analysis of equal gain combining in energy detection for cognitive radio over nakagami channels

8/2/2019 Analysis of Equal Gain Combining in Energy Detection for Cognitive Radio Over Nakagami Channels

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Analysis of Equal Gain Combining in Energy

Detection for Cognitive Radio over Nakagami

Channels

Sanjeewa P. Herath, Nandana Rajatheva

Telecommunications Field of Study, School of Engineering and Technology,

Asian Institute of Technology, Klong Luang, Pathumthani 12120, Thailand

[email protected], [email protected]

AbstractThis paper addresses the problem of energy detec-tion of unknown deterministic signal of a primary user in acognitive radio environment. As an extension to the previousworks, we focus on equal gain combining technique when thewireless channel is modeled as Nakagami-m. We derive seriesform exact expressions for probability of detection and falsealarm when the number of diversity branches are 1, 2, 3 andL 4. Finally, performance variation is shown against thenumber of diversity branches and the time bandwidth productin decision statistic with the aid of numerical results.

I. INTRODUCTION

Detection of spectrum holes in cognitive radio environment

is a problem of significant interest. To best utilize the spectrum

holes by secondary users, the detection of such spectrum holes

quickly and accurately is of great importance. Mathematically

speaking, achieving higher probability of detection with very

low probability of false alarm by means of few number of

samples is the objective of a secondary user system designer.

Detection of a primary user while a secondary user experi-

ences a deep fade is also expected and hence the detectingmechanism should be able to tolerate the low signal to noise

ratio (SNR) conditions. Since the secondary user has very

little knowledge about the primary user, the detection of an

unknown deterministic signal of that user by means of an

energy detector is a proposed mechanism for a secondary user

to identify the presence of primary user.

The detection of an unknown deterministic signal in the

presence of noise through an energy detector is first studied

by Urkowitz [1]. In his original work, the receiver consists of

an energy detector which measures the energy of the received

signal over a period of time. Then the detection problem

reduces to a binary hypothesis test with the decision variable

having statistics of Chi-square (2) distribution under eachhypothesis [1]. Further, by means of the lowpass equivalent

representation of a bandpass process, statistics of the decision

variable of bandpass signal are also shown to follow the 2

distribution.

Kostylev [2] further analyzes the problem formulated in [1]

and extends the work for Rayleigh, Rician and Nakagami

fading channels. In his work, closed form expression for

probability of false alarm (Pf) and probability of detection(Pd) are derived for Rayleigh fading channel. However, the

calculation of Pd requires numerical integration for Nakagami

fading channel and an infinite summation for the Rician fading

channel.

An alternative approach for calculating Pd for aforemen-

tioned fading channels is presented in [3] and closed form

expressions are derived for Pd under each fading environment.

Further, derivations in [3] are cross checked with the work in[2]. More importantly, closed form expressions for Pd when

diversity techniques are used and channel is under Rayleigh

fading is introduced in [3] including maximal ratio, selection

and switch and stay combining. Recent results in [4] present

closed form expressions for Pd and Pf over AWGN channels

and Nakagami Fading channel. In this work, closed form

expressions for Pd and Pf are obtained for both correlated and

i.i.d. Rayleigh fading channels when Square Law Combining

(SLC) technique is used. Expressions for Pd and Pf over

Rayleigh fading channel for Square Law Selection (SLS)

combining is also presented therein. Authors in [5] derive the

expressions for Pd and Pf when maximum ratio and selection

processing diversity combining is used.In this paper, we discuss the same detection problem deriv-

ing series form expression for Pd and Pf when the wireless

channel is modeled by Nakagami-m fading and an equal

gain combining receiver, assuming the statistics of diversity

branches to be i.i.d. Our work considers the number of i.i.d.

branches (L) 1,2,3 and L 4. Finally numerical results arepresented showing the variation of Pd against fixed value of

Pf over various parameters of interest.

I I . DETECTION OVER AWGN CHANNEL

To be consistent with the previous works and to minimize

the consequences of different notations, we follow the nota-tions similar to [3] given below.

s(t) : Unknown deterministic signal waveformn(t) : Noise waveform - White Gaussian random processr(t) : Received signalT : Observation time interval

W : One sided bandwidth

u = T W: Time bandwidth productN01 : One sided noise power spectral density

Es : Signal energy over the interval T

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedin

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: Energy threshold of the receiver

L : Number of branches of the receiver combiner

H0 : Hypothesis 0; no s(t) presentH1 : Hypothesis 1; s(t) present2 : Central Chi-square distribution with

degrees freedom

2() : Non central Chi-square distribution with degrees freedom and non centrality parameter

Energy detector decision is a result of binary hypothesis

test. The test can be written as below [1].

y(t) =

n(t) : H0s(t) + n(t) : H1

Hence, the decision statistic Y, can be represented as 22uunder H0 and

22u() under H1 as given below,

fY(y) =

1

2u(u)yu1e

y2 : H0

12(

y2 )

u12 e

2+y2 Iu1(

2y) : H1

where (.) is gamma function and I(.) is the th ordermodified Bessel function of the first kind [1],[3]. Hence, Pdand Pf over AWGN channel can be given as in (1) and (2),

Pd = Qu

2,

(1)

Pf =

u, 2

(u)

(2)

where Qu(., .) is the generalized Marcum Q-function and(., .) is the incomplete gamma function [3].

III . AVERAGE DETECTION PROBABILITY (Pd) OVERNAKAGAMI-m FADING CHANNEL WITH EQUAL GAI N

COMBINING

If the number of branches in the equal gain combiner is

L, then the combined signal amplitude is given by [8, eqn.

6.32, pp.285],

=Li=1

i. (3)

The SNR of the composite signal () is defined by

= Es2

LN01[8, eqn.6.33, pp.285]. This follows, average SNR

per branch E() = =EsN01 where = E(

2i ) and E(.) is

the mathematical expectation.

By means of the alternative representation of Marcum Q-

function given in [7, eqn.4.63, pp.101], we can write

Qu (a

, b) = 1 ea2+ b2

2

n=u

b

a

nIn (ab

) (4)

where a =

2 and b =

. Hence, the probability of detec-

tion when i number of branches are used (Pd,i, i = 1, 2, . . ,L)

can be calculated averaging (4) over the probability density

function (PDF) of ( 0) i.e. gi(),

Pd,i =

0

Qu(a

, b) gi() d

= 1 eb2

2

n=u

b

a

n 0

n2 e

a22 In (ab

) gi()d

(5)

where we have used the fact that

0gi()d = 1.

A. Single Branch (L = 1)

When the received signal follows Nakagami-m distribution,

PDF of the received signal amplitude is given by,

f() =2mm2m1

(m)me

m2

, 0 (6)

where (.) is gamma function and m is fading figure. Hence,g1() can be written as in (7).

g1() =1

(m)

m

mm1e

m , 0 (7)

By (5) and (7), Pd,1 can be expressed as,

Pd,1 = 1e

b2

2

(m)

m

m n=u

b

a

n

0

em1n2 In(ab

)d

(8)

where = 1 + m

. Using [9, eqn.6.643-2, pp.709], we can

show Pd,1 for m > 0 as in (9),

Pd,1 = 1 e2

m

m n=u

2

n1

(n + 1)

1F1m; n + 1; 2(9)

where we have use the relation given in (10),

Mk,(x) = x+ 1

2 ex2 1F1

1

2+ k; 1 + 2; x

. (10)

Here, Mk,(.) is the Whittaker function and 1F1(., .; .) is theconfluent hypergeometric function of the first kind.

This result is numerically equivalent to the result [3,

eqn.(20)] for integer m. Since m is not restricted to integers in

(9), this can be treated as a more general result for Nakagami-

m fading channel compared to the result in [3, eqn.(20)].

B. Two i.i.d. branches (L = 2)

When the received signal follows Nakagami-m distribution,PDF of combined signal amplitude is given by [6, eqn.(4)].

Hence by following the same procedure for (7), g2() can bederived as given in (11),

g2() =2

2m1emL

24m1(2m)

2(m)(2m + 12)

mL

2m

1F1

2m; 2m +1

2;

mL

2

, 0.

(11)


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By (5) and (11), Pd,2 can be expressed as,

Pd,2 = 1 2(m)eb2

2

L

2m n=u

b

a

n 0

2mn21

emL+ a

2

2

In(ab

)1F1

2m; 2m +

1

2;

mL

2

d

(12)

where 2(m) defined by (13),

2(m) =2

(2m)m2m

2(m)(2m + 12)24m1

. (13)

The hypergeometric series expansion of the form 1F1(.; .; .) isgiven by (14) [10, pp.202],

1F1(a; b; x) =k=0

(a)k xk

(b)k k!. (14)

Here, (.)k is Pochhammer symbols [10, pp.202] which isdefined by (15).

(a)k = (a + k)(k)

(15)

Using [9, eqn.6.643-2, pp.709], (12),(13),(14) and (15), Pd,2for m > 0 can be evaluated as,

Pd,2 = 1

e2

n=u

k=0

2

nmL

2m+k

2(m,n,k)24m+k2 k!

1F1(2m + k; n + 1;

2)

(16)

where 2(m,n,k) is defined by (17) and = 1 +mL

.

2(m,n,k) = 2(2m + k)

2(m)(n + 1)(2m + k + 12)(17)

C. Three i.i.d. branches (L = 3)

When the received signal follows Nakagami-m distribution,

PDF of combined signal amplitude is given by, [6, eqn.(8)] and

by following the same procedure used for (7), g3(), 0can be expressed as,

g3() =4

(2m)emL

3(m)24m1

n=0

(2m + n)

(2m + n + 12)

(4m + 2n) 3m+n1

(6m + 2n)(n + 1) 2nmL

3m+n

2F2

2m, 4m + 2n; 3m + n +1

2, 3m + n;

mL

2

.

(18)

The Hypergeometric series expansion of generalized hyper-

geometric function [10, pp.202] is given in (19).

pFq(a1,...,ap; b1,...,bq; x) =

n=0

(a1)n...(ap)n(b1)n...(bp)n

xn

n!(19)

Using the form of 2F2(a1, a2; b1, b2; x) in (19) and using (5)and (18), Pd,3 can be shown as in (20),

Pd,3 = 13(m)eb2

2

n=u

p=0

k=0

b

a

n m2

p+k L

3m+p+

(2m +p)(4m + 2p)(6m + 2p)(p + 1)(2m +p + 12)

(2m)k (4m + 2p)k(3m +p)k

3m +p + 12

k

k!

0

3m+p+kn21 e

mL+ a

2

2

In(ab

)d

(20)

where 3(m) is defined by (21),

3(m) =4

(2m)m3m

3(m)24m1. (21)

Using [9, eqn.6.643-2, pp.709] and the identity given in (10),

Pd,3 for m > 0 can be computed as,

Pd,3 = 1e2

n=u

p=0

k=0

2n

mL

3m+p+k

3(m,n,p,k)24m+p+k3 k!

1F1

3m +p + k; n + 1;

2

(22)

where 3(m,n,p,k) is defined by (23) below and = 1+mL

.

3(m,n,p,k) =(2m +p)(2m + k)(3m +p)

3(m)(n + 1)(p + 1)(2m +p + 12)

(3m +p +12)(4m + 2p + k)

(3m +p + k + 12)(6m + 2p)(23)

IV. FOUR OR MORE I.I.D. BRANCHES (L 4)When the received signal follows Nakagami-m distribution,

PDF of combined signal amplitude of four branch EGC is

given by [6, eqn.(9)] and by following the same procedure used

for (7), g4() can be expressed as given in (24). Following asimilar procedure in deriving (22), Pd,4 for m > 0 can beevaluated as given in (25) where 4(m ,n,p ,q ,k) is definedby (26) and = 1 + mL

.

The PDF of combined signal amplitude of Nakagami-m

faded L 4 branch EGC is given in [6, eqn.(10)]. Byfollowing the same line of arguments for (7), gL() can be

evaluated as given in (27). Following similar procedure inderiving (22), Pd,L for m > 0 can be evaluated as given in(28) where = 1 + mL

. Replacing k1 by p, k2 by q, p by k

and L = 4 in (28), we can easily verify (25).Probability of false alarm depends only on the hypothesis

H0. Hence the statistics of each sample taken from combined

signal follows central Chi-square distribution and further de-

cision variable also follows the central Chi-square distribution

with 2u degree freedom (22u) [1],[3]. This leads to the samePf,i for i = 1, 2,...,L as given by (2).


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g4() =8

(2m)m4memL

4(m)24m1

p=0

q=0

(2m +p)(2m + q)(4m + 2p + q)

(p + 1)(q + 1)(2m +p + 12)(8m + 2p + 2q)4m+p+q1 2qp mp+q

L

4m+p+q2F2(2m, 6m + 2p + 2q; 4m +p + q +

1

2, 4m +p + q;

mL

2)

(24)

Pd,4 = 1

e2

n=u

p=0

q=0

k=0

2

nmL

4m+p+q+k4(m ,n,p ,q ,k)

24m+p+kq4 k!1F1

4m +p + q + k; n + 1;

2

(25)

4(m ,n,p ,q ,k) =(2m +p)(2m + q)(2m + k)(4m +p + q)(4m + 2p + q)(4m +p + q + 12)(6m + 2p + 2q + k)

4(m)(q + 1)(p + 1)(n + 1)(2m +p + 12)(4m +p + q + k +12)(6m + 2p + 2q)(8m + 2p + 2q)

(26)

gL() =2L1

(2m) mLm e

mL

24m1 L(m)

k1=0

k2=0

...

kL2

L2i=1

(2m + ki)

(1 + ki)

(4m + 2k1 + k2) (6m + 2k1 + 2k2 + k3) ...

2m(L 2) + 2L3i=1 ki + kL22m(L 1) + 2L2i=1 ki

(2m + k1 +12) (6m + 2k1 + 2k2) (8m + 2k1 + 2k2 + 2k3) ...

2Lm + 2

L2i=1 ki

2m(L 2) + 2L3i=1

ki + kL2

m(

L2i=1 ki)2(

L2i=2 kik1)(Lm+

L2i=1 ki1)

L

(Lm+L2i=1 ki)

2F2

2m, 2m(L 1) + 2 L

2i=1

ki; Lm +L

2i=1

ki +12

, Lm +L

2i=1

ki;mL

2

, L 4

(27)

Pd,L = 1

e2

n=u

k1=0

k2=0

...

kL2

p=0

L2i=1

(2m + ki)

(1 + ki)

(4m + 2k1 + k2)(6m + 2k1 + 2k2 + k3) ...

2m(L 3) + 2L4i=1 ki + kL32 2m(L 2) + 2L3i=1 ki + kL2

(2m + k1 +12)(6m + 2k1 + 2k2)(8m + 2k1 + 2k2 + 2k3) ... 2Lm + 2L2i=1 ki

(2m +p)

Lm +

L2i=1 ki

Lm +L2

i=1 ki +12

2m(L 1) + 2L2i=1 ki +pL(m) (n + 1)

mL +

L2i=1 ki +p +

12

2

n mL

(Lm+L2i=1 ki+p)2(4m+k1

L2i=2 kiL+p) p!

1F1

p + Lm +

L2i=1

ki; n + 1;

2

, L 4

(28)


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0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Average SNR(dB)

Pd

Fading m=1.0

EGC L=2 m=0.5

EGC L=2 m=1.0

EGC L=2 m=1.5

EGC L=2 m=2.0

EGC L=2 m=2.5

EGC L=2 m=3.0

Fig. 1. Pd,i for i=1,2 when Pf = 0.001 and u = 1.

V. NUMERICAL RESULTS

We describe the performance of the receiver by means of

Pd,i vs. SNR curves for pre-specified Pf,i value. First, energy

threshold of the receiver () is calculated for the specified Pf,ivalue. Then for that , Pd,i is evaluated.

Fig.1 shows the Pd,i performance improvement of dual

branch EGC receiver over different values of m when u = 1and Pf,i = 0.001. As it is expected, higher values of mand SNR result in better performance. However, the highest

Pd,i improvement is observed over lower range of m values.

Fig.2 plots the similar analysis including the three branch EGC

receiver.

Fig.3 shows the Pd,i performance variation for u = 1, 4

when m = 2, L = 1, 2, 3 and Pf,i = 0.01. This plot shows theperformance reduction for higher value of u in each case, i.e.no-diversity fading channel and L = 2, 3. Comparing suitablecurves of Fig.1, Fig.2 and Fig.3, it is possible to observe the

performance reduction of Pd,i for higher Pf,i requirement.

However, the SNR requirement for higher detection probability

is moderate in all cases.

VI . CONCLUSION

We consider the problem of primary user detection in

cognitive radio over the Nakagami-m fading channel with

the equal gain combining diversity receiver. Expressions are

derived for exact probability of detection when the number

of diversity branches are 1, 2, 3 and L 4. Interestingly,all the expressions could be expressed in terms of confluenthypergeometric function of the first kind 1F1(.; .; .). Theseresults could be readily used in deciding the number of

diversity branches and the energy threshold value to achieve

a specified false alarm rate of equal gain combiner energy

detector receiver in cognitive radio.

REFERENCES

[1] H. Urkowitz, Energy detection of Unknown Deterministic Signals,Proc IEEE, vol. 55, no. 4, pp.523-531,Apr. 1967.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Average SNR(dB)

Pd

Fading u=1 m=1

Fading u=1 m=3

EGC L=2 u=1 m=1

EGC L=2 u=1 m=3

EGC L=3 u=1 m=1

EGC L=3 u=1 m=3

Fig. 2. Pd,i for i=1,2,3 when Pf = 0.01 and u = 1.

0 5 10 15 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Average SNR(dB)

Pd

Fading u=1 m=2

Fading u=4 m=2

EGC L=2 u=1 m=2

EGC L=2 u=4 m=2

EGC L=3 u=1 m=2

EGC L=3 u=4 m=2

Fig. 3. Pd,i for i = 1, 2, 3 when Pf = 0.01.

[2] V.I. Kostylev, Energy detection of a signal with random amplitude,IEEE Int. Conf. ICC 2002, vol 3, pp.1606 - 1610, Apr-May 2002.

[3] Fadel F. Digham, Mohamed-Slim Alouni and Marvin K. Simon, On theEnergy Detection of Unknown Signals Over Fading Channels, IEEE Int.Conf. ICC03, vol 5, pp. 3575 - 3579, May 2003.

[4] Fadel F. Digham, Mohamed-Slim Alouni and Marvin K. Simon, On theEnergy Detection of Unknown Signals Over Fading Channels, IEEETrans. Commun., vol 55, no.1,pp.21-24, Jan. 2007.

[5] Ashish Pandharipande and Jean-Paul M.G.Linnartz, Performance analy-sis of primary user detection in a multiple antenna cognitive radio,

IEEE Int. Conf. ICC 07, pp. 6482 - 6486, June 2007.[6] Prathapasinghe Dharmawansa, Nandana Rajatheva and Kazi

Ahmed, On the distribution of the sum of Nakagami-m randomvariable, IEEE trans. on commun., vol. 55, no. 7, pp.1407-1416, july2007.

[7] Marvin K.Simon and Mohamed-Slim Alouni, Digital CommunicationOver Fading Channels, 2nd Edition.

[8] Gordon L. Stuber Principles of Mobile Communication, 2nd Edition,Kluwer Academic Publishers.

[9] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, andProducts, 7th edition, 2007.

[10] Harry Bateman, Higher Transcendental Functions, Volume I, 1953.


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analysis of equal gain combining in energy detection for cognitive radio over nakagami channels

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