cooperative sensing specrum

7232019 Cooperative Sensing Specrum

httpslidepdfcomreaderfullcooperative-sensing-specrum 15

Cooperative Spectrum Sensing with an Improved

Energy Detector in Cognitive Radio Network Ajay Singh Manav R Bhatnagar and Ranjan K Mallik

Department of Electrical Engineering

Indian Institute of Technology DelhiHauz Khas IN-110016 New Delhi India

Email ajaysingh8226gmailcommanavrkmallikeeiitdacin

Abstractmdash In this paper performance of cooperative spectrumsensing with an improved energy detector is discussed TheCognitive radios (CRs) utilize an improved energy detector fortaking a decision of the presence of the primary user (PU) Theimproved energy detector uses an arbitrary positive power

of the amplitude of the received samples of the primary userrsquossignals The decision of each CR is orthogonally forwarded toa fusion center (FC) which takes final decision of the presenceof the PU We minimize sum of the probability of false alarmand missed detection in cooperative spectrum sensing to obtainan optimized number of CRs for detecting a spectrum hole An

optimized value of and sensing threshold of each CR is alsoobtained by minimizing the total probability of error

I INTRODUCTION

Detecting an unused spectrum and sharing it without inter-

fering the primary users (PUs) is a key requirement of the

cognitive radio network For detection of a spectrum hole the

cognitive communication network relies upon the cognitive

radios (CRs) or secondary users [1] Spectrum sensing is

a procedure in which a CR captures the information of a

band available for transmission and then shares the frequency

band without interfering the primary users Cooperation among

the CRs provides improvement in detection of the primarysignals [2] Due to the random nature of the wireless channel

the signals sensed by the CRs are also random in nature

Therefore sometimes a CR can falsely decide the presence

or absence of the PU [3] In [3] a cooperation based spec-

trum sensing scheme is proposed to detect the PU with an

optimal linear combination of the received energies from the

cooperating CRs in a fusion center (FC) The cooperative

spectrum sensing scheme provides better immunity to fading

noise uncertainty and shadowing as compared to a cognitive

network that does not utilize cooperation among the CRs [4]

Probability of miss and probability of false alarm 1038389 play

important role in describing a reliable communication We

can fix either of the probabilities at a particular value and

then optimize the other one for improving the probability of

detection of the primary signals [5] In [6] optimization of

the total probability of error is performed by minimizing sum

of the probability of false alarm and the probability of missed

detection of an cooperative spectrum sensing scheme It is

explored in [6] that how many CRs are needed for optimum

detection of the primary user in a cooperative spectrum sensing

based scheme such that the total error probability does not

exceed a fixed value

For the detection of unknown deterministic signals cor-

rupted by the additive white Gaussian noise (AWGN) an

energy detector is derived in [7] which is also called conven-

tional energy detector The conventional energy detector [7]

utilizes square of the magnitude of the received data sample

which is equal to the energy of the received sample It is

shown in [8] that the performance of a cognitive radio network

can be improved by utilizing an improved energy detector in

the CRs where the conventional energy detector is modified

by replacing the squaring operation of the received signalamplitude with an arbitrary positive power 1038389

In this paper we consider optimization of the cooperative

spectrum sensing scheme with an improved energy detector to

minimize the sum of the probability of false alarm and missed

detection We derive a closed form analytical expression to find

an optimal number of the CRs required for cooperation while

utilizing the improved energy detector in each CR Moreover

we also perform optimization of the threshold and the energy

detector to minimize the total error rate

I I SYSTEM M ODEL

We consider a system consisting of 1103925 number of CRs one

primary user and a fusion center There are two hypothesis907317 0 and 907317 1 in the -th CR for the detection of the spectrum

hole

907317 0 1103925() = 1103925()983084 if PU is absent983084 (1)

907317 1 1103925() = () + 1103925()983084 if PU is present983084 (2)

where = 1983084 29830869830869830869830861103925 () sim (0983084 2907317) where 2

907317 is the

average transmitted power of the PU denotes a zero mean

Gaussian signal transmitted by the PU and 1103925() sim (0983084 2)

is additive white Gaussian noise (AWGN) with zero mean

and 2 variance The variance of the signal received at each

secondary user under 907317 1 will be 2907317 + 2

It is assumed that

each CR contains an improved energy detector [8] The -th

CR utilizes the following statistic for deciding of the presence

of the PU [8]

=∣ 1103925 ∣ 983084 1038389 gt 0983086 (3)

It can be seen from (3) that for 1038389 = 2 reduces to statistics

corresponding to the conventional energy detector [7] In a

cooperative sensing scheme multiple CRs exists in a cognitive

radio network such that each CR makes independent decision

regarding the presence or absence of PU We consider a

978-1-61284-091-811$2600 copy2011 IEEE



cooperative scheme in which each secondary user sends its

binary decision 1103925 (0 or 1) to the fusion center by using an

improved energy detector over error free orthogonal channels

The fusion center combines these binary decisions to find the

presence or absence of the PU as follows

=

sum1103925=1

1103925983084 (4)

where D is the sum of the all 1-bit decisions from the CRs Let

le 1103925 corresponds to a number of cooperating CRs out of

1103925 CRs The FC uses a majority rule for deciding the presence

or absence of the PU As per the majority decision rule if

is greater than then hypothesis ℋ1 holds and if is smaller

than then hypothesis ℋ0 will be true The hypothesis ℋ0

and ℋ1 can be written as

ℋ0 lt 983084 if PU is absent983084 (5)

ℋ1 ge 983084 if PU is present983086 (6)

III PROBABILITY OF FALSE A LARM AND MISSED

DETECTION IN THE C RS

The cumulative distribution function (cdf) of the improved

energy detector can be written as

Pr(∣1103925∣ le ) =Pr(∣1103925∣ le 1 )

=Pr(minus1 le 1103925 le

1 )

=Pr(1103925 le 1 ) minus Pr(1103925 lt minus

1 )983084 (7)

where Pr() denotes the probability For finding the probability

density function (pdf) of the decision statistics we need

to differentiate (8) with respect to (wrt) to get

() = 1

1038389

1minus

1038389(

1 ) +

1

1038389

1minus

1038389(minus

1 )983084 (8)

where 1038389() is the pdf of received signal at the FC under

hypothesis 907317 0 or 907317 1 depending upon the presence or absence

of the primary signal Let 1038389∣11039250() and 1038389∣11039250

() be the

pdfs of the received signal under hypothesis 907317 0 and 907317 1

respectively From (5) and (6) it can be deduced that

1038389∣11039250() =

1radic 22

exp

1048616minus 2

22

1048617983084 (9)

1038389∣11039251() =

1

radic 2(2

907317 + 2)

exp

1048616minus 2

2(2907317 + 2

)

1048617983086 (10)

The pdf of under hypothesis 907317 0 is given from (8) and (9)as follows

∣ 0() =

radic 2

1minus exp(minus

2

2907317)

1038389radic

983086 (11)

From (8) and (10) the pdf of under hypothesis 907317 1 can

be obtained as

∣ 1() =

radic 2

1minus exp(minus

2

2(2907317+2)

)

1038389radic

radic

(2907317 + 2

)983086 (12)

The probability of false alarm 1038389 is defined as [10 Eq (41)

Chapter 2]

1038389 ≜

int 1

∣ 0()983084 (13)

where 1 is the decision region corresponding to hypothesis

907317 1 Let us assume that the decision threshold in each CR is

given as The probability of false alarm 1038389 in each CR is

derived in Appendix I as

1038389 = 1radic

9830803

2983084

2

22

983081983084 (14)

where () is the gamma function [9 Eq (611)]

and (983084 ) is the incomplete gamma function

(983084 )=991787 infin minus1 exp(minus) [9 Eq (653)] The probability

of miss is defined by [10 Eq (41) Chapter 2]

≜

int 0

∣ 1()983084 (15)

where 0 is the decision region corresponding to the hypothe-

sis 907317 0 In Appendix I it is shown that the probability of miss

in each CR will be

= 1radic

9830803

2983084

2

2(2907317 + 2

)

983081983084 (16)

where (983084 ) is the incomplete gamma function

(983084 )=991787 0 minus1 exp(minus) [9 Eq (652)]

IV OPTIMIZATION OF COOPERATIVE SPECTRUM SENSING

SCHEME

The probability of false alarm of the FC for cooperative

sensing will be

= Pr(

ℋ1

∣ℋ0)983084 (17)

From (5) (6) and (17) the probability of false alarm of the

FC can be written as

= sum=

1048616 1103925

1048617 1038389 (1 minus 1038389 )

minus983086 (18)

In the FC the probability of missed detection will be

= Pr(ℋ0∣ℋ1)983086 (19)

The probability of missed detection of the FC can be obtained

from (5) (6) and (19) as follows

= 1 minus

sum=

1048616 1103925

1048617 (1 minus )( ) minus983086 (20)

Let us define which denotes the total error rate in the

cooperative scheme as

≜ +

=1 + sum=

1048616 1103925

1048617 1038389 (1 minus 1038389 )

minus

minus sum=

1048616 1103925

1048617(1 minus )( ) minus983084 (21)



0 1 2 3 4 5 6 7 8 9 1010

minus4

10minus3

10minus2

10minus1

100

p

T o t a l e

r r o r p r o b a b i l i t y

n=1

n=2

n=3

n=4

n=5

n=6

Fig 1 Total probability of error versus for different number of CRs for1038389 = 10 and SNR=10 dB

It can be seen from (21) that denotes the total error rate in

the case when and are equiprobable

A Optimization of Number of Cooperating CRs

In a cognitive network with very large number of cooper-

ative CRs large delays are incurred in deciding the presence

of the spectrum hole Therefore it is expedient to find an

optimized number of CRs which significantly contribute in

deciding the presence of the PU

Theorem 1 Optimal number of CRs (n) for a fixed value

of the sensing threshold the positive power 1038389 of the

amplitude of received sample of PU and the signal-to-noise

ratio (SNR) between the PU-CR link will be

lowast = min

10486161103925983084lceil

1103925 1 +

rceil1048617983084 (22)

where

=

ln

1radic

98308032 983084

2

22907317

983081

1minus 1radic

98308032 983084

2

2(2+2907317)

983081

ln

1radic

98308032 983084

2

2(2+2907317)

983081

1minus 1radic

98308032 983084

2

22907317

983081983084 (23)

and lceil983086rceil denotes the ceiling function

Proof In order to obtain an optimized number of the CRs

we need to differentiate (21) wrt and set the result equal

to zero to get 1048616 1103925

10486171048667

minus(1 minus )minus 1038389 (1 minus 1038389 )

minus1048669

= 0983086 (24)

After some algebra and with the help of results given in [6]

we get (22) from (24)

0 1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

45

5

p

O p t i m a l n u m b e r o f C R s

λ =10λ =20

λ =50

Fig 2 Optimal number of CRs versus for 1038389 = 10983084 20983084 50 andSNR=10 dB

B Optimization of the Improved Energy Detector

In order to get an optimized value of 1038389 we need todifferentiate (21) wrt 1038389 keeping and SNR fixed and

set the result to zero After some algebra we get

part

part1038389 = minus

sum=

1048616 1103925

1048617(1103925 minus ) 1038389

(1 minus 1038389 ) minusminus1 part 1038389

part1038389

+ sum=

1048616 1103925

1048617 1038389

(1 minus 1038389 ) minus part 1038389

part1038389

minus sum=

1048616 1103925

1048617(1103925 minus )

minusminus1(1 minus )part

part1038389

+ sum=

1048616 1103925

1048617

minus(1 minus )minus1part

part1038389 983084 (25)

where part part and part

part are given as

part 1038389

part1038389 =

3 exp(minus

2

22907317)

radic 2103838923

983084 (26)

part

part1038389 = minus

3 exp(

minus

2

2(2907317

+2

) )

radic 210383892(2 + 2

907317)32

983086 (27)

It is difficult to obtain a closed form solution of 1038389 however we

can obtain an optimized value of 1038389 numerically by using (25)

C Optimization of Sensing Threshold

An optimal value of for given 1038389 and SNR is obtained

by differentiating (21) wrt and setting the derivative equal



0 1 2 3 4 5 6 7 8 9 1010

minus4

10minus3

10minus2

10minus1

SNR (dB)

T o t

a l e r r o r r a t e

p=175

p=200

p=225

p=250

Fig 3 Total error rate versus SNR for = 198308675983084 298308600983084 298308625983084 298308650 1103925 = 1and sensing threshold 1038389 = 50

to zero After some algebra we get

part

part = minus sum=

1048616 1103925

1048617

(1103925 minus ) 1038389 (1 minus 1038389 ) minusminus1part 1038389

part

+ sum=

1048616 1103925

1048617 1038389

(1 minus 1038389 ) minus part 1038389

part

minus sum=

1048616 1103925

1048617(1103925 minus )


part

+ sum=

1048616 1103925

1048617


part 983084 (28)


part can be obtained as follow

part 1038389

part = minus

3minus1 exp(minus

2

22907317)

radic 2103838923

983084 (29)

part

part = minus

3minus1 exp(minus

2

2(2907317+2)

)radic

21038389(2 + 2

907317)32

983086 (30)

The optimized value of can be found numerically from (28)

D Probability of Detection of Spectrum Hole

Probability of detection of spectrum hole in the FC can

be obtain from (20) as follows

=1 minus

=

sum=

1048616 1103925

10486179831311 minus 1radic

9830803

2983084

2

2(2907317 + 2

)

983081983133

times983131

1radic

9830803

2983084

2

2(2907317 + 2

)

983081983133 minus983086 (31)

0 2 4 6 8 10 12 14 16 18 20

10minus0006

10minus0005

10minus0004

10minus0003

10minus0002

10minus0001

SNR (dB)

P r o b a b

i l i t y o f d e t e c t i o n

p=175

p=200

p=225

p=250

Fig 4 Probability of detection versus SNR for = 198308675983084 298308600983084 298308625983084 2983086501103925 = 1 and sensing threshold 1038389 = 50

V NUMERICAL R ESULTS

In Fig 1 we have plotted the total error rate versus 1038389 plotsfor different number of cooperative CRs = 1983084 2983084 3983084 4983084 5983084 6

= 10 and SNR=10 dB It can be seen from Fig 1 that the

total error rate is a convex function of 1038389 for a fixed value of

and SNR It can be seen from Fig 1 that for = 10and SNR=10 dB the total error rate is minimum for = 2

It implies that for = 10 and SNR=10 dB only two CRs

are required for deciding the presence of the PU by using an

improved energy detector We have shown plots of the optimal

number of CRs required for cooperation versus 1038389 for =10983084 20983084 50 and SNR=10 dB in Fig 2 It can be seen from

Fig 2 that the optimal number of the cooperative CRs varies

with the value of 1038389 and at SNR=10 dB For example for

1038389 = 3 the optimal number of the CRs is 2 3 and 4 for = 50983084 20983084 and 10 respectively Fig 3 shows that the total

error rate versus SNR plot with 1038389 = 198308675983084 2983084 298308625983084 298308650 = 1

and = 50 It can be seen from Fig 3 that the total error rate

decreases by increasing the SNR and the value of 1038389 Further it

can be seen from Fig 3 that the improved energy detector with

1038389 = 298308650 significantly outperforms the conventional energy

detector ie when 1038389 = 2 For 1038389 = 198308675983084 2983084 298308625983084 298308650 = 1

and = 50 the probability of detection of a spectrum hole

versus SNR curve is plotted in Fig 4 It is shown in Fig 4

that the probability of detection of a spectrum hole increases

with increase in the value of 1038389

Fig 5 illustrates that for a fixed range of and 1038389 there

exists an optimal number of CRs which minimizes the total

error rate In Fig 6 we have plotted the total error rate versus

1038389 and plots for different number of cooperative CRs at 10 dB

SNR It can be seen from Fig 6 that there exists an optimal

value of 1038389 and for which the total error rate is minimized for

a fixed number of CRs The plots of Figs 5 and 6 suggest that

in order to optimize the overall performance of the cooperative

cognitive system utilizing the improved energy detector we

need to jointly optimize the number of cooperative CRs the

energy detector and the sensing threshold in the CR



Fig 5 Optimal number of CRs versus 1038389 and at SNR=10 dB

VI CONCLUSIONS

We have discussed how to choose an optimal number of

cooperating CRs in order to minimize the total error rate of

a cooperative cognitive communication system by utilizing an

improved energy detector It is shown that the performance of

the cooperative spectrum sensing scheme depends upon the

choice of the power of the absolute value of the received data

sample and the sensing threshold of the cooperating CRs as

well We have obtained an analytical expression of the optimal

number of CRs required for taking the decision of the presence

of the primary user when the CRs utilize the improved energy

detector It is shown by simulations that an optimal value of the

total error rate can be obtained by utilizing a joint optimization

of the number of cooperating CRs threshold in each CR and

the energy detector

APPENDIX IPROOF OF (14) A ND (16)

From (11) and (13) the probability of false alarm 1038389 in

each CR can be obtained as

1038389 =

int infin

983087 0()

=

int infin

radic 2

1minus exp(minus

2

2907317)

1038389radic

983086 (32)

After applying some algebra in (32) we get

1038389 = 1radic

int infin

2

22907317

radic exp(minus)983086 (33)

By comparing (33) with the expression of the incomplete

gamma function [9 Eq (653)] we get (14)

From (12) and (15) the probability of miss in each CR

will be

=

int infin

983087 1()

=

int infin

radic 2

1minus exp(minus

2

2(2907317+2)

)

1038389radic

radic

(2907317 + 2

)983086 (34)

Fig 6 Total error rate versus 1038389 and for different number cooperating users1103925 = 1983084 2983084 3983084 4983084 5983084 6 at SNR=10 dB

After some algebra we get

= 1radic

int

2

2(2+2907317)

0


From (35) and [9 Eq (652)] we get (16)

REFERENCES

[1] S Haykin ldquoCognitive Radio Brain-empowered wireless communica-tionsrdquo IEEE J Sel Areas Commun vol 23 pp 201-220 Feb 2005

[2] G Ganesan and YLi ldquoCooperative spectrum sensing in cognitive radioPart II multiuser networksrdquo IEEE Trans on Wireless Commun vol6pp2214-2222 June 2007

[3] Z Quan S Cui and A H Sayed ldquoOptimal linear cooperation forspectrum sensing in cognitive radio networksrdquo IEEE J Sel Topics inSig Proc vol 2 no 1 pp 28-40 Feb 2008

[4] J Unnikrishnan and V V Veeravalli ldquoCooperative sensing for primarydetection in cognitive radiordquo IEEE J Sel Topics in Sig Proc vol 2 no1 pp 18-27 Feb 2008

[5] E Peh and Y-C Liang ldquoOptimization for cooperative sensing in cog-nitive radio networksrdquo In Proc IEEE Wireless Communications and

Networking Conference (WCNC) pp 27-32 Mar 2007 Hong Kong[6] W Zhang R K Mallik and K B Letaief ldquoCooperative spectrum sensing

optimization in cognitive radio networksrdquo In Proc IEEE InternationalConference on Communication (ICC) pp 3411-3415 May 2008 BeijingChina

[7] H Urkowitz ldquoEnergy detection of unknown deterministic signalsrdquo InProc IEEE vol 55 no 4 pp 523-531 Apr 1967

[8] Y Chen ldquoImproved energy detector for random signals in Gaussiannoiserdquo IEEE Trans Wireless Commun vol 9 no 2 pp 558-563 Feb2010

[9] M Abramowitz and I A Stegun Handbook of Mathematical Functionswith Formulas Graphs and Mathematical Tables 9th ed New York Denver 1970

[10] H L Van Trees ldquoDetection Estimation and Modulation Theoryrdquo NewYork NY Wiley 1968 part 1



cooperative scheme in which each secondary user sends its

binary decision 1103925 (0 or 1) to the fusion center by using an

improved energy detector over error free orthogonal channels

The fusion center combines these binary decisions to find the

presence or absence of the PU as follows

=

sum1103925=1

1103925983084 (4)

where D is the sum of the all 1-bit decisions from the CRs Let

le 1103925 corresponds to a number of cooperating CRs out of

1103925 CRs The FC uses a majority rule for deciding the presence

or absence of the PU As per the majority decision rule if

is greater than then hypothesis ℋ1 holds and if is smaller

than then hypothesis ℋ0 will be true The hypothesis ℋ0

and ℋ1 can be written as

ℋ0 lt 983084 if PU is absent983084 (5)

ℋ1 ge 983084 if PU is present983086 (6)

III PROBABILITY OF FALSE A LARM AND MISSED

DETECTION IN THE C RS

The cumulative distribution function (cdf) of the improved

energy detector can be written as

Pr(∣1103925∣ le ) =Pr(∣1103925∣ le 1 )

=Pr(minus1 le 1103925 le

1 )

=Pr(1103925 le 1 ) minus Pr(1103925 lt minus

1 )983084 (7)

where Pr() denotes the probability For finding the probability

density function (pdf) of the decision statistics we need

to differentiate (8) with respect to (wrt) to get

() = 1

1038389

1minus

1038389(

1 ) +

1

1038389

1minus

1038389(minus

1 )983084 (8)

where 1038389() is the pdf of received signal at the FC under

hypothesis 907317 0 or 907317 1 depending upon the presence or absence

of the primary signal Let 1038389∣11039250() and 1038389∣11039250

() be the

pdfs of the received signal under hypothesis 907317 0 and 907317 1

respectively From (5) and (6) it can be deduced that

1038389∣11039250() =

1radic 22

exp

1048616minus 2

22

1048617983084 (9)

1038389∣11039251() =

1

radic 2(2

907317 + 2)

exp

1048616minus 2

2(2907317 + 2

)

1048617983086 (10)

The pdf of under hypothesis 907317 0 is given from (8) and (9)as follows

∣ 0() =

radic 2

1minus exp(minus

2

2907317)

1038389radic

983086 (11)

From (8) and (10) the pdf of under hypothesis 907317 1 can

be obtained as

∣ 1() =

radic 2

1minus exp(minus

2

2(2907317+2)

)

1038389radic

radic

(2907317 + 2

)983086 (12)

The probability of false alarm 1038389 is defined as [10 Eq (41)

Chapter 2]

1038389 ≜

int 1

∣ 0()983084 (13)

where 1 is the decision region corresponding to hypothesis

907317 1 Let us assume that the decision threshold in each CR is

given as The probability of false alarm 1038389 in each CR is

derived in Appendix I as

1038389 = 1radic

9830803

2983084

2

22

983081983084 (14)

where () is the gamma function [9 Eq (611)]

and (983084 ) is the incomplete gamma function

(983084 )=991787 infin minus1 exp(minus) [9 Eq (653)] The probability

of miss is defined by [10 Eq (41) Chapter 2]

≜

int 0

∣ 1()983084 (15)

where 0 is the decision region corresponding to the hypothe-

sis 907317 0 In Appendix I it is shown that the probability of miss

in each CR will be

= 1radic

9830803

2983084

2

2(2907317 + 2

)

983081983084 (16)

where (983084 ) is the incomplete gamma function

(983084 )=991787 0 minus1 exp(minus) [9 Eq (652)]

IV OPTIMIZATION OF COOPERATIVE SPECTRUM SENSING

SCHEME

The probability of false alarm of the FC for cooperative

sensing will be

= Pr(

ℋ1

∣ℋ0)983084 (17)

From (5) (6) and (17) the probability of false alarm of the

FC can be written as

= sum=

1048616 1103925

1048617 1038389 (1 minus 1038389 )

minus983086 (18)

In the FC the probability of missed detection will be

= Pr(ℋ0∣ℋ1)983086 (19)

The probability of missed detection of the FC can be obtained

from (5) (6) and (19) as follows

= 1 minus

sum=

1048616 1103925

1048617 (1 minus )( ) minus983086 (20)

Let us define which denotes the total error rate in the

cooperative scheme as

≜ +

=1 + sum=

1048616 1103925

1048617 1038389 (1 minus 1038389 )

minus

minus sum=

1048616 1103925

1048617(1 minus )( ) minus983084 (21)



0 1 2 3 4 5 6 7 8 9 1010

minus4

10minus3

10minus2

10minus1

100

p

T o t a l e


n=1

n=2

n=3

n=4

n=5

n=6














lowast = min

10486161103925983084lceil

1103925 1 +

rceil1048617983084 (22)

where

=

ln

1radic

98308032 983084

2

22907317

983081

1minus 1radic

98308032 983084

2

2(2+2907317)

983081

ln

1radic

98308032 983084

2

2(2+2907317)

983081

1minus 1radic

98308032 983084

2

22907317

983081983084 (23)




to zero to get 1048616 1103925

10486171048667


minus1048669

= 0983086 (24)



0 1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

45

5

p


λ =10λ =20

λ =50





part

part1038389 = minus

sum=

1048616 1103925

1048617(1103925 minus ) 1038389


part1038389

+ sum=

1048616 1103925

1048617 1038389

(1 minus 1038389 ) minus part 1038389

part1038389

minus sum=

1048616 1103925

1048617(1103925 minus )


part1038389

+ sum=

1048616 1103925

1048617


part1038389 983084 (25)


part are given as

part 1038389

part1038389 =

3 exp(minus

2

22907317)

radic 2103838923

983084 (26)

part

part1038389 = minus

3 exp(

minus

2

2(2907317

+2

) )

radic 210383892(2 + 2

907317)32

983086 (27)








0 1 2 3 4 5 6 7 8 9 1010

minus4

10minus3

10minus2

10minus1

SNR (dB)

T o t


p=175

p=200

p=225

p=250



part

part = minus sum=

1048616 1103925

1048617


part

+ sum=

1048616 1103925

1048617 1038389

(1 minus 1038389 ) minus part 1038389

part

minus sum=

1048616 1103925

1048617(1103925 minus )


part

+ sum=

1048616 1103925

1048617


part 983084 (28)



part 1038389

part = minus

3minus1 exp(minus

2

22907317)

radic 2103838923

983084 (29)

part

part = minus

3minus1 exp(minus

2

2(2907317+2)

)radic

21038389(2 + 2

907317)32

983086 (30)





=1 minus

=

sum=

1048616 1103925

10486179831311 minus 1radic

9830803

2983084

2

2(2907317 + 2

)

983081983133

times983131

1radic

9830803

2983084

2

2(2907317 + 2

)

983081983133 minus983086 (31)

0 2 4 6 8 10 12 14 16 18 20

10minus0006

10minus0005

10minus0004

10minus0003

10minus0002

10minus0001

SNR (dB)

P r o b a b


p=175

p=200

p=225

p=250






































VI CONCLUSIONS














the energy detector




1038389 =

int infin

983087 0()

=

int infin

radic 2

1minus exp(minus

2

2907317)

1038389radic

983086 (32)


1038389 = 1radic

int infin

2

22907317





will be

=

int infin

983087 1()

=

int infin

radic 2

1minus exp(minus

2

2(2907317+2)

)

1038389radic

radic

(2907317 + 2

)983086 (34)



= 1radic

int

2

2(2+2907317)

0


From (35) and [9 Eq (652)] we get (16)

REFERENCES














0 1 2 3 4 5 6 7 8 9 1010

minus4

10minus3

10minus2

10minus1

100

p

T o t a l e


n=1

n=2

n=3

n=4

n=5

n=6














lowast = min

10486161103925983084lceil

1103925 1 +

rceil1048617983084 (22)

where

=

ln

1radic

98308032 983084

2

22907317

983081

1minus 1radic

98308032 983084

2

2(2+2907317)

983081

ln

1radic

98308032 983084

2

2(2+2907317)

983081

1minus 1radic

98308032 983084

2

22907317

983081983084 (23)




to zero to get 1048616 1103925

10486171048667


minus1048669

= 0983086 (24)



0 1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

45

5

p


λ =10λ =20

λ =50





part

part1038389 = minus

sum=

1048616 1103925

1048617(1103925 minus ) 1038389


part1038389

+ sum=

1048616 1103925

1048617 1038389

(1 minus 1038389 ) minus part 1038389

part1038389

minus sum=

1048616 1103925

1048617(1103925 minus )


part1038389

+ sum=

1048616 1103925

1048617


part1038389 983084 (25)


part are given as

part 1038389

part1038389 =

3 exp(minus

2

22907317)

radic 2103838923

983084 (26)

part

part1038389 = minus

3 exp(

minus

2

2(2907317

+2

) )

radic 210383892(2 + 2

907317)32

983086 (27)








0 1 2 3 4 5 6 7 8 9 1010

minus4

10minus3

10minus2

10minus1

SNR (dB)

T o t


p=175

p=200

p=225

p=250



part

part = minus sum=

1048616 1103925

1048617


part

+ sum=

1048616 1103925

1048617 1038389

(1 minus 1038389 ) minus part 1038389

part

minus sum=

1048616 1103925

1048617(1103925 minus )


part

+ sum=

1048616 1103925

1048617


part 983084 (28)



part 1038389

part = minus

3minus1 exp(minus

2

22907317)

radic 2103838923

983084 (29)

part

part = minus

3minus1 exp(minus

2

2(2907317+2)

)radic

21038389(2 + 2

907317)32

983086 (30)





=1 minus

=

sum=

1048616 1103925

10486179831311 minus 1radic

9830803

2983084

2

2(2907317 + 2

)

983081983133

times983131

1radic

9830803

2983084

2

2(2907317 + 2

)

983081983133 minus983086 (31)

0 2 4 6 8 10 12 14 16 18 20

10minus0006

10minus0005

10minus0004

10minus0003

10minus0002

10minus0001

SNR (dB)

P r o b a b


p=175

p=200

p=225

p=250






































VI CONCLUSIONS














the energy detector




1038389 =

int infin

983087 0()

=

int infin

radic 2

1minus exp(minus

2

2907317)

1038389radic

983086 (32)


1038389 = 1radic

int infin

2

22907317





will be

=

int infin

983087 1()

=

int infin

radic 2

1minus exp(minus

2

2(2907317+2)

)

1038389radic

radic

(2907317 + 2

)983086 (34)



= 1radic

int

2

2(2+2907317)

0


From (35) and [9 Eq (652)] we get (16)

REFERENCES














0 1 2 3 4 5 6 7 8 9 1010

minus4

10minus3

10minus2

10minus1

SNR (dB)

T o t


p=175

p=200

p=225

p=250



part

part = minus sum=

1048616 1103925

1048617


part

+ sum=

1048616 1103925

1048617 1038389

(1 minus 1038389 ) minus part 1038389

part

minus sum=

1048616 1103925

1048617(1103925 minus )


part

+ sum=

1048616 1103925

1048617


part 983084 (28)



part 1038389

part = minus

3minus1 exp(minus

2

22907317)

radic 2103838923

983084 (29)

part

part = minus

3minus1 exp(minus

2

2(2907317+2)

)radic

21038389(2 + 2

907317)32

983086 (30)





=1 minus

=

sum=

1048616 1103925

10486179831311 minus 1radic

9830803

2983084

2

2(2907317 + 2

)

983081983133

times983131

1radic

9830803

2983084

2

2(2907317 + 2

)

983081983133 minus983086 (31)

0 2 4 6 8 10 12 14 16 18 20

10minus0006

10minus0005

10minus0004

10minus0003

10minus0002

10minus0001

SNR (dB)

P r o b a b


p=175

p=200

p=225

p=250






































VI CONCLUSIONS














the energy detector




1038389 =

int infin

983087 0()

=

int infin

radic 2

1minus exp(minus

2

2907317)

1038389radic

983086 (32)


1038389 = 1radic

int infin

2

22907317





will be

=

int infin

983087 1()

=

int infin

radic 2

1minus exp(minus

2

2(2907317+2)

)

1038389radic

radic

(2907317 + 2

)983086 (34)



= 1radic

int

2

2(2+2907317)

0


From (35) and [9 Eq (652)] we get (16)

REFERENCES















VI CONCLUSIONS














the energy detector




1038389 =

int infin

983087 0()

=

int infin

radic 2

1minus exp(minus

2

2907317)

1038389radic

983086 (32)


1038389 = 1radic

int infin

2

22907317





will be

=

int infin

983087 1()

=

int infin

radic 2

1minus exp(minus

2

2(2907317+2)

)

1038389radic

radic

(2907317 + 2

)983086 (34)



= 1radic

int

2

2(2+2907317)

0


From (35) and [9 Eq (652)] we get (16)

REFERENCES












cooperative sensing specrum

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