text: Caballero-Garc´ıa et al.: Activity from V 0332+53 broadly in agreement with those previously obtained (Reig of this spectral analysis and the derived unabsorbed ﬂuxes & Nespoli, 2013). and luminosities are shown in Tab. 3,4 and in Fig. 6. 3.3.3. Spectral analysis In addition, we ﬁtted the spectra obtained with Suzaku, during much a lower luminosity period (see Fig. 7 and We ﬁtted the brightest background-subtracted spectra Tab. 5). Spectra from XIS1, XIS3 and PIN were ﬁtted in the with standard spectral models in XSPEC 12.8.2 (Arnaud, 0.8-10,14-25 keV energy range, with the 1.6-2.3 keV energy 1996). All errors quoted in this work are 68% (1σ) conﬁ- range ignored due to the uncertainties in calibration asso- dence. The spectral ﬁts were limited to the 1-10, 4.5-28, ciated with the instrumental Si K edge16. The spectra were 26-60 keV range for the XRT, PCA and the HEXTE, re- grouped to have at least 50,250 counts for the two XIS and spectively, where the calibration of the instruments is the PIN, respectively. In this case a model composed by pho- best. The spectra were grouped in order to have at least toelectric absorption (phabs) and a power-law with an ex- 100, 50, 50 counts for the XRT, PCA and HEXTE, respec- ponential high-energy cutoﬀ of the type “highecut” was a tively for each background-subtracted spectral channel and rather better description than “cutoffpl” (i.e. ∆ χ2 = 15 to avoid oversampling of the intrinsic energy resolution. for the same number of d.o.f and better residuals). With As it is a common procedure in observational stud- this model we obtained a statistics of χ2/ν = 1.4 and ies of NS systems we used empirical models to describe systematic residuals at E < 4 KeV, that we improved by the X-ray spectra. To compare our results with previous adding a blackbody spectrum with normalization propor- studies we used the same (or very similar) spectral compo- tional to the surface area (bbodyrad), obtaining a ﬁt statis- nents. To ﬁt the spectral continuum we used a model com- tics of (χ2/ν = 1.3 for the same number of d.o.f.). Because posed by photoelectric absorption (Balucinska-Church & the normalization of the black-body component was not McCammon 1992; phabs in XSPEC) and a power-law with well constrained we tried to improve it and changed it by an a high-energy exponential cutoﬀ (cutoffpl). The absorp- emission component describing an accretion disc consisting tion component was not used in Obs. 1-4 and 11 (we did not of multiple blackbody components (diskbb; Mitsuda et al. ﬁt the data from Obs. 12,13 because of the lack of counts), 1984; Makishima et al. 1986). We obtained a very similar ≥ since PCA spectra are not aﬀected (i.e. at E 4.5 keV). In ﬁt statistics (see Tab. 5). A plausible origin for the ﬁrst the case of XRT observations, with lower energy coverage model component (bbodyrad) is the NS surface or perhaps ≥ (i.e. at E 1 keV) the absorption component was included the boundary layer (if present). Low-level accretion onto in all our ﬁts. This model provided signiﬁcant wavy resid- the surface of a neutron star can indeed produce a black- uals in the whole spectra of all the observations with non body like spectrum (Zampieri et al., 1995). Still, a disc ori- ≈ − acceptable chi-squared values (χ2/ν 5 60; with ν being gin cannot be ruled out since it was also possible to ﬁt the number of degrees of freedom, i.e. d.o.f.). the data with a multicolor disc blackbody. Similar results for a set of low-accreting NS have been recently obtained The ﬁts and the residuals of the whole spectra improved ≈ − (χ2/ν 1.5 6) by the addition of a cyclotron line scatter- (Armas≈Padilla−et al., 2013), but the temperatures found are kT = 0.5 0.7 keV, thus much lower than in our case ing feature (CRSF; hereafter referred to as “cyclotron fea- ≈ ± (kT = 2.4 0.9 keV). The higher value we obtain might ture”). It was accounted for with a gaussian absorption line be indicative of the much higher accretion rate in our case. (gabs). The use of this model component and/or the ob- Nevertheless, with the current data it is not possible to tained line parameters are consistent with previous studies discern which model component is better to describe the (Kreykenbohm et al., 2005; Tsygankov et al., 2006; Reig data. et al., 2006; Reig & Nespoli, 2013; Lutovinov et al., 2015). The parameters were let to be free in the HEXTE data and To account for the uncertainties in the absolute ﬂux tied to those in the XRT and PCA data. normalization between XIS1, XIS3 and PIN we introduced a multiplicative constant that was ﬁxed to 1 for XIS1 and The ﬁts still show signiﬁcant residuals at 7-10 keV. This let to vary freely for XIS3 and PIN. The most relevant feature is known as the “10 keV feature”, and its origin is results of this spectral analysis and the derived unabsorbed uncertain (Coburn et al., 2002). We modeled it as a gaus- ﬂuxes and luminosities are shown in Tab. 5. sian absorpt≈ion li−ne (gabs), thus improving the ﬁts substan- tially (χ2/ν 1.0 1.2). In this case the parameters were let 4. Discussion to be free in the PCA data and tied to those in the XRT 4.1. Long-term Be-disc/neutron star interaction and HEXTE data. The values obtained are within those expected (Coburn et al., 2002) but with broader dispersion From the long-term observations of V 0332+53 we can than those obtained e.g. for XTE J1946+274 by Mu¨ller see that V 0332+53 spends most of its time in the optical et al. (2012). brightening phases (Fig 2). The most signiﬁcant one is the Although the beneath of the spectral ﬁts, there are still period between 1993-2005. It started approx. 4 years after peaked positive residuals in the region 6-7 keV of the PCA the giant outburst that the system exhibited in 1989 and spectra. We accounted for them adding a gaussian line pro- peaked simultaneously with the X-ray maximum in 2004. ﬁle (gaussian) at 6.4-6.97 keV in order to account for the Indeed all of the optical brightening episodes of BQ Cam Fe Kα ﬂuorescence line. The PCA residuals ﬂattened and are accompanied with the X-ray activities. It is important we obtained the ﬁnal ﬁtting solution in this way. to note that the brightness of the optical companion de- To account for uncertainties in the absolute ﬂux normal- creases after the X-ray outbursts. This behavior can be well ization between PCA and HEXTE we introduced a multi- explained by the weakening of the decretion disc during the plicative constant that was ﬁxed to 1 for the PCA and let mass transfer to the NS. to vary freely for HEXTE. In the case of XRT, PCA and HEXTE data we ﬁxed the multiplicative constant of PCA 16 http://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/ to 1 and the rest to vary freely. The most relevant results sical.html 6 ##LTLine##

text: Caballero-Garc´ıa

text: et

text: al.:

text: Activity

text: from

text: V

text: broadly

text: in

text: agreement

text: with

text: those

text: previously

text: obtained

text: (Reig

text: of

text: this

text: spectral

text: analysis

text: and

text: the

text: derived

text: unabsorbed

text: ﬂuxes

text: Nespoli,

text: and

text: luminosities

text: are

text: shown

text: in

text: Tab.

text: and

text: in

text: Fig.

text: Spectral

text: analysis

text: In

text: addition,

text: we

text: ﬁtted

text: the

text: spectra

text: obtained

text: with

text: Suzaku,

text: during

text: much

text: a

text: lower

text: luminosity

text: period

text: (see

text: Fig.

text: and

text: We

text: ﬁtted

text: the

text: brightest

text: background-subtracted

text: spectra

text: Tab.

text: Spectra

text: from

text: and

text: PIN

text: were

text: ﬁtted

text: in

text: the

text: with

text: standard

text: spectral

text: models

text: in

text: XSPEC

text: (Arnaud,

text: keV

text: energy

text: range,

text: with

text: the

text: keV

text: energy

text: All

text: errors

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text: this

text: work

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text: uncertainties

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text: calibration

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text: dence.

text: The

text: spectral

text: ﬁts

text: were

text: limited

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text: the

text: ciated

text: with

text: the

text: instrumental

text: Si

text: K

text: The

text: spectra

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text: keV

text: range

text: for

text: the

text: XRT,

text: PCA

text: and

text: the

text: HEXTE,

text: re-

text: grouped

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text: at

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text: two

text: XIS

text: and

text: spectively,

text: where

text: the

text: calibration

text: of

text: the

text: instruments

text: is

text: the

text: PIN,

text: respectively.

text: In

text: this

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text: model

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text: pho-

text: best.

text: The

text: spectra

text: were

text: grouped

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text: toelectric

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text: (phabs)

text: and

text: a

text: power-law

text: with

text: an

text: ex-

text: counts

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text: the

text: XRT,

text: PCA

text: and

text: HEXTE,

text: respec-

text: ponential

text: high-energy

text: cutoﬀ

text: of

text: the

text: type

text: “highecut”

text: was

text: a

text: tively

text: for

text: each

text: background-subtracted

text: spectral

text: channel

text: and

text: rather

text: better

text: description

text: than

text: “cutoffpl”

text: (i.e.

text: χ2

text: to

text: avoid

text: oversampling

text: of

text: the

text: intrinsic

text: energy

text: resolution.

text: for

text: the

text: same

text: number

text: of

text: d.o.f

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text: better

text: residuals).

text: With

text: As

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text: a

text: common

text: procedure

text: in

text: observational

text: stud-

text: this

text: model

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text: a

text: statistics

text: of

text: χ2/ν

text: and

text: ies

text: of

text: NS

text: systems

text: we

text: used

text: empirical

text: models

text: to

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text: systematic

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text: E

text: KeV,

text: that

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text: by

text: the

text: X-ray

text: spectra.

text: To

text: compare

text: our

text: results

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text: studies

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text: obtaining

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text: nents.

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text: spectral

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text: tics

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text: for

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text: d.o.f.).

text: Because

text: posed

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text: (Balucinska-Church

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text: black-body

text: component

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text: McCammon

text: phabs

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text: XSPEC)

text: and

text: a

text: power-law

text: with

text: well

text: constrained

text: we

text: tried

text: to

text: improve

text: it

text: and

text: changed

text: it

text: by

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text: high-energy

text: exponential

text: cutoﬀ

text: (cutoffpl).

text: The

text: absorp-

text: emission

text: component

text: describing

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text: accretion

text: disc

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text: not

text: used

text: in

text: Obs.

text: and

text: (we

text: did

text: not

text: of

text: multiple

text: blackbody

text: components

text: (diskbb;

text: Mitsuda

text: et

text: al.

text: ﬁt

text: the

text: data

text: from

text: Obs.

text: because

text: of

text: the

text: lack

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text: counts),

text: Makishima

text: et

text: al.

text: We

text: obtained

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text: very

text: similar

text: since

text: PCA

text: spectra

text: are

text: not

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text: (i.e.

text: at

text: E

text: keV).

text: In

text: ﬁt

text: statistics

text: (see

text: Tab.

text: A

text: plausible

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text: ﬁrst

text: the

text: case

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text: observations,

text: with

text: lower

text: energy

text: coverage

text: model

text: component

text: (bbodyrad)

text: is

text: the

text: NS

text: surface

text: or

text: perhaps

text: (i.e.

text: at

text: E

text: keV)

text: the

text: absorption

text: component

text: was

text: included

text: the

text: boundary

text: layer

text: (if

text: present).

text: Low-level

text: accretion

text: onto

text: in

text: all

text: our

text: ﬁts.

text: This

text: model

text: provided

text: signiﬁcant

text: wavy

text: resid-

text: the

text: surface

text: of

text: a

text: neutron

text: star

text: can

text: indeed

text: produce

text: a

text: black-

text: uals

text: in

text: the

text: whole

text: spectra

text: of

text: all

text: the

text: observations

text: with

text: non

text: body

text: like

text: spectrum

text: (Zampieri

text: et

text: al.,

text: Still,

text: a

text: disc

text: ori-

text: acceptable

text: chi-squared

text: values

text: with

text: ν

text: being

text: gin

text: cannot

text: be

text: ruled

text: out

text: since

text: it

text: was

text: also

text: possible

text: to

text: ﬁt

text: the

text: number

text: of

text: degrees

text: of

text: freedom,

text: i.e.

text: d.o.f.).

text: the

text: data

text: with

text: a

text: multicolor

text: disc

text: blackbody.

text: Similar

text: results

text: for

text: a

text: set

text: of

text: low-accreting

text: NS

text: have

text: been

text: recently

text: obtained

text: The

text: ﬁts

text: and

text: the

text: residuals

text: of

text: the

text: whole

text: spectra

text: improved

text: by

text: the

text: addition

text: of

text: a

text: cyclotron

text: line

text: scatter-

text: (Armas≈Padilla−et

text: al.,

text: but

text: the

text: temperatures

text: found

text: are

text: kT

text: keV,

text: thus

text: much

text: lower

text: than

text: in

text: our

text: case

text: ing

text: feature

text: (CRSF;

text: hereafter

text: referred

text: to

text: as

text: “cyclotron

text: fea-

text: (kT

text: keV).

text: The

text: higher

text: value

text: we

text: obtain

text: might

text: ture”).

text: It

text: was

text: accounted

text: for

text: with

text: a

text: gaussian

text: absorption

text: line

text: be

text: indicative

text: of

text: the

text: much

text: higher

text: accretion

text: rate

text: in

text: our

text: case.

text: (gabs).

text: The

text: use

text: of

text: this

text: model

text: component

text: and/or

text: the

text: ob-

text: Nevertheless,

text: with

text: the

text: current

text: data

text: it

text: is

text: not

text: possible

text: to

text: tained

text: line

text: parameters

text: are

text: consistent

text: with

text: previous

text: studies

text: discern

text: which

text: model

text: component

text: is

text: better

text: to

text: describe

text: the

text: (Kreykenbohm

text: et

text: al.,

text: Tsygankov

text: et

text: al.,

text: Reig

text: data.

text: et

text: al.,

text: Reig

text: Nespoli,

text: Lutovinov

text: et

text: al.,

text: The

text: parameters

text: were

text: let

text: to

text: be

text: free

text: in

text: the

text: HEXTE

text: data

text: and

text: To

text: account

text: for

text: the

text: uncertainties

text: in

text: the

text: absolute

text: ﬂux

text: tied

text: to

text: those

text: in

text: the

text: XRT

text: and

text: PCA

text: data.

text: normalization

text: between

text: and

text: PIN

text: we

text: introduced

text: a

text: multiplicative

text: constant

text: that

text: was

text: ﬁxed

text: to

text: for

text: and

text: The

text: ﬁts

text: still

text: show

text: signiﬁcant

text: residuals

text: at

text: keV.

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text: let

text: to

text: vary

text: freely

text: for

text: and

text: PIN.

text: The

text: most

text: relevant

text: feature

text: is

text: known

text: as

text: the

text: keV

text: feature”,

text: and

text: its

text: origin

text: is

text: results

text: of

text: this

text: spectral

text: analysis

text: and

text: the

text: derived

text: unabsorbed

text: uncertain

text: (Coburn

text: et

text: al.,

text: We

text: modeled

text: it

text: as

text: a

text: gaus-

text: ﬂuxes

text: and

text: luminosities

text: are

text: shown

text: in

text: Tab.

text: sian

text: absorpt≈ion

text: li−ne

text: (gabs),

text: thus

text: improving

text: the

text: ﬁts

text: substan-

text: tially

text: In

text: this

text: case

text: the

text: parameters

text: were

text: let

text: Discussion

text: to

text: be

text: free

text: in

text: the

text: PCA

text: data

text: and

text: tied

text: to

text: those

text: in

text: the

text: XRT

text: Long-term

text: Be-disc/neutron

text: star

text: interaction

text: and

text: HEXTE

text: data.

text: The

text: values

text: obtained

text: are

text: within

text: those

text: expected

text: (Coburn

text: et

text: al.,

text: but

text: with

text: broader

text: dispersion

text: From

text: the

text: long-term

text: observations

text: of

text: V

text: we

text: can

text: than

text: those

text: obtained

text: e.g.

text: for

text: XTE

text: by

text: Mu¨ller

text: see

text: that

text: V

text: spends

text: most

text: of

text: its

text: time

text: in

text: the

text: optical

text: et

text: al.

text: brightening

text: phases

text: (Fig

text: The

text: most

text: signiﬁcant

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text: spectral

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text: period

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text: It

text: started

text: approx.

text: years

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text: in

text: the

text: region

text: keV

text: of

text: the

text: PCA

text: the

text: giant

text: outburst

text: that

text: the

text: system

text: exhibited

text: in

text: and

text: spectra.

text: We

text: accounted

text: for

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text: adding

text: a

text: gaussian

text: line

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text: peaked

text: simultaneously

text: with

text: the

text: X-ray

text: maximum

text: in

text: ﬁle

text: (gaussian)

text: at

text: keV

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text: order

text: to

text: account

text: for

text: the

text: Indeed

text: all

text: of

text: the

text: optical

text: brightening

text: episodes

text: of

text: BQ

text: Cam

text: Fe

text: Kα

text: ﬂuorescence

text: line.

text: The

text: PCA

text: residuals

text: ﬂattened

text: and

text: are

text: accompanied

text: with

text: the

text: X-ray

text: activities.

text: It

text: is

text: important

text: we

text: obtained

text: the

text: ﬁnal

text: ﬁtting

text: solution

text: in

text: this

text: way.

text: to

text: note

text: that

text: the

text: brightness

text: of

text: the

text: optical

text: companion

text: de-

text: To

text: account

text: for

text: uncertainties

text: in

text: the

text: absolute

text: ﬂux

text: normal-

text: creases

text: after

text: the

text: X-ray

text: outbursts.

text: This

text: behavior

text: can

text: be

text: well

text: ization

text: between

text: PCA

text: and

text: HEXTE

text: we

text: introduced

text: a

text: multi-

text: explained

text: by

text: the

text: weakening

text: of

text: the

text: decretion

text: disc

text: during

text: the

text: plicative

text: constant

text: that

text: was

text: ﬁxed

text: to

text: for

text: the

text: PCA

text: and

text: let

text: mass

text: transfer

text: to

text: the

text: NS.

text: to

text: vary

text: freely

text: for

text: HEXTE.

text: In

text: the

text: case

text: of

text: XRT,

text: PCA

text: and

text: HEXTE

text: data

text: we

text: ﬁxed

text: the

text: multiplicative

text: constant

text: of

text: PCA

text: to

text: and

text: the

text: rest

text: to

text: vary

text: freely.

text: The

text: most

text: relevant

text: results

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text: 27 Pµ,v (A, B; r) =1 − κE(cid:32)µµEE2(−11(cid:115)γ+EκµEE2+)1µeE2µ+E1κE ∞∫0 γE(cid:33)µE2−1 e(cid:32)−µ(cid:112)E(1γ+EκE)γE (cid:115) (cid:33) × κ (1 + κ ) γ 2µ (1 + κ ) γ I − 2µ E E E Q 2κ µ , M M E dγ µE 1 E γ µM M M γ E E M − =1 P . 0 (38) From [35, eq. 3.16], we have, (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:88) − −A2r + B2r 1 v A m AB       P (A, B; r) = P (A, B; r) + exp I µ,v 0,0 (cid:88) 2R − (cid:88)Br m R    m= µ     × µ v + k − − − − − v j − − − rv k+1R v k 1 rj 1R j 1 . (39) k + m m k=1 j=1 From [35, eq. 3.5], we have, (cid:18) (cid:19) (cid:20) (cid:21) (cid:18) (cid:19) (cid:0) (cid:1) √ Ar √ B − − −A2r2 + B2 ABr P (A, B; r) = Q , 1 + r2 1 exp I . (40) 0,0 1 + r2 1 + r2 2 (1 + r2) 0 1 + r2 Letting P´ = P (A, B; r) and combining (38), (39) and (40) we obtain (22). Note that [35] 0,0 uses lower-case symbols (a, b) and we use upper-case symbols (A, B). This is because we deﬁne a = 1 and b = 1 . γ¯M γ¯E REFERENCES [1] O. Vermesan and P. Friess, Internet of Things-From Research and Innovation to Market Deployment. Aalborg, Denmark: River Publishers, 2014. [2] S. L. Cotton, “Human body shadowing in cellular device-to-device communications: Channel modeling using the shadowed κ-µ fading model,” IEEE J. Sel. Areas Commun., vol. 33, no. 1, pp. 111–119, 2015. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Pµ,v

label: equation

text: (A,

text: B;

text: r)

text: ∞∫0

text: κ

text: κ

text: γ

text: 2µ

text: κ

text: γ

text: I

text: 2µ

text: E

text: E

text: E

text: Q

text: 2κ

text: µ

text: M

text: M

text: E

text: dγ

text: µE

text: E

text: γ

text: µM

text: M

text: M

text: γ

text: E

text: E

text: M

text: P

text: From

text: eq.

text: we

text: have,

text: v

text: A

text: m

text: AB

text: P

text: (A,

text: B;

text: r)

text: P

text: (A,

text: B;

text: r)

text: exp

text: I

text: µ,v

text: m

text: R

text: m=

text: µ

text: µ

text: v

text: k

text: v

text: j

text: rv

text: v

text: k

text: rj

text: j

text: k

text: m

text: m

text: From

text: eq.

text: we

text: have,

text: Ar

text: B

text: ABr

text: P

text: (A,

text: B;

text: r)

text: Q

text: exp

text: I

text: Letting

text: P´

text: P

text: (A,

text: B;

text: r)

text: and

text: combining

text: and

text: we

text: obtain

text: Note

text: that

text: uses

text: lower-case

text: symbols

text: (a,

text: b)

text: and

text: we

text: use

text: upper-case

text: symbols

text: (A,

text: B).

text: This

text: is

text: because

text: we

text: deﬁne

text: a

text: and

text: b

text: γ¯M

text: γ¯E

text: REFERENCES

text: O.

text: Vermesan

text: and

text: P.

text: Friess,

text: Internet

text: of

text: Things-From

text: Research

text: and

text: Innovation

text: to

text: Market

text: Deployment.

text: Aalborg,

text: Denmark:

text: River

text: Publishers,

text: S.

text: L.

text: Cotton,

text: “Human

text: body

text: shadowing

text: in

text: cellular

text: device-to-device

text: communications:

text: Channel

text: modeling

text: using

text: the

text: shadowed

text: κ-µ

text: fading

text: model,”

text: IEEE

text: J.

text: Sel.

text: Areas

text: Commun.,

text: vol.

text: no.

text: pp.

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text: 7 Fig. 1. The proposed system model. power in the eavesdropper’s channel respectively, then, the corresponding instantaneous SNR’s | | | | at the legitimate receiver and the eavesdropper are given by, γ = P hM 2 and γ = P hE 2 , M NM E NE E | | E | | while the average SNR’s are given by, γ¯ = P ( hM 2) and γ¯ = P ( hE 2). Now let us consider M NM E NE the channel components of Bob and Eve which are assumed to be i.n.i.d. random variables with { } { } parameters κ , µ , γ¯ and κ , µ , γ¯ , respectively. The PDF’s of γ and γ can be M M M E E E M E re-written from (2) as, (cid:32) (cid:115) (cid:33) fγM (γM) = µM(1 + κκMM)µMµ2M−2+1 γ1 γMMµMµ2M+2−1 e1 µeM−µκMM(1γ+MκM)γM IµM−1 2µM κM (1 +γMκM) γM (7) (cid:32) (cid:115) (cid:33) fγE (γE) = µE(1 + κκEE)µEµ2E−2+1 1γγEEµEµ2E+2−1 1eeµE−κµEE(1γ+EκE)γE IµE−1 2µE κE (1 +γEκE) γE . (8) Likewise, the CDF’s of γ and γ can(cid:34)be re-writte(cid:115)n from (3) as, (cid:35) M E (cid:112) − 2 (1 + κ ) µ γ F (γ ) = 1 Q 2κ µ , M M M (9) γM M µM M M γ M ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fig.

text: The

label: caption

text: proposed

text: system

text: model.

text: power

text: in

text: the

text: eavesdropper’s

text: channel

text: respectively,

text: then,

text: the

text: corresponding

text: instantaneous

text: SNR’s

text: at

text: the

text: legitimate

text: receiver

text: and

text: the

text: eavesdropper

text: are

text: given

text: by,

text: γ

text: P

text: hM

text: and

text: γ

text: P

text: hE

text: M

text: NM

text: E

text: NE

text: E

text: E

text: while

text: the

text: average

text: SNR’s

text: are

text: given

text: by,

text: γ¯

text: P

text: hM

text: and

text: γ¯

text: P

text: hE

text: Now

text: let

text: us

text: consider

text: M

text: NM

text: E

text: NE

text: the

text: channel

text: components

text: of

text: Bob

text: and

text: Eve

text: which

text: are

text: assumed

text: to

text: be

text: i.n.i.d.

text: random

text: variables

text: with

text: parameters

text: κ

text: µ

text: γ¯

text: and

text: κ

text: µ

text: γ¯

text: respectively.

text: The

text: PDF’s

text: of

text: γ

text: and

text: γ

text: can

text: be

text: M

text: M

text: M

text: E

text: E

text: E

text: M

text: E

text: re-written

text: from

text: as,

text: fγM

text: (γM)

text: γ1

text: κM

text: +γMκM)

text: γM

text: fγE

text: (γE)

text: κE

text: +γEκE)

text: γE

text: Likewise,

text: the

text: CDF’s

text: of

text: γ

text: and

text: γ

text: from

text: as,

text: M

text: E

text: κ

text: µ

text: γ

text: F

text: (γ

text: Q

text: 2κ

text: µ

text: M

text: M

text: M

text: γM

text: M

text: µM

text: M

text: M

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