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Spectrum Sensing via Energy Detector in Low SNR

Saman Atapattu, Chintha Tellambura, and Hai Jiang

Department of Electrical and Computer Engineering, University of Alberta, Canada

Email: {atapattu, chintha, hai.jiang}@ece.ualberta.ca

Abstract—As required in the IEEE 802.22 proposal, spectrum

sensing techniques should be capable enough to sense the primary

signal with very low receiver sensitivity such as at -116 dBm. In

this paper, the detection performance of an energy detector used

for spectrum sensing in cognitive radio networks is investigated

under such very low signal-to-noise ratio (SNR) levels. The

analysis focuses on the derivation of a closed-form expression for

the average missed-detection probability over Rayleigh fading

and Nakagami-m fading channels. Subsequently, the detection

threshold is optimized for minimizing the total error rate. The

analysis is validated by numerical and simulation results. The

sensing requirements defined in IEEE 802.22 are also discussed

with numerical examples.

Index Terms—Cognitive radio, energy detection, spectrum

sensing, threshold selection.

I. INTRODUCTION

One of the most challenging tasks in cognitive radio net-

works is spectrum sensing. In the IEEE 802.22 wireless re-

gional area networks (WRAN) proposal, no specific spectrum

sensing technique is given. So designers have freedom to select

any spectrum sensing technique to meet the specified sensing

requirements [1]. Among the available spectrum sensing tech-

niques such as matched filter, cyclostationary feature detection

and eigenvalue detection, energy detection has gained renewed

interests in recent research efforts due to its low complexity.

The conventional energy detector measures the energy associ-

ated with the received signal over a specified time period and

a bandwidth. The decision of an energy detector is feasible

even when little prior knowledge of the transmitted signal

is available. The decision statistic of an energy detector is

a measure of the received signal energy after proper filtering,

sampling, squaring and integration.

Assuming a deterministic signal is transmitted over a flat

band-limited Gaussian noise channel, a basic mathematical

model of the decision statistic is given in [2] in order

to calculate the detection probability (Pd) and false alarm

probability (Pf). Subsequently, the performance of an energy

detector in terms of the average detection probability, the

receiver operating characteristic (ROC), and the area under

the ROC curve (AUC) over different fading channels, diversity

techniques and cooperative relay networks has been analyzed

in [3]–[10]. In spectrum sensing of cognitive radio networks,

the secondary user has no a priori knowledge of the primary

signal. The information bearing signal can have different

possible waveforms with random data sequences. Therefore,

it is appropriate to treat the received signal samples as a

random process. When both signal and noise follow Gaussian

processes, the decision statistic is modeled with a Gaussian

distribution using the central limit theorem (CLT) [11]–[13].

The Gaussian model is popular in the parameter optimiza-

tion problems, e.g., optimizing the operating threshold or the

power allocation so as to achieve the maximal throughput or

minimal error rate. This model often gives a more convenient

cost function which may result in a convex optimization

problem. However, based on the Gaussian model, the analysis

of the average detection performance of an energy detector

over different fading scenarios is not available in the open

literature because of the involved mathematical complexity.

The existing analytical results are limited to the additive white

Gaussian noise (AWGN) channel, and performance over other

fading scenarios are obtained only by simulations.

In this paper, we derive the average missed-detection proba-

bility of an energy detector in low signal-to-noise ratio (SNR)

region over Rayleigh fading and Nakagami-m fading channels.

The low SNR assumption is fairly reasonable because, as

in IEEE 802.22 WRAN, the spectrum sensing technique

should be able to detect the primary signal with the missed-

detection and the false alarm probabilities less than 0.1 and

the receiver sensitivity being -116 dBm [1], [14], [15]. More

importantly, we determine the optimal detection threshold of

the energy detector to minimize the total error rate. Based on

the analytical results, some numerical examples are given to

meet the IEEE 802.22 WRAN requirements.

The rest of this paper is organized as follows. Section II

briefly discusses energy detection and its low SNR model.

Section III gives the average missed-detection probability.

Sections IV is devoted to the analysis of the optimal detection

threshold. Section V presents numerical and simulation results,

followed by concluding remarks in Section VI.

II. ENERGY DETECTION AND LOW SNR APPROXIMATION

The spectrum sensing in cognitive radio networks follows

a binary hypothesis testing problem: hypothesis H0 (signal

absent) and hypothesis H1 (signal present). The received

signal for the binary hypothesis can be given as

?

where x(t) is the transmitted signal, h is the wireless channel

gain, and w(t) is the additive white Gaussian noise (AWGN)

which is assumed to be a circularly symmetric complex Gaus-

sian (CSCG) random variable with mean zero and variance

σ2

pre-filter followed by a square-law device and a finite time

y(t) =

w(t)

hx(t) + w(t)

: H0

: H1

w. The conventional analog energy detector consists of a

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integrator. The output of the integrator is called the decision

statistic. It may be proportional to the received signal energy

and can be given as Λ(y) =?N

is appropriate to treat the signal samples as a random process.

Thus, the sample of the transmitted signal x(n) follows an

independent and identically distributed (i.i.d.) random process

with mean zero and variance σ2

the detector is γ =

σ2

w

sample number N is large enough, using CLT, the probability

density function (PDF) of Λ(y) under H0, fΛ(y,H0), is

a normal distribution with Nσ2

Similarly, the PDF of Λ(y) under H1, fΛ(y,H1), is a normal

distribution with Nσ2

for a complex-valued phase-shift keying (PSK) modulated

signal. The false alarm probability, Pf, and the detection

probability for given h, Pd|h(γ), are given as

??

??

respectively, where λ is the threshold and Q(·) is the standard

Q-function.

Since IEEE 802.22 WRAN standard is interested in the

spectrum sensing in very low SNR region (<-20 dB), we

consider a low SNR energy detection model. Under the low

SNR assumption (i.e., σ2

on the variance of the decision statistic under H1. Therefore,

PDF fΛ(y,H1) is Gaussian distributed with Nσ2

mean and Nσ4

function (CDF) of the exact and its low SNR approximation

for three different low SNR values when σw = 1 and

N = 2000. The approximated and the exact CDFs are very

close. Therefore, for low SNR, Pf and Pd|h(γ) can be given

in alternative forms as

Pf=1

2Erfc

?λ − Nσ2

respectively, where Erfc(·) is the complementary error function

which is defined as Erfc(z) =

Q(z) =1

√2

III. AVERAGE MISSED-DETECTION PROBABILITY

The instantaneous detection probability given in (3) is

equivalent to the average detection probability over AWGN

channel when γ is replaced by the average SNR ¯ γ. Here,

we are interested in the missed-detection probability (as it is

one requirement in IEEE 802.22 WRAN), Pmd|h(γ), which

is given as Pmd|h(γ) = 1 − Pd|h(γ). When the SNR dis-

tribution is fγ(x), the average missed-detection probability,

Pmd, is Pmd=?∞

n=1|y(n)|2where N is the

number of samples. When the signal has an unknown form, it

s. Then, the received SNR at

for the given channel h. When the

|h|2σ2

s

wmean and Nσ4

wvariance.

w(1+γ) mean and Nσ4

w(1+2γ) variance

Pf= Q

λ

Nσ2

w

− 1

?√N

??

?

(1)

Pd|h(γ) = Q

λ

Nσ2

w

− 1 − γ

N

(1 + 2γ)

?

,

(2)

s≈ σ2

w), the signal has a little impact

w(1 + γ)

wvariance. Fig. 1 shows cumulative distribution

?λ − Nσ2

w

√2Nσ2

w

?

Pd|h(γ) =1

2Erfc

w(1 + γ)

√2Nσ2

w

?

,

(3)

2

√π

?∞

z

e−t2dt and we have

2Erfc

?

z

?

.

0Pmd|h(x)fγ(x)dx. We consider Rayleigh

1800 190020002100

y

220023002400

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CDF (FΛ(y,H1))

Exact

Low SNR approximation

− 20 dB

− 15 dB

− 10 dB

Fig. 1. CDF of the decision statistic under hypothesis H1.

fading and Nakagami-m fading channels, which can model

a variety of fading effects. If the signal amplitude follows

a Nakagami-m distribution, the SNR distribution is given as

fγNak(x) =(m

parameter, and it follows Rayleigh fading when m = 1. With

the aid of the identity Erfc(−x) = 2 − Erfc(x), and after

some algebraic manipulations, the average missed-detection

probability over Nakagami-m fading channel, PNak

given as

?

2Γ(m)

0

¯ γ)

Γ(m)xm−1e−m

m

¯ γx[16] where m is the fading

md, can be

PNak

md

=

m

¯ γ

?m

?∞

xm−1e−mx

¯ γErfc

? ?

N

2x +Nσ2−λ

√2Nσ2

?

dx. (4)

In the following, we define an integral expression, I(n,p,a,b),

as [17, eq. (2.8.9.1)]

?∞

?

I(n,p,a,b) ?

0

xne−pxErfc(ax + b)dx

= (−1)n

∂n

Erfc(b)−e

p2+4pab

4a2

Erfc(b+p

2a)

p

?

∂pn

where n is a positive integer, Re[p]> 0, c > 0, and

the nth order partial derivative with respect to p. For integer

m, PNak

md

given in (4) can be evaluated as

?

∂n

∂pn[·] is

PNak

md

=

m

¯ γ

?m

2Γ(m)I

?

m − 1,p,

?

N

2,Nσ2− λ

√2Nσ2

??????

p=m

¯ γ

.

(5)

When m = 1, the average missed-detection probability over

Rayleigh fading channel, PRay

md, is

PRay

md=1

2

?

Erfc

?Nσ2− λ

× Erfc

√2Nσ2

?

− e

1

¯ γ2+4

¯ γ

?

Nσ2−λ

√2Nσ2

2N

?√

N

2

?Nσ2− λ

√2Nσ2+

1

¯ γ√2N

??

.

(6)

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Page 3

When a random signal is present, the results in (5) and (6)

are novel closed-form expressions for the average missed-

detection probability.

IV. THRESHOLD SELECTION

The threshold, λ, which varies form 0 to ∞ is a common

parameter for the false alarm, the detection and the missed-

detection probabilities which are denoted as Pf(λ), Pd(λ)

and Pmd(λ), respectively. The common practice of setting the

threshold is based on the false alarm probability. For given N,

σw, and the constant false alarm probability (CFAP=¯Pf), the

selected threshold is λ =

Although achieving a high Pd(λ) while keeping Pf(λ) low

is preferable (e.g., in IEEE 802.22 WRAN recommendations,

¯Pf ≤ 0.1 and¯Pd ≥ 0.9), such a threshold selection is not

always possible in practice. Therefore, the threshold selection

can be viewed as an optimization problem. Some research

work has been done for this problem based on different

objectives [11], [18]–[23]. The minimization of the total

error rate which is defined as Pe(λ) ? Pf(λ) + Pmd(λ)

is a possible way of selecting λ [18]. We consider different

fading scenarios in the following subsections, and the optimal

threshold, λ∗, is derived such that the total error is minimized,

i.e., λ∗= argmin

λ

A. AWGN Channel

For an AWGN fading channel, the optimal threshold is given

using (1) and (2) as

?

−1

?√2Erfc−1(2¯Pf) +√N

?√Nσ2

w.

Pe(λ).

λ∗= argmin

λ

1+1

2Erfc

?λ − Nσ2

?

w

√2Nσ2

λ − Nσ2

?2N(1 + 2γ)σ2

w

?

2Erfc

w(1 + γ)

w

??

.

Therefore, the optimal threshold for any SNR value can be

derived as (see the Appendix)

?

In low SNR, i.e., γ ? 1, thus 1+2γ ≈ 1, the optimal threshold

can be well-approximated as

?

B. Rayleigh Channel

For a Rayleigh fading channel, the total error is Pe= (Pf+

PRay

md

is given in (6). When∂Pe

simplified as1

?

Erfc

√2N¯ γ

It is complicated to derive the exact solution for λ∗with

this non-linear equation. The solution can be obtained nu-

merically. However, with the assumption of very low SNR

λ∗=Nσ2

w

2

1 +

?

1+ 2γ

?

1+(1 + 2γ)ln(1 + 2γ)

Nγ2

??

.

(7)

λ∗≈Nσ2

w

2

1 +

?1 + 2γ

?

≈ Nσ2

w.

(8)

md) where PRay

∂λ= 0, it can be

e

1

√2N¯ γ−λ−Nσ2

w

√2Nσ2

w

?2

?

1

−λ − Nσ2

w

√2Nσ2

w

?

=

?

2N

π

¯ γ.

(9)

1Due to space limitation, the detailed derivation is omitted.

and the observation made in the AWGN channel, we can

find an approximated optimal value. When ¯ γ ? 1, the right

hand side of (9) approaches a very small value. We take

α =

√2N¯ γ−

left hand side of (9) also approaches a very small value due

to lim

(9) is satisfied for λ around Nσ2

that the second order derivative of Pe(λ),

λ ≈ Nσ2

around λ ≈ Nσ2

the numerical results given in Section V.

Note that the optimal threshold selection with a Nakagami-

m fading channel is analytically complicated because Pe =

(Pf+PNak

m fading (when 1 < m < ∞) is varying between the

Rayleigh fading and the Gaussian fading, we can also claim

that the optimal threshold is around λ ≈ Nσ2

exact solution can be obtained numerically using mathematical

software packages such as MATHEMATICA and MATLAB.

It is possible that we cannot achieve the recommended error

rate requirements (Pf ≤ 0.1 and Pmd ≤ 0.1) even at the

optimal threshold value. One possible way of achieving the

requirements is by increasing the number of samples N. Since

N ≈ τfswhere τ is the sensing time and fsis the sampling

frequency, the sensing time also increases when N increases.

However, there is a limitation for the allowable sensing time,

i.e., τ ≤ 2 seconds in accordance with IEEE 802.22 WRAN

[1], [15]. This is a main drawback in spectrum sensing with

energy detection in low SNR. This issue is discussed with

numerical examples in the following section.

1

λ−Nσ2

√2Nσ2

w

w. When ¯ γ ? 1 and λ → Nσ2

w, the

α→0eα2Erfc(α) → 0. Thus, we can say that the equation

w. Further, it can be shown

∂2Pe(λ)

∂λ2

md) has a minimum

> 0 when

w. Therefore, Pe = (Pf+ PRay

w. This observation is also concluded with

md) has highly non-linear behavior. Since Nakagami-

w. However, the

V. NUMERICAL/SIMULATION RESULTS AND DISCUSSION

This section provides numerical and simulation results. We

defined the normalized threshold asˆλ ?

is normalized by the number of samples.2The noise variance

is set to σ2

Pe(λ∗), P∗

One of the main contributions of this research is to derive

a closed-form expression for the average missed-detection

probability over Nakagami-m fading channel in low SNR.

Fig. 2 shows ROC curves (i.e., Pd versus Pf) for three

different fading scenarios such as AWGN, Nakagami-4 and

Rayleigh fading channels. The numerical results which are

based on expressions (5) and (6) are represented by curves,

while simulation results are represented by discrete marks. The

ROC curves are plotted by varyingˆλ from 0.95 to 1.50 for the

average SNR ¯ γ=-20 dB and ¯ γ=-15 dB when N = 2000. The

numerical results closely match with the simulation results

for all three fading scenarios at -15 dB and -20 dB, which

confirms the accuracy of the approximation. The multipath

fading parameter m has a negligible impact on the energy

detection at -20 dB, because the faded replicas of different

λ

N, i.e., the threshold

w= 1, unless specified otherwise. We denote P∗

f= Pf(λ∗), and P∗

e=

md= Pmd(λ∗).

2This is the threshold if the decision statistic is selected as Λ(y) =

?N

1

N

n=1|y(n)|2.

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Page 4

00.10.20.3 0.40.5

Pf

0.6 0.70.80.91

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pd

AWGN

Nakagami − 4

Rayleigh

−15 dB

−20 dB

Fig. 2.ROC curves of an energy detector over fading channels.

0.650.70.75 0.80.85

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Threshold

Pe

AWGN

Nakagami−5

Rayleigh

− 20 dB

− 15 dB

Fig. 3.Total error rate versus normalized threshold.

multipaths have no significant contribution to increase the

effective SNR at very low SNR.

We show that the total error rate Pe= Pf+ Pmdcan be

minimized at the optimal threshold value λ∗. The total error

rate versus normalized threshold for AWGN, Nakagami-5 and

Rayleigh fading channels is shown in Fig. 3 at ¯ γ = −15 dB

and ¯ γ = −20 dB when N = 2000. It shows thatˆλ∗≈ 0.755

at ¯ γ= -20 dB, andˆλ∗≈ 0.760 at ¯ γ= -15 dB when σ2

for all three different fading scenarios. As in expression (8),

the optimal normalized threshold is given asˆλ∗=

σ2

w

2

Rayleigh and Nakagami-m fading channels. So the analytical

results in Section IV is confirmed. Further, we can see that one

possible way of minimizing the total error rate is by increasing

the average SNR, e.g., P∗

by increasing SNR from -20 dB to -15 dB in AWGN channel.

Since our main focus is on low SNR region, another possible

way of minimizing the total error rate is by increasing the

number of samples. In the spectrum sensing, IEEE 802.22

w= 0.75

λ

N

∗≈

?1 +√1 + 2γ

?≈ σ2

wfor AWGN, and it is also valid for

ecan be reduced from 0.82 to 0.48

10

3

10

N

4

10

5

10

−1

Error rates

Pe

Pf

Pmd

Pe

Pf

Pmd

* at −15 dB

* at −15 dB

*

at −15 dB

* at −20 dB

* at −20 dB

*

at −20 dB

Fig. 4. Minimum error rates versus number of samples in AWGN channel.

10

3

10

4

10

5

10

6

10

−1

N

Error rates

Pe

Pf

Pmd

Pe

Pf

Pmd

* at −15 dB

* at −15 dB

*

at −15 dB

* at −20 dB

* at −20 dB

*

at −20 dB

Fig. 5.Minimum error rates versus number of samples in Rayleigh channel.

WRAN expects Pf ≤ 0.1 and Pmd ≤ 0.1, and channel

detection time (CDT) τ ≤ 2 seconds under any detection

technique. Since N ≈ τfswhere fsis the sampling frequency

which may depend on the sampling rate of the analog to digital

converter and FFT (fast Fourier transform) bin resolution, we

cannot increase N beyond τfs. We take fs= 62.5 kHz which

is a typical FFT bin resolution of an experimental energy

detection implementation [24]. We consider the minimum

error rate requirements as Pf ≤ 0.1 and Pmd ≤ 0.1. Fig.

4 and Fig. 5 show minimum error rates (P∗

versus N for AWGN and Rayleigh channels, respectively. For

-15 dB, the requirements can be achieved when N ≥ 7000

with P∗

processing delays, the minimum CDT is 0.112 seconds. For

-20 dB, the requirements can be achieved when N ≥ 67000

with P∗

The Rayleigh fading represents the effect of heavily built-

up urban environments on radio signals. As in Fig. 5, neither

at -15 dB nor at -20 dB can the requirements be met within 2

e, P∗

f, and P∗

md)

e ≈ 0.193 in AWGN channel. If we neglect other

e≈ 0.197 within 1.072 seconds in AWGN channel.

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Page 5

seconds (or equivalently at N ≤ 125000), e.g., P∗

when N = 125000 at -20 dB. It needs more than 3×106sam-

ples to achieve P∗

way of detecting the signal within 2 seconds is by increasing

the sampling frequency fsbeyond 1.5 MHz. Another possible

way is to use a diversity technique or cooperative spectrum

sensing topology to increase the number of effective decision

statistics while maintaining a small number of samples per

node or per branch.

VI. CONCLUSION

Detection performance of the energy detector is stud-

ied in the low SNR regime. The average missed-detection

probability is derived in a closed-form over Rayleigh and

Nakagami-m fading channels. For minimizing the total error

rate, the optimal detection threshold is exactly derived for

AWGN channel, and it is also approximated for Rayleigh

and Nakagami-m fading channels. The false alarm and the

missed-detection probabilities may not be satisfied even at the

optimal detection threshold. Increasing the number of samples

until the maximum allowable sensing time is achieved, and

increasing the sampling frequency are two possible ways to

meet the requirements of the false-alarm and the missed-

detection probabilities. The research findings help to design

an energy detector in an implementable way as fulfilling

fundamental sensing requirements proposed in IEEE 802.22

WRAN (channel detection time ≤ 2 seconds, missed-detection

and false alarm probabilities ≤ 0.1).

APPENDIX

The optimal threshold, λ∗, is achieved when

0. With the aid of (1), (2), Q(x) =

?= −2e

∂Pe(λ)

∂λ

md≈ 0.293

md≤ 0.1 at -20 dB. Therefore, one possible

∂Pe(λ)

∂λ

x

√2

=

1

2Erfc

?

?

, and

∂

∂xErfc?x−a

b

−(x−a)2

b√π

b2

, we have

=

e

−(λ−N(1+γ)σ2

?2πN(1 + 2γ)σ2

w)2

2N(1+2γ)σ4

w

w

−e

−(λ−Nσ2

√2πNσ2

w)2

2Nσ4

w

w

= 0.

(10)

After some algebraic manipulations and taking the logarithm,

(10) can be simplified into a quadratic equation of λ as

?

Thus, the solution for λ is

?

Since λ ≥ 0, λ∗can be selected as (7). In low SNR, i.e., γ ?

1, 1+2γ ≈ 1 and thus ln(1+2γ) ≈ 0, the optimal threshold

can be well approximated as (8). Now we can consider the

second order derivative which is given as

λ2− Nσ2

wλ −Nσ4

w

2

Nγ +(1 + 2γ)ln(1 + 2γ)

γ

?

= 0.

λ =Nσ2

w

2

1 ±

?

1 + 2γ

?

1 +(1 + 2γ)ln(1 + 2γ)

Nγ2

??

.

∂2Pe(λ)

∂λ2

=(λ − Nσ2

w)e

−(λ−Nσ2

w)2

2Nσ4

w

√2πN3/2σ6

w

−(λ − N(1 + γ)σ2

w)e

w(1 + 2γ)3/2

−(λ−N(1+γ)σ2

w)2

2N(1+2γ)σ4

w

√2πN3/2σ6

.

Using (7), it is easy to show that

therefore, there is a global minimum at λ = λ∗.

∂2Pe(λ∗)

∂λ2

> 0, and

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings