On the Energy Detection of Unknown Signalsover Fading Channels+

Fadel F. Digham, Mohamed-Slim Alouini, and Marvin K. SimonDepartment of Electrical and Computer Engineering

University of MinnesotaMinneapolis, MN 55455, USA

E-mails: Jet Propulsion Laboratory (JPL)

California Institute of Technology (Caltech)Pasadena, CA 91109-8099, USA.

E-mail:

Abstract This paper presents another look at the problemof energy detection of unknown signals over different fadingchannels. We start with the no diversity case and present somealternative closed-form expressions for the probability of detec-tion (Pd) to those recently reported in [1]. We then investigatethe system performance when different diversity schemes areemployed. It is shown that there is not much improvementin the probability of detection when either the probability offalse alarm (Pf ) exceeds 0.1 or the average signal-to-noiseratio (SNR) exceeds 20 dB. In addition, receiver operatingcharacteristic (ROC) curves comparing the performance of equal-gain combining (EGC), selection combining (SC), and switchand stay combining (SSC) are presented. As an example, EGCintroduces a gain of two orders of magnitude from the probabilityof miss perspective compared to the no diversity case while bothSC and SSC introduce a gain of about one order of magnitude.

I. INTRODUCTIONIn many wireless applications, it is of great interest to check

the presence and availability of an active communication linkwhen the transmitted signal is unknown. In such scenarios,one appropriate choice consists of using an energy detectorwhich measures the energy in the received waveform over anobservation time window. This interesting problem was studiedin the classic paper by Urkowitz [2] assuming deterministicsignals transmission over a flat band-limited Gaussian noisechannel. More specifically, relying on the sampling theoremto approximate the received signal energy and on chi-squarestatistics of the resulting sum of squared Gaussian randomvariables, Urkowitz derived both the probability of detection(Pd) and the probability of false alarm (Pf ). The receiver op-erating characteristic (ROC) [3] was then obtained by plottingPd versus Pf for different system parameters of interest.

This energy detection problem has been revisited recentlyby Kostylev [1] for signals operating over a variety of fadingchannels. While the probability of detection has been derivedin closed-form for Rayleigh fading channels, the results in-volve a numerical integration and an infinite summation forNakagami and Rician fading channels, respectively. In this

+This work was supported by the ARL Communications and NetworksCTA, under cooperative agreement DAAD19-01-2-0011.

paper, we have another look at the problem. Our contributionis two-fold. First, we present an alternative analytical approachto the one presented in [1] and obtain closed-form expressionsfor the probability of detection not only over Rayleigh fadingbut also over Nakagami and Rician fading channels. Second,and more importantly, we quantify the improvement in theprobability of detection when equal gain combining (EGC),selection combining (SC), and switch and stay combining(SSC) diversity schemes are used.

The rest of this paper is organized as follows. In Section II,the system model under consideration is described and impor-tant notations are listed. Section III evaluates the conditional(on the fading) probabilities of detection and of false alarm(or equivalently Pd and Pf over additive white Gaussian noise(AWGN) channels). While Section IV deduces the detectionprobability over various fading channels, Section V studiesthe impact of diversity on this detection probability. Finally,numerical examples and concluding remarks are offered inSection VI.

II. SYSTEM MODEL AND NOTATIONSBefore describing the system model, we first list the main

notations that are going to be used in this paper for additionalclarity and to avoid any kind of confusion when going backto [2] and [1]. s(t) : signal waveform. n(t) : noise waveform which is modelled as a zero-mean

white Gaussian random process. N01 : one-sided noise power spectral density, i.e., N01

N0 in [1]. N02 = N012 : two-sided noise power spectral density. Es : signal energy=

T0 s

2(t)dt. = EsN01 : signal-to-noise ratio (SNR), i.e., 2 in [2]. : average SNR, i.e., d2 in [1]. : energy threshold used by the energy detector. T : observation time interval, seconds. W : one-sided bandwidth (Hz), i.e. positive bandwidth of

the low-pass (LP) signal. u=TW : time bandwidth product, i.e., u B2 1 in [1].

35750-7803-7802-4/03/$17.00 2003 IEEE

fc : carrier frequency. Pd : probability of detection. Pf : probability of false alarm. Pm = 1 Pd : probability of missing. H0 : hypothesis 0 corresponding to no signal transmitted. H1 hypothesis 1 corresponding to signal transmitted. N(, 2) : a Gaussian variate with mean and variance

2. 2 : a central chi-square variate with degrees of

freedom. 2() : a noncentral chi-square variate with degrees

of freedom and noncentrality parameter .The received signal r(t) takes the form

r(t) = h s(t) + n(t), (1)where h=0 or 1 under hypotheses H0 or H1, respectively. Asdescribed in [2], the received signal is first pre-filtered by anideal bandpass filter with transfer function

H(f) ={ 2

N01, |f fc| W,

0, |f fc| > W, (2)

to limit the average noise power and normalize the noisevariance. The output of this filter is then squared and integratedover a time interval T to finally produce a measure of theenergy of the received waveform. The output of the integratordenoted by Y will act as the test statistic to test the twohypotheses H0 and H1. Although this process is of band-passtype, one can still deal with its low-pass equivalent form andeventually translate it back to its band-pass type [4]. Besides,it has been verified in [2] that both low-pass and band-passprocesses are equivalent from the decision statistics perspec-tive which is our main concern. Therefore, for convenience,we address in this paper the problem for a low-pass process.According to the sampling theorem, the noise process can beexpressed as [5]

n(t) =

i=ni sinc(2Wt i), (3)

where sinc(x) = sin(pix)pix and ni = n(

i2W

). One can easily

check that

ni N(0, N01W ), for all i. (4)Over the time interval (0, T ), the noise energy can be approx-imated as [2] T

0n2(t) dt =

12W

2ui=1

n2i , (5)

where u = TW . We assume that T and W are chosen torestrict u to integer values. If we define

ni =niN01W

, (6)

then, the test or decision statistic Y can be written as [2]

Y =2ui=1

ni2. (7)

Y can be viewed as the sum of the squares of 2u standardGaussian variates with zero mean and unit variance. Therefore,Y follows a central chi-square (2) distribution with 2udegrees of freedom.

The same approach is applied when the signal s(t) ispresent with the replacement of each ni by ni + si wheresi = s

(i

2W

). The decision statistic Y in this case will have

a noncentral 2 distribution with 2u degrees of freedom anda non centrality parameter 2 [2]. Following the short-handnotations mentioned in the beginning of this section, we candescribe the decision statistic as

Y {

22u, H0,22u(2), H1.

(8)

The probability density function (PDF) of Y can then bewritten as

fY (y) =

1

2u(u)yu1e

y2 , H0,

12

(y2

)u12

e2+y

2 Iu1(2y), H1,

(9)

where (.) is the gamma function [6, Section 8.31] and I(.)is the th-order modified Bessel function of the first kind [6,Section 8.43].

III. DETECTION AND FALSE ALARM PROBABILITIES OVERAWGN CHANNELS

An approximate expression for Pd over AWGN channelswas presented in [2]. In this section we present exact closed-form expressions for both Pd and Pf .

The probability of detection and false alarm can be generallycomputed by

Pd = Pr(Y > |H1), (10)

Pf = Pr(Y > |H0), (11)where is the decision threshold. Using (9) to evaluate (11)yields

Pf =(u, 2

)(u)

, (12)

where (., .) is the incomplete gamma function [6]. This resultmatches the one obtained in [1, (19)] after replacing each Bby B/21 since each B should be equal to 2u in our notation.Furthermore, the parameter in [1] should be equal to B orequivalently 2u.

On the other hand, the probability of detection can beobtained from (9) to evaluate (10). More specifically makinguse of [4, Eq. (2.1-124)], the cumulative distribution function(CDF) of Y can be evaluated (for an even number of degreesof freedom which is 2u in our case) as

FY (y) = 1Qu(

2,y), (13)

where Qu(a, b) is the generalized Marcum Q-function [7].Hence,

Pd = Qu(

2,). (14)

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IV. AVERAGE DETECTION PROBABILITY OVER FADINGCHANNELS WITH NO DIVERSITY

In this section, we derive the average detection probabilityover Rayleigh, Nakagami, and Rician fading channels. Weprovide alternative expressions to those obtained in [1]. Ourexpressions are in closed form and are based on a differentapproach by averaging the conditional Pd in the AWGN caseas given by (14) over the SNR fading distribution. Of course,Pf of (12) will remain the same under any fading channelsince Pf is considered for the case of no signal transmissionand as such is independent of SNR.

A. Rayleigh ChannelsIf the signal amplitude follows a Rayleigh distribution, then

the SNR follows an exponential PDF given by

f() =1exp

(

), 0, (15)

The average Pd in this case, PdRay , can now be evaluated byaveraging (14) over (15) while making the change of variablex =

2 and making use of [7, Eq. (30)] yielding

PdRay= e2

u2n=0

1n!

(

2

)n+(1 +

)u1[e

2(1+) e2

u2n=0

1n!

2(1 + )

].(16)

This expression is equivalent to that given in [1, Eq. (20)]taking into account the following observations and correctionsin [1, Eq. (20)]: the normalized incomplete gamma functionP (a, b) = (a, b)/(a) can be expressed in its series form [6,Eq. (8.352.1)], setting d2 = , and correcting two typos: eachB should be replaced by B/2 1 and a minus sign in theexponential argument should be inserted.

B. Nakagami ChannelsIf the signal amplitude follows a Nakagami distribution,

then the PDF of follows a gamma PDF given by

f() =1

(m)

(m

)mm1 exp

(m

), 0, (17)

where m is the Nakagami parameter. The average Pd in thecase of Nakagami channels PdNak can now be obtained byaveraging (14) over (17) and then using again the change ofvariable x =

2 yielding

PdNak = 0

x2m1 exp(mx

2

2

)Qu

(x,)dx, (18)

where

=1

(m)2m1

(m

)m. (19)

Evaluating the integral in (18) as described in Appendix A,PdNak can be written as

PdNak=

[G1+

u1n=1

(/2)n

2(n!) 1F1

(m;n+ 1;

2

m+

)],(20)

where 1F1(.; .; .) is the confluent hypergeometric function ((., .; .)) [6, Section 9.2],

= (m)(

2m+

)me/2, (21)

and

G1 = 0

x2m1 exp(mx

2

2

)Q(x,)dx, (22)

where Q(., .)=Q1(., .) is the first-order Marcum Q-function.G1 can be evaluated for integer m with the aid of [7, Eq. (25)]as

G1=2m1(m 1)!(

m

)m m+ e2 mm+[(

1 +m

)(m

m+

)m1

Lm1(2

m+

)+m2n=0

(m

m+

)nLn

(2

m+

)],

(23)where Ln(.) is the Laguerre polynomial of degree n [6,8.970]. As a byproduct, we obtain an alternative expressionfor PdRay when setting m = 1 in (20) and this expressionis numerically equivalent to the one obtained in (16). Whencomparing our expression in (20) with the one including anumerical integration in [1, Eq. (33)], it seems that there is anerror in the Kostylevs expression since it does not reduce tothe special case of Rayleigh. Moreover, it is equal to zero forthe special case of u = 1 (equivalently B = 0) and m = 1.

C. Rician ChannelIf the signal strength follows a Rician distribution, the PDF

of will be

f()=K + 1

exp

(K (K + 1)

)I0

(2

K(K + 1)

),

0, (24)where K is the Rician factor. The average Pd in the case ofa Rician channel, PdRic, is then obtained by averaging (14)over (24) and substituting x for 2. The resulting expressioncan be solved for u = 1 using [8, Eq. (45)] to yield

PdRic|u=1 = Q(

2KK + 1 +

,

(K + 1)K + 1 +

). (25)

For K = 0, this expression reduces to the Rayleigh expressionwith u = 1.

V. AVERAGE DETECTION PROBABILITY WITH DIVERSITYRECEPTION

In this section, we address the energy detection performancewhen EGC, SC, or dual SSC diversity schemes are employed.For simplicity, we focus on the case in which the diversitypaths are independent and identically distributed (IID) and aresubject to Rayleigh fading.

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A. Equal Gain CombiningThe output SNR, t, of the EGC combiner is the sum of

the SNRs on all branches, i.e., t =L

l=1 l where L isthe number of diversity branches. Adding L IID noncentral2 variates with 2u degrees of freedom and non-centralityparameter 2l each results in another noncentral 2 variatewith 2Lu degrees of freedom and non-centrality parameterL

l=1 2l [4, Eq. (2.1-117)]. Hence, the Pd at the EGC outputfor AWGN channels can be evaluated by analogy to (14) as

PdEGC = QLu(

2t,). (26)

The PDF of t for IID Rayleigh branches is known to be givenby

f(t) =1

(L 1)!L L1t exp (t/). (27)

The average Pd for the EGC diversity scheme, PdEGC , canthen be obtained by averaging (26) over (27). One can noticethat the PDF in (27) is similar to that in (17) when replacingeach m by L and each by L. This is intuitively correct sincethe Nakagami parameter m can be viewed as a diversity order.Hence, PdEGC is equivalent to PdNak in (20) after replacingeach m by L, each by L, and each u by Lu.

B. Selection CombiningIn the SC diversity scheme, the branch with maximum SNR,

max, is to be selected. The PDF of max for IID Rayleighbranches is known to be given by

fmax() =L

(1 e/

)L1e/ . (28)

This PDF can be rewritten as

fmax() = LL1i=0

(1)ii+ 1

(L 1

i

)1

/(i+ 1)e

/(i+1) . (29)

The PDF in (29) represents a weighted sum of exponentialvariates each with parameter i+1 . Hence, the average Pd forthe SC diversity scheme, PdSC , can be evaluated as

PdSC = LL1i=0

(1)ii+ 1

(L 1

i

)PdRay

(

i+ 1

), (30)

where PdRay(

i+1

)is the PdRay obtained in (16) with the

replacement of each by i+1 .

C. Switch and Stay CombiningWe address in this section the evaluation of the average Pd

for the dual SSC diversity scheme [9]. The PDF of the SNR atthe output of the SSC with dual IID Rayleigh branches, SSC ,is [9]

fssc() =

{1 e/ (1 eT /) , < T ,

1 e/ (2 eT /) , T , (31)

where T is the switching threshold. The average Pd forthe dual SSC diversity scheme, PdSSC , can be obtained byaveraging (14) over the PDF in (31) yielding

PdSSC =(1 eT /

)PdRay

+ T

Qu(

2,)

1e/d, (32)

where PdRay is given in (16). Relying one more time on thechange of variable x =

2 in the integral part of (32) and

making use of [7, Eq. (32)], PdSSC can be obtained in closedform as

PdSSC =(1 eT /

)PdRay + e

T /Qu(

2T ,)

+(1 +

)u1e

2(1+)

[1Qu

(2T

1 +

,

1 +

)]. (33)

The optimal threshold T which maximizes the PdSSC can beobtained by solving PdSSCT = 0 in (32) yielding

T =12

[Q1u (PdRay,

)]2

, (34)where Q1u (., .) denotes the inverse uth-order Marcum Q-function with respect to its first argument.

VI. NUMERICAL EXAMPLES AND CONCLUSIONSWe are interested in describing the receiver performance

through its ROC curves (Pd versus Pf ) or complementaryROC curves (Pm versus Pf ) for different situations of interest.Figure 1 illustrates the complementary ROC over a Rayleighchannel for different average SNR, , values and a time-bandwidth product u equal to 5. In Figure 1, one can generallyinfer that the Pm Pf curves have low slopes for Pf < 0.1.One can also notice that there is an improvement of about ahalf order of magnitude from the Pm perspective with eachstep of 5 dB increase in values starting from 10 to 25 dB.

Figure 2 shows the effect of the Nakagami parameter onthe complementary ROC curves for a Nakagami channel with = 20 dB and u = 5. There is an improvement of roughly oneorder of magnitude from the probability of miss perspective form = 2 compared to the Rayleigh case and this improvementstarts to diminish as m increases.

The performance of different diversity systems with dualbranches over IID Rayleigh channels is illustrated in Figure 3for = 20 dB and u = 5. There is an obvious gainfrom the Pm perspective in the case of diversity systemscompared to the no diversity one. This gain can be quantifiedas approximately one order of magnitude for both SC and SSCschemes, and approximately two orders of magnitude for theEGC scheme.

Finally, the effect of number of diversity branches is illus-trated in Figure 4 wherein the SC diversity is considered sinceit represents a good compromise between system performanceand receiver complexity. It seems that there is a gain ofroughly one order of magnitude from the Pm perspective witha diminishing effect as L increases.

3578

104 103 102 101 100105

104

103

102

101

100

Probability of False Alarm Pf

Prob

abbi

lity o

f Miss

ing

P m

=10 dB=15 dB=20 dB=25 dB

Fig. 1. Complementary ROC curves for Rayleigh channel at different values and u = 5.

APPENDIX AEVALUATION OF GM =

0 x

ep2x2/2QM (ax, b)dx

With the aid of [7, Eq. (29)], GM can be recursivelyevaluated as

GM = GM1 + CM1FM , for > 1, (35)where

CM1 =(+12

) (b2

2

)M1eb

2/2

2(M 1)!(p2+a2

2

) +12

, (36)

and

FM =1F1

(+ 12

;M ;b2

2a2

p2 + a2

). (37)

One can evaluate GM iteratively as follows

GM = GM1 + CM1FM= GM2 + CM2FM2 + CM1FM1.

.

.

= G1 +M1n=1

CnFn+1. (38)

REFERENCES[1] V. I. Kostylev, Energy detection of a signal with random amplitude, in

Proc. IEEE Int. Conf. on Commun. (ICC02), New York City, New York,pp. 16061610, May 2002.

[2] H. Urkowitz, Energy detection of unknown deterministic signals, Proc.IEEE, vol. 55, pp. 523531, April 1967.

[3] H. V. Trees, Detection, Estimation, and Modulation Theory. New York,NY: John Wiley & Sons, 1968.

[4] J. G. Proakis, Digital Communications. McGraw-Hill, fourth ed., 2001.[5] C. E. Shannon, Communication in the presence of noise, Proc. IRE,

vol. 37, pp. 1021, January 1949.[6] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and

Products. San Diego, CA: Academic Press, sixth ed., 2000.[7] A. H. Nuttall, Some integrals involving the QM -function, Naval

Underwater Systems Center (NUSC) technical report, May 1974.[8] A. H. Nuttall, Some integrals involving the Q-function, Naval Under-

water Systems Center (NUSC) technical report, April 1972.[9] G. L. Stuber, Principles of Mobile Communications. Norwell, MA:

Kluwer Academic Publishers, second ed., 2000.

104 103 102 101 100107

106

105

104

103

102

101

100

Probability of False Alarm Pf

Prob

abbi

lity o

f Miss

Pm

m=3m=2m=1

Fig. 2. Complementary ROC curves for Nakagami channel at different mvalues ( = 20 dB and u = 5).

104 103 102 101 100108

107

106

105

104

103

102

101

100

Probability of False Alarm Pf

Prob

abbi

lity o

f Miss

Pm

No DiversitySSCSCEGC

Fig. 3. Complementary ROC curves for dual-branch diversity systems overRayleigh channel (u = 5 and = 20 dB).

104 103 102 101 1001012

1010

108

106

104

102

100

Probability of False Alarm Pf

Prob

abbi

lity o

f Miss

Pm

L=1

L=2

L=3

L=4

L=5

Fig. 4. Complementary ROC curves for L-branch SC diversity receptionover a Rayleigh channel ( = 20 dB and u = 5).

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