Ratio of Two Envelopes Taken from Variates and Some PracticalApplications

Da Silva, C. R. N., Bhargav, N., Leonardo, E. J., Cotton, S. L., & Yacoub, M. D. (2019). Ratio of Two EnvelopesTaken from Variates and Some Practical Applications. IEEE Access, 7, 54449-54463. [8703806].https://doi.org/10.1109/ACCESS.2019.2907891

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Received February 11, 2019, accepted March 20, 2019, date of publication May 1, 2019, date of current version May 6, 2019.

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Ratio of Two Envelopes Taken Fromα-µ, η-µ, and κ-µ Variates andSome Practical ApplicationsCARLOS RAFAEL NOGUEIRA DA SILVA 1, NIDHI SIMMONS 2, ELVIO J. LEONARDO 3,SIMON L. COTTON 2, (Senior Member, IEEE), AND MICHEL DAOUD YACOUB 11Wireless Technology Laboratory, School of Electrical and Computer Engineering, State University of Campinas, Campinas 13083-970, Brazil2Wireless 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 partby 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 cumulativedistribution functions of the ratio of random envelopes involving the α-µ, η-µ, and κ-µ fading distributionsare derived. The expressions are obtained in terms of easily computable infinite series and also in terms ofthe 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 oneof the η-µ, and their reciprocals are found in novel exact closed-form expressions. In addition, simpleclosed-form expressions for the asymptotes of the probability density functions and cumulative distributionfunctions 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-vehiclecommunications 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, spectrumsharing, spectrum sensing, V2V.

I. INTRODUCTIONIn wireless systems, a number of applications make use ofthe ratio of random variables (RVs) representing the envelopeor power of the various short term fading processes. Forinstance, 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 physicallayer security, the probability of a positive secrecy capacityinvolves the ratio of RVs. Other applications can be foundin wireless networks with technologies such as multihopcommunications, spectrum sharing and spectrum sensing toname but a few.

Among the myriad of possible fading scenarios, thoseinvolving the α-µ, η-µ, and κ-µ processes emerge asespecially attractive given their generality, mathematical

The associate editor coordinating the review of this manuscript andapproving it for publication was Ivan Wang-Hei Ho.

tractability, and, more importantly, practical applicability.The α-µ distribution [1] is defined in terms of two fadingparameters, namely α 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, suchas the Weibull and Nakagami-m. The κ-µ distribution [2]is also written in terms of two fading parameters, namelyκ and µ, the former related to the ratio between the totalpower of the dominant waves and the total power of thescattered waves, and the latter as before. It includes as specialcases popular distributions such as the Rice and Nakagami-m.The η-µ distribution [2] is again defined in terms of twofading parameters, namely η and µ, the former describingtwo different phenomena, as revisited next, and the latter asbefore. In Format 1, η is given by the power ratio betweenthe quadrature and in-phase scattered waves. In Format 2,η represents the correlation coefficient between these two

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C. R. N. da Silva et al.: Ratio of Two Envelopes Taken From α-µ, η-µ, and κ-µ Variates

quadrature components. The η-µ distribution includes asspecial cases Hoyt and Nakagami-m.

In this paper, a number of fundamental statistics,which will underpin the characterization of general fadingchannels, are derived. These are the probability densityfunctions (PDFs) and the cumulative distribution func-tions (CDFs) for the ratio of two independent and arbitraryRVs involving the α-µ, κ-µ, and η-µ distributions.1 Exactexpressions are obtained for the PDFs and CDFs in termsof the multivariable Fox H-function. Exact, reasonably sim-ple, and rapidly convergent series representations for thesestatistics are also provided. The motivation to use the FoxH-function comes from the fact that it is able to compactlyrepresent a diversity of combined functions. Although notyet implemented in commercial mathematical software tools,it has gained widespread popularity for the evaluation ofwireless communications performance. For instance, the sin-gle variable Fox H-function is used to describe the cascadedchannel over generalized Nakagami-m fading in [4]. In [5],the Fox H-function is applied to model spherically invariantrandom processes, whereas in [6]–[9], expressions for thecapacities and bit error probabilities for different modulationschemes over fading channels are given. The authors in [10],thoroughly analyze integral transformations involving theFoxH-function within the scope of wireless communications,whereas in [11] and [12], the multivariable Fox H-function isused to obtain expressions for the symbol error rate in thecontext of maximal ratio combining and equal gain combin-ing, for the α-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 distributionsaccounting for the path loss, shadowing, and multipath fadingas well as communication links covering random distances.Nonetheless, the overwhelmingmajority of the fadingmodelsthat we address in our work are not considered in [13].

One of the key challenges which presides over the use ofthe Fox H-function is how to evaluate it, whereby computa-tion of the multivariable Fox H-function is still an open issue.To this end, thework reported in [12] provides a useful Pythonimplementation, which, according to the authors, is preciseand efficient for up to four variables. Alternatively, calculusof 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 variablesgrow beyond four. It is important to mention that, in our case,the use of the Fox-H function has not been an end in itselfbut 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 (obtainedby means of the use of the Fox-H function). This meansthat it is possible to avoid dealing with the multivariable

1In [3], these same statistics, namely PDF and CDF, are found for thecross-product of α-µ, η-µ, and κ-µ fading variates.

Fox H-function altogether when computing the ratios of theaforementioned RVs.

The ratio of some of these general variates have beenused mainly in the context of co-channel interference andfor specific scenarios. For instance, in [14] and [15], the µparameter 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 expressionsare obtained in the analysis of co-channel interference forthe κ-µ/κ-µ scenario. In [19], the outage is obtained usingnested summations for the α-µ scenario. In [20], the authorcalculates the PDF of the ratio of α-µ variates given in termsof the Fox H-function, which is a result equivalent to theone presented here. However, no expression for the CDF isoffered. Also, in [21], the authors calculate the PDF and theCDF of the ratio of α-µRVs, but express their results either asan infinite power series or, with some reasonable restrictionon 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 functions of theratio of random envelopes involving the α-µ, κ-µand η-µ fading distributions are derived. The expres-sions are obtained in terms of the multivariable FoxH-function and also in terms of easily computableinfinite series;

2) Novel closed-form expressions for many of the ratiomixtures 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 withinteger µ for the η-µ variate, η-µ/η-µ with integer µfor only one of the η-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 signalingchannel for all ratios are obtained in a high signal-to-noise ratio (SNR) condition;

5) To demonstrate some practical applications for theresults presented in this paper, we employ the ratiosof the aforementioned RVs to estimate the secrecycapacity of a Gaussian wire tap channel used fordevice-to-device (D2D) and V2V communicationsusing data obtained from field measurements con-ducted at 5.8 GHz.

The remainder of this paper is organized as fol-lows. Section II reviews the statistical models considered.Section III provides exact expressions for the PDF and CDFfor all combinations of ratios involving α-µ, η-µ, and κ-µRVs along with asymptotic expressions for the lower andupper tails. In the same way, asymptotic expressions for theBER of a binary signaling channel in high SNR condition areobtained. Section IV depicts some plots for the ratio PDFs.

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Some practical applications are then presented in Section Vusing data obtained from field measurements. Section VIprovides some concluding remarks.

II. PRELIMINARIESA. THE α-µ DISTRIBUTIONThe PDF of a fading envelope R > 0 following an α-µdistribution, with α-root mean square r̂ , α

√E[Rα] is given

by [1]

fR(r) =αrαµ−1

0(µ)Aαµexp

(−rα

Aα

)(1)

in which E[·] represents the expectation operator, α > 0is associated with the non-linearities of the wireless radiochannel, µ > 0 is given by µ = E2[Rα]/V[Rα] and it isrelated to the number of multipath clusters,0(·) is the Gammafunction [22, Eq. (6.1.1)],V[·] is the variance operator and theconstant A is defined as

A =r̂

µ1/α . (2)

B. THE κ-µ DISTRIBUTIONThe PDF of a fading envelope R > 0 following a κ-µ distri-bution, with rms value r̂ ,

√E[R2], is written, alternatively

to [2, Eq. (1)] by using the identity [22, Eq. (9.6.47)], as

fR(r) =2 r2µ−1

eκµK2µ exp(−r2

K2

)0F̃1

(µ; κµ

r2

K2

)(3)

in which κ > 0 is the ratio between the total power of thedominant components and the total power of the scatteredwaves, µ > 0 is related to the number of multipath clustersand can be obtained in terms of the physical parameters byµ = E2[R2]/V[R2] × (1 + 2κ)/(1 + κ)2, 0F̃1(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 isdefined as

K =r̂

√µ(1+ κ)

. (4)

C. THE η-µ DISTRIBUTIONThe PDF of a fading envelope R > 0 following an η-µ distri-bution, with rms value r̂ ,

√E[R2], is given, alternatively to

[2, Eq. (17)] by using the identity [22, Eq. (9.6.47)] and someminor algebraic manipulations, as

fR(r)=2hµr4µ−1

0(2µ)E4µ exp(−r2hE2

)0F1

(µ+

12;r4H2

4E4

)(5)

in which µ > 0 is related to the number of multipathclusters and is given in terms of the physical parameters byµ = E2[R2]/V[R2] × (1 + (H/h)2)/2, and the constant E isdefined as

E =r̂√2µ. (6)

The parameters h and H are functions of η and are definedaccording to the chosen format. In Format 1, 0 < η < ∞

is the scattered-wave power ratio between in-phase andquadrature components, with h = (2 + η−1 + η)/4 andH = (η−1 − η)/4. In Format 2, −1 < η < 1 is the cor-relation coefficient between the scattered-wave in-phase andquadrature components, with h = (1− η2)−1 and H = ηh.

III. PDFS AND CDFS OF THE RATIO DISTRIBUTIONIn this section, the PDF and CDF of the random variableZ , X/Y , with X and Y being independent and arbitrarilydistributed RVs following either an α-µ, κ-µ or η-µ distri-bution, are presented. While these results can be expressedin terms of the multivariate Fox H-function, unfortunatelythe multivariate Fox H-function does not provide any seg-regation in the number of integrations as well as making itquite difficult to infer changes in the curves by modifyingthe parameters. It is also recognized that while works on thesoftware implementation of the multivariate Fox H-functionare on-going, it may be a number of years before these arereadily available in popular mathematical packages such asMATLAB andMathematica. Acknowledging this, any resultscomputed in terms of the Fox H-function are relegated toAppendix A. Instead we focus on providing computableseries representations for all of the statistics of interest, whichyield much greater insights into the interaction of numeratorand denominator RVs.

Tables 1 and 2 show series representation for the PDFand CDF, respectively, of the ratio distribution involving theα-µ, κ-µ and η-µ fading models, in which (a)n is 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 witha, b ∈ {α, κ, η}, are defined as

uαα =Ax

Ay, uακ =

Ax

Ky, uαη = Ax

√hyEy

,

uκκ =Kx

Ky, uκη = Kx

√hyEy

, uηη =Ex√hx

√hyEy

,

uκα =Kx

Ay, uηα =

ExAy√hx, uηκ =

ExKy√hx,

and vab =z2

z2 + u2ab. (7)

The corresponding mathematical derivation can be foundin Appendix C. We maintain that all of the series derivedhere are convergent. The mathematical proof for this canbe done by following the standard procedure presentedin [24, Section 1.7 and 2.9]. It is noteworthy that in [25],the statistics for the ratio of two α-µ variates were obtained inclosed form as a sum of a hypergeometric function providedthat 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 andCDF of the ratio of twoα-µ variatesprovided in Tables 1 and 2 have no restriction whatsoever.Although the convergence of these series is dependent onthe condition αx < αy this presents no restriction on theirapplicability as the reciprocal distribution can be used toevaluate the required statistics. It is important to remark

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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 oftwo 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 onan ordinary desktop computer.2 The series involving the α-µ

2‘‘Rapidly’’ refers to the time taken for an ordinary desktop computer toevaluate the series. In particular, for those not involving the α-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 choiceof the parameters. For those involving the α-µ variates, the number ofterms for the same accuracy were around 20 and the time taken variedfrom 0.6s to 5s. The stopping criterion for the calculations is governed bygenerating sufficient terms to achieve an accuracy of 10−8. In general, thisis accomplished when the nth term is at the order of the required precision.

distribution are still troublesome as they involve the singlevariable Fox H-function and therefore are dependent on theefficiency of the Fox H-function implementation. Although aclever computational implementation of the Fox-H functioncan be found in [4], when α > 2 and in the vicinity of z = 0it may yield incorrect results. A useful implementation ofthe Fox H-function, customized for the needs of this paper,is shown in Appendix D. It is also worthwhile remarking thatsome 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 haveshown that, surprisingly, the overall convergence speed of thedouble summation is faster than the single one. Of course,

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by changing the order of summation of the residues, otherseries expansions can be found, but those given here havebeen proved efficient.

A. SERIES REPRESENTATIONS: RATIO OFκ-µ/α-µ, η-µ/α-µ AND η-µ/κ-µIn the previous subsections, the PDFs and CDFs for the ratiosα-µ/κ-µ, α-µ/η-µ and κ-µ/η-µwere presented. The PDF forthe reciprocal distribution, i.e., Z̃ = 1/Z , may be obtainedfollowing a standard statistical procedure as

fZ̃ (z) =1z2fZ

(1z

), (8)

and its CDF as

FZ̃ (z) = 1− FZ

(1z

). (9)

Hence, Tables 1 and 2 can be used directly to obtain thesestatistics.

B. SOME SPECIAL CASES: NEWCLOSED-FORM EXPRESSIONSFor some combinations of parameters, the ratio PDF maybe obtained as a closed-form expression given in terms ofthe hypergeometric function. For instance, by setting µx =µy = 1/2 in the η-µ distribution (thus reducing it to the Hoytdistribution), using [26, Eq. (6.8.1.6)] along with the lineartransformation given by [26, Eq. (7.3.1.3)], and after somealgebraic manipulations, the PDF is given as

fZ (z) =2√hxhy(r̂x r̂y)2(hyr̂2x + z

2hx r̂2y )z

(hyr̂4x + 2z2hxhyr̂2x r̂2y + z4hx r̂4y )3/2

×2F1

(34,54; 1;

4z4H2xH

2y (r̂x r̂y)

4

(hyr̂4x + 2z2hxhyr̂2x r̂2y + z4hx r̂4y )2

).

(10)

Ratios involving the Nakagami-m distribution can also beobtained in terms of simpler functions. By setting αx = 2 inthe α-µ distribution (thus reducing it to the Nakagami-m dis-tribution), and after some algebraic manipulations, the PDFof the ratios of Nakagami-m/κ-µ, and Nakagami-m/η-µ aregiven, respectively as

fZ (z) =2vµxακ (1− vακ) µy

zeκyµyB(µx , µy

)×1F1

(µx + µy;µy; κyµy(1− vακ )

), and (11)

fZ (z) =2vµxαη

(1− vαη

)2µy

zB(µx , 2µy

)hµyy

×2F1

(µx + 2µy

2,1+ µx + 2µy

2;

1+ 2µy2;H2y

h2y

(1− vαη

)2). (12)

Another interesting particular case involving the η-µ dis-tribution is found in closed-form for µ ∈ Z. By writingthe 2F1(·, ·; ·; ·) function, found in the results involvingthe 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-µand η-µ/η-µ are obtained and given, respectively, by (13)and (14), as shown at the bottom of this page, in whichGa =

∣∣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 over κ-µ/η-µ and η-µ/η-µ. Some importantclosed-form expressions for some particular cases of the ratioκ-µ/κ-µ are derived in [18]

Of course, the inverse of these ratios (κ-µ andNakagami-m, η-µ and Nakagami-m, and η-µ and κ-µ) can beeasily obtained as hinted earlier. The PDF for other importantratio distributions that are special cases of the Nakagami-m

fZ (z) =21−µy

(1− vκη

)µy vµxκηzeκxµx

∣∣Hy∣∣µyµy−1∑k=0

(1− µy

)k

(µy)k

k!

((−1)k (1− Gκ)−µx−µy+k (2Gκ )−k

B(µx , µy

) (1− µx − µy

)k

×1F1

(−k + µx + µy;µx;

κxµxvκη(1− Gκ)

)+

(−1)µyµx (1+ Gκ)−µx−µy(µx + µy

)B(µy, µx + µy

)×(Gκ − 1)k (2Gκ )−k(1+ µx + µy

)k

2F2

(1+ µx , µx + 2µy;µx , 1+ k + µx + µy;

κxµxvκη(1+ Gκ)

)). (13)

fZ (z) =21−µy

(1− vηη

)µyv2µxηη

zhµxx∣∣Hy∣∣ µy

µy−1∑k=0

(1− µy

)k

(µy)k

k!

(Gη − 12Gη

)k ( (1− Gη

)−ξ1B(2µx , µy

)(1− ξ1)k

×2F1

(ξ1 − k

2,ξ1 − k + 1

2;12+ µx;

H2x v

2ηη

h2x(1− Gη

)2

)+

2(−1)µyµx(1+ Gη

)−ξ1ξ1B

(µy, ξ1

)(1+ ξ1)k

×3F2

(1+ µx , µx + µy,

12+ µx + µy;

1+ k + ξ12

,2+ k + ξ1

2;

H2x v

2ηη

h2x(1+ Gη

)2)). (14)

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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 bymaking the appropriate parameter substitutions.

C. ASYMPTOTIC EXPRESSIONS1) PDF AND CDFTo obtain closed-form expressions for the behavior of thePDF for both the lower and upper tails, the following proce-dure is performed. For the cases involving the α-µ distribu-tion, the Fox H-function is expanded in a power series usingthe residue theorem. By taking the poles over the 0(µx + t),the variable z will have a positive exponent. Now, consider zsmall enough so that it is possible to ignore all terms in thesum other than the first one resulting in an expression for thelower tail PDF. On the other hand, by taking the residues overthe other set of poles, a negative exponent on z arises. In thiscase, when z is high enough, all terms of the sum but the firstmay be ignored resulting in the upper tail PDF.

In the other scenarios, for small values of z, we mayconsider vab ∼= z2/u2ab and 1 − vab ∼= 1. As z is in thevicinity of zero, summation indexes different from zero canbe ignored. That is, for the lower tail asymptotic behavior,only the first term in the sum is used. Proceeding like thisand after some algebraic manipulations, the lower tail PDFspresented 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

Table 1 and performing the required summation, the upper tail

PDFs given in Table 3 are obtained. Of course, by integratingthese expressions, the CDFs of the asymptotic behavior canalso be obtained, and these are given in Table 4. Note thatthe upper tail CDFs are indeed one, although the expressionsprovided in Table 4 can be used to achieve those for the lowertails of the reciprocal distribution by using (9).

2) BIT ERROR RATEIn 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 themodulation and detection schemes3 and 0(·, ·) is the comple-mentary Gamma function [22, Eq. (6.5.3)]. Now consider abinary signal over a ratio channel Z = h1/h2 where h1 , Xand h2 , Y , with X and Y as defined before. The SNR isobtained as

0 =EsN0

(h1h2

)2

. (16)

in which Es is the average energy per symbol and N0 is thenoise power spectral density. The average SNR is given as

γ̄ =EsN0

E[h21]E[h−22 ]. (17)

3The authors in [28, Table 8.1] provide the range of possible values fora and b.

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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 SNRis given as

f0(γ ) =

√E[Z2]

2√γ γ̄

fZ

√γE [Z2]

γ̄

(18)

The average BER is obtained by averaging (15) over theSNR as

P̄ =∫∞

0P(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 specialfunction therein to obtain a PDF in the form of

∑i ai(γ /γ̄ )

bi .To obtain the asymptotic behavior for the BER in a high SNRregime, higher exponents in the PDF can be ignored takingonly i = 0, which means that the BER behavior at high SNRlevels depends only on the lower tail of the channel PDF. Thatis, to obtain an asymptotic equation for the BER at high SNRover a ratio channel, replace fZ (z) by the respective channellower tail provided in Table 3. Table 5 gives the asymptoticbehavior of the BER over a ratio channel for all possiblecombinations of ratios.

IV. NUMERICAL RESULTSAs an illustration, Fig. 1 shows 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 noteworthythat the slope (angular coefficient) of the lower tail dependssolely 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 thedistribution of the denominator, specifically the parameterµ (and α if the α-µ distribution is involved). This can beobserved in the expressions of Tables 3 and 4 and in theillustrations of Fig. 1(a), (b), (c). On the other hand, the linearcoefficient (shift on the y axis) in the lower tail is moreinfluenced by the numerator, whereas in the upper tail thisis by the denominator (see Fig. 1(c)). It is worth highlightingthat the α-µ, κ-µ, and η-µ distributions have a finite non-zero value at z = 0 when αµ = 1, 2µ = 1 and 4µ = 1,respectively. The same effect is encountered for the ratiodistributions if the RV on the numerator also satisfies thecondition for finite, non-zero value at z = 0.In Fig. 2, the BER is depicted for the κ-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 analyticalcurves. Additionally, it can be seen from Fig. 2 that theasymptotes start to coincide with the exact curve earlier whenthe numerator parameter µ (or αµ if the numerator is α-µdistributed) is small.

V. SOME PRACTICAL APPLICATIONSIn this section, we demonstrate the practical utility of the newformulations derived here by analyzing the secrecy capacityof a Gaussian wire-tap channel [29], [30], for D2D and V2Vcommunications, using data obtained from field measure-ments. The secrecy capacity for a Gaussian wire-tap channel

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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+ γm1+ γw

), if γm > γw

0, if γm 6 γw

(20)

in which γm , X2 and γw , Y 2 are the SNR for the mainand the wire-tap channels, respectively, and are given by γi =Ph2i /Ni, i = (m,w), in which P and Ni are the transmission

FIGURE 2. BER of the ratio of κ-µ and η-µ RVs with µy = 2, ηy = 0.75,κx = 2, and µx = {0.25,1,2.5} along with their asymptotic behavior forhigh SNR.

and noise power, respectively, and hi is the channel gain. Theprobability of positive secrecy capacity is given as

Pr[Cs > 0] = 1− Pr

[hmhw

<

√NmNw

]

= 1− FZ

√ γ̄wE[h2m]γ̄mE[h2w]

=FZ̃√ γ̄mE[h2w]

γ̄wE[h2m]

(21)

in which: (i) FZ (·) is obtained from (24) with parameterstaken from Table 8 and evaluated using the exact series givenin Table 2 in accordance with the chosen fading model for hmand hw; (ii) FZ̃ is the CDF for the reciprocal distribution givenin (9); and (iii) γ̄m and γ̄w are the mean values of the SNR forthe main and wire-tap channels, respectively.

A. D2D COMMUNICATIONSTheD2Dmeasurements were conducted at 5.8 GHzwithin anindoor seminar room. The exact details of the measurementsetup, experiments, and data analysis can be found in [31].The measurement trial considered three persons who carriedthe hypothetical user equipments (UEs) A, B and E4 and werepositioned at points X, Y and Z respectively (see Fig. 3). Allthree persons had the UEs positioned at their heads, and wereinitially stationary. The persons at positions Y and Z werethen instructed to walk around randomly within a circle ofradius 0.5 m from their starting positions whilst imitating avoice call. It should be noted that the channel between UEsA and B is referred to as the main channel whilst the channelbetween UEs A and E is referred to as the wire-tap channel.

For the analysis, the data sets were normalized totheir respective local means prior to parameter estimation.To determine an appropriate window size for the extractionof the local mean signal, the raw data was visually inspectedand overlaid with the local mean signal for differing win-dow sizes. In this case, a smoothing window 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 encoun-tered in eavesdropping analyses of wireless security applications.

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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 the α-µ, κ-µ and η-µfading models fitted to the D2D fading data for the main andthe wire-tap channels, respectively. A total of 74763 samplesof the received signal power were obtained and used forparameter estimation. The parameter estimates were obtainedusing the lsqnonlin function available in the optimizationtoolbox of MATLAB along with the α-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 Table 6.

From Fig. 4, we observe that both the α-µ and κ-µ PDFsprovide a very good approximation to the measured data,whilst some disparity is noticed between the lower tail ofthe empirical PDF and the theoretical η-µ PDF. To select thecandidate model, from the α-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 uponthe ranking performed using the computed AIC, it was foundthat the α-µ distribution was the most likely model for themain channel, whilst the κ-µ distribution was the most likelymodel for the wire-tap channel. The α parameter estimatefor the main channel was found to be higher than that foran equivalent Nakagami-m fading channel (α = 2, µ = m),while µ was found to be 0.68 suggesting a tendency towardssignificant fading for the main channel. This is understand-able due to the constantly changing orientation and postureof both the test subjects. For the wire-tap channel, we see thatthe κ parameter estimate was greater than 1, suggesting thata perceptible dominant component existed. The µ parameterestimate for the wire-tap channel was found to be relativelyclose 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 fieldtrials (see Table 6), Fig. 5 depicts the estimated proba-bility of positive secrecy capacity versus γ̄w for a rangeof γ̄m, when the main and the wire-tap channels experi-ence α-µ and κ-µ fading, respectively. We now adopt thefollowing 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 the α-µ, κ-µ and η-µ PDFs for the D2Dchannel measurements.

TABLE 6. Parameter estimates for the α-µ, κ-µ and η-µ fading models fitted to the D2D measured data along with the AIC rank.

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FIGURE 5. Probability of positive secrecy capacity versus γ̄w consideringa range of γ̄m for the D2D channel measurements when the main andwire-tap channels are assumed to undergo α-µ and κ-µ fading,respectively. Here, PSC indicates positive secrecy capacity.

secrecy capacity. We perform our analysis when γ̄m is fixed at10 dB and for two different levels of positive secrecy capacity:0.10 (10% level) and 0.50 (50% level), which are indicativeof low and mid-range levels of positive secrecy capacity,respectively. From Fig. 5, we observe that if the wire-tappercan improve her average SNR (γ̄w) from 5 dB to 10 dB,the probability of positive secrecy capacity will decreasefrom 77% to 49%. Furthermore, to ensure a positive secrecycapacity level of at least 10%, we find that the wire-tappersaverage SNR must not exceed 19 dB. Likewise, to ensurea mid-range positive secrecy capacity level of at least 50%,γ̄w must not exceed 10 dB.

B. V2V COMMUNICATIONSThe V2V channel measurements considered in this workwere also conducted at 5.8 GHz. The exact details of themeasurement setup and experiments can be found in [31].Specifically, the transmitter, the legitimate receiver and thewire-tapper were positioned on the center of the dashboardswithin vehicles A, B and E, respectively (see [31, Fig. 10]).

Initially, vehicles A and B drove towards each other at a speedof 30 mph whilst the vehicle containing node E remainedparked on the side of the road. The fading data used in theanalysis presented here, considered the channel acquisitionsobtained when vehicles A and B drove past each other andcontinued their onward journey. As before, it should be notedthat the channel between nodes A and B is referred to as themain channel whilst the channel between nodes A and E isreferred to as the wire-tap channel.

Similar to the D2D channel measurements, the data setswere normalized to their respective local means prior toparameter estimation. In this case, a smoothing windowof 200 samples was used. Figs. 6(a) and (b) show the PDFof the α-µ, κ-µ and η-µ fading models fitted to the V2Vdata for the main and the wire-tap channels, respectively.A total of 56579 samples of the received signal power wereobtained and used for parameter estimation. When com-pared to the α-µ and η-µ PDFs, we observe that the κ-µdistribution provides a better approximation around the lowertail of the empirical PDFs in Figs. 6(a) and (b). On thecontrary, the η-µ PDF provides a better approximation thanthe α-µ and κ-µ distributions around the median (where thegreatest number of fade levels occur) and also the upper tailof the empirical PDF. Accordingly, from the computed AICrankings (see Table 6), it was found that the η-µ distributionwas the most likely model for both the main and wire-tapchannels. Inspecting the estimated η parameter for the V2Vchannel measurements (see Table 6), we observe that underthe assumption of Format 1 for the η-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 of η-µ fading, low estimates of µwerealso found for both themain andwire-tap channels, indicatingthat minimal environmental multipath contributions offeredby the surrounding environment.

Now, using the parameter estimates from V2V field tri-als (see Table 6), Fig. 7 depicts the estimated probabilityof positive secrecy capacity versus γ̄w for a range of γ̄m,

FIGURE 6. Empirical envelope PDF of (a) the main channel and (b) the wire-tapchannel compared to the α-µ, κ-µ and η-µ PDFs for the V2Vchannel measurements.

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FIGURE 7. Probability of positive secrecy capacity versus γ̄w consideringa range of γ̄m for the V2V when it is assumed to undergo η-µ fading. Here,PSC indicates positive secrecy capacity.

when the main and the wire-tap channel experiences η-µfading. Following a similar approach to theD2D analysis withγ̄m fixed at 10 dB, we observe that increasing γ̄w from 5 dBto 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 γ̄w must not exceed 15 dB and 10 dB,respectively.

VI. CONCLUSIONSExact, computationally efficient series representations forthe PDF and CDF of the ratio of two fading envelopesarbitrarily taken from α-µ, κ-µ, and η-µ distributions areprovided. Novel and compact expression for these ratios arealso 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 withinteger µ for only one of the η-µ, and their reciprocals arefound. Simple closed-form expressions for the asymptotes ofthe PDFs and CDFs of all ratios, both for lower and upper tailsof the distributions as well as for BER of a binary signalingchannel have been derived. Applications for these resultsinclude, for instance, multihop systems, spectrum sharing,the characterization of co-channel interference and physicallayer security, among other areas of wireless communica-tions. Multipath-shadowing composite fading statistics canalso be obtained as special cases of the ratio distribution andthe various combinations given here can be used to charac-terize a plethora of fading environments. Interestingly, and aswell known, the sum of independent identically distributed(i.i.d.) κ-µ powers is another κ-µ. In the sameway, the sum ofi.i.d. η-µ is another η-µ. Therefore, the results provided herecan be directly applied to maximal ratio combining systemsin the presence of interference. In the same way, the sumof i.i.d. α-µ, κ-µ, and η-µ envelopes can be well approx-imated by the respective distributions, then rendering theresults useful in the study of equal gain combining systems.

Similar comments concerning the sum of independent non-identically distributed variates (power and envelope) alsoapply. Finally, a practical application for the novel resultsderived here was demonstrated by analyzing the secrecycapacity of a Gaussian wire-tap channel for D2D and V2Vcommunications, using data from field measurements withexperiments 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 mathematicaltechnique. In spite of its unquestionable utility, accordingto [32] and [33], it is not always the most efficient way toobtain a particular mathematical result. In our experience,with the expressions presented in this paper, using Mathe-matica and the same set of parameters, numerical integrationtakes a significantly larger amount of time to compute thanpower series do. In particular, for the ratios involving η-µand/or κ-µ, as far as computation time is concerned, there is aclear advantage in favor of the series representations over thecorresponding integral representations (roughly on the othervarying from 10 to 100 times). Of course, the time and thenumber of terms to achieve a certain accuracy depend onthe choice of the parameters. On the other hand, as alreadymentioned in the paper, the performance of the series specif-ically involving the α-µ distribution is highly dependenton the Fox H-function implementation. Although the seriesrequire very few terms for convergence (around 20 terms),at present, there is no clear advantage between evaluating theexpressions using the integral or the series representation asfar as computational time is concerned. However, the seriesexpressions find a clear advantage over their counterpartintegral representations because, from the series, it is possibleto obtain their asymptotes and, then, the asymptotic perfor-mance behavior of the systemmodeled under the correspond-ing scenarios. This has been successfully attained in terms ofsimple closed-form expressions for all the ratios in this paper.

APPENDIX ARESULTS IN TERMS OF THE MULTIVARIATEFOX H-FUNCTIONIn Section III, the PDF and CDF for the ratio distributionsare given in terms of infinite power series. These resultswere obtained through the sum of residues of several FoxH-functions with two and three variables. Here, we presentthe PDF and CDF in terms of the multivariable FoxH-function5 defined in [34]6

H[x; (β,B); (δ,D),L] =(

12π j

)N ∮L2(s)x−sds, (22)

5The integral form of the Fox H-function on several variables was notused to evaluate the PDF and CDF analysed here. As a matter of fact, theseresults were a tool used to derive the series expansion provided in Section III.An interesting Python implementation for the multivariable Fox H-functionis found in [12].

6There are several notations for the multivariable Fox H-function.We believe that the notation by Hai and Srivastava [34] is the most concise.

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in which j =√−1 is the imaginary unit; s = [s1, . . . , sN ],

x = [x1, . . . , xN ], β = [β1, . . . , βm] and δ = [δ1, . . . , δn]denote vectors of complex numbers and B = (bi,j)m×N andD = (di,j)n×N are matrices of real numbers. Also x−s =∏N

i=1 x−sii , ds =

∏Ni=1 dsi, L is an appropriate contour on

the complex space, and

2(s) =

m∏i=10

(βi +

N∑k=1

bi,ksk

)n∏i=10

(δi +

N∑k=1

di,ksk

) . (23)

Interestingly, both the exact PDFs and the exact CDFs forthe ratio of variates can be compactly written in a generalform given as

g(z) = CH[x; (β,B); (δ,D),L] (24)

Tables 7 and 8 show the values for the parameters appearingin (24) for, respectively, the ratio PDF and ratio CDF for eachcombination of the random variables X and Y considered inthis work. The conditions for convergence of the integral in(22) are thoroughly discussed in [34], [35]. We highlight thatall the necessary convergence conditions in our paper are metin accordance with these references.

Also, I3 denotes the identity matrix of order three. Thegeneral steps followed in order to arrive at (24) can be foundin the Appendix B.

APPENDIX BDERIVATION OF THE RATIO STATISTICSLet Z , X/Y be the ratio between two independent andarbitrarily distributed random variates with positive support.Then its PDF can be obtained through standard probabilityprocedures as

fZ (z) =

∞∫0

yfX (zy)fY (y)dy. (25)

Consider now that both X and Y follow the κ-µ distribu-tion. Then replacing (3) in (25) and performing some minoralgebraic manipulations, fZ (z) becomes

fZ (z) =4z2µx−1

exp(κxµx + κyµy

)K2µxx K2µy

y

×

∞∫0

y−1+2µx+2µy exp

(−y2

(z2

K2x+

1K2y

))

× 0F̃1

(µx;

κxµx(zy)2

K2x

)0F̃1

(µy;

κyµyy2

K2y

)dy

(26)

Using [26, Eq. (7.2.3.12)] to write the hypergeometricfunctions with their Mellin-Barnes contour integral

representation we obtain

fZ (z) =4z2µx−1

exp(κxµx + κyµy

)K2µxx K2µy

y

(12π j

)2

×

∞∫0

y−1+2µx+2µy exp

(−y2

(z2

K2x+

1K2y

))

×

∫∫©L

0(τ1)0(τ2)

0 (−τ1 + µx) 0(−τ2 + µy

)×

(−κxµx(zy)2

K2x

)−τ1 (−κyµyy2

K2y

)−τ2dτ1dτ2dy.

(27)

By changing the order of integration and performing the inte-gral on the variable y, the inner integral can be solved using[22, Eq. (6.1.1)]. Then, after some algebraic manipulations,fZ (z) reduces to

fZ (z) =2 vµxκκ (1− vκκ)µy

z exp(κxµx + κyµy

) ( 12π i

)2

×

∫∫©L

0(−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 possibledue to convergence of both inner integrals. The only penaltyis a distortion in the complex contour, which is not reallyan issue. Finally, by performing the variable transformationt1 = τ1 + µx and t2 = τ2 + µy and after some algebraicmanipulations (28) will be in the form of (22) with parametersx, β, B, δ and D provided on the fourth line of Table 7.

Following similar steps as for the κ-µ/κ-µ described abovethe ratio PDF for each combination of fading distribution isobtained. To summarize, the closed-form expression for theratio PDF is obtained by

1) If applicable, write the 0F1 function by its contourintegral representation using [26, Eq. (7.2.3.12)];

2) If the ratio involves α-µ, write one exponentialfunction 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 innerintegral using [22, Eq. (6.1.1)];

4) After some algebraic manipulation and variable trans-formation, interpret the integral as Fox H-functionusing (22).

The ratio CDF can be readily obtained through its defini-tion by writing the PDF as contour integral using (22) andchanging the order of integration. The inner integral will bein the form

∫ x0 τ

k−1dτ = 0(k)/0(k + 1)xk , and after somealgebraic manipulations, the result can be interpreted as a FoxH-function with the parameters provided in Table 8.

APPENDIX CDERIVATION OF THE SERIES REPRESENTATIONSeries representations for the ratio PDF and CDF may beobtained through the sum of residues of the Fox H-function.

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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 Gammafunction 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. Takingthe residues of (28) over τ1 and τ2, fZ (z) can be written as adouble sum given by

fZ (z) =2vµxκκ (1− vκκ)µy

zeκxµx+κyµy

×

∞∑i=0

∞∑j=0

(vκκκxµx)i((1− vκκ) κyµy

)ji!j!B

(i+ µx , j+ µy

) . (30)

Performing the sum over the index j the series for the ratioPDF for the κ-µ/κ-µ in Table 1 is obtained. Through a similarprocess, the other series in Tables 1 and 2 can be obtained.

APPENDIX DMATHEMATICA IMPLEMENTATION FOR THESINGLE FOX H-FUNCTIONThe following Mathematica program is a simple imple-mentation for the Fox H-function used to evaluate the FoxH-function present in Tables 1 and 2, in which the parameter

limit is used to speed up the computation of the numericalintegration and it can be chosen as large as possible7.8

7 In particular, it has been found that for the Fox-H function appearing inthe PDFs, such limit is set to infinity. As for the CDF, such limit has been setto 50. These have been used for all of our computations.

8This implementation is, by no means, a general one, but it is sufficient toevaluate the Fox H-functions presented here.

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CARLOS RAFAEL NOGUEIRA DA SILVA wasborn in Brazil, in 1986. He received the B.S. degreein electrical engineering from the Federal Uni-versity of Itajubá, MG, Brazil, in 2010, the M.S.degree in electrical engineering from the NationalInstitute of Telecommunications, MG, in 2012,and the Ph.D. degree from the School of Electricaland Computer Engineering, State University ofCampinas, UNICAMP, in 2018. He is currentlyholding a Postdoctoral FAPESP Scholarship with

the State University of Campinas, UNICAMP. His research interests includewireless channel modeling, spectrum sensing, cognitive radio, compositechannel characterization and applications, and wireless communications ingeneral.

NIDHI SIMMONS received the B.E. degree(Hons.) in telecommunications engineering fromVisvesvaraya Technological University, Belgaum,India, in 2011, the M.Sc. degree (Hons.) in wire-less communications and signal processing fromthe University of Bristol, U.K., in 2012, and thePh.D. degree in electrical engineering from theQueen’s University of Belfast, U.K., in 2018. Sheis 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-centriccommunications.

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C. R. N. da Silva et al.: Ratio of Two Envelopes Taken From α-µ, η-µ, and κ-µ Variates

ELVIO J. LEONARDO was born in Brazil.He received the B.Eng., M.Sc., and Ph.D. degreesfrom the State University of Campinas, Brazil,in 1984, 1992, and 2013, respectively, all in elec-trical engineering. In 1980, he was involved inthe 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 toBrazil, where he started his academic career. He is currently an AdjunctProfessor with the State University ofMaringa. His research interests includewireless communications and embedded systems.

SIMON L. COTTON (S’04–M’07–SM’14)received the B.Eng. degree in electronics andsoftware from Ulster University, Ulster, U.K.,in 2004, and the Ph.D. degree in electrical andelectronic engineering from the Queen’s Univer-sity of Belfast, Belfast, U.K., in 2007. He is cur-rently a Reader in wireless communications andthe Director of the Centre for Wireless Innova-tion (CWI), Queen’s University Belfast. He hasauthored and coauthored over 140 publications

in major IEEE/IET journals and refereed international conferences, twobook chapters, and two patents. Among his research interests are cellulardevice-to-device, vehicular, and body-centric communications. His otherresearch interests include radio channel characterization and modeling, andthe simulation of wireless channels. He was a recipient of the H. A. WheelerPrize, in 2010, from the IEEE Antennas and Propagation Society for thebest applications journal paper in the IEEE TRANSACTIONS ON ANTENNAS AND

PROPAGATION, in 2009, and the Sir George Macfarlane Award from the U.K.Royal Academy of Engineering in recognition of his technical and scientificattainment 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, Campinas, SP, Brazil, in 1978and 1983, respectively, and the Ph.D. degree fromthe University of Essex, U.K., in 1988. From 1978to 1985, he was a Research Specialist in the devel-opment of the Tropico digital exchange familywith the Research and Development Center of

Telebrás, Brazil. In 1989, he joined the School of Electrical and ComputerEngineering, UNICAMP, where he is currently a Full Professor. He consultsfor several operating companies and industries in the wireless communica-tions area. He is the author of Foundations of Mobile Radio Engineering(Boca Raton, FL: CRC, 1993), Wireless Technology: Protocols, Standards,and Techniques (Boca Raton, FL: CRC, 2001), and the coauthor of theTelecommunications: Principles and Trends (São Paulo, Brasil: Erica, 1997,in Portuguese). He holds two patents. His general research interest includeswireless communications.

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