research article a spectrum sensing scheme for partially...

Research ArticleA Spectrum Sensing Scheme for Partially Polarized Waves over120572-120583 Generalized Gamma Fading Channels

Mohamed A Hankal Islam A Eshrah and Hazim Tawfik

Electronics and Electrical Communications Engineering Department Faculty of Engineering Cairo University Giza 12613 Egypt

Correspondence should be addressed to Mohamed A Hankal mohamedhankalgmailcom

Received 9 November 2013 Accepted 19 December 2013 Published 9 February 2014

Academic Editors N Bouguila and C-M Kuo

Copyright copy 2014 Mohamed A Hankal et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Schemes for spectrum holes sensing for cognitive radio based on the estimation of the Stokes parameters of monochromatic andquasimonochromatic polarized electromagnetic waves are developed Statistical information that includes the variations of thepolarization state in both cases (present and absent) of PrimaryUser (PU) is accounted for A detector based on the fluctuation of theStokes parameters is analyzed and its performance is compared with that of energy detectors which use only the scalar amplitudeinformation to sense the PU signal The cooperative spectrum sensing based on the polarization in which the reporting channelsare noisy will be investigated The cluster technique is proposed to reduce the bit error probability due to channel impairment Aclosed-form expression for the polarization detection is derived using 120572-120583 generalized fading model which provides directly anexpression for the special cases of Nakagami-m andWeibull models as well as their derivativesThese expressions are verified usingsimulation The results show that the polarization spectrum sensing gives superior performance for a wide range of SNR over theconventional energy detection method

1 Introduction

Cognitive radio (CR) technology has witnessed a growinginterest over the past decade as it promises more efficientuse of the available spectrum [1 2] A key stage in CR isspectrum sensing in which the Secondary User (SU) mustdetect the presence of a Primary User (PU) in a certainchannel and thus deems this part of the spectrum unusedand make the decision to share it This entails a sequenceof functions that the CR system should perform such aspower control [3] and spectrum management [4] Severaltechniques were proposed to improve spectrum sensing suchas energy detection [5] cyclostationary feature detection[6] sensing based on smart antennas [7 8] and widebandspectrum sensing [9 10] These techniques primarily makeuse of the amplitude frequency and phase information of thePU signal

It is possible however to improve the spectrum sensingprocess by exploiting the polarization state of the signalIn radar systems [11] the polarization state was used toimprove the detection capability of the system The new

polarization-dependent detection statistics which use thepower and relative phase of the two orthogonal polarizationcomponents was proposed to enhance radar detection inhomogenous channels [12]The radar detection performancewas enhanced based on the polarization difference betweenthe clutter and the target [11] The sine of the relative phasebetween two orthogonally polarized received signals hasbeen proposed and tested as detection statistic in radarsystems [13] New statistics use two orthogonal polarizationcomponent powers and their relative phase to enhance targetdetection [12] They are thus fundamentally different fromthe well-known Marcum-Swerling envelope detector [14]which operates on only a single polarization component ofthe received power and the pseudocoherent detector [13]which operates on only the relative phase of the polarizationcomponents

Recent research on polarization detection in CR wasconcerned with completely polarized waves A Virtual Polar-ization Detection (VPD) method based on the vector signalprocessing was presented for effective spectrum sensing ofcognitive radios [15] Polarization spectrum hole sensing was

Hindawi Publishing CorporationISRN Signal ProcessingVolume 2014 Article ID 140545 12 pageshttpdxdoiorg1011552014140545

2 ISRN Signal Processing

proposed for cognitive radio to optimize the received polar-ization at the SU in order to protect the PU from interferencecaused by the SU and to reduce the interference from PU toSU [16]Optimal PolarizationReception (OPR)was proposedfor CR to improve the SINR [17] A new blind spectrumsensing method based on the polarization characteristic ofthe received signal which is completely represented by theorientation of a polarization vector was proposed [18] anda closed-form expression for the probability of false alarmand probability of detection under Additive White GaussianNoise (AWGN) and Rayleigh-fading channels was derivedThe fading and noisy nature of a wireless communicationchannel places a major challenge for spectrum sensing Sincesensing decisions based on a single SU measurements maybe unreliable the idea of collaborative spectrum sensinghas attracted a lot of research interest [19] However thepolarization state is changed with spectrum sensing timewhich was not considered in this work

In this paper the above constraint is addressed byproposing a polarization-based spectrum sensing schemewhere a statistical model for the PU signal polarizationparameters is adopted This model takes into account thechannel backscatter and the partial polarization nature of thePU signal with the assumption that the channel experiencesslow fading Cooperative polarization spectrum sensing isproposed to mitigate the effects of fading and shadowingwhich can seriously degrade the sensing performance Acluster-based cooperation scheme is proposed to decrease biterror probability All SUs are grouped into few clusters andone cluster head is set for each cluster to collect the sensingresults make cluster decisions and forwardmeasurements tothe central unitThus the bit error probability will be reducedgreatly because most of SUs will be closer to the clusterheads than to the central unit Analytical results show thatsignificant improvement can be achieved with our proposedmethod

The rest of this paper is organized as follows In Sec-tion 2 the method of characterizing the signal polarizationis provided The spectrum sensing models are developedin Section 3 Section 4 describes the cooperative spectrumsensing mechanism and the cluster technique of the coop-erative sensing is shown in Section 5 Simulation resultsare illustrated in Section 6 followed by the conclusion inSection 7

2 The Method of Characterizingthe Signal Polarization

The polarization of a monochromatic plane wave is com-pletely specified by constant amplitude and relative phase ofthe two orthogonal electric-field components

In a cognitive radio system the SUuses orthogonally dualpolarized antennas to detect the PU signal which can becompletely described in vector form as

= [

119864ℎ

119864V] =

[

[

119864119894

ℎ+ 119895119864

119902

ℎ

119864119894

V + 119895119864119902

V

]

]

(1)

where 119894 and 119902 indicate the in-phase and quadrate phasecomponents respectively Typically the polarization of amonochromatic wave is determined by the Jones vector 119869 ofthe signal defined as [20]

119869 = [

cos 120577119890119895120579 sin 120577

] (2)

where 120577 = tanminus1(|119864V||119864ℎ|) and 120579 = tanminus1(|119864119902

V ||119864119894

V|) minus tanminus1

(|119864119902

ℎ||119864

119894

ℎ|)

Alternatively the polarization is determined by the geo-metrical parameters namely the ellipticity angle 120591 and orien-tation angle 120601 which can be uniquely represented by a pointon the Poincare sphere The field component parameters 120577

and 120579 and the geometrical parameters 120591 and 120601 are describedin the following set of equations [21]

sin (2120577) cos (120579) = sin (2120601) cos (2120591)

sin (2120577) sin (120579) = sin (2120591)

cos (2120577) = cos (2120601) cos (2120591)

(3)

However if the amplitudes and phases encounter slowfluctuations with time which is the case if the polarization ofprimary signal suffers from either noise or fading the compo-nents of the Jones vector are said to be quasimonochromaticor narrowband The polarization of a quasimonochromaticwave is quantified using an average polarization state vectorwhich may be defined in terms of four measurable compo-nents namely = [1199090 1199091 1199092 1199093]

119879 These components areknown as the Stokes parameters (SPs) and are given by [12]

1199090 =

1

119873

119873

sum

119899=1

[(119864119894

ℎ(119899119879119904))

2

+ (119864119902

ℎ(119899119879119904))

2

+(119864119894

V (119899119879119904))2

+ (119864119902

V (119899119879119904))2]

1199091 =

1

119873

119873

sum

119899=1

[(119864119894

ℎ(119899119879119904))

2

+ (119864119902

ℎ(119899119879119904))

2

minus(119864119894

V (119899119879119904))2

minus (119864119902

V (119899119879119904))2]

1199092 =

2

119873

119873

sum

119899=1

[119864119894

ℎ(119899119879119904) 119864

119894

V (119899119879119904) + 119864119902

ℎ(119899119879119904) 119864

119902

V (119899119879119904)]

1199093 =

2

119873

119873

sum

119899=1

[119864119894

ℎ(119899119879119904) 119864

119902

V (119899119879119904) minus 119864119894

V (119899119879119904) 119864119902

ℎ(119899119879119904)]

(4)

where119879119904 is the sampling time and119873 is the number of sampleswhich is restricted by the bandwidth of a narrowband filter1199090 1199091 1199092 and 1199093 are physically recognizable quantities asfollows

(1) 1199090 is the sum of the power in the ℎ and V electric-fieldcomponents and thus represents the total power of thereceived signal

(2) 1199091 is the difference between the power in the ℎ and Velectric-field components

ISRN Signal Processing 3

(3) 1199092 is the difference between the power in the twoorthogonal electric-field components whose axes arerotated 45∘ relative to the ℎ and V axes

(4) 1199093 is the difference between the right-hand and theleft-hand circularly polarized power

The component 1199090 satisfies the relation

1199090 ge 1199091 + 1199092 + 1199093 (5)

Alternatively the average polarization state can beobtained from the coherence matrix C of the received fieldsuch that

C = ⟨[

[

119864ℎ119864lowast

ℎ119864ℎ119864

lowast

V

119864V119864lowast

ℎ119864V119864

lowast

V

]

]

⟩ (6)

where ⟨sdot⟩ denotes averaging over 119873119879119904 and lowast representsthe complex conjugate The coherence matrix C is a linearcombination of the SPs such that

C = 1199090F0 + 1199091F1 + 1199092F2 + 1199093F3 (7)

where F0 = 05 [1 00 1

] F1 = 05 [1 00 minus1

] F2 = 05 [0 11 0

] andF3 = 05 [

0 1minus1 0

]The relationship of the SPs to the geometrical parameters

(120601 120591) is shown in the following set of equations

1199091 = cos (2120601) cos (2120591)

1199092 = sin (2120601) cos (2120591)

1199093 = sin (2120591)

(8)

Thus far the orientation of a polarization vector on theunit Poincare sphere can completely represent the polariza-tion state of the received signal

3 Spectrum Sensing Based on the Polarization

Spectrum sensing is essentially a binary hypothesis testingproblem which indicates the PUrsquos absence or presencerespectively such that

119884 (119905) =

ℎ119864 (119905) + 119882 (119905) 1198671

119882(119905) 1198670

(9)

where 119884(119905) is the observed signal at the CR 119864(119905) is thePU signal 119882(119905) is the zero mean Gaussian random processwith identical autocorrelation and power spectral density1198730

WattsHz and ℎ is the amplitude gain of the channel havingmean-square valueΩ = ℎ

2

and Probability Density Function(PDF) 119891ℎ(ℎ) The received instantaneous signal power ismodulated by ℎ2 and consequently the instantaneous Signal-to-Noise Ratio (SNR) can be expressed as 120574 = ℎ

2(1198641199041198730)

with an average 120574 = Ω(1198641199041198730) where 119864119904 is the signal energyaccumulated over the observation period

In this paper we propose two schemes which use thedegree of polarization and the axial ratio to detect the PUsignal

31 SpectrumSensing Based on theDegree of Polarization Thedegree of polarization (119863) is a quantity used to describe thepolarized portion of an electromagnetic wave A perfectlypolarized wave has 119863 equal to 1 whereas an unpolarizedwave has 119863 equal to 0 A wave which is partially polarizedcan therefore be represented by superposition of a polarizedand unpolarized component This implies a119863 somewhere inbetween 0 and 1 Alternatively the degree of polarization119863 isdefined as the ratio of the polarized power to the total powerin the wave that is

119863 =

radic1199092

1+ 119909

2

2+ 119909

2

3

1199090

(10)

The estimate of the ratio of the polarized power to thetotal power in the received signal is used as detection statisticsin radar systems [12] Figure 1 depicts a block diagram of theproposed spectrum sensing system where 119863119878 = 119863 There aretwo orthogonal antennas which detect the horizontal andvertical components 119864ℎ and 119864V of the signal 119864 respectivelyThe Stokes vector estimation block delivers the Stokes vector to the polarization degree estimator which in turnproduces the detection statistic 119863 This statistic serves as theinput to the threshold detector to decide whether the PU ispresent or not

The distribution that governs the statistic 119863 can be usedto estimate the probability of detection and the probability offalse alarm Here the in-phase and quadrature componentsof the quasimonochromatic wave are assumed to be zero-mean Gaussian random processes The estimation of the SPand the elements of the sample correlation random processis equivalent which is called Wishart distribution [22] Thusthe probability density function of the detection statistic119863 isgiven by [12]

119875 (119909) =

2Γ (119873 minus 12)

radic120587Γ (119873 minus 1)

[1 minus 1198632

infin]

119873

[1 minus 1199092]

119873minus2

[1 minus 1198632infin1199092]2119873minus1

times

119873

sum

119896=1

Γ (2119873)1198632119896minus2

1199041199092119896

Γ (2119896) Γ (2119873 minus 2119896 + 1)

(11)

where119863infin and 1199040 1199041 1199042 1199043 are the actual degree of polariza-tion and the actual Stokes vector components respectivelyThey are obtained in the limit as the number of independentsamples approaches infinity For a fixed threshold 120582 theconditional probability of false alarm119875119891 and detection119875119889 canbe expressed as [12]

119875119891 =

1

2

1198681minus1205822 (119873 minus 1

3

2

)

119875119889 = 1 minus 119860[1 minus (

120574

1 + 120574

)

2

]

119873

times

119873

sum

119896=1

119861119896(

120574

1 + 120574

)

2119896minus2 infin

sum

119899=0

119862119896119899(

120574

1 + 120574

)

2119899

(12)

where 120574 is the Signal-to-Noise Ratio (SNR) 1198681minus1205822(119899 + 12119873)

is the incomplete beta function 119860 = 2Γ(119873 + 12)radic120587


Figure 1 The proposed spectrum sensing system model

119861119896 = 1Γ(2119896)Γ(2119873minus2119896+1) and119862119896119899 = Γ(2119873+119899minus1)Γ(119899+119896minus

12)(Γ(119899+1)Γ(119873+119899+119896minus12))1198681205822(119896+119899+12119873minus1)The SNR120574 can be affected by fading that in turn affects 119875119889 and hence inorder to incorporate its influence 119875119889 must be averaged overall possible values of 120574 according to

119875119889 = int

infin

0

119875119889 (120574) 119891120574 (120574) 119889120574(13)

where 119891(sdot) represents the PDF of the channel For a fadingsignal with envelope ℎ an arbitrary parameter 120572 gt 0 and a120572-root mean value ℎ =

120572radic119864(ℎ

120572) the 120572-120583 PDF 119891ℎ(ℎ) is given

by [23]

119891ℎ (ℎ) =

120572120583120583ℎ120572120583minus1

119890minus120583(ℎℎ)

120572

Γ (120583) ℎ

minus120572120583 (14)

where 120583 = E2(ℎ

120572)V(ℎ

120572) and E(sdot) and V(sdot) are the

expectation and variance operators respectively [23] ThePDF of the SNR is obtained by a change of variables as shownin [24]

119891120574 (120574) =

(1205722) 1205831205831205741205721205832minus1

Γ (120583) 1205741205721205832

119890minus120583(120574120574)

1205722

(15)

By substituting (15) in (13) 119875119889 can be written as

119875119889 = 1 minus 1198601015840

119873

sum

119896=1

119861119896

infin

sum

119899=0

119862119896119899 int

infin

0

119892119899 (120574) 119889120574 (16)

where

119892119899 (120574) = [1 minus (

120574

1 + 120574

)

2

]

119873

times (

120574

1 + 120574

)

2119899+2119896minus2

1205741205722120583minus1

119890minus120583(120574120574)

1205722

1198601015840=

1205722120583120583

Γ (120583) 120574minus1205721205832

119860

(17)

Using [25] one can get the average probability of detec-tion 119875119889 as follows

119875119889 = 1 minus

119873

sum

119896=1

119861119896

infin

sum

119899=0

119862119896119899

119873

sum

119903=0

(

119873

119903) (minus1)

119903

times

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(18)

where

119860 (V 120583 119911) =1

Γ (V)

times [11198652

(V1

2

minus

120583

2

1 minus

120583

2

minus

119911

4

) Γ (120583) Γ (V)]

times11198652

(V +120583

2

1

2

120583

2

+ 1 minus

119911

4

)

1199111205832

2

times Γ (minus

120583

2

) Γ (V +120583

2

)

minus11198652

(V +120583

2

+

1

2

3

2

120583

2

+

3

2

minus

119911

4

)

119911 =

120583

120574

V = 2 (119873 + 119899 + 119896 minus 1)

11198652 (V1

2

minus

120583

2

1 minus

120583

2

minus

119911

4

)

(19)

is the Hypergeometric function 119866(sdot ) is the Meijer 119866

function and 119880 is the Hypergeometric 119880 function [26]

32 Spectrum Sensing Based on the Axial Ratio of a Polariza-tion Ellipse Thepolarized portion of the PU signal representsa net polarization ellipse traced by the electric field vector asa function of time The ellipse has a magnitude (119877) such that

119877 =

10038161003816100381610038161003816100381610038161003816

1199093

1199090

10038161003816100381610038161003816100381610038161003816

(20)

The ellipticity is the ratio of theminor to themajor axis ofthe corresponding electric field polarization ellipse and varies


from 0 for linearly polarized wave to 1 for circularly polarizedwave The polarization ellipse is alternatively described by itseccentricity which is zero for a circularly polarized wave andincreases as the ellipse becomes thinner It then becomes onefor a linearly polarized wave Alternatively 119877 is defined as theratio of the polarized power to the total power in the wave

Figure 1 depicts a block diagram of the proposed spec-trum sensing system where 119863119878 = 119877 This statistic serves asthe input to the threshold detector to decide whether the PUis present or not The distribution that governs the statistic 119877is to be determined to estimate the probability of detectionand the probability of false alarm [12] which is given by

119875119877 (119909)

=

Γ (119873 + 12)

radic120587Γ (119873)

[1 minus 1198632

infin]

2

[1 minus 1199092]

119873minus1

times

[1 minus (11990431199091199040)]

[(1 minus (11990431199091199040))2minus (119904

2

1+ 119904

2

2119904

2

0) (1 minus 119909

2)2]

119873+12

+

[1 + 11990431199091199040]

[(1 + 11990431199091199040)2+ ((119904

2

1+ 119904

2

2) 119904

2

0) (1 minus 119909

2)2]

119873+12

(21)

For a fixed threshold120582 the conditional probability of falsealarm 119875119891 can be expressed as

119875119891 =

1

2

1198681minus1205822 (119873

1

2

) (22)

and the probability of detection 119875119889 can be given by

119875119889 = 1 minus [1 minus (

120574

1 + 120574

)

2

]

119873

times

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

1198681205822 (119899 +

1

2

119873)(

120574

1 + 120574

)

2119899

(23)

119875119889 must be averaged over all possible values of 120574 asfollows

119875119889 = int

infin

0

119901119889 (120574) 119891120574 (120574) 119889120574 (24)

where 119891120574(120574) represents the 120572-120583 probability density functionFollowing a similar approach to that of Section 31 and bysubstituting (15) in (24) 119875119889 can be written as

119875119889 = 1 minus 119896

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

1198681205822 (119899 +

1

2

119873)int

infin

0

119892119899 (120574) 119889120574

(25)

where

119892119899 (120574) = [1 minus (

120574

1 + 120574

)

2

]

119873

(

120574

1 + 120574

)

2119899

1205741205722120583minus1

119890minus120583(120574120574)

1205722

119896 =

120572120583120583

2Γ (120583) 1205741205721205832

(26)

Using binomial theorem and [25] 119875119889 will be given by

119875119889 = 1 minus

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

times 119868119884th(119899 +

1

2

119873)

119873

sum

119903=0

(

119873

119903) (minus1)

119903

times

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+(12)

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(27)

It is worthmentioning that the restriction for this schemeis that the polarization information of the primary signalmust be known a priori If the PU signal is linearly polarizedthen 119877 will be close to 0 for high SNR and if the PU signal iscircularly polarized then 119877 will be close to 1 for high SNR

4 Spectrum Sensing Based onCooperative Polarization Detection

The fading and noisy nature of a wireless communicationchannel places amajor challenge on the accuracy of spectrumsensing Sensing decisions that is based on measurements ofa single SUmay be unreliable Cooperative spectrum sensingis one possible solution to overcome this unreliability InFigure 2 a number of SUs which are distributed in differentlocations independently can detect the PU and make thedecision whether the signal exists or not According to theinformation received from various SUs the central unitmakes the final decision based on some rules A generalfusion rule is when a final decision of 1 is taken when 119898-out-of -119873 SUs report 1 When 119898 = 1 the 119898-out-of -119873 ruleis equivalent to the OR rule When 119898 = 119873 the decision rulebecomes the AND rule By selecting different values of 119898different detection performances are obtained

Two cases are considered namely cooperative spectrumsensing with perfect and imperfect reporting channels

41 Perfect Reporting Channels If the channels between eachSU and the central unit are noise free then the overallprobability of false alarm 119876119865 and the overall probability ofdetection119876119863 of the cooperative spectrum sensing for119898-out-of -119873 rule of the cooperative spectrum sensing are given by[27]

119876119865 =

119873

sum

119894=119898

(

119873

119894) (119875119891119894)

119894

(1 minus 119875119891119894)

119873minus119894

119876119863 =

119873

sum

119894=119898

(

119873

119894) (119875119889119894)

119894(1 minus 119875119889119894)

119873minus119894

(28)

where the 119875119891119894 and 119875119889119894 are the probabilities of false alarmand detection of the degree of polarization and axial ratio asderived in (14) (18) (22) and (27) respectively


SU1

SU3

SU2

SU4

PU

Central unit

Figure 2 System model of the cooperative network

42 Imperfect Reporting Channels In practical systems thereporting channels between the SUs and the central unit willexperience fading This in turn will degrade transmissionreliability of the sensing results that are reported from the SUsto the central unit Let 119875119864119894 denote the probability of receiving1198671 at the central unit when the 119894th Secondary User sends1198670

and the probability of receiving 1198670 at the central unit whenthe 119894th SecondaryUser sends1198671 which can be calculated over120572-120583 fading channel as follows

119875119864119894 = int

infin

0

1

2

erfc (radic120574)

(1205722) 120583120583120574(1205721205832)minus1

Γ (120583) 1205741205721205832

119890minus120583(120574120574)

1205722

119889120574 (29)

where (12) erfc(radic120574) is the probability of error for BPSK inan 119860119882119866119873 channel The shown integral can be put in theform of Laplace transform and hence using [25] (372ndash18)and substituting V = 120583 minus 1 119886 = radic120574(1120583)

2120572 1198972119896 = 1120572the probability of error can be calculated as

119875119864119894 =

120572

4Γ (120583)

times[

[

radic2119897120583minus12

(2120587)(119896+119897minus1)2

119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ(119897minus120583+1)1

Δ(119896

1

2

) 0

)]

]

(30)

where Δ(119896 119886) = 119886119896 (119886 + 1)119896 (119886 + 119896 minus 1)119896

Consequently the overall probability of false alarm 119876119865 isobtained as

119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[119867

CU0

119867PU0

]

119873minus119894

(31)

119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[1 minus

119867CU1

119867PU0

]

119873minus119894

(32)

Prob119867

CU1

119867PU0

= [Prob119867

CU1

119867SU1

times Prob119867

SU1

119867PU0

]

+ Prob119867

CU1

119867SU0

sdot Prob119867

SU0

119867PU0

(33)

119876119865 =

119873

sum

119894=119898

(

119873

119894) [(1 minus 119875119864119894) 119875119865119894 + 119875119864119894 (1 minus 119875119865119894)]

119894

times [(1 minus 119875119864119894) (1 minus 119875119865119894) + 119875119864119894119875119865119894]119873minus119894

(34)

where 119867CU1

119867PU1

and 119867SU1

are the hypothesis in the centralunit PU and SU respectively


Substituting (12) and (30) into (34) yields

119876119865 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898)

)

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus

Γ (119898 1205822)

Γ (119898)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583)

Γ (119898 1205822)

Γ (119898)

119873minus119894

(35)

where

119869 (119896 119897 120572 120583) =


(2120587)(119896+119897minus1)2

times 119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ (119897 minus120583+1) 1

Δ (119896

1

2

) 0)

(36)

Similarly the detection probability 119876119863 can be computed as

119876119863 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

+

120572

8Γ (120583)

119869 (119896 119897 120572 120583)119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

119873minus119894

(37)

where 119875119863119894 is the probability of detection of the polarizationobtained in (18) (38) for degree detection and axial ratiodetection methods respectively

Therefore for AND rule the overall probability of falsealarm and detection are given by

119876119865 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+120572

4Γ (120583)119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898))

119894

(38)

119876119863 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+120572

4Γ (120583)119869 (119896 119897 120572 120583)(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

(39)

5 Cluster-Based CooperativeSpectrum Sensing

The clustering method is proposed in the cooperative spec-trum sensing scheme in order to improve the sensing per-formance by decreasing the reporting channel error which isproved to exploit a selection diversity gain [28] In this paperwe assume that the instantaneous channel state informationof the channel is available for the Secondary Users and thechannel between any two SecondaryUsers in the same clusteris perfect because they are in the vicinity of each otherFigure 3 shows the systemmodel of cluster-based cooperativespectrum sensing We assume that there are 119870 SUs who are

divided into 119871 clusters The 119894th cluster has an integer number119873119894 of SUs which satisfies

119871

sum

119894=119897

119873119894 = 119870 (40)

Firstly all SUs are assumed to belong to few clusters bythe distributed clustering algorithms [29] Secondly theSecondary User who has the largest instantaneous reportingchannel gain will be selected as the cluster head Then thecooperative sensing is carried out through the following stepsFirstly all SUs perform the local spectrum sensing Every SUsends a decision to the cluster head Secondly the cluster headreceives those local decisions from the SUs in the same clusterand then makes the decision according to certain fusion ruleAll cluster heads send their decisions to the central unit Inthe end the central unit makes the final decision accordingto the fusion rule The cluster head and central unit makethe decision according to the OR-rule in order to limit theinterference from the SUs to the PU The119898-out-of -119873 fusionrule is adopted in the cluster head and the OR fusion ruleis adopted in the central unit [30] If each cluster has 119870119871

SUs then the global false alarm probability and detectionprobability will be given by

119876119865 = 1

minus(

1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(

Γ (119898 1205822)

Γ (119898)

)

119870

(1 minus

Γ (119898 1205822)

Γ (119898)

)

119870119871minus119870)

119871

(41)


SU1

SU1

SU2

SU2

SU3

SU3

PU

Central unit

Cluster head

Cluster head

Figure 3 Cluster-based spectrum sensing mode

119876119863 = 1 minus (1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)(119875119889)119870(1 minus 119901119889)

119870119871minus119870)

119871

(42)

Substituting 119875119889 for the degree detection in (18) into (42)yields

119876119863 = 1 minus

(

(

(

(

1minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870

(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870119871minus119870

)

)

)

)

119871

(43)

where119880 (V 120583 119911)

=

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(44)

and substituting 119875119889 for the axial ratio detection in (27) into(42) yields

119876119863 = 1 minus

(

(

(

(

(

(

(

1 minus

119870119871

sum119898=119897

((119870

119871)

119898

)

times(

1 minus

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870

(

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870119871minus119870

)

)

)

)

)

)

)

119871

(45)

6 Results and Discussion

Results obtained using the proposed methods are presentedMonte Carlo simulation consisting of 100000 independenttrials was performed The degree of freedom is set to 35 tomake fair comparison between the proposed methods andthe energy detection method [31] The equal and normalizedpolarization states of the PUs are randomly generated with120577 isin (0 1205872) 120601 isin (minus1205872 1205872) It is assumed that SUs usedual polarized antennas which is necessary to detect thehorizontal and vertical components of the wave

61 The Degree of Polarization (DoP) Method Figure 4compares the Receiver Operating Characteristic (ROC) ofthe DoP method with the directional method [18] It canbe noticed from this figure that the DoP method improvesthe 119875119889 at low SNR since the signal is polarized and thenoise is unpolarized wave This makes the signal detectionin particular at low SNR more efficient which is the range ofinterest for CR systems Therefore it cannot be used in highSNR but it offers a baseline for comparison

Figure 5 shows theROCunderAWGNcomparedwith theenergy detection (ED)method It is clear from this figure thatthe proposed method improves the probability of detectionover a wide range of SNR (from minus5 dB to 5 dB) For examplein the case of SNR = 0 dB 119875119889 improved from 22 to 88Moreover it is observed that 119875119889 at SNR = 0 dB for the DoPmethod is better than 119875119889 for the ED method at SNR = 5 dBmeaning the improvement of more than +5 dB for the DoPmethod relative to the ED method It is also noticed thatat SNR = 10 dB the DoP method offers 100 detectionprobability for all values of 119875119891 This improvement is due tothe more accurate detection of the polarized portion

The case of Rayleigh fading channel is considered inFigure 6 which illustrates that fading has a stronger impacton the ED than on the proposed DoP method This isattributed to the fact that the DoP is based on a secondorder statistic that is the improvement in the probability of


04

05

06

07

08

09

0 5 10 15 20

1

minus20 minus15 minus10 minus5

AWGN-directional method (Guo et al 2013) AWGN-DoP method

Prob

abili

ty o

f det

ectio

n

SNR (dB)

Figure 4 Probability of detection versus SNR for directional andDoP methods

detection for SNR = 5 dB is approximately 30 at 119875119891 = 02A better performance for the DoP is reported for Nakagamifading as shown in Figure 7 where at SNR = 0 dB theimprovement is approximately 20 when the 119875119891 = 02Moreover the 119875119889 at SNR = 5 dB for the proposed methodis greater than the 119875119889 for the EDmethod at SNR = 10 dB thismeans that the improvement is nearly +7 dB

It can be noticed that at fading conditions the depo-larization effect increases which means higher dispersionof polarization states Therefore the detection performancedecreases Thus the smaller the depolarization effect on theprimary signal the more constant the polarization state andthus the better the detection performance

62 The Axial Ratio (AR) Method To demonstrate theperformance of the axial ratio (AR) method Figure 8 showsthe characteristics for different values of 120572 and 120583 For 120572 = 2

and 120583 = 2 the probability of detection is 91 at 119875119891 = 02which is a superior value relative to the ED method Thiscan be easily explained since at strong fading less energy iscoupled between the cross-polarized channels which resultsin small dispersion of the polarization state leading to betterdetection performance

Figure 9 shows the superior performance of the ARmethod relative to the ED method over a wide range of SNRunder Rayleigh fading channel In Figure 10 whereNakagamifading is considered for SNR = minus5 dB an improvement ofapproximately 60 in the probability of detection is achievedfor 119875119891 = 02 and 70 for SNR = 5 dB relative to theED method Hence the AR method is the most robust andapplicable detection method in the case of fading conditionandor presence of noise power uncertainty

Prob

abili

ty o

f det

ectio

n

00

1

1Probability of false alarm

DoP method AWGN SNR = 0dBDoP method AWGN SNR = minus5dB

DoP method AWGN SNR = 5dB

04

05

06

07

08

09

01

02

03

Energy method AWGN SNR = minus5dBEnergy method AWGN SNR = 0dBEnergy method AWGN SNR = 5dB

04 06 0802

Figure 5 ROC of the DoP method in AWGN

0 10

1

Prob

abili

ty o

f det

ectio

n

Probability of false alarm04 06 0802

04

05

06

07

08

09

01

02

03

DoP method SNR = 0dBDoP method SNR = 5dBDoP method SNR = 10dB

Energy method SNR = 10dB

Energy method SNR = 0dBEnergy method SNR = 5dB

Figure 6 ROC of the DoP method under Rayleigh fading channel


010

1

Probability of false alarm

Prob

abili

ty o

f det

ectio

n




04 06 0802

04

05

06

07

08

09

01

02

03

Figure 7 ROCof theDoPmethod underNakagami fading channel

0 1

1


Prob

abili

ty o

f det

ectio

n

120572 = 2 120583 = 2

120572 = 2 120583 = 1

120572 = 1 120583 = 1

120572 = 1 120583 = 2

04 06 08

09

02

04

05

06

07

08

Figure 8 ROC of the DoP method under different 120572 120583 at SNR =

5 dB

0 10

1


Prob

abili

ty o

f det

ectio

n

AR method SNR = minus5dBAR method SNR = 0dBAR method SNR = 5dBAR method SNR = 10dB

04

06

08

09

01

02

03

04

05

06

07

0802



Energy method SNR = minus5dB

Figure 9 ROC of the AR method under Rayleigh fading channel





04

05

06

07

08

09

01

02

03

04 06 08020 10

1


Prob

abili

ty o

f det

ectio

n

Figure 10 ROC of the ARmethod under Nakagami fading channel


The DoP methodThe AR method

0 10

1


Prob

abili

ty o

f det

ectio

n

04 06 0802

04

05

06

07

08

09

01

02

03

Figure 11 ROC of the two methods (AR DoP) at SNR=0 dB underNakagami fading channel

A comparison between both approaches is drawn inFigure 11 showing the superiority of theARmethod relative tothe DoP where for 119875119891 = 02 the AR method gives 119875119889 = 073

compared to 119875119889 = 058 for the DoP It can be seen that thesensing performance of theAR is better than theDoPmethoddue to the fact that the AR method resists the depolarizationeffect on the primary signal

63The Cooperative Scheme Figure 12 shows the ROC of thecooperative scheme for the DoP method over a wide rangeof SNR under imperfect Nakagami fading channel It is clearthat the119875119889 is very high at low119875119891 especially at SNR = 5 dB and10 dB It can be observed that the 119875119889 increase rapidly with theincrease of the SNRWhen the SNR = 10 dB the 119875119889 is almost100 for all values of 119875119891

7 Conclusions

Spectrum sensing based on Stokes parameters was thor-oughly analyzed using new detection statistics namely thedegree of polarization and the axial ratio The proposedapproaches were studied under different fading scenariosand the obtained results demonstrated superior performancerelative to the conventional energy detection method Anextensive study is reported on the two methods and theirperformance under 120572-120583 fading channels The results demon-strated that the proposed algorithms are particularly appli-cable for the case of unknown primary polarization andorpresence of noise power uncertainty In general the proposedmethods are better than energy detection in different rangesof the SNR and under different fading conditions They are


Prob

abili

ty o

f det

ectio

n

0 10

1

SNR = 0dBSNR = 5dBSNR = 10dB

04 06 0802

04

05

06

07

08

09

01

02

03

Figure 12 ROC of the cooperative scheme of the DoP methodunder imperfect Nakagami fading channel

however more complex in terms of implementation due tothe use of two orthogonally polarized antennas It is alsoworth noting that the axial ratio method is generally betterthan the degree of polarization method

Cooperative spectrum sensing was then considered andshown to be a powerful method for dealing with the hiddenterminal problem Simulation results show that the clusterrule gives superior performance for a wide range of SNR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005

[2] A Sahai and D Cabric ldquoSpectrum sensing fundamental limitsand practical challengesrdquo in Proceedings of the IEEE Interna-tional Symposium onNew Frontiers in Dynamic SpectrumAccessNetwork (DySPAN rsquo05) Baltimore Md USA November 2005

[3] W Wei P Tao and W Wenbo ldquoOptimal power controlunder interference temperature constraints in cognitive radionetworkrdquo in Proceedings of the IEEE Wireless Communicationsand Networking Conference (WCNC rsquo07) pp 116ndash120 March2007

[4] I F AkyildizW-Y LeeM C Vuran and SMohanty ldquoA surveyon spectrum management in cognitive radio networksrdquo IEEECommunications Magazine vol 46 no 4 pp 40ndash48 2008


[5] F F Digham M-S Alouini and M K Simon ldquoOn the energydetection of unknown signals over fading channelsrdquo IEEETransactions on Communications vol 55 no 1 pp 21ndash24 2007

[6] A V Dandawate and G B Giannakis ldquoStatistical tests forpresence of cyclostationarityrdquo IEEE Transactions on SignalProcessing vol 42 no 9 pp 2355ndash2369 1994

[7] L Zhang Y-C Liang Y Xin and H V Poor ldquoRobust cognitivebeamforming with partial channel state informationrdquo IEEETransactions on Wireless Communications vol 8 no 8 pp4143ndash4153 2009

[8] H Kim J Kim S Yang M Hong and Y Shin ldquoAn effectiveMIMO-OFDM system for IEEE 80222WRAN channelsrdquo IEEETransactions on Circuits and Systems II vol 55 no 8 pp 821ndash825 2008

[9] Z Quan S Cui A H Sayed and H V Poor ldquoWidebandspectrum sensing in cognitive radio networksrdquo in Proceedingsof the IEEE International Conference on Communications (ICCrsquo08) pp 901ndash906 May 2008

[10] Z Tian ldquoCompressed wideband sensing in cooperative cog-nitive radio networksrdquo in Proceedings of the IEEE GlobalTelecommunications Conference (GLOBECOM rsquo08) pp 3756ndash3760 December 2008

[11] A J Poelman ldquoOn using orthogonally polarized noncoherentreceiving channels to detect target echoes in gaussian noiserdquoIEEE Transactions on Aerospace and Electronic Systems vol 11no 4 pp 660ndash663 1975

[12] G M Vachula and R M Barnes ldquoPolarization detection of afluctuating radar targetrdquo IEEE Transactions on Aerospace andElectronic Systems vol 19 no 2 pp 250ndash257 1983

[13] R E Stovall ldquoA gaussian noise analysis of the pseudo-coherentdiscriminantrdquo Tech Rep Note 1978-46 MIT Lincoln Labora-tory 1978

[14] D PMeyer andH AMayer Radar Target Detection Handbookof Theory and Practice Academic Press New York NY USA1973

[15] F Liu C Feng C Guo Y Wang and D Wei ldquoPolarizationspectrum sensing scheme for cognitive radiosrdquo in Proceedingsof the 5th International Conference onWireless CommunicationsNetworking and Mobile Computing (WiCOM rsquo09) September2009

[16] DWei C Guo F Liu and Z Zeng ldquoA SINR improving schemebased on optimal polarization receiving for the cognitiveradiosrdquo in Proceedings of the IEEE International Conference onNetwork Infrastructure and Digital Content (IC-NIDC rsquo09) pp100ndash104 November 2009

[17] F Liu C Feng C Guo Y Wang and D Wei ldquoVirtual polar-ization detection a vector signal sensing method for cognitiveradiosrdquo in Proceedings of the 71st IEEE Vehicular TechnologyConference (VTC-Spring rsquo10) pp 1ndash5 Taipei Taiwan 2010

[18] C Guo X Wu C Feng and Z Zeng ldquoSpectrum sensing forcognitive radios based on directional statistics of polarizationvectorsrdquo IEEE Journal on Selected Areas in Communications vol31 no 3 pp 379ndash393 2013

[19] K B Letaief and W Zhang ldquoCooperative communications forcognitive radio networksrdquo Proceedings of the IEEE vol 97 no 5pp 878ndash893 2009

[20] L P Murza ldquoThe non coherent polarimetry of noise likeradiationrdquo Radio Engineering and Electronic Physics vol 23 no7 pp 57ndash63 1978

[21] G Deschamps ldquoTechniques for handling elliptically polarizedwaves with special reference to antennas part IImdashgeometrical

representation of the polarization of a plane electromagneticwaverdquo Proceedings of the IRE vol 39 no 5 pp 540ndash544 1951

[22] N Goodman ldquoStatistical analysis based on a certain multivari-ate complex gaussian distribution (an introduction)rdquo Annals ofMathematical Statistics vol 34 no 1 pp 152ndash177 1963

[23] M D Yacoub ldquoThe 120572-120583 distribution a physical fading modelfor the Stacy distributionrdquo IEEE Transactions on VehicularTechnology vol 56 no 1 pp 27ndash34 2007

[24] M K Simon and M S Alouini Digital Communication overFading Channels vol 86 Wiley-IEEE Press 2004

[25] A Prudnikov Y A Brychkov and O Marichev Integrals andSeries vol 4 Gordon and Breach Science 1986

[26] F W Olver D W Lozier R F Boisvert and C W Clark NISTHandbook of Mathematical Functions Cambridge UniversityPress 2010

[27] A Ghasemi and E S Sousa ldquoCollaborative spectrum sensingfor opportunistic access in fading environmentsrdquo in Proceedingsof the 1st IEEE International Symposium on New Frontiers inDynamic Spectrum Access Networks (DySPAN rsquo05) pp 131ndash136November 2005

[28] C Sun W Zhang and K B Letaief ldquoCluster-based cooperativespectrum sensing in cognitive radio systemsrdquo in Proceedingsof the IEEE International Conference on Communications (ICCrsquo07) pp 2511ndash2515 June 2007

[29] O Younis and S Fahmy ldquoDistributed clustering in ad-hocsensor networks a hybrid energy-efficient approachrdquo in Pro-ceedings of the 23rd Annual Joint Conference of the IEEEComputer and Communications Societies (INFOCOM rsquo04) vol1 pp 629ndash640 March 2004

[30] A Ghasemi and E S Sousa ldquoSpectrum sensing in cognitiveradio networks the cooperation-processing tradeoffrdquo WirelessCommunications and Mobile Computing vol 7 no 9 pp 1049ndash1060 2007

[31] J Ma G Zhao and Y Li ldquoSoft combination and detectionfor cooperative spectrum sensing in cognitive radio networksrdquoIEEETransactions onWireless Communications vol 7 no 11 pp4502ndash4507 2008

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Submit your manuscripts athttpwwwhindawicom

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation



DistributedSensor Networks



proposed for cognitive radio to optimize the received polar-ization at the SU in order to protect the PU from interferencecaused by the SU and to reduce the interference from PU toSU [16]Optimal PolarizationReception (OPR)was proposedfor CR to improve the SINR [17] A new blind spectrumsensing method based on the polarization characteristic ofthe received signal which is completely represented by theorientation of a polarization vector was proposed [18] anda closed-form expression for the probability of false alarmand probability of detection under Additive White GaussianNoise (AWGN) and Rayleigh-fading channels was derivedThe fading and noisy nature of a wireless communicationchannel places a major challenge for spectrum sensing Sincesensing decisions based on a single SU measurements maybe unreliable the idea of collaborative spectrum sensinghas attracted a lot of research interest [19] However thepolarization state is changed with spectrum sensing timewhich was not considered in this work

In this paper the above constraint is addressed byproposing a polarization-based spectrum sensing schemewhere a statistical model for the PU signal polarizationparameters is adopted This model takes into account thechannel backscatter and the partial polarization nature of thePU signal with the assumption that the channel experiencesslow fading Cooperative polarization spectrum sensing isproposed to mitigate the effects of fading and shadowingwhich can seriously degrade the sensing performance Acluster-based cooperation scheme is proposed to decrease biterror probability All SUs are grouped into few clusters andone cluster head is set for each cluster to collect the sensingresults make cluster decisions and forwardmeasurements tothe central unitThus the bit error probability will be reducedgreatly because most of SUs will be closer to the clusterheads than to the central unit Analytical results show thatsignificant improvement can be achieved with our proposedmethod

The rest of this paper is organized as follows In Sec-tion 2 the method of characterizing the signal polarizationis provided The spectrum sensing models are developedin Section 3 Section 4 describes the cooperative spectrumsensing mechanism and the cluster technique of the coop-erative sensing is shown in Section 5 Simulation resultsare illustrated in Section 6 followed by the conclusion inSection 7

2 The Method of Characterizingthe Signal Polarization

The polarization of a monochromatic plane wave is com-pletely specified by constant amplitude and relative phase ofthe two orthogonal electric-field components

In a cognitive radio system the SUuses orthogonally dualpolarized antennas to detect the PU signal which can becompletely described in vector form as

= [

119864ℎ

119864V] =

[

[

119864119894

ℎ+ 119895119864

119902

ℎ

119864119894

V + 119895119864119902

V

]

]

(1)

where 119894 and 119902 indicate the in-phase and quadrate phasecomponents respectively Typically the polarization of amonochromatic wave is determined by the Jones vector 119869 ofthe signal defined as [20]

119869 = [

cos 120577119890119895120579 sin 120577

] (2)

where 120577 = tanminus1(|119864V||119864ℎ|) and 120579 = tanminus1(|119864119902

V ||119864119894

V|) minus tanminus1

(|119864119902

ℎ||119864

119894

ℎ|)

Alternatively the polarization is determined by the geo-metrical parameters namely the ellipticity angle 120591 and orien-tation angle 120601 which can be uniquely represented by a pointon the Poincare sphere The field component parameters 120577

and 120579 and the geometrical parameters 120591 and 120601 are describedin the following set of equations [21]

sin (2120577) cos (120579) = sin (2120601) cos (2120591)

sin (2120577) sin (120579) = sin (2120591)

cos (2120577) = cos (2120601) cos (2120591)

(3)

However if the amplitudes and phases encounter slowfluctuations with time which is the case if the polarization ofprimary signal suffers from either noise or fading the compo-nents of the Jones vector are said to be quasimonochromaticor narrowband The polarization of a quasimonochromaticwave is quantified using an average polarization state vectorwhich may be defined in terms of four measurable compo-nents namely = [1199090 1199091 1199092 1199093]

119879 These components areknown as the Stokes parameters (SPs) and are given by [12]

1199090 =

1

119873

119873

sum

119899=1

[(119864119894

ℎ(119899119879119904))

2

+ (119864119902

ℎ(119899119879119904))

2

+(119864119894

V (119899119879119904))2

+ (119864119902

V (119899119879119904))2]

1199091 =

1

119873

119873

sum

119899=1

[(119864119894

ℎ(119899119879119904))

2

+ (119864119902

ℎ(119899119879119904))

2

minus(119864119894

V (119899119879119904))2

minus (119864119902

V (119899119879119904))2]

1199092 =

2

119873

119873

sum

119899=1

[119864119894

ℎ(119899119879119904) 119864

119894

V (119899119879119904) + 119864119902

ℎ(119899119879119904) 119864

119902

V (119899119879119904)]

1199093 =

2

119873

119873

sum

119899=1

[119864119894

ℎ(119899119879119904) 119864

119902

V (119899119879119904) minus 119864119894

V (119899119879119904) 119864119902

ℎ(119899119879119904)]

(4)

where119879119904 is the sampling time and119873 is the number of sampleswhich is restricted by the bandwidth of a narrowband filter1199090 1199091 1199092 and 1199093 are physically recognizable quantities asfollows

(1) 1199090 is the sum of the power in the ℎ and V electric-fieldcomponents and thus represents the total power of thereceived signal

(2) 1199091 is the difference between the power in the ℎ and Velectric-field components





1199090 ge 1199091 + 1199092 + 1199093 (5)


C = ⟨[

[

119864ℎ119864lowast

ℎ119864ℎ119864

lowast

V

119864V119864lowast

ℎ119864V119864

lowast

V

]

]

⟩ (6)


C = 1199090F0 + 1199091F1 + 1199092F2 + 1199093F3 (7)

where F0 = 05 [1 00 1

] F1 = 05 [1 00 minus1

] F2 = 05 [0 11 0

] andF3 = 05 [

0 1minus1 0



1199091 = cos (2120601) cos (2120591)

1199092 = sin (2120601) cos (2120591)

1199093 = sin (2120591)

(8)




119884 (119905) =

ℎ119864 (119905) + 119882 (119905) 1198671

119882(119905) 1198670

(9)



2


2(1198641199041198730)




119863 =

radic1199092

1+ 119909

2

2+ 119909

2

3

1199090

(10)



119875 (119909) =

2Γ (119873 minus 12)

radic120587Γ (119873 minus 1)

[1 minus 1198632

infin]

119873

[1 minus 1199092]

119873minus2


times

119873

sum

119896=1

Γ (2119873)1198632119896minus2

1199041199092119896

Γ (2119896) Γ (2119873 minus 2119896 + 1)

(11)


119875119891 =

1

2

1198681minus1205822 (119873 minus 1

3

2

)

119875119889 = 1 minus 119860[1 minus (

120574

1 + 120574

)

2

]

119873

times

119873

sum

119896=1

119861119896(

120574

1 + 120574

)

2119896minus2 infin

sum

119899=0

119862119896119899(

120574

1 + 120574

)

2119899

(12)







119875119889 = int

infin

0

119875119889 (120574) 119891120574 (120574) 119889120574(13)


120572radic119864(ℎ


by [23]

119891ℎ (ℎ) =

120572120583120583ℎ120572120583minus1

119890minus120583(ℎℎ)

120572

Γ (120583) ℎ

minus120572120583 (14)

where 120583 = E2(ℎ

120572)V(ℎ



119891120574 (120574) =

(1205722) 1205831205831205741205721205832minus1

Γ (120583) 1205741205721205832

119890minus120583(120574120574)

1205722

(15)


119875119889 = 1 minus 1198601015840

119873

sum

119896=1

119861119896

infin

sum

119899=0

119862119896119899 int

infin

0

119892119899 (120574) 119889120574 (16)

where

119892119899 (120574) = [1 minus (

120574

1 + 120574

)

2

]

119873

times (

120574

1 + 120574

)

2119899+2119896minus2

1205741205722120583minus1

119890minus120583(120574120574)

1205722

1198601015840=

1205722120583120583

Γ (120583) 120574minus1205721205832

119860

(17)


119875119889 = 1 minus

119873

sum

119896=1

119861119896

infin

sum

119899=0

119862119896119899

119873

sum

119903=0

(

119873

119903) (minus1)

119903

times

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(18)

where

119860 (V 120583 119911) =1

Γ (V)

times [11198652

(V1

2

minus

120583

2

1 minus

120583

2

minus

119911

4

) Γ (120583) Γ (V)]

times11198652

(V +120583

2

1

2

120583

2

+ 1 minus

119911

4

)

1199111205832

2

times Γ (minus

120583

2

) Γ (V +120583

2

)

minus11198652

(V +120583

2

+

1

2

3

2

120583

2

+

3

2

minus

119911

4

)

119911 =

120583

120574

V = 2 (119873 + 119899 + 119896 minus 1)

11198652 (V1

2

minus

120583

2

1 minus

120583

2

minus

119911

4

)

(19)




119877 =

10038161003816100381610038161003816100381610038161003816

1199093

1199090

10038161003816100381610038161003816100381610038161003816

(20)





119875119877 (119909)

=

Γ (119873 + 12)

radic120587Γ (119873)

[1 minus 1198632

infin]

2

[1 minus 1199092]

119873minus1

times

[1 minus (11990431199091199040)]

[(1 minus (11990431199091199040))2minus (119904

2

1+ 119904

2

2119904

2

0) (1 minus 119909

2)2]

119873+12

+

[1 + 11990431199091199040]

[(1 + 11990431199091199040)2+ ((119904

2

1+ 119904

2

2) 119904

2

0) (1 minus 119909

2)2]

119873+12

(21)


119875119891 =

1

2

1198681minus1205822 (119873

1

2

) (22)


119875119889 = 1 minus [1 minus (

120574

1 + 120574

)

2

]

119873

times

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

1198681205822 (119899 +

1

2

119873)(

120574

1 + 120574

)

2119899

(23)


119875119889 = int

infin

0

119901119889 (120574) 119891120574 (120574) 119889120574 (24)


119875119889 = 1 minus 119896

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

1198681205822 (119899 +

1

2

119873)int

infin

0

119892119899 (120574) 119889120574

(25)

where

119892119899 (120574) = [1 minus (

120574

1 + 120574

)

2

]

119873

(

120574

1 + 120574

)

2119899

1205741205722120583minus1

119890minus120583(120574120574)

1205722

119896 =

120572120583120583

2Γ (120583) 1205741205721205832

(26)


119875119889 = 1 minus

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

times 119868119884th(119899 +

1

2

119873)

119873

sum

119903=0

(

119873

119903) (minus1)

119903

times

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+(12)

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(27)






119876119865 =

119873

sum

119894=119898

(

119873

119894) (119875119891119894)

119894

(1 minus 119875119891119894)

119873minus119894

119876119863 =

119873

sum

119894=119898

(

119873

119894) (119875119889119894)

119894(1 minus 119875119889119894)

119873minus119894

(28)



SU1

SU3

SU2

SU4

PU

Central unit




119875119864119894 = int

infin

0

1

2

erfc (radic120574)

(1205722) 120583120583120574(1205721205832)minus1

Γ (120583) 1205741205721205832

119890minus120583(120574120574)

1205722

119889120574 (29)



119875119864119894 =

120572

4Γ (120583)

times[

[


(2120587)(119896+119897minus1)2

119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ(119897minus120583+1)1

Δ(119896

1

2

) 0

)]

]

(30)

where Δ(119896 119886) = 119886119896 (119886 + 1)119896 (119886 + 119896 minus 1)119896


119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[119867

CU0

119867PU0

]

119873minus119894

(31)

119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[1 minus

119867CU1

119867PU0

]

119873minus119894

(32)

Prob119867

CU1

119867PU0

= [Prob119867

CU1

119867SU1

times Prob119867

SU1

119867PU0

]

+ Prob119867

CU1

119867SU0

sdot Prob119867

SU0

119867PU0

(33)

119876119865 =

119873

sum

119894=119898

(

119873

119894) [(1 minus 119875119864119894) 119875119865119894 + 119875119864119894 (1 minus 119875119865119894)]

119894


(34)

where 119867CU1

119867PU1

and 119867SU1




119876119865 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898)

)

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus

Γ (119898 1205822)

Γ (119898)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583)

Γ (119898 1205822)

Γ (119898)

119873minus119894

(35)

where

119869 (119896 119897 120572 120583) =


(2120587)(119896+119897minus1)2

times 119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ (119897 minus120583+1) 1

Δ (119896

1

2

) 0)

(36)


119876119863 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

+

120572

8Γ (120583)

119869 (119896 119897 120572 120583)119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

119873minus119894

(37)



119876119865 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+120572

4Γ (120583)119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898))

119894

(38)

119876119863 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+120572

4Γ (120583)119869 (119896 119897 120572 120583)(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

(39)




119871

sum

119894=119897

119873119894 = 119870 (40)



119876119865 = 1

minus(

1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(

Γ (119898 1205822)

Γ (119898)

)

119870

(1 minus

Γ (119898 1205822)

Γ (119898)

)

119870119871minus119870)

119871

(41)


SU1

SU1

SU2

SU2

SU3

SU3

PU

Central unit

Cluster head

Cluster head


119876119863 = 1 minus (1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)(119875119889)119870(1 minus 119901119889)

119870119871minus119870)

119871

(42)


119876119863 = 1 minus

(

(

(

(

1minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870

(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870119871minus119870

)

)

)

)

119871

(43)

where119880 (V 120583 119911)

=

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(44)


119876119863 = 1 minus

(

(

(

(

(

(

(

1 minus

119870119871

sum119898=119897

((119870

119871)

119898

)

times(

1 minus

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870

(

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870119871minus119870

)

)

)

)

)

)

)

119871

(45)







04

05

06

07

08

09

0 5 10 15 20

1



Prob

abili

ty o

f det

ectio

n

SNR (dB)







Prob

abili

ty o

f det

ectio

n

00

1




04

05

06

07

08

09

01

02

03


04 06 0802


0 10

1

Prob

abili

ty o

f det

ectio

n


04

05

06

07

08

09

01

02

03






010

1


Prob

abili

ty o

f det

ectio

n




04 06 0802

04

05

06

07

08

09

01

02

03


0 1

1


Prob

abili

ty o

f det

ectio

n

120572 = 2 120583 = 2

120572 = 2 120583 = 1

120572 = 1 120583 = 1

120572 = 1 120583 = 2

04 06 08

09

02

04

05

06

07

08


5 dB

0 10

1


Prob

abili

ty o

f det

ectio

n


04

06

08

09

01

02

03

04

05

06

07

0802









04

05

06

07

08

09

01

02

03

04 06 08020 10

1


Prob

abili

ty o

f det

ectio

n




0 10

1


Prob

abili

ty o

f det

ectio

n

04 06 0802

04

05

06

07

08

09

01

02

03





7 Conclusions



Prob

abili

ty o

f det

ectio

n

0 10

1


04 06 0802

04

05

06

07

08

09

01

02

03






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1199090 ge 1199091 + 1199092 + 1199093 (5)


C = ⟨[

[

119864ℎ119864lowast

ℎ119864ℎ119864

lowast

V

119864V119864lowast

ℎ119864V119864

lowast

V

]

]

⟩ (6)


C = 1199090F0 + 1199091F1 + 1199092F2 + 1199093F3 (7)

where F0 = 05 [1 00 1

] F1 = 05 [1 00 minus1

] F2 = 05 [0 11 0

] andF3 = 05 [

0 1minus1 0



1199091 = cos (2120601) cos (2120591)

1199092 = sin (2120601) cos (2120591)

1199093 = sin (2120591)

(8)




119884 (119905) =

ℎ119864 (119905) + 119882 (119905) 1198671

119882(119905) 1198670

(9)



2


2(1198641199041198730)




119863 =

radic1199092

1+ 119909

2

2+ 119909

2

3

1199090

(10)



119875 (119909) =

2Γ (119873 minus 12)

radic120587Γ (119873 minus 1)

[1 minus 1198632

infin]

119873

[1 minus 1199092]

119873minus2


times

119873

sum

119896=1

Γ (2119873)1198632119896minus2

1199041199092119896

Γ (2119896) Γ (2119873 minus 2119896 + 1)

(11)


119875119891 =

1

2

1198681minus1205822 (119873 minus 1

3

2

)

119875119889 = 1 minus 119860[1 minus (

120574

1 + 120574

)

2

]

119873

times

119873

sum

119896=1

119861119896(

120574

1 + 120574

)

2119896minus2 infin

sum

119899=0

119862119896119899(

120574

1 + 120574

)

2119899

(12)







119875119889 = int

infin

0

119875119889 (120574) 119891120574 (120574) 119889120574(13)


120572radic119864(ℎ


by [23]

119891ℎ (ℎ) =

120572120583120583ℎ120572120583minus1

119890minus120583(ℎℎ)

120572

Γ (120583) ℎ

minus120572120583 (14)

where 120583 = E2(ℎ

120572)V(ℎ



119891120574 (120574) =

(1205722) 1205831205831205741205721205832minus1

Γ (120583) 1205741205721205832

119890minus120583(120574120574)

1205722

(15)


119875119889 = 1 minus 1198601015840

119873

sum

119896=1

119861119896

infin

sum

119899=0

119862119896119899 int

infin

0

119892119899 (120574) 119889120574 (16)

where

119892119899 (120574) = [1 minus (

120574

1 + 120574

)

2

]

119873

times (

120574

1 + 120574

)

2119899+2119896minus2

1205741205722120583minus1

119890minus120583(120574120574)

1205722

1198601015840=

1205722120583120583

Γ (120583) 120574minus1205721205832

119860

(17)


119875119889 = 1 minus

119873

sum

119896=1

119861119896

infin

sum

119899=0

119862119896119899

119873

sum

119903=0

(

119873

119903) (minus1)

119903

times

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(18)

where

119860 (V 120583 119911) =1

Γ (V)

times [11198652

(V1

2

minus

120583

2

1 minus

120583

2

minus

119911

4

) Γ (120583) Γ (V)]

times11198652

(V +120583

2

1

2

120583

2

+ 1 minus

119911

4

)

1199111205832

2

times Γ (minus

120583

2

) Γ (V +120583

2

)

minus11198652

(V +120583

2

+

1

2

3

2

120583

2

+

3

2

minus

119911

4

)

119911 =

120583

120574

V = 2 (119873 + 119899 + 119896 minus 1)

11198652 (V1

2

minus

120583

2

1 minus

120583

2

minus

119911

4

)

(19)




119877 =

10038161003816100381610038161003816100381610038161003816

1199093

1199090

10038161003816100381610038161003816100381610038161003816

(20)





119875119877 (119909)

=

Γ (119873 + 12)

radic120587Γ (119873)

[1 minus 1198632

infin]

2

[1 minus 1199092]

119873minus1

times

[1 minus (11990431199091199040)]

[(1 minus (11990431199091199040))2minus (119904

2

1+ 119904

2

2119904

2

0) (1 minus 119909

2)2]

119873+12

+

[1 + 11990431199091199040]

[(1 + 11990431199091199040)2+ ((119904

2

1+ 119904

2

2) 119904

2

0) (1 minus 119909

2)2]

119873+12

(21)


119875119891 =

1

2

1198681minus1205822 (119873

1

2

) (22)


119875119889 = 1 minus [1 minus (

120574

1 + 120574

)

2

]

119873

times

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

1198681205822 (119899 +

1

2

119873)(

120574

1 + 120574

)

2119899

(23)


119875119889 = int

infin

0

119901119889 (120574) 119891120574 (120574) 119889120574 (24)


119875119889 = 1 minus 119896

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

1198681205822 (119899 +

1

2

119873)int

infin

0

119892119899 (120574) 119889120574

(25)

where

119892119899 (120574) = [1 minus (

120574

1 + 120574

)

2

]

119873

(

120574

1 + 120574

)

2119899

1205741205722120583minus1

119890minus120583(120574120574)

1205722

119896 =

120572120583120583

2Γ (120583) 1205741205721205832

(26)


119875119889 = 1 minus

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

times 119868119884th(119899 +

1

2

119873)

119873

sum

119903=0

(

119873

119903) (minus1)

119903

times

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+(12)

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(27)






119876119865 =

119873

sum

119894=119898

(

119873

119894) (119875119891119894)

119894

(1 minus 119875119891119894)

119873minus119894

119876119863 =

119873

sum

119894=119898

(

119873

119894) (119875119889119894)

119894(1 minus 119875119889119894)

119873minus119894

(28)



SU1

SU3

SU2

SU4

PU

Central unit




119875119864119894 = int

infin

0

1

2

erfc (radic120574)

(1205722) 120583120583120574(1205721205832)minus1

Γ (120583) 1205741205721205832

119890minus120583(120574120574)

1205722

119889120574 (29)



119875119864119894 =

120572

4Γ (120583)

times[

[


(2120587)(119896+119897minus1)2

119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ(119897minus120583+1)1

Δ(119896

1

2

) 0

)]

]

(30)

where Δ(119896 119886) = 119886119896 (119886 + 1)119896 (119886 + 119896 minus 1)119896


119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[119867

CU0

119867PU0

]

119873minus119894

(31)

119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[1 minus

119867CU1

119867PU0

]

119873minus119894

(32)

Prob119867

CU1

119867PU0

= [Prob119867

CU1

119867SU1

times Prob119867

SU1

119867PU0

]

+ Prob119867

CU1

119867SU0

sdot Prob119867

SU0

119867PU0

(33)

119876119865 =

119873

sum

119894=119898

(

119873

119894) [(1 minus 119875119864119894) 119875119865119894 + 119875119864119894 (1 minus 119875119865119894)]

119894


(34)

where 119867CU1

119867PU1

and 119867SU1




119876119865 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898)

)

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus

Γ (119898 1205822)

Γ (119898)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583)

Γ (119898 1205822)

Γ (119898)

119873minus119894

(35)

where

119869 (119896 119897 120572 120583) =


(2120587)(119896+119897minus1)2

times 119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ (119897 minus120583+1) 1

Δ (119896

1

2

) 0)

(36)


119876119863 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

+

120572

8Γ (120583)

119869 (119896 119897 120572 120583)119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

119873minus119894

(37)



119876119865 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+120572

4Γ (120583)119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898))

119894

(38)

119876119863 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+120572

4Γ (120583)119869 (119896 119897 120572 120583)(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

(39)




119871

sum

119894=119897

119873119894 = 119870 (40)



119876119865 = 1

minus(

1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(

Γ (119898 1205822)

Γ (119898)

)

119870

(1 minus

Γ (119898 1205822)

Γ (119898)

)

119870119871minus119870)

119871

(41)


SU1

SU1

SU2

SU2

SU3

SU3

PU

Central unit

Cluster head

Cluster head


119876119863 = 1 minus (1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)(119875119889)119870(1 minus 119901119889)

119870119871minus119870)

119871

(42)


119876119863 = 1 minus

(

(

(

(

1minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870

(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870119871minus119870

)

)

)

)

119871

(43)

where119880 (V 120583 119911)

=

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(44)


119876119863 = 1 minus

(

(

(

(

(

(

(

1 minus

119870119871

sum119898=119897

((119870

119871)

119898

)

times(

1 minus

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870

(

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870119871minus119870

)

)

)

)

)

)

)

119871

(45)







04

05

06

07

08

09

0 5 10 15 20

1



Prob

abili

ty o

f det

ectio

n

SNR (dB)







Prob

abili

ty o

f det

ectio

n

00

1




04

05

06

07

08

09

01

02

03


04 06 0802


0 10

1

Prob

abili

ty o

f det

ectio

n


04

05

06

07

08

09

01

02

03






010

1


Prob

abili

ty o

f det

ectio

n




04 06 0802

04

05

06

07

08

09

01

02

03


0 1

1


Prob

abili

ty o

f det

ectio

n

120572 = 2 120583 = 2

120572 = 2 120583 = 1

120572 = 1 120583 = 1

120572 = 1 120583 = 2

04 06 08

09

02

04

05

06

07

08


5 dB

0 10

1


Prob

abili

ty o

f det

ectio

n


04

06

08

09

01

02

03

04

05

06

07

0802









04

05

06

07

08

09

01

02

03

04 06 08020 10

1


Prob

abili

ty o

f det

ectio

n




0 10

1


Prob

abili

ty o

f det

ectio

n

04 06 0802

04

05

06

07

08

09

01

02

03





7 Conclusions



Prob

abili

ty o

f det

ectio

n

0 10

1


04 06 0802

04

05

06

07

08

09

01

02

03






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119875119889 = int

infin

0

119875119889 (120574) 119891120574 (120574) 119889120574(13)


120572radic119864(ℎ


by [23]

119891ℎ (ℎ) =

120572120583120583ℎ120572120583minus1

119890minus120583(ℎℎ)

120572

Γ (120583) ℎ

minus120572120583 (14)

where 120583 = E2(ℎ

120572)V(ℎ



119891120574 (120574) =

(1205722) 1205831205831205741205721205832minus1

Γ (120583) 1205741205721205832

119890minus120583(120574120574)

1205722

(15)


119875119889 = 1 minus 1198601015840

119873

sum

119896=1

119861119896

infin

sum

119899=0

119862119896119899 int

infin

0

119892119899 (120574) 119889120574 (16)

where

119892119899 (120574) = [1 minus (

120574

1 + 120574

)

2

]

119873

times (

120574

1 + 120574

)

2119899+2119896minus2

1205741205722120583minus1

119890minus120583(120574120574)

1205722

1198601015840=

1205722120583120583

Γ (120583) 120574minus1205721205832

119860

(17)


119875119889 = 1 minus

119873

sum

119896=1

119861119896

infin

sum

119899=0

119862119896119899

119873

sum

119903=0

(

119873

119903) (minus1)

119903

times

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(18)

where

119860 (V 120583 119911) =1

Γ (V)

times [11198652

(V1

2

minus

120583

2

1 minus

120583

2

minus

119911

4

) Γ (120583) Γ (V)]

times11198652

(V +120583

2

1

2

120583

2

+ 1 minus

119911

4

)

1199111205832

2

times Γ (minus

120583

2

) Γ (V +120583

2

)

minus11198652

(V +120583

2

+

1

2

3

2

120583

2

+

3

2

minus

119911

4

)

119911 =

120583

120574

V = 2 (119873 + 119899 + 119896 minus 1)

11198652 (V1

2

minus

120583

2

1 minus

120583

2

minus

119911

4

)

(19)




119877 =

10038161003816100381610038161003816100381610038161003816

1199093

1199090

10038161003816100381610038161003816100381610038161003816

(20)





119875119877 (119909)

=

Γ (119873 + 12)

radic120587Γ (119873)

[1 minus 1198632

infin]

2

[1 minus 1199092]

119873minus1

times

[1 minus (11990431199091199040)]

[(1 minus (11990431199091199040))2minus (119904

2

1+ 119904

2

2119904

2

0) (1 minus 119909

2)2]

119873+12

+

[1 + 11990431199091199040]

[(1 + 11990431199091199040)2+ ((119904

2

1+ 119904

2

2) 119904

2

0) (1 minus 119909

2)2]

119873+12

(21)


119875119891 =

1

2

1198681minus1205822 (119873

1

2

) (22)


119875119889 = 1 minus [1 minus (

120574

1 + 120574

)

2

]

119873

times

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

1198681205822 (119899 +

1

2

119873)(

120574

1 + 120574

)

2119899

(23)


119875119889 = int

infin

0

119901119889 (120574) 119891120574 (120574) 119889120574 (24)


119875119889 = 1 minus 119896

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

1198681205822 (119899 +

1

2

119873)int

infin

0

119892119899 (120574) 119889120574

(25)

where

119892119899 (120574) = [1 minus (

120574

1 + 120574

)

2

]

119873

(

120574

1 + 120574

)

2119899

1205741205722120583minus1

119890minus120583(120574120574)

1205722

119896 =

120572120583120583

2Γ (120583) 1205741205721205832

(26)


119875119889 = 1 minus

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

times 119868119884th(119899 +

1

2

119873)

119873

sum

119903=0

(

119873

119903) (minus1)

119903

times

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+(12)

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(27)






119876119865 =

119873

sum

119894=119898

(

119873

119894) (119875119891119894)

119894

(1 minus 119875119891119894)

119873minus119894

119876119863 =

119873

sum

119894=119898

(

119873

119894) (119875119889119894)

119894(1 minus 119875119889119894)

119873minus119894

(28)



SU1

SU3

SU2

SU4

PU

Central unit




119875119864119894 = int

infin

0

1

2

erfc (radic120574)

(1205722) 120583120583120574(1205721205832)minus1

Γ (120583) 1205741205721205832

119890minus120583(120574120574)

1205722

119889120574 (29)



119875119864119894 =

120572

4Γ (120583)

times[

[


(2120587)(119896+119897minus1)2

119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ(119897minus120583+1)1

Δ(119896

1

2

) 0

)]

]

(30)

where Δ(119896 119886) = 119886119896 (119886 + 1)119896 (119886 + 119896 minus 1)119896


119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[119867

CU0

119867PU0

]

119873minus119894

(31)

119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[1 minus

119867CU1

119867PU0

]

119873minus119894

(32)

Prob119867

CU1

119867PU0

= [Prob119867

CU1

119867SU1

times Prob119867

SU1

119867PU0

]

+ Prob119867

CU1

119867SU0

sdot Prob119867

SU0

119867PU0

(33)

119876119865 =

119873

sum

119894=119898

(

119873

119894) [(1 minus 119875119864119894) 119875119865119894 + 119875119864119894 (1 minus 119875119865119894)]

119894


(34)

where 119867CU1

119867PU1

and 119867SU1




119876119865 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898)

)

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus

Γ (119898 1205822)

Γ (119898)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583)

Γ (119898 1205822)

Γ (119898)

119873minus119894

(35)

where

119869 (119896 119897 120572 120583) =


(2120587)(119896+119897minus1)2

times 119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ (119897 minus120583+1) 1

Δ (119896

1

2

) 0)

(36)


119876119863 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

+

120572

8Γ (120583)

119869 (119896 119897 120572 120583)119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

119873minus119894

(37)



119876119865 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+120572

4Γ (120583)119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898))

119894

(38)

119876119863 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+120572

4Γ (120583)119869 (119896 119897 120572 120583)(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

(39)




119871

sum

119894=119897

119873119894 = 119870 (40)



119876119865 = 1

minus(

1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(

Γ (119898 1205822)

Γ (119898)

)

119870

(1 minus

Γ (119898 1205822)

Γ (119898)

)

119870119871minus119870)

119871

(41)


SU1

SU1

SU2

SU2

SU3

SU3

PU

Central unit

Cluster head

Cluster head


119876119863 = 1 minus (1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)(119875119889)119870(1 minus 119901119889)

119870119871minus119870)

119871

(42)


119876119863 = 1 minus

(

(

(

(

1minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870

(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870119871minus119870

)

)

)

)

119871

(43)

where119880 (V 120583 119911)

=

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(44)


119876119863 = 1 minus

(

(

(

(

(

(

(

1 minus

119870119871

sum119898=119897

((119870

119871)

119898

)

times(

1 minus

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870

(

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870119871minus119870

)

)

)

)

)

)

)

119871

(45)







04

05

06

07

08

09

0 5 10 15 20

1



Prob

abili

ty o

f det

ectio

n

SNR (dB)







Prob

abili

ty o

f det

ectio

n

00

1




04

05

06

07

08

09

01

02

03


04 06 0802


0 10

1

Prob

abili

ty o

f det

ectio

n


04

05

06

07

08

09

01

02

03






010

1


Prob

abili

ty o

f det

ectio

n




04 06 0802

04

05

06

07

08

09

01

02

03


0 1

1


Prob

abili

ty o

f det

ectio

n

120572 = 2 120583 = 2

120572 = 2 120583 = 1

120572 = 1 120583 = 1

120572 = 1 120583 = 2

04 06 08

09

02

04

05

06

07

08


5 dB

0 10

1


Prob

abili

ty o

f det

ectio

n


04

06

08

09

01

02

03

04

05

06

07

0802









04

05

06

07

08

09

01

02

03

04 06 08020 10

1


Prob

abili

ty o

f det

ectio

n




0 10

1


Prob

abili

ty o

f det

ectio

n

04 06 0802

04

05

06

07

08

09

01

02

03





7 Conclusions



Prob

abili

ty o

f det

ectio

n

0 10

1


04 06 0802

04

05

06

07

08

09

01

02

03






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119875119877 (119909)

=

Γ (119873 + 12)

radic120587Γ (119873)

[1 minus 1198632

infin]

2

[1 minus 1199092]

119873minus1

times

[1 minus (11990431199091199040)]

[(1 minus (11990431199091199040))2minus (119904

2

1+ 119904

2

2119904

2

0) (1 minus 119909

2)2]

119873+12

+

[1 + 11990431199091199040]

[(1 + 11990431199091199040)2+ ((119904

2

1+ 119904

2

2) 119904

2

0) (1 minus 119909

2)2]

119873+12

(21)


119875119891 =

1

2

1198681minus1205822 (119873

1

2

) (22)


119875119889 = 1 minus [1 minus (

120574

1 + 120574

)

2

]

119873

times

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

1198681205822 (119899 +

1

2

119873)(

120574

1 + 120574

)

2119899

(23)


119875119889 = int

infin

0

119901119889 (120574) 119891120574 (120574) 119889120574 (24)


119875119889 = 1 minus 119896

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

1198681205822 (119899 +

1

2

119873)int

infin

0

119892119899 (120574) 119889120574

(25)

where

119892119899 (120574) = [1 minus (

120574

1 + 120574

)

2

]

119873

(

120574

1 + 120574

)

2119899

1205741205722120583minus1

119890minus120583(120574120574)

1205722

119896 =

120572120583120583

2Γ (120583) 1205741205721205832

(26)


119875119889 = 1 minus

infin

sum

119899=0

Γ (119873 + 119899)

Γ (119899 + 1) Γ (119873)

times 119868119884th(119899 +

1

2

119873)

119873

sum

119903=0

(

119873

119903) (minus1)

119903

times

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+(12)

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(27)






119876119865 =

119873

sum

119894=119898

(

119873

119894) (119875119891119894)

119894

(1 minus 119875119891119894)

119873minus119894

119876119863 =

119873

sum

119894=119898

(

119873

119894) (119875119889119894)

119894(1 minus 119875119889119894)

119873minus119894

(28)



SU1

SU3

SU2

SU4

PU

Central unit




119875119864119894 = int

infin

0

1

2

erfc (radic120574)

(1205722) 120583120583120574(1205721205832)minus1

Γ (120583) 1205741205721205832

119890minus120583(120574120574)

1205722

119889120574 (29)



119875119864119894 =

120572

4Γ (120583)

times[

[


(2120587)(119896+119897minus1)2

119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ(119897minus120583+1)1

Δ(119896

1

2

) 0

)]

]

(30)

where Δ(119896 119886) = 119886119896 (119886 + 1)119896 (119886 + 119896 minus 1)119896


119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[119867

CU0

119867PU0

]

119873minus119894

(31)

119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[1 minus

119867CU1

119867PU0

]

119873minus119894

(32)

Prob119867

CU1

119867PU0

= [Prob119867

CU1

119867SU1

times Prob119867

SU1

119867PU0

]

+ Prob119867

CU1

119867SU0

sdot Prob119867

SU0

119867PU0

(33)

119876119865 =

119873

sum

119894=119898

(

119873

119894) [(1 minus 119875119864119894) 119875119865119894 + 119875119864119894 (1 minus 119875119865119894)]

119894


(34)

where 119867CU1

119867PU1

and 119867SU1




119876119865 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898)

)

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus

Γ (119898 1205822)

Γ (119898)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583)

Γ (119898 1205822)

Γ (119898)

119873minus119894

(35)

where

119869 (119896 119897 120572 120583) =


(2120587)(119896+119897minus1)2

times 119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ (119897 minus120583+1) 1

Δ (119896

1

2

) 0)

(36)


119876119863 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

+

120572

8Γ (120583)

119869 (119896 119897 120572 120583)119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

119873minus119894

(37)



119876119865 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+120572

4Γ (120583)119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898))

119894

(38)

119876119863 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+120572

4Γ (120583)119869 (119896 119897 120572 120583)(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

(39)




119871

sum

119894=119897

119873119894 = 119870 (40)



119876119865 = 1

minus(

1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(

Γ (119898 1205822)

Γ (119898)

)

119870

(1 minus

Γ (119898 1205822)

Γ (119898)

)

119870119871minus119870)

119871

(41)


SU1

SU1

SU2

SU2

SU3

SU3

PU

Central unit

Cluster head

Cluster head


119876119863 = 1 minus (1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)(119875119889)119870(1 minus 119901119889)

119870119871minus119870)

119871

(42)


119876119863 = 1 minus

(

(

(

(

1minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870

(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870119871minus119870

)

)

)

)

119871

(43)

where119880 (V 120583 119911)

=

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(44)


119876119863 = 1 minus

(

(

(

(

(

(

(

1 minus

119870119871

sum119898=119897

((119870

119871)

119898

)

times(

1 minus

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870

(

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870119871minus119870

)

)

)

)

)

)

)

119871

(45)







04

05

06

07

08

09

0 5 10 15 20

1



Prob

abili

ty o

f det

ectio

n

SNR (dB)







Prob

abili

ty o

f det

ectio

n

00

1




04

05

06

07

08

09

01

02

03


04 06 0802


0 10

1

Prob

abili

ty o

f det

ectio

n


04

05

06

07

08

09

01

02

03






010

1


Prob

abili

ty o

f det

ectio

n




04 06 0802

04

05

06

07

08

09

01

02

03


0 1

1


Prob

abili

ty o

f det

ectio

n

120572 = 2 120583 = 2

120572 = 2 120583 = 1

120572 = 1 120583 = 1

120572 = 1 120583 = 2

04 06 08

09

02

04

05

06

07

08


5 dB

0 10

1


Prob

abili

ty o

f det

ectio

n


04

06

08

09

01

02

03

04

05

06

07

0802









04

05

06

07

08

09

01

02

03

04 06 08020 10

1


Prob

abili

ty o

f det

ectio

n




0 10

1


Prob

abili

ty o

f det

ectio

n

04 06 0802

04

05

06

07

08

09

01

02

03





7 Conclusions



Prob

abili

ty o

f det

ectio

n

0 10

1


04 06 0802

04

05

06

07

08

09

01

02

03






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










SU1

SU3

SU2

SU4

PU

Central unit




119875119864119894 = int

infin

0

1

2

erfc (radic120574)

(1205722) 120583120583120574(1205721205832)minus1

Γ (120583) 1205741205721205832

119890minus120583(120574120574)

1205722

119889120574 (29)



119875119864119894 =

120572

4Γ (120583)

times[

[


(2120587)(119896+119897minus1)2

119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ(119897minus120583+1)1

Δ(119896

1

2

) 0

)]

]

(30)

where Δ(119896 119886) = 119886119896 (119886 + 1)119896 (119886 + 119896 minus 1)119896


119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[119867

CU0

119867PU0

]

119873minus119894

(31)

119876119865 =

119873

sum

119894=119898

(

119873

119894)Prob[

119867CU1

119867PU0

]

119894

times Prob[1 minus

119867CU1

119867PU0

]

119873minus119894

(32)

Prob119867

CU1

119867PU0

= [Prob119867

CU1

119867SU1

times Prob119867

SU1

119867PU0

]

+ Prob119867

CU1

119867SU0

sdot Prob119867

SU0

119867PU0

(33)

119876119865 =

119873

sum

119894=119898

(

119873

119894) [(1 minus 119875119864119894) 119875119865119894 + 119875119864119894 (1 minus 119875119865119894)]

119894


(34)

where 119867CU1

119867PU1

and 119867SU1




119876119865 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898)

)

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus

Γ (119898 1205822)

Γ (119898)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583)

Γ (119898 1205822)

Γ (119898)

119873minus119894

(35)

where

119869 (119896 119897 120572 120583) =


(2120587)(119896+119897minus1)2

times 119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ (119897 minus120583+1) 1

Δ (119896

1

2

) 0)

(36)


119876119863 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

+

120572

8Γ (120583)

119869 (119896 119897 120572 120583)119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

119873minus119894

(37)



119876119865 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+120572

4Γ (120583)119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898))

119894

(38)

119876119863 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+120572

4Γ (120583)119869 (119896 119897 120572 120583)(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

(39)




119871

sum

119894=119897

119873119894 = 119870 (40)



119876119865 = 1

minus(

1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(

Γ (119898 1205822)

Γ (119898)

)

119870

(1 minus

Γ (119898 1205822)

Γ (119898)

)

119870119871minus119870)

119871

(41)


SU1

SU1

SU2

SU2

SU3

SU3

PU

Central unit

Cluster head

Cluster head


119876119863 = 1 minus (1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)(119875119889)119870(1 minus 119901119889)

119870119871minus119870)

119871

(42)


119876119863 = 1 minus

(

(

(

(

1minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870

(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870119871minus119870

)

)

)

)

119871

(43)

where119880 (V 120583 119911)

=

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(44)


119876119863 = 1 minus

(

(

(

(

(

(

(

1 minus

119870119871

sum119898=119897

((119870

119871)

119898

)

times(

1 minus

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870

(

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870119871minus119870

)

)

)

)

)

)

)

119871

(45)







04

05

06

07

08

09

0 5 10 15 20

1



Prob

abili

ty o

f det

ectio

n

SNR (dB)







Prob

abili

ty o

f det

ectio

n

00

1




04

05

06

07

08

09

01

02

03


04 06 0802


0 10

1

Prob

abili

ty o

f det

ectio

n


04

05

06

07

08

09

01

02

03






010

1


Prob

abili

ty o

f det

ectio

n




04 06 0802

04

05

06

07

08

09

01

02

03


0 1

1


Prob

abili

ty o

f det

ectio

n

120572 = 2 120583 = 2

120572 = 2 120583 = 1

120572 = 1 120583 = 1

120572 = 1 120583 = 2

04 06 08

09

02

04

05

06

07

08


5 dB

0 10

1


Prob

abili

ty o

f det

ectio

n


04

06

08

09

01

02

03

04

05

06

07

0802









04

05

06

07

08

09

01

02

03

04 06 08020 10

1


Prob

abili

ty o

f det

ectio

n




0 10

1


Prob

abili

ty o

f det

ectio

n

04 06 0802

04

05

06

07

08

09

01

02

03





7 Conclusions



Prob

abili

ty o

f det

ectio

n

0 10

1


04 06 0802

04

05

06

07

08

09

01

02

03






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119876119865 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898)

)

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus

Γ (119898 1205822)

Γ (119898)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583)

Γ (119898 1205822)

Γ (119898)

119873minus119894

(35)

where

119869 (119896 119897 120572 120583) =


(2120587)(119896+119897minus1)2

times 119866119896+1119897

119897+1119896+1((

120574(1120583)2120572

119896

)

119896

119897119897|

Δ (119897 minus120583+1) 1

Δ (119896

1

2

) 0)

(36)


119876119863 =

119873

sum

119894=119898

(

119873

119894)

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+

120572

4Γ (120583)

119869 (119896 119897 120572 120583) (1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

times

(1 minus

120572

4Γ (120583)

119869 (119896 119897 120572 120583))(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

+

120572

8Γ (120583)

119869 (119896 119897 120572 120583)119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

119873minus119894

(37)



119876119865 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))

Γ (119898 1205822)

Γ (119898)

+120572

4Γ (120583)119869 (119896 119897 120572 120583) (1 minus

Γ (119898 1205822)

Γ (119898))

119894

(38)

119876119863 =

119873

prod

119894=0

(1 minus120572

4Γ (120583)119869 (119896 119897 120572 120583))119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

)

+120572

4Γ (120583)119869 (119896 119897 120572 120583)(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119894

(39)




119871

sum

119894=119897

119873119894 = 119870 (40)



119876119865 = 1

minus(

1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(

Γ (119898 1205822)

Γ (119898)

)

119870

(1 minus

Γ (119898 1205822)

Γ (119898)

)

119870119871minus119870)

119871

(41)


SU1

SU1

SU2

SU2

SU3

SU3

PU

Central unit

Cluster head

Cluster head


119876119863 = 1 minus (1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)(119875119889)119870(1 minus 119901119889)

119870119871minus119870)

119871

(42)


119876119863 = 1 minus

(

(

(

(

1minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870

(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870119871minus119870

)

)

)

)

119871

(43)

where119880 (V 120583 119911)

=

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(44)


119876119863 = 1 minus

(

(

(

(

(

(

(

1 minus

119870119871

sum119898=119897

((119870

119871)

119898

)

times(

1 minus

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870

(

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870119871minus119870

)

)

)

)

)

)

)

119871

(45)







04

05

06

07

08

09

0 5 10 15 20

1



Prob

abili

ty o

f det

ectio

n

SNR (dB)







Prob

abili

ty o

f det

ectio

n

00

1




04

05

06

07

08

09

01

02

03


04 06 0802


0 10

1

Prob

abili

ty o

f det

ectio

n


04

05

06

07

08

09

01

02

03






010

1


Prob

abili

ty o

f det

ectio

n




04 06 0802

04

05

06

07

08

09

01

02

03


0 1

1


Prob

abili

ty o

f det

ectio

n

120572 = 2 120583 = 2

120572 = 2 120583 = 1

120572 = 1 120583 = 1

120572 = 1 120583 = 2

04 06 08

09

02

04

05

06

07

08


5 dB

0 10

1


Prob

abili

ty o

f det

ectio

n


04

06

08

09

01

02

03

04

05

06

07

0802









04

05

06

07

08

09

01

02

03

04 06 08020 10

1


Prob

abili

ty o

f det

ectio

n




0 10

1


Prob

abili

ty o

f det

ectio

n

04 06 0802

04

05

06

07

08

09

01

02

03





7 Conclusions



Prob

abili

ty o

f det

ectio

n

0 10

1


04 06 0802

04

05

06

07

08

09

01

02

03






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










SU1

SU1

SU2

SU2

SU3

SU3

PU

Central unit

Cluster head

Cluster head


119876119863 = 1 minus (1 minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)(119875119889)119870(1 minus 119901119889)

119870119871minus119870)

119871

(42)


119876119863 = 1 minus

(

(

(

(

1minus

119870119871

sum

119898=119897

((

119870

119871

)

119898

)

times(119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870

(1 minus 119860

infin

sum

119899=0

119886119899119866119896119897

119897119896(119878

Δ (119897 minusV)Δ (119896 0)

))

119870119871minus119870

)

)

)

)

119871

(43)

where119880 (V 120583 119911)

=

119860(V 120583 119911) 120572 = 1

Γ (V + 120583)119880 (V 1 minus 120583 119911) 120572 = 2

(119911119896)V119897120583+12

radic2120587

119866119896119896+119897

119896+119897119896

times[

2

119911119896|

Δ (119897 minus120583) Δ (119896 V + 1)

Δ (119896 0)] 120572 = 1 2

(44)


119876119863 = 1 minus

(

(

(

(

(

(

(

1 minus

119870119871

sum119898=119897

((119870

119871)

119898

)

times(

1 minus

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870

(

infin

sum119899=0

Γ (119873+ 119899)

Γ (119899 + 1) Γ (119873)119868119884th

(119899 +1

2119873)

times

119873

sum119903=0

(119873

119903) (minus1)

119903119880(V 120583 119911)

)

119870119871minus119870

)

)

)

)

)

)

)

119871

(45)







04

05

06

07

08

09

0 5 10 15 20

1



Prob

abili

ty o

f det

ectio

n

SNR (dB)







Prob

abili

ty o

f det

ectio

n

00

1




04

05

06

07

08

09

01

02

03


04 06 0802


0 10

1

Prob

abili

ty o

f det

ectio

n


04

05

06

07

08

09

01

02

03






010

1


Prob

abili

ty o

f det

ectio

n




04 06 0802

04

05

06

07

08

09

01

02

03


0 1

1


Prob

abili

ty o

f det

ectio

n

120572 = 2 120583 = 2

120572 = 2 120583 = 1

120572 = 1 120583 = 1

120572 = 1 120583 = 2

04 06 08

09

02

04

05

06

07

08


5 dB

0 10

1


Prob

abili

ty o

f det

ectio

n


04

06

08

09

01

02

03

04

05

06

07

0802









04

05

06

07

08

09

01

02

03

04 06 08020 10

1


Prob

abili

ty o

f det

ectio

n




0 10

1


Prob

abili

ty o

f det

ectio

n

04 06 0802

04

05

06

07

08

09

01

02

03





7 Conclusions



Prob

abili

ty o

f det

ectio

n

0 10

1


04 06 0802

04

05

06

07

08

09

01

02

03






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










04

05

06

07

08

09

0 5 10 15 20

1



Prob

abili

ty o

f det

ectio

n

SNR (dB)







Prob

abili

ty o

f det

ectio

n

00

1




04

05

06

07

08

09

01

02

03


04 06 0802


0 10

1

Prob

abili

ty o

f det

ectio

n


04

05

06

07

08

09

01

02

03






010

1


Prob

abili

ty o

f det

ectio

n




04 06 0802

04

05

06

07

08

09

01

02

03


0 1

1


Prob

abili

ty o

f det

ectio

n

120572 = 2 120583 = 2

120572 = 2 120583 = 1

120572 = 1 120583 = 1

120572 = 1 120583 = 2

04 06 08

09

02

04

05

06

07

08


5 dB

0 10

1


Prob

abili

ty o

f det

ectio

n


04

06

08

09

01

02

03

04

05

06

07

0802









04

05

06

07

08

09

01

02

03

04 06 08020 10

1


Prob

abili

ty o

f det

ectio

n




0 10

1


Prob

abili

ty o

f det

ectio

n

04 06 0802

04

05

06

07

08

09

01

02

03





7 Conclusions



Prob

abili

ty o

f det

ectio

n

0 10

1


04 06 0802

04

05

06

07

08

09

01

02

03






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










010

1


Prob

abili

ty o

f det

ectio

n




04 06 0802

04

05

06

07

08

09

01

02

03


0 1

1


Prob

abili

ty o

f det

ectio

n

120572 = 2 120583 = 2

120572 = 2 120583 = 1

120572 = 1 120583 = 1

120572 = 1 120583 = 2

04 06 08

09

02

04

05

06

07

08


5 dB

0 10

1


Prob

abili

ty o

f det

ectio

n


04

06

08

09

01

02

03

04

05

06

07

0802









04

05

06

07

08

09

01

02

03

04 06 08020 10

1


Prob

abili

ty o

f det

ectio

n




0 10

1


Prob

abili

ty o

f det

ectio

n

04 06 0802

04

05

06

07

08

09

01

02

03





7 Conclusions



Prob

abili

ty o

f det

ectio

n

0 10

1


04 06 0802

04

05

06

07

08

09

01

02

03






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0 10

1


Prob

abili

ty o

f det

ectio

n

04 06 0802

04

05

06

07

08

09

01

02

03





7 Conclusions



Prob

abili

ty o

f det

ectio

n

0 10

1


04 06 0802

04

05

06

07

08

09

01

02

03






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article a spectrum sensing scheme for partially...

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