Collaborative Autocorrelation-Based SpectrumSensing of OFDM signals in Cognitive Radios

Sachin Chaudhari, Jarmo Lunden, Visa KoivunenSMARAD CoE, Signal Processing Laboratory

Helsinki Univ. of Technology, FinlandEmail: {sachin, jrlunden, visa}@wooster.hut.fi

Abstract—A simple and efficient spectrum sensing scheme forOrthogonal Frequency Division Multiplexing (OFDM) signals ofprimary user in cognitive radio systems is proposed in this paper.A detector exploiting the well-known autocorrelation propertyof cyclic prefix (CP) based OFDM signals is developed. Theproposed scheme is then extended to the case of many secondaryusers collaborating in order to detect the primary user in the faceof shadowing and fading. The amount of information each usersends to other users or fusion center is constrained by censor-ing scheme where only informative decision statistics are sent.Censoring allows reducing the power consumption in batteryoperated mobile terminals. The statistical properties of the deci-sion statistics are established. Limits on the censoring region arefound under constraints on false-alarm and transmission rates.The distribution of the test statistics for cooperative detectionwith censoring is approximated using characteristic functions.The performance of the scheme is studied by simulations.

I. INTRODUCTION

Spectrum sensing is needed in cognitive radios in orderto find opportunities for agile use of spectrum. Moreover,it is crucial for managing the level of interference causedto primary users (PUs) of the spectrum. Sensing providesawareness of the radio operating environment. A cognitiveradio may then adapt its parameters such as carrier frequency,power and waveforms dynamically in order to provide the bestavailable connection and to meet the user’s needs within theconstraints on interference. Spectrum sensing techniques basedon cyclostationarity have been proposed in [1], [2] in additionto classical energy detectors.

OFDM will be a key technology in broadband wireless com-munication systems. Among the OFDM systems [3] are: IEEE802.11a/g Wireless LANs (WLANs), IEEE 802.16 or WiMAXWireless MANs, IEEE 802.20 or Mobile Broadband WirelessAccess (MBWA) systems, Long Term Evolution (LTE), DVBterrestrial digital TV systems DVB-T, DVB-H, T-DMB andISDB-T, the WiMedia Alliance’s Ultra wideband (UWB)implementation etc. Therefore it is reasonable to assume thatthe primary users will often use OFDM transmission. Hence,the problem of detecting OFDM signals is very relevant.

The presence of cyclic prefix (CP) gives OFDM basedsystems the following well-known, convenient property. Theautocorrelation function is non-zero at delays τ = ±Td, whereTd is the useful symbol length. This information has been em-ployed in synchronization (see, e.g. [4]) and blind equalizationin OFDM systems, for example. In this paper we proposea collaborative detection scheme that exploits this property.

The autocorrelation based decision statistics from multiplesecondary users are combined in order to detect a primary userreliably in the face of shadowing and fading. Consequently,the hidden node problem is avoided. User cooperation maybe used to improve the performance, facilitate the use ofsimpler detectors and increase the coverage in a cognitive radionetwork. Transmission of decision statistics is constrainedso that each user is sharing its decision statistics only ifit is considered to be informative. This operation is calledcensoring. It allows for lowering power consumption since lessdata needs to be transmitted. Censoring has been employed inenergy efficient sensor networks [5], [6]. Cooperative detectionwith censoring for cyclostationary based test statistics has beenproposed in [7]. The threshold value of the detector and theboundaries of no-send region are optimized under constraintson both false-alarm and transmission rates. We establish thedistributions of the test statistics and analyse the performanceof the proposed detector in the low SNR regime.

This paper is organised as follows. In Section II, detectionscheme based on the well-known autocorrelation property ofOFDM signals is proposed and statistical properties of thetest statistics are established. Cooperative detection schemebased on censoring where each secondary user is transmittingautocorrelation-based test statistics only if it is informative isdescribed in Section III. Simulations in Section IV demonstratethe performance of the autocorrelation based detection for bothsingle user and cooperative detection with censoring. Finally,Section V concludes the paper.

II. SPECTRUM SENSING USING AUTOCORRELATION

In this section, we will consider individual detector based onthe non-zero autocorrelation property. Formally, this propertymay be stated as follows. Assume that length of the usefulsymbol data is Td and cyclic prefix is Tc for an OFDM system,the autocorrelation coefficient for the system at τ = ±Td willbe µ = Tc

Tc+Td. Now, we will derive the test statistic and

establish its distribution for single detector.Let H0 be the null hypothesis, i.e., there is no OFDM based

primary user transmission present, and H1 be the alternatehypothesis, i.e., OFDM based primary user is active. We willexploit the autocorrelation property of OFDM systems usingCP for the detection of primary users. The autocorrelation may

191978-1-4244-2247-0/08/$25.00 ©2008 IEEE.

be estimated as

R(τ) =1

M

M∑

t=1

x(t)x∗(t + τ), (1)

wherex(t) = s(t) + w(t) (2)

is the received OFDM signal, s(t) is the transmitted OFDMsignal, w(t) is additive white gaussian noise (AWGN) and Mis the number of samples used in autocorrelation estimation.Central Limit Theorem (CLT) states that the sum of many in-dependent and identically-distributed (i.i.d.) random variables,under the assumption that the sum of the variables has a finitevariance, approaches gaussian distribution for sufficiently largenumber of random variables. Assuming sufficiently large In-verse Fast Fourier Transform (IFFT) size for the OFDM signal,and invoking the CLT, we have

s(t) ∼ Nc(0, σ2s)

w(t) ∼ Nc(0, σ2n) (3)

x(t) ∼ Nc(0, σ2s + σ2

n),

where it has been assumed that s(t) and w(t) are independentof each other and Nc(.) denotes distribution for complex gaus-sian random variable. Since the data is not known and cyclicprefix will change from symbol to symbol r = R(τ)

∣

∣

τ=±Td

will also be a random variable with mean and variance.Under H1, the mean of r is given by

E[r|H1] = E[1

M

M∑

t=1

x(t)x∗(t + Td)]

=1

M

M∑

t=1

E[x(t)x∗(t + Td)]

=1

M

M∑

t=1

E[(s(t) + w(t))(s∗(t + Td) + w∗(t + Td))]

=1

M

M∑

t=1

E[s(t)s∗(t + Td)] + E[w(t)w∗(t + Td)]

=1

M

M∑

t=1

E[s(t)s∗(t + Td)]

= µσ2s . (4)

Second moment of r under H1 can be calculated as

E[|r|2|H1] = E

∣

∣

∣

∣

∣

1

M

M∑

t=1

x(t)x∗(t + Td)

∣

∣

∣

∣

∣

2

=1

M2

M∑

t1=1

M∑

t2=1

E[x(t1)x∗(t1 + Td)x

∗(t2)x(t2 + Td)]

=1

M2

M∑

t1=1

M∑

t2=1

E[(

s(t1) + w(t1))(

s∗(t2) + w∗(t2))

(

s∗(t1 + Td) + w∗(t1 + Td))(

s(t2 + Td) + w(t2 + Td))]

.

(5)

If a,b,c, and d are jointly Gaussian (complex or real) randomvariables, then [8]

E[abcd] = E[ab]E[cd] + E[ac]E[bd] + E[ad]E[bc] −2E[a]E[b]E[c]E[d]. (6)

Using independence of the signal samples s(t1) and s(t2) andnoise samples w(t1) and w(t2) where t1 6= t2 and (6), we canexpand (5) to get

E[|r|2|H1] =(σ2

s + σ2n)2 + 3µ2

1

M(7)

where µ1 = µσ2s .

Therefore, the variance of r under H1 is given by

Var(r|H1) = E[|r|2|H1] − |E[r|H1]|2

=(σ2

s + σ2n)2 + 2µ2

1

M. (8)

Under H0, independence of w(t) and w∗(t + Td) givesE[r|H0] = 0 and the variance is

Var(r|H0) =(σ2

n)2

M. (9)

Now we need the distribution of r which is sum of productsof two complex gaussian random variables. Here we derive thedistribution of sum of products of two real gaussian randomvariables and show that the distribution can be closely approx-imated by gaussian distribution based on CLT for sufficientlylarge M . In [9], the characteristic function of Y = X1X2,where X1 and X2 are i.i.d. real gaussian random variableswith distribution Nr(0, σ

2), is given by

φY (ω) =(

√

1 + σ4ω2)−1

. (10)

Let us define Z =∑M

i=1 Yi where Yi are product of twogaussian random variables. Also Yi are i.i.d.. Therefore, thecharacteristic function for Z is given by

φZ(ω) =

M∏

i=1

φYi(ω)

= φMY (ω)

=(

√

1 + σ4ω2)−M

. (11)

The pdf of Z is calculated by inverting the characteristicfunction to get

fZ(z) =2

3−M2

√πσ(−1−M)|z|M−1

2 K 1−M2

( |z|σ2 )

2πΓ(M2 )

(12)

where Kn(.) is the modified bessel function of the secondkind and Γ(.) is the gamma function. The first moment andsecond moment are evaluated from characteristic function andgiven as

E(Z) = 0

E(Z2) = Var(Z) = Mσ2. (13)

192

−40 −30 −20 −10 0 10 20 30 400

0.05

0.1

0.15

0.2

0.25

z

f Z(z

)

M=5Gaussian M=5M=50Gaussian M=50

Figure 1. Plots of theoretical pdf and gaussian approximation for sum ofproduct of two Gaussian random variables : As M increases the approximatedGaussian pdf tends to the theoretical pdf.

Using first and second moments, pdf of Z can also beapproximated to be gaussian based on the CLT. Also theproducts of gaussian random variables in the sum are i.i.d. Inour case, the number of samples M over which the test statisticis calculated is sufficiently large (typically > 1000 samples).Thus based on CLT we may assume that Z is Gaussian

Z ∼ Nr(0,Mσ2). (14)

Fig. 1 shows plots of theoretical pdf (12) and approximatedgaussian pdf (13). As M increases the pdf of Z tends to thegaussian pdf.

Similarly, using the CLT, r can also be approximated asgaussian random variable as

r ∼ Nc(0, σ20) under H0

r ∼ Nc(µ1, σ21) under H1 (15)

where

µ1 = µσ2s ;

σ21 =

(σ2s + σ2

n)2 + 2µ21

M(16)

σ20 =

σ4n

M.

The pdf for r under H0 and H1 are given by

f0(r) =1

πσ20

e− rr∗

σ20 under H0 (17)

f1(r) =1

πσ21

e− (r−µ1)(r−µ1)∗

σ21 under H1.

Therefore the likelihood ratio test for the detection problemrepresented by (15) is given by

f1(r)

f0(r)

H1

≷H0

γ (18)

where γ is the threshold dependent on the type of detectorwe want to design. Since we have exponential distribution

models, it is convenient to use log-likelihood ratios (LLRs).Taking natural log of both sides, we get Log Likelihood RatioTest (LLRT)

2 ln

(

σ0

σ1

)

+

(

1

σ20

− 1

σ21

)

|r|2 +2µ1rr

σ21

− µ21

σ21

H1

≷H0

ln(γ) (19)

where rr = r+r∗

2 is the real part of r.

Moving constant terms 2 ln(

σ0

σ1

)

and µ21

σ21

on right hand sidewe have,

(

1

σ20

− 1

σ21

)

|r|2 +2µ1rr

σ21

H1

≷H0

γ1 (20)

where

γ1 = ln(γ) − 2 ln

(

σ0

σ1

)

+µ2

1

σ21

. (21)

Thus from the above equation, it can be seen that teststatistic for LLRT consist of two terms:

1) The first term(

1σ20− 1

σ21

)

|r|2 uses difference in Var(r)

under hypothesis H1 and H0, i.e, σ21 and σ2

0 .2) The second term 2µ1rr

σ21

uses difference in mean of E[r]hypothesis H1 and H0, i.e., µ1 and 0.

Now we can devise a scheme that uses either one of them orcombines both of them for the test statistic. Note that for lowSNR, the second term will be dominant. While for high SNR,the first term will be dominant. Using both of them may givegood result for all values of SNR, but it requires knowledge ofσ2

s and σ2n. Also it is computationally complex as compared to

the case when we use only one of them. Since we are aiming atgood performance at low SNR and computationally efficientalgorithm, we can use the second term, i.e. the differencesin mean of the two hypotheses. Using the second term givesalmost the same performance as we would have achieved usingboth the terms for low SNR.

Hence, in order to employ Log Likelihood Ratio Test(LLRT) for the detection problem based on the second term2µ1rr

σ21

, we propose to use rr = real(r) as the test statistic for

detection for low SNRs (σ2n ≫ σ2

s ). The distribution for thetest statistics in low SNR can be approximated as

rr ∼ Nr(0, σ2) under H0

rr ∼ Nr(µ1, σ2) under H1, (22)

where

σ2 = (σ2s + σ2

n)2/(2M) = (Var(x))2/(2M). (23)

Note that σ2 ≈ σ21/2 ≈ σ2

0/2 for low SNRs.Since, now the distribution is known, we can maximize the

detection probability under a given false-alarm rate constraint.The false alarm α = PFA = P (rr > η|H0) given by

α =1

2erfc(

η√2σ

) (24)

where erfc(.) is complementary error function. The probabilityof detection Pd = P (rr > η|H1) is given by

Pd =1

2erfc(

η − µ1√2σ

). (25)

193

Note that this scheme requires knowledge of Td which isa reasonable requirement. Even if it is not known, we cancalculate R(τ) for various practical values of τ = Td. From(23), σ2 is calculated from the received signal by the secondaryuser. Thus this scheme provides a simple and computationallyefficient detector based on minimalist information about theprimary user.

III. COOPERATIVE DETECTION WITH CENSORING

In cognitive radio systems, there are typically multiplegeographically distributed secondary users that need to detectwhether the primary user is active or not. The cooperationmay be then coordinated by a dedicated fusion center or oneof the users. To derive a test for the Fusion Center (FC), weassume that the sensors are independent conditioned whetherhypothesis H0 or H1 is true.

The disadvantage of cooperative detection is that the over-head traffic increases the consumption of power in batteryoperated mobile terminals. This motivates censoring approachwhere the amount of information sent to fusion center isconstrained. In censoring only informative test statistics aresent. Generally very small or very large values of the teststatistic are considered to be informative.

Censoring region for the collaborating users can be de-termined under the constraints on data rate and false-alarmprobabilities. The optimal censoring region was found to be asingle interval of the likelihood ratio in [6]. In this paper wehave found the censoring thresholds as suggested in [7], [10].The upper threshold η1,i of the censoring region for user i isgiven by the communication rate constraint of the form

p(rri > η1,i|H0) ≤ κi ∀ i = 1, . . . , L (26)

where κi ≤ 1 is the send rate of user i, and L is total number ofcollaborating users. For simplicity we assume that κ = κi,∀i.The lower limit is chosen to be η0,i = −∞. The thresholdvalues must be communicated to the FC or user making theglobal decision.

In censoring scenario, let us define the test statistic send tofusion center for user i by random variable rci such that

rci =

{

rri, rri > η1,i

ϕ, otherwise

where ϕ denotes a null, i.e., that the test statistic is not send tothe FC when rri is below censoring threshold. We have usedthe following test statistic employed in [7] to combine the teststatistics send by individual secondary users at the FC

RL =L

∑

i=1

rci

=

K∑

i=1

rri +

L−K∑

i=1

rri (27)

where K is the number of users who are transmitting and rri

is the average value of test statistics in the no-send region.In this test, test statistics of the users in no-send region are

replaced by the average value of the test statistics in the no-send region. Under the null hypothesis, the values for rri areobtained by calculating the mean values of gaussian randomvariables limited to the no-send regions.

To design and evaluate the performance of a detector weneed to establish the distribution of the test statistic RL.Censoring affects the distribution of the global statistics. Thedistribution of rci is given as follows

f(rci|H0) = e

−r2ci

2σ2√2πσ2·(1−G(η1,i))

rci 6= ϕ, (28)

where G(·) denotes the cumulative distribution function ofthe Gaussian distribution. The characteristic function for rci

is given by

φrci(ω) = E[ejωrci ]

=e−

σ2ω2

2 (1 + erf(−η1,i+jσ2ω√2σ

))

1 + erf(−η1,i√2σ

)(29)

where erf(.) denotes the error function. Since the sensors areassumed to be conditionally independent, and rri is determin-istic, the characteristic function for RL for a given K = k isgiven by

φRL(ω) = ejω(L−k)rriφk

rci(ω). (30)

In [11], cumulative distribution function F (x) of a randomvariable X with zero mean and unit variance is approximatedby

F (x) ≈ 1

2+

λx

2π−

N−1∑

ν=1−Nν 6=0

φX(λν)

2πjνe−jλνx (31)

where φ(·) is the characteristic function of X . N definesthe number of points 2N − 1 at which the distribution isapproximated and λ is a constant chosen such that the fullrange of the distribution is represented, i.e., includes both 0and 1. Thus we can approximate the distributions of RL byfirst normalizing the characteristic function given by (30) andthen numerically inverting it [7].

IV. SIMULATION RESULTS

For all simulation cases, the results are averaged over 1000realizations. The limit on false-alarm rate is set to PFA = 0.05.No oversampling has been used. All the simulations are carriedout for Additive White Gaussian Noise (AWGN) channel.

For single user case, detector uses 100 OFDM symbols.Size of FFT is chosen to be 32. Therefore Td = 32. Cyclicprefix is chosen as one fourth of Td, i.e., Tc = 8. ThereforeM = 100(Td+Tc). Fig.2 shows plots for detection probability(Pd) vs SNR for theory and simulation. The SNR is definedas SNR = 10 ln

(

σ2s

σ2n

)

. Fig.3 shows Receiver OperatingCharacteristic (ROC) curves for simulated and theoretical case.Theoretical values of Pd for different SNRs and differentPFA are calculated using (25) under the assumptions that thestatistics are known. From the figures, it is clear that theoryand simulation results are in par at low SNR regime validating

194

−25 −20 −15 −10 −5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pro

babi

lity

of d

etec

tion

Pd

PFA

=0.05

TheorySimulation

Figure 2. Probability of detection vs SNR for single user case. Theoreticalanalysis and simulation results are close.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Probability of false alarm PFA

Pro

babi

lity

of D

etec

tion

Pd

SNR= −7 dB

SimulationTheory

Figure 3. ROC for single user case for SNR=-7dB. Theoretical analysis andsimulation results are close.

the theoretical analysis. Although the analysis is done for lowSNR values, the detector performance remains good at highSNRs.

Next, the performance of the proposed scheme for thefollowing existing wireless standards with different CP lengthsis presented

• DVB-T: Td=8192, Tc=1024, Detection period = 5 Sym-bols, M = 5(Td + Tc), µ = 0.1111

• LTE: Td=512, Tc=36, Detection period = 100 Symbols,M = 100(Td + Tc), µ = 0.0657

• WLAN: Td=52, Tc=13, Detection period = 1000 Sym-bols, M = 1000(Td + Tc), µ = 0.2.

Detection period for all of them is chosen such that detectionperiod in number of samples M is approximately of the sameorder (approximately 50000). Fig.4 shows the performance ofthe proposed detector for these wireless standards. It is evidentfrom the figure that as µ increases, the performance of thedetector improves for detecting the PU of the corresponding

−30 −25 −20 −15 −10 −5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pro

babi

lity

of d

etec

tion

Pd

PFA

=0.05

DVB M=5*9216LTE M=100*548WLAN M=1000*65

Figure 4. Probability of detection vs SNR for the three standards : As µ valueincreases, performance improves.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of False Alarm PFA

Pro

babi

lity

of D

etec

tion

Pd

SNR = −8 (dB)

1 cyclic freq.2 cyclic freqs. D

S

Autocorr.

Figure 5. ROC curves for the proposed scheme and MCF. Performance ofautocorrelation based proposed scheme is better than single cyclic frequencycase and comparable to MCF detector based on 2 cyclic frequencies.

OFDM based wireless standard assuming that M for them isof the same order.

Next, the performance of the proposed scheme is comparedto cyclostationarity based multicycle detector presented in [2].The parameters chosen for comparison are taken from [2].Detector uses 100 OFDM symbols. Size of FFT was chosento be 32. Therefore Td = 32. Cyclic prefix is chosen as onefourth of Td, i.e., Tc = 8. Therefore M = 100(Td +Tc). Fig.5compares the detection performance for the two methods. Itis observed that the proposed autocorrelation based schemeis better than single cyclic frequency method and comparableto MCF detector based on two cyclic frequencies. It shouldbe noted that the proposed method has significantly lowercomplexity compared to MCF method [2].

The performance of collaborative detection with censoringis studied next. Here we have used L = 10, Td = 32, Tc = 8,N = 1000 and λ = 0.01. Average number of users which

195

−25 −20 −15 −10 −5 00

1

2

3

4

5

6

7

8

9

10

SNR (dB)

Ave

rage

No.

of T

x U

sers

PFA

=0.05, L=10

κ = 0.01κ = 0.1κ = 0.5κ = 1

Figure 6. Average number of secondary users transmitting test statistics vsSNR for RL. Average number of secondary users transmitting test statisticsto FC under null hypothesis are in accordance with rate constraints imposed.

−25 −20 −15 −10 −5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pro

babi

lity

of d

etec

tion

Pd

PFA

=0.05, L = 10

κ=0.01κ=0.1κ=0.5κ=1

Figure 7. Probability of detection vs SNR for different values of rateconstraints for RL test statistic. Performance loss due to censoring is notsignificant .

are transmitting their test statistics to the Fusion Center orthe user making the decision as a function of SNR is shownin fig.6. It can be seen that the average number of userstransmitting their test statistics to the fusion center at verylow SNR corresponds to the rate constraint imposed. Also notethat the reductions in transmission rate are highest in low SNRregion and when primary users are inactive. Fig.7 shows thedetection performance of RL under various communicationconstraints. The performance loss remains small even withvery strict communication constraints.

V. CONCLUSION

In this paper, we have proposed a simple and computa-tionally efficient spectrum sensing scheme based on the well-known autocorrelation property of OFDM signals with CP. Theperformance of the scheme is studied using theoretical analysisand then validated by simulations. The performance of the

detector is also shown comparable with the existing schemesbased on cyclostationarity while the computational complexityis quite low. Collaborative Detection is performed by multipledisplaced secondary users in order to mitigate shadowing andfading. In order to save energy, censoring approach where onlyinformative decision statistics are shared is adopted. A test isdeveloped and the statistical properties of the test statisticsare established for individual secondary user as well as forcollaborative decision making. The performance of the collab-orative detector is optimized under constraints on false-alarmand communication rates. The simulation studies demonstratethe reliable performance of the proposed method at very lowSNR regime. Performance loss caused by censoring is minoreven for strict communication rate constraints.

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