples was used. As an example of the data fitting process, 4A, B and E are analogous to Alice, Bob and Eve as commonly
FIGURE 3. Seminar room environment showing the position of the UEs A, B and E for the D2D scenario.
Figs. 4(a) and (b) show the PDF of theα-µ, κ-µ and η-µ fading models fitted to the D2D fading data for the main and the wire-tap channels, respectively. A total of 74763 samples of the received signal power were obtained and used for parameter estimation. The parameter estimates were obtained using the lsqnonlin function available in the optimization toolbox of MATLAB along with theα-µ, κ-µ and η-µ PDFs. To allow the reader to reproduce these plots, parameter esti-mates for all of measurement scenarios are given in Table6.
From Fig.4, we observe that both the α-µ and κ-µ PDFs provide a very good approximation to the measured data, whilst some disparity is noticed between the lower tail of the empirical PDF and the theoreticalη-µ PDF. To select the candidate model, from theα-µ, κ-µ and η-µ distributions, most likely to have been responsible for generating the fading, the Akaike information criterion (AIC) was used. Based upon the ranking performed using the computed AIC, it was found that theα-µ distribution was the most likely model for the main channel, whilst theκ-µ distribution was the most likely model for the wire-tap channel. Theα parameter estimate for the main channel was found to be higher than that for an equivalent Nakagami-m fading channel (α = 2, µ = m), whileµ was found to be 0.68 suggesting a tendency towards significant fading for the main channel. This is understand-able due to the constantly changing orientation and posture of both the test subjects. For the wire-tap channel, we see that theκ parameter estimate was greater than 1, suggesting that a perceptible dominant component existed. Theµ parameter estimate for the wire-tap channel was found to be relatively close to 1, indicating that a single cluster of scattered multi-path contributes to the signals received in the D2D (indoor) scenario.
Utilizing the parameter estimates from the D2D field trials (see Table 6), Fig. 5 depicts the estimated proba-bility of positive secrecy capacity versus ¯γw for a range
of ¯γm, when the main and the wire-tap channels
experi-ence α-µ and κ-µ fading, respectively. We now adopt the following approach to analyze the probability of positive
FIGURE 4. Empirical envelope PDF of (a) the main channel and (b) the wire-tap channel compared to theα-µ, κ-µ and η-µ PDFs for the D2D channel measurements.
FIGURE 5. Probability of positive secrecy capacity versus ¯γwconsidering a range of ¯γmfor the D2D channel measurements when the main and
wire-tap channels are assumed to undergoα-µ and κ-µ fading, respectively. Here, PSC indicates positive secrecy capacity.
secrecy capacity. We perform our analysis when ¯γmis fixed at
10 dB and for two different levels of positive secrecy capacity: 0.10 (10% level) and 0.50 (50% level), which are indicative of low and mid-range levels of positive secrecy capacity, respectively. From Fig.5, we observe that if the wire-tapper can improve her average SNR ( ¯γw) from 5 dB to 10 dB,
the probability of positive secrecy capacity will decrease from 77% to 49%. Furthermore, to ensure a positive secrecy capacity level of at least 10%, we find that the wire-tappers average SNR must not exceed 19 dB. Likewise, to ensure a mid-range positive secrecy capacity level of at least 50%,
¯
γwmust not exceed 10 dB.
B. V2V COMMUNICATIONS
The V2V channel measurements considered in this work were also conducted at 5.8 GHz. The exact details of the measurement setup and experiments can be found in [31]. Specifically, the transmitter, the legitimate receiver and the wire-tapper were positioned on the center of the dashboards within vehicles A, B and E, respectively (see [31, Fig. 10]).
Initially, vehicles A and B drove towards each other at a speed of 30 mph whilst the vehicle containing node E remained parked on the side of the road. The fading data used in the analysis presented here, considered the channel acquisitions obtained when vehicles A and B drove past each other and continued their onward journey. As before, it should be noted that the channel between nodes A and B is referred to as the main channel whilst the channel between nodes A and E is referred to as the wire-tap channel.
Similar to the D2D channel measurements, the data sets were normalized to their respective local means prior to parameter estimation. In this case, a smoothing window of 200 samples was used. Figs.6(a) and (b) show the PDF of the α-µ, κ-µ and η-µ fading models fitted to the V2V data for the main and the wire-tap channels, respectively. A total of 56579 samples of the received signal power were obtained and used for parameter estimation. When com-pared to theα-µ and η-µ PDFs, we observe that the κ-µ distribution provides a better approximation around the lower tail of the empirical PDFs in Figs. 6(a) and (b). On the contrary, theη-µ PDF provides a better approximation than theα-µ and κ-µ distributions around the median (where the greatest number of fade levels occur) and also the upper tail of the empirical PDF. Accordingly, from the computed AIC rankings (see Table6), it was found that theη-µ distribution was the most likely model for both the main and wire-tap channels. Inspecting the estimatedη parameter for the V2V channel measurements (see Table6), we observe that under the assumption of Format 1 for theη-µ fading model [2], a power imbalance existed between the in-phase and quadra-ture components for both the main and wire-tap channels. Under the assumption ofη-µ fading, low estimates of µ were also found for both the main and wire-tap channels, indicating that minimal environmental multipath contributions offered by the surrounding environment.
Now, using the parameter estimates from V2V field tri-als (see Table 6), Fig. 7 depicts the estimated probability of positive secrecy capacity versus ¯γw for a range of ¯γm,
FIGURE 6. Empirical envelope PDF of (a) the main channel and (b) the wire-tapchannel compared to theα-µ, κ-µ and η-µ PDFs for the V2V channel measurements.
FIGURE 7. Probability of positive secrecy capacity versus ¯γwconsidering a range of ¯γmfor the V2V when it is assumed to undergoη-µ fading. Here,
PSC indicates positive secrecy capacity.
when the main and the wire-tap channel experiences η-µ fading. Following a similar approach to the D2D analysis with
¯
γmfixed at 10 dB, we observe that increasing ¯γwfrom 5 dB
to 10 dB causes the probability of positive secrecy capac-ity to significantly decrease from 89% to 50%. Moreover, to ensure a positive secrecy capacity level of at least 10% or 50%, we find that ¯γwmust not exceed 15 dB and 10 dB,
respectively.
VI. CONCLUSIONS
Exact, computationally efficient series representations for the PDF and CDF of the ratio of two fading envelopes arbitrarily taken from α-µ, κ-µ, and η-µ distributions are provided. Novel and compact expression for these ratios are also provided in terms of the Fox H-function. New, sim-ple, exact, closed-form expressions are obtained for the spe-cial cases Hoyt/Hoyt, η-µ/Nakagami-m, κ-µ/Nakagami-m, κ-µ/η-µ with integer µ for the η-µ variate, η-µ/η-µ with integerµ for only one of the η-µ, and their reciprocals are found. Simple closed-form expressions for the asymptotes of the PDFs and CDFs of all ratios, both for lower and upper tails of the distributions as well as for BER of a binary signaling channel have been derived. Applications for these results include, for instance, multihop systems, spectrum sharing, the characterization of co-channel interference and physical layer security, among other areas of wireless communica-tions. Multipath-shadowing composite fading statistics can also be obtained as special cases of the ratio distribution and the various combinations given here can be used to charac-terize a plethora of fading environments. Interestingly, and as well known, the sum of independent identically distributed (i.i.d.)κ-µ powers is another κ-µ. In the same way, the sum of i.i.d.η-µ is another η-µ. Therefore, the results provided here can be directly applied to maximal ratio combining systems in the presence of interference. In the same way, the sum of i.i.d. α-µ, κ-µ, and η-µ envelopes can be well approx-imated by the respective distributions, then rendering the results useful in the study of equal gain combining systems.
Similar comments concerning the sum of independent non-identically distributed variates (power and envelope) also apply. Finally, a practical application for the novel results derived here was demonstrated by analyzing the secrecy capacity of a Gaussian wire-tap channel for D2D and V2V communications, using data from field measurements with experiments conducted at 5.8 GHz.
A final comment concerning the convenience of evalu-ating integrals or infinite series is due. Numerical integra-tion is a very useful and widely employed mathematical technique. In spite of its unquestionable utility, according to [32] and [33], it is not always the most efficient way to obtain a particular mathematical result. In our experience, with the expressions presented in this paper, using Mathe-matica and the same set of parameters, numerical integration takes a significantly larger amount of time to compute than power series do. In particular, for the ratios involvingη-µ and/orκ-µ, as far as computation time is concerned, there is a clear advantage in favor of the series representations over the corresponding integral representations (roughly on the other varying from 10 to 100 times). Of course, the time and the number of terms to achieve a certain accuracy depend on the choice of the parameters. On the other hand, as already mentioned in the paper, the performance of the series specif-ically involving the α-µ distribution is highly dependent on the Fox H-function implementation. Although the series require very few terms for convergence (around 20 terms), at present, there is no clear advantage between evaluating the expressions using the integral or the series representation as far as computational time is concerned. However, the series expressions find a clear advantage over their counterpart integral representations because, from the series, it is possible to obtain their asymptotes and, then, the asymptotic perfor-mance behavior of the system modeled under the correspond-ing scenarios. This has been successfully attained in terms of
simple closed-form expressions for all the ratiosin this paper.
APPENDIX A
RESULTS IN TERMS OF THE MULTIVARIATE FOX H-FUNCTION
In SectionIII, the PDF and CDF for the ratio distributions are given in terms of infinite power series. These results were obtained through the sum of residues of several Fox H-functions with two and three variables. Here, we present the PDF and CDF in terms of the multivariable Fox H-function5defined in [34]6 H[x; (β, B); (δ, D), L] = 1 2πj NI L2(s)x −sd s, (22)
5The integral form of the Fox H-function on several variables was not
used to evaluate the PDF and CDF analysed here. As a matter of fact, these results were a tool used to derive the series expansion provided in SectionIII. An interesting Python implementation for the multivariable Fox H-function is found in [12].
6There are several notations for the multivariable Fox H-function.
in which j = √−1 is the imaginary unit; s = [s1, . . . , sN],
x = [x1, . . . , xN],β = [β1, . . . , βm] andδ = [δ1, . . . , δn]
denote vectors of complex numbers and B = (bi,j)m×N and
D = (di,j)n×N are matrices of real numbers. Also x−s =
QN i=1x
−si
i , d s =
QN
i=1dsi, L is an appropriate contour on
the complex space, and
2(s) = m Q i=1 0βi+ N P k=1 bi,ksk n Q i=1 0 δi+ N P k=1 di,ksk . (23)
Interestingly, both the exact PDFs and the exact CDFs for the ratio of variates can be compactly written in a general form given as
g(z) = CH[x; (β, B); (δ, D), L] (24) Tables7and8show the values for the parameters appearing in (24) for, respectively, the ratio PDF and ratio CDF for each combination of the random variables X and Y considered in this work. The conditions for convergence of the integral in (22) are thoroughly discussed in [34], [35]. We highlight that all the necessary convergence conditions in our paper are met in accordance with these references.
Also, I3 denotes the identity matrix of order three. The
general steps followed in order to arrive at (24) can be found in the Appendix B.
APPENDIX B
DERIVATION OF THE RATIO STATISTICS
Let Z , X /Y be the ratio between two independent and arbitrarily distributed random variates with positive support. Then its PDF can be obtained through standard probability procedures as fZ(z) = ∞ Z 0 yfX(zy)fY(y)dy. (25)
Consider now that both X and Y follow theκ-µ distribu-tion. Then replacing (3) in (25) and performing some minor algebraic manipulations, fZ(z) becomes
fZ(z) = 4z2µx−1 exp κxµx+κyµy Kx2µxK2µy y × ∞ Z 0 y−1+2µx+2µyexp −y2 z 2 K2 x + 1 K2 y !! ×0F˜1 µx; κxµx(zy)2 K2 x 0F˜1 µy; κyµyy2 K2 y ! dy (26) Using [26, Eq. (7.2.3.12)] to write the hypergeometric functions with their Mellin-Barnes contour integral
representation we obtain fZ(z) = 4z2µx−1 exp κxµx+κyµy K2xµxK 2µy y 1 2πj 2 × ∞ Z 0 y−1+2µx+2µyexp −y2 z 2 K2 x + 1 K2 y !! × ZZ L 0(τ1)0(τ2) 0 (−τ1+µx) 0 −τ2+µy × −κxµx(zy) 2 K2 x −τ1 −κyµyy 2 K2 y !−τ2 dτ1dτ2dy. (27) By changing the order of integration and performing the inte-gral on the variable y, the inner inteinte-gral can be solved using [22, Eq. (6.1.1)]. Then, after some algebraic manipulations,
fZ(z) reduces to fZ(z) = 2 vµx κκ(1 − vκκ)µy zexp κxµx+κyµy 1 2πi 2 × ZZ L0 −τ1 −τ2+µx+µy 0 (τ1) 0 (τ2) 0 (µx−τ1) 0 µy−τ2 ×(−κxµxvκκ)−x1 −κyµy(1 − vκκ)−x2dτ1dτ2. (28) Interestingly, the change in the order of integration is possible due to convergence of both inner integrals. The only penalty is a distortion in the complex contour, which is not really an issue. Finally, by performing the variable transformation
t1 = τ1+µx and t2 = τ2+µy and after some algebraic
manipulations (28) will be in the form of (22) with parameters
x,β, B, δ and D provided on the fourth line of Table7. Following similar steps as for theκ-µ/κ-µ described above the ratio PDF for each combination of fading distribution is obtained. To summarize, the closed-form expression for the ratio PDF is obtained by
1) If applicable, write the 0F1 function by its contour
integral representation using [26, Eq. (7.2.3.12)]; 2) If the ratio involves α-µ, write one exponential
function by its contour integral representation using [26, Eq. (8.4.3.1)] with [26, Eq. (8.2.1.1)];
3) Change the order of integration and solve the inner integral using [22, Eq. (6.1.1)];
4) After some algebraic manipulation and variable trans-formation, interpret the integral as Fox H-function using (22).
The ratio CDF can be readily obtained through its defini-tion by writing the PDF as contour integral using (22) and changing the order of integration. The inner integral will be in the formR0xτk−1dτ = 0(k)/ 0(k + 1)xk, and after some algebraic manipulations, the result can be interpreted as a Fox H-function with the parameters provided in Table8.
APPENDIX C
DERIVATION OF THE SERIES REPRESENTATION
Series representations for the ratio PDF and CDF may be obtained through the sum of residues of the Fox H-function.
TABLE 7. Parameters for the fox H-Function of the ratio PDF.
TABLE 8. Parameters for the fox H-Function of the ratio CDF.
From the residue theorem [36], the residue of the Gamma function around its poles is
res−i0(x)f (x) =
(−1)i
i! f(−i), i ∈ Z ∧ i ≥ 0 (29)
considering f (x) has no poles on negative integer i. Taking the residues of (28) overτ1andτ2, fZ(z) can be written as a
double sum given by
fZ(z) = 2vµx κκ(1 − vκκ)µy zeκxµx+κyµy × ∞ X i=0 ∞ X j=0 (vκκκxµx)i (1 − vκκ) κyµy j i!j!B i +µx, j + µy . (30) Performing the sum over the index j the series for the ratio PDF for theκ-µ/κ-µ in Table1is obtained. Through a similar process, the other series in Tables1and2can be obtained.
APPENDIX D
MATHEMATICA IMPLEMENTATION FOR THE SINGLE FOX H-FUNCTION
The following Mathematica program is a simple imple-mentation for the Fox H-function used to evaluate the Fox H-function present in Tables1and2, in which the parameter
limitis used to speed up the computation of the numerical integration and it can be chosen as large as possible7.8
7In particular, it has been found that for the Fox-H function appearing in
the PDFs, such limit is set to infinity. As for the CDF, such limit has been set to 50. These have been used for all of our computations.
8This implementation is, by no means, a general one, but it is sufficient to
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CARLOS RAFAEL NOGUEIRA DA SILVA was born in Brazil, in 1986. He received the B.S. degree in electrical engineering from the Federal Uni-versity of Itajubá, MG, Brazil, in 2010, the M.S. degree in electrical engineering from the National Institute of Telecommunications, MG, in 2012, and the Ph.D. degree from the School of Electrical and Computer Engineering, State University of Campinas, UNICAMP, in 2018. He is currently holding a Postdoctoral FAPESP Scholarship with the State University of Campinas, UNICAMP. His research interests include wireless channel modeling, spectrum sensing, cognitive radio, composite channel characterization and applications, and wireless communications in general.
NIDHI SIMMONS received the B.E. degree (Hons.) in telecommunications engineering from Visvesvaraya Technological University, Belgaum, India, in 2011, the M.Sc. degree (Hons.) in wire-less communications and signal processing from the University of Bristol, U.K., in 2012, and the Ph.D. degree in electrical engineering from the Queen’s University of Belfast, U.K., in 2018. She is currently a Research Fellow with the Cen-tre for Wireless Innovation, Queen’s University Belfast. Her research interests include physical-layer security, channel char-acterization and modeling, and cellular device-to-device and body-centric communications.
ELVIO J. LEONARDO was born in Brazil. He received the B.Eng., M.Sc., and Ph.D. degrees from the State University of Campinas, Brazil, in 1984, 1992, and 2013, respectively, all in elec-trical engineering. In 1980, he was involved in the development of telecommunications equip-ment with the Telebras Research and Develop-ment Centre. In 1992, he moved to Australia, where he was with The University of Sydney. In 1995, he joined the Motorola’s Software Centre in Adelaide, Australia, where he was involved in projects in the communi-cations area. Since 1998, he has been with Motorola, he moved to Chicago, where he spent the next four years. In 2002, he decided to move back to Brazil, where he started his academic career. He is currently an Adjunct Professor with the State University of Maringa. His research interests include wireless communications and embedded systems.
SIMON L. COTTON (S’04–M’07–SM’14) received the B.Eng. degree in electronics and software from Ulster University, Ulster, U.K., in 2004, and the Ph.D. degree in electrical and electronic engineering from the Queen’s Univer-sity of Belfast, Belfast, U.K., in 2007. He is cur-rently a Reader in wireless communications and the Director of the Centre for Wireless Innova-tion (CWI), Queen’s University Belfast. He has authored and coauthored over 140 publications in major IEEE/IET journals and refereed international conferences, two book chapters, and two patents. Among his research interests are cellular device-to-device, vehicular, and body-centric communications. His other research interests include radio channel characterization and modeling, and the simulation of wireless channels. He was a recipient of the H. A. Wheeler Prize, in 2010, from the IEEE Antennas and Propagation Society for the best applications journal paper in the IEEE TRANSACTIONS ONANTENNAS AND
PROPAGATION, in 2009, and the Sir George Macfarlane Award from the U.K.
Royal Academy of Engineering in recognition of his technical and scientific attainment since graduating from his first degree in engineering, in 2011.
MICHEL DAOUD YACOUB was born in Brazil, in 1955. He received the B.S.E.E. and M.Sc. degrees from the School of Electrical and Com-puter Engineering, State University of Camp-inas, UNICAMP, CampCamp-inas, SP, Brazil, in 1978 and 1983, respectively, and the Ph.D. degree from the University of Essex, U.K., in 1988. From 1978 to 1985, he was a Research Specialist in the devel-opment of the Tropico digital exchange family with the Research and Development Center of Telebrás, Brazil. In 1989, he joined the School of Electrical and Computer Engineering, UNICAMP, where he is currently a Full Professor. He consults for several operating companies and industries in the wireless communica-tions area. He is the author of Foundacommunica-tions of Mobile Radio Engineering (Boca Raton, FL: CRC, 1993), Wireless Technology: Protocols, Standards, and Techniques(Boca Raton, FL: CRC, 2001), and the coauthor of the Telecommunications: Principles and Trends(São Paulo, Brasil: Erica, 1997, in Portuguese). He holds two patents. His general research interest includes wireless communications.
