spectrum-sensing algorithms for cognitive radio based on statistical covariances

1804 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009

Spectrum-Sensing Algorithms for Cognitive RadioBased on Statistical Covariances

Yonghong Zeng, Senior Member, IEEE, and Ying-Chang Liang, Senior Member, IEEE

Abstract—Spectrum sensing, i.e., detecting the presence of pri-mary users in a licensed spectrum, is a fundamental problem incognitive radio. Since the statistical covariances of the receivedsignal and noise are usually different, they can be used to differ-entiate the case where the primary user’s signal is present fromthe case where there is only noise. In this paper, spectrum-sensingalgorithms are proposed based on the sample covariance matrixcalculated from a limited number of received signal samples.Two test statistics are then extracted from the sample covariancematrix. A decision on the signal presence is made by comparing thetwo test statistics. Theoretical analysis for the proposed algorithmsis given. Detection probability and the associated threshold arefound based on the statistical theory. The methods do not need anyinformation about the signal, channel, and noise power a priori.In addition, no synchronization is needed. Simulations based onnarrow-band signals, captured digital television (DTV) signals,and multiple antenna signals are presented to verify the methods.

Index Terms—Communication, communication channels,covariance matrices, signal detection, signal processing.

I. INTRODUCTION

CONVENTIONAL fixed spectrum-allocation policieslead to low spectrum usage in many frequency bands.

Cognitive radio, which was first proposed in [1], is a promis-ing technology for exploiting the underutilized spectrum inan opportunistic manner [2]–[5]. One application of cognitiveradio is spectral reuse, which allows secondary networks/usersto use the spectrum allocated/licensed to the primary userswhen they are not active [6]. To do so, the secondary users arerequired to frequently perform spectrum sensing, i.e., detect-ing the presence of the primary users. If the primary usersare detected to be inactive, the secondary users can use thespectrum for communications. On the other hand, whenever theprimary users become active, the secondary users have to detectthe presence of those users in high probability and vacate thechannel within a certain amount of time. One communicationsystem using the spectrum reuse concept is the IEEE 802.22wireless regional area networks [7], which operate on the veryhigh-frequency/ultrahigh-frequency bands that are currently al-located for TV broadcasting services and other services, suchas wireless microphones. Cognitive radio is also an emergingtechnology for vehicular devices. For example, in [8], cognitive

Manuscript received January 30, 2008; revised June 30, 2008. First publishedSeptember 3, 2008; current version published April 22, 2009. The review of thispaper was coordinated by Prof. H.-C. Wu.

The authors are with the Institute for Infocomm Research, A*STAR,Singapore 138632 (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TVT.2008.2005267

radio is proposed for underwater vehicles to fully use thelimited underwater acoustic bandwidth, and in [9], it is usedfor autonomous vehicular communications.

Spectrum sensing is a fundamental task for cognitive radio.However, there are several factors that make spectrum sensingpractically challenging. First, the signal-to-noise ratio (SNR)of the primary users may be very low. For example, the wire-less microphones operating in TV bands only transmit signalswith a power of about 50 mW and a bandwidth of 200 kHz.If the secondary users are several hundred meters away fromthe microphone devices, the received SNR may be well below−20 dB. Second, multipath fading and time dispersion of thewireless channels make the sensing problem more difficult.Multipath fading may cause signal power fluctuation of aslarge as 20–30 dB. On the other hand, coherent detectionmay not be possible when the time-dispersed channel is un-known, particularly when the primary users are legacy systems,which do not cooperate with the secondary users. Third, thenoise/interference level may change with time, which yieldsnoise uncertainty. There are two types of noise uncertainty:1) receiver device noise uncertainty and 2) environment noiseuncertainty. The receiver device noise uncertainty comes from[10]–[12] the nonlinearity of components and the time-varyingthermal noise in the components. The environment noise uncer-tainty may be caused by the transmissions of other users, eitherunintentionally or intentionally. Because of noise uncertainty,in practice, it is very difficult to obtain accurate noise power.

There have been several sensing methods, including the like-lihood ratio test (LRT) [13], energy detection method [10]–[15],matched filtering (MF)-based method [11], [13], [15], [16],and cyclostationary detection method [17]–[19], each of whichhas different requirements and advantages/disadvantages. Al-though LRT is proven to be optimal, it is very difficult touse, because it requires exact channel information and dis-tributions of the source signal and noise. To use LRT fordetection, we need to obtain the channels and signal andnoise distributions first, which are practically intractable. TheMF-based method requires perfect knowledge of the channelresponses from the primary user to the receiver and accuratesynchronization (otherwise, its performance will dramaticallybe reduced) [15], [16]. As mentioned earlier, this may notbe possible if the primary users do not cooperate with thesecondary users. The cyclostationary detection method needsto know the cyclic frequencies of the primary users, which maynot be realistic for many spectrum reuse applications. Further-more, this method demands excessive analog-to-digital (A/D)converter requirements and signal processing capabilities [11].

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ZENG AND LIANG: SPECTRUM-SENSING ALGORITHM FOR COGNITIVE RADIO BASED ON STATISTICAL COVARIANCE 1805

Energy detection, unlike the two other methods, does not needany information of the signal to be detected and is robustto unknown dispersed channels and fading. However, energydetection requires perfect knowledge of the noise power. Wrongestimation of the noise power leads to an SNR wall and highprobability of false alarm [10]–[12], [15], [20]. As pointed outearlier, the estimated noise power could be quite inaccurate dueto noise uncertainty. Thus, the main drawback for the energy de-tection is its sensitiveness to noise uncertainty [10]–[12], [15].Furthermore, while energy detection is optimal for detecting anindependent and identically distributed (i.i.d.) signal [13], it isnot optimal for detecting a correlated signal, which is the casefor most practical applications.

In this paper, to overcome the shortcoming of energy detec-tion, we propose new methods based on the statistical covari-ances or autocorrelations of the received signal. The statisticalcovariance matrices or autocorrelations of signal and noise aregenerally different. Thus, this difference is used in the proposedmethods to differentiate the signal component from backgroundnoise. In practice, there are only a limited number of signalsamples. Hence, the detection methods are based on the samplecovariance matrix. The steps of the proposed methods are givenas follows: First, the sample covariance matrix of the receivedsignal is computed based on the received signal samples. Then,two test statistics are extracted from the sample covariancematrix. Finally, a decision on the presence of the signal ismade by comparing the ratio of two test statistics with athreshold. Theoretical analysis for the proposed algorithms isgiven. Detection probability and the associated threshold forthe decision are found based on the statistical theory. Themethods do not need any information of the signal, channel,and noise power a priori. In addition, no synchronization isneeded. Simulations based on narrow-band signals, captureddigital television (DTV) signals, and multiple antenna signalsare presented to evaluate the performance of the proposedmethods.

The rest of this paper is organized as follows: The detectionalgorithms and theoretical analysis are presented in Section II.Section III gives the performance analysis and finds thresholdsfor the algorithms. A theoretical comparison with the energydetection is also discussed in this section. Simulation results forvarious types of signals are given in Section IV. Conclusions aredrawn in Section V. Finally, a prewhitening technique is givenin the Appendix.

Some of the notation we use is as follows: Boldface lettersare used to denote matrices and vectors, superscript (·)T standsfor transpose, Iq denotes the identity matrix of order q, and E[·]stands for expectation operation.

II. COVARIANCE-BASED DETECTIONS

Let xc(t) = sc(t) + ηc(t) be the continuous-time receivedsignal, where sc(t) is the possible primary user’s signal andηc(t) is the noise. ηc(t) is assumed to be a stationary processsatisfying E(ηc(t)) = 0, E(η2

c (t)) = σ2η, and E(ηc(t)ηc(t +

τ)) = 0 for any τ �= 0. Assume that we are interested in thefrequency band with central frequency fc and bandwidth W .We sample the received signal at a sampling rate fs, where

fs ≥ W . Let Ts = 1/fs be the sampling period. For notationsimplicity, we define x(n) Δ= xc(nTs), s(n) Δ= sc(nTs), andη(n) Δ= ηc(nTs). There are two hypotheses: 1) H0, i.e., thesignal does not exist, and 2) H1, i.e., the signal exists. Thereceived signal samples under the two hypotheses are given by[11], [12], [15], and [16]

H0 : x(n) = η(n) (1)

H1 : x(n) = s(n) + η(n) (2)

respectively, where s(n) is the transmitted signal samples thatpassed through a wireless channel consisting of path loss, mul-tipath fading, and time dispersion effects; and η(n) is the whitenoise, which is i.i.d., having mean zero and variance σ2

η. Notethat s(n) can be the superposition of the received signals frommultiple primary users. No synchronization is needed here.

Two probabilities are of interest for spectrum sensing:1) probability of detection Pd, which defines, at hypothesis H1,the probability of the sensing algorithm having detected thepresence of the primary signal, and 2) probability of false alarmPfa, which defines, at hypothesis H0, the probability of thesensing algorithm claiming the presence of the primary signal.

A. CAV Detection

Let us consider L consecutive samples and define the follow-ing vectors:

x(n) = [x(n) x(n − 1) · · · x(n − L + 1) ]T (3)

s(n) = [ s(n) s(n − 1) · · · s(n − L + 1) ]T (4)

η(n) = [ η(n) η(n − 1) · · · η(n − L + 1) ]T . (5)

Parameter L is called the smoothing factor in the following.Considering the statistical covariance matrices of the signal andnoise defined as

Rx = E[x(n)xT (n)

](6)

Rs =E[s(n)sT (n)

](7)

we can verify that

Rx = Rs + σ2ηIL. (8)

If signal s(n) is not present, Rs = 0. Hence, the off-diagonalelements of Rx are all zeros. If there is a signal and the signalsamples are correlated, Rs is not a diagonal matrix. Hence,some of the off-diagonal elements of Rx should be nonzeros.Denote rnm as the element of matrix Rx at the nth row andmth column, and let

T1 =1L

L∑n=1

L∑m=1

|rnm| (9)

T2 =1L

L∑n=1

|rnn|. (10)

Then, if there is no signal, T1/T2 = 1. If the signal is present,T1/T2 > 1. Hence, ratio T1/T2 can be used to detect thepresence of the signal.



In practice, the statistical covariance matrix can only becalculated using a limited number of signal samples. Define thesample autocorrelations of the received signal as

λ(l)=1

Ns

Ns−1∑m=0

x(m)x(m − l), l=0, 1, . . . , L − 1 (11)

where Ns is the number of available samples. Statistical covari-ance matrix Rx can be approximated by the sample covariancematrix defined as

Rx(Ns) =

⎡⎢⎢⎣

λ(0) λ(1) · · · λ(L − 1)λ(1) λ(0) · · · λ(L − 2)

......

. . ....

λ(L − 1) λ(L − 2) · · · λ(0)

⎤⎥⎥⎦ . (12)

Note that the sample covariance matrix is symmetric andToeplitz. Based on the sample covariance matrix, we proposethe following signal detection method:

Algorithm 1: Covariance Absolute Value (CAV) DetectionAlgorithm

Step 1) Sample the received signal, as previously described.Step 2) Choose a smoothing factor L and a threshold γ1,

where γ1 should be chosen to meet the requirementfor the probability of false alarm. This will be dis-cussed in the next section.

Step 3) Compute the autocorrelations of the received signalλ(l), l = 0, 1, . . . , L − 1, and form the sample co-variance matrix.

Step 4) Compute

T1(Ns) =1L

L∑n=1

L∑m=1

|rnm(Ns)| (13)

T2(Ns) =1L

L∑n=1

|rnn(Ns)| (14)

where rnm(Ns) are the elements of the samplecovariance matrix Rx(Ns).

Step 5) Determine the presence of the signal based onT1(Ns), T2(Ns), and threshold γ1. That is, ifT1(Ns)/T2(Ns) > γ1, the signal exists; otherwise,the signal does not exist.

Remark: The statistics in the algorithm can directly becalculated from autocorrelations λ(l). However, for betterunderstanding and easing the mathematical derivation for theprewhitening later in the Appendix, here, we choose to use thecovariance matrix expression.

B. Theoretical Analysis for the CAV Algorithm

The proposed method only uses the received signal samples.It does not need any information of the signal, channel, andnoise power a priori. In addition, no synchronization is needed.

The validity of the proposed CAV algorithm relies on theassumption that the signal samples are correlated, i.e., Rs isnot a diagonal matrix. (Some of the off-diagonal elements ofRs should be nonzeros.) Obviously, if signal samples s(n) arei.i.d., then Rs = σ2

sIL. In this case, the assumption is invalid,and the algorithm cannot detect the presence of the signal.

However, usually, the signal samples should be correlateddue to three reasons.

1) The signal is oversampled. Let T0 be the Nyquist sam-pling period of signal sc(t) and sc(nT0) be the sampledsignal based on the Nyquist sampling rate. Based on thesampling theorem, signal sc(t) can be expressed as

sc(t) =∞∑

n=−∞sc(nT0)g(t − nT0) (15)

where g(t) is an interpolation function. Hence, the signalsamples s(n) = sc(nTs) are only related to sc(nT0). Ifthe sampling rate at the receiver fs > 1/T0, i.e., Ts < T0,then, s(n) = sc(nTs) must be correlated. An exampleof this is a narrow-band signal, such as the wirelessmicrophone signal. In a 6-MHz-bandwidth TV band, awireless microphone signal only occupies about 200 kHz.When we sample the received signal with a sampling rateof not lower than 6 MHz, the wireless microphone signalis actually oversampled and, therefore, highly correlated.

2) The propagation channel has time dispersion; thus, theactual signal component at the receiver is given by

sc(t) =

∞∫−∞

h(τ)s0(t − τ)dτ (16)

where s0(t) is the original transmitted signal, and h(t)is the response of the time-dispersive channel. Sincesampling period Ts is usually very small, the integration(16) can be approximated as

sc(t) ≈ Ts

∞∑k=−∞

h(kTs)s0(t − kTs). (17)

Hence

sc(nTs) ≈ Ts

K1∑k=K0

h(kTs)s0 ((n − k)Ts) (18)

where [K0Ts,K1Ts] is the support of channel responseh(t), i.e., h(t) = 0 for t �∈ [K0Ts,K1Ts]. For the time-dispersive channel, K1 > K0; thus, the signal samplessc(nTs) are correlated, even if the original signal sampless0(nTs) could be i.i.d.

3) The original signal is correlated. In this case, even if thechannel is a flat-fading channel and there is no oversam-pling, the received signal samples are correlated.

Another assumption for the algorithm is that the noisesamples are i.i.d. This is usually true if no filtering is used.However, if a narrow-band filter is used at the receiver, the



noise samples will sometimes be correlated. To deal with thiscase, we need to prewhiten the noise samples or pretransformthe covariance matrix. A method is given in the Appendix tosolve this problem.

The computational complexity of the algorithm is given asfollows: Computing the autocorrelations of the received signalrequires about LNs multiplications and additions. ComputingT1(Ns) and T2(Ns) requires about L2 additions. Therefore,the total number of multiplications and additions is aboutLNs + L2.

C. Generalized Covariance-Based Algorithms

Based on the same principle as CAV, generalized covariance-based methods can be designed to detect the signal. Let ψ1

and ψ2 be two nonnegative functions with multiple variables.Assume that

ψ1(a) > 0, for a �= 0, ψ1(0) = 0

ψ2(b) > 0, for b �= 0, ψ2(0) = 0.

Then, the following method can be used for signal detection:

Algorithm 2: Generalized Covariance-Based DetectionStep 1) Sample the received signal, as previously described.Step 2) Choose a smoothing factor L and a threshold γ2,

where γ2 should be chosen to meet the requirementfor the probability of false alarm.

Step 3) Compute sample covariance matrix Rx(Ns).Step 4) Compute

T4(Ns) = ψ2 (rnn(Ns), n = 1, . . . , L) (19)

T3(Ns) = T4(Ns) + ψ1 (rnm(Ns), n �= m) . (20)

Step 5) Determine the presence of the signal based onT3(Ns), T4(Ns), and threshold γ2. That is, ifT3(Ns)/T4(Ns) > γ2, the signal exists; otherwise,the signal does not exist.

Obviously, the CAV algorithm is a special case of thegeneralized method when ψ1 and ψ2 are absolute summationfunctions. As another example, we can choose ψ1(a) = aT aand ψ2(b) = bT b. For this choice

T3(Ns) =1L

L∑n=1

L∑m=1

|rnm(Ns)|2 (21)

T4(Ns) =1L

L∑n=1

|rnn(Ns)|2 . (22)

D. Spectrum Sensing Using Multiple Antennas

Multiple-antenna systems have widely been used to increasethe channel capacity or improve the transmission reliabilityin wireless communications. In the following, we assume thatthere are M > 1 antennas at the receiver and exploit the re-ceived signals from these antennas for spectrum sensing. In thiscase, the received signal at antenna i is given by

H0 : xi(n) = ηi(n) (23)

H1 : xi(n) = si(n) + ηi(n). (24)

In hypothesis H1, si(n) is the signal component received byantenna i. Since all si(n)’s are generated from the same sourcesignal, the si(n)’s are correlated for i. It is assumed that theηi(n)’s are i.i.d. for n and i.

Let us combine all the signals from the M antennas and de-fine the vectors in (25)–(27), as shown in the bottom of the page.Note that (3)–(5) are a special case (M = 1) of the precedingequations. Defining the statistical covariance matrices in thesame way as those in (6) and (7), we obtain

Rx = Rs + σ2ηIML. (28)

Except for the different matrix dimensions, the precedingequation is the same as (8). Hence, the CAV algorithm andgeneralized covariance-based method previously described candirectly be used for the multiple-antenna case.

Let s0(n) be the source signal. The received signal atantenna i is

si(n) =Ni∑

k=0

hi(k)s0(n − k) + ηi(n), i = 1, 2, . . . ,M

(29)

where hi(k) is the channel responses from the source user toantenna i at the receiver. Define

h(n) = [h1(n), h2(n), . . . , hM (n)]T (30)

H =

⎡⎢⎣

h(0) · · · · · · h(N) · · · 0. . .

. . .0 · · · h(0) · · · · · · h(N)

⎤⎥⎦ (31)

where N = maxi(Ni), and hi(n) is zero padded if Ni < N .Note that the dimension of H is ML × (N + L). We have

Rs = HRs0HT (32)

x(n) = [x1(n) · · · xM (n) x1(n − 1) · · · xM (n − 1) · · · x1(n − L + 1) · · · xM (n − L + 1) ]T (25)

s(n) = [ s1(n) · · · sM (n) s1(n − 1) · · · sM (n − 1) · · · s1(n − L + 1) · · · sM (n − L + 1) ]T (26)

η(n) = [ η1(n) · · · ηM (n) η1(n − 1) · · · ηM (n − 1) · · · η1(n − L + 1) · · · ηM (n − L + 1) ]T (27)



where Rs0 = E(s0sT0 ) is the statistical covariance matrix of the

source signal, where

s0 = [s0(n) s0(n−1) · · · s0(n−N−L + 1)]T . (33)

Note that the received signals at different antennas are cor-related. Hence, using multiple antennas, increase the correla-tions among the signal samples at the receiver and make thealgorithms valid at all cases. In fact, at worst case, when allthe channels are flat fading, i.e., N1 = N2 = · · · = NM = 0,and the source signal sample s0(n) is i.i.d., we haveRs = σ2

sHHT , where H is an ML × L matrix, as previously

defined. Obviously, Rs is not a diagonal matrix, and the algo-rithms can work.

III. PERFORMANCE ANALYSIS AND

THRESHOLD DETERMINATION

For a good detection algorithm, Pd should be high, and Pfa

should be low. The choice of threshold γ is a compromisebetween Pd and Pfa. Since we have no information on the signal(we do not even know if there is a signal or not), it is difficultto set the threshold based on Pd. Hence, usually, we choose thethreshold based on Pfa. The steps are given as follows: First,we set a value for Pfa. Then, we find a threshold γ to meet therequired Pfa. To find the threshold based on the required Pfa, wecan use either theoretical derivation or computer simulation. Ifsimulation is used to find the threshold, we can generate whiteGaussian noises as the input (no signal) and adjust the thresholdto meet the Pfa requirement. Note that the threshold here isrelated to the number of samples used for computing the sampleautocorrelations and the smoothing factor L but is not relatedto the noise power. If theoretical derivation is used, we needto find the statistical distribution of T1(Ns)/T2(Ns), which isgenerally a difficult task. In this section, using the central limittheorem, we will find the approximations for the distribution ofthis random variable and provide the theoretical estimations forthe two probabilities Pd and Pfa and the threshold associatedwith these probabilities.

A. Statistics Computation

Based on the symmetric property of the covariance matrix,we can rewrite T1(Ns) and T2(Ns) in (13) and (14) as

T1(Ns) =λ(0) +2L

L−1∑l=1

(L − l) |λ(l)| (34)

T2(Ns) =λ(0). (35)

Define

Xl = [x(Ns − 1 − l) · · · x(−l)]T (36)

Sl = [s(Ns − 1 − l) · · · s(−l)]T (37)

ηl = [η(Ns − 1 − l) · · · η(−l)]T . (38)

Let the normalized correlation among the signal samples be

αl = E [s(n)s(n − l)] /σ2s (39)

where σ2s is the signal power, i.e., σ2

s = E[s2(n)]. |αl| de-fines the correlation strength among the signal samples; here,0 � |αl| � 1. Based on the notations, we have

λ(l) =1

NsXT

0 Xl =1

Ns

(ST

0 + ηT0

)(Sl + ηl)

=1

Ns

(ST

0 Sl + ST0 ηl + ηT

0 Sl + ηT0 ηl

). (40)

Obviously

E (λ(0)) =σ2s + σ2

η (41)

E (λ(l)) =αlσ2s , l = 1, 2, . . . , L − 1. (42)

Now, we need to find the variance of λ(l). Since

λ2(l) =1

N2s

(ST

0 Sl + ST0 ηl + ηT

0 Sl + ηT0 ηl

)2

=1

N2s

[ (ST

0 Sl

)2+

(ST

0 ηL

)2+

(ηT

0 Sl

)2+

(ηT

0 ηL

)2

+ 2(ST

0 Sl

) (ST

0 ηl

)+ 2

(ST

0 Sl

) (ηT

0 Sl

)+ 2

(ST

0 Sl

) (ηT

0 ηl

)+ 2

(ST

0 ηl

) (ηT

0 Sl

)+ 2

(ST

0 ηl

) (ηT

0 ηl

)+ 2

(ηT

0 Sl

) (ηT

0 ηl

) ](43)

it can be verified that

E((

ST0 ηl

)2)

= E((

ηT0 Sl

)2)

= Nsσ2sσ2

η (44)

E((

ηT0 η0

)2)

=(N2

s + 2Ns

)σ4

η (45)

E((

ηT0 ηl

)2)

=Nsσ4η, l = 1, . . . , L − 1 (46)

E((

ST0 Sl

) (ST

0 ηl

))= E

((ST

0 Sl

) (ηT

0 Sl

))= E

((ST

0 ηl

) (ηT

0 ηl

))= E

((ηT

0 Sl

) (ηT

0 ηl

))= 0 (47)

E((

ST0 S0

) (ηT

0 η0

))= N2

s σ2sσ2

η (48)

E((

ST0 Sl

) (ηT

0 ηl

))= 0, l = 1, 2, . . . , L − 1 (49)

E((

ST0 ηl

) (ηT

0 Sl

))= α2l(Ns − l)σ2

sσ2η (50)

E(ST

0 Sl

)= αlσ

2s . (51)

Based on these results, we can easily obtain the following twolemmas.

Lemma 1: When there is no signal, we have

E (λ(0)) =σ2η var (λ(0)) =

2Ns

σ4η (52)

E (λ(l)) = 0 var (λ(l)) =1

Nsσ4

η, l = 1, . . . , L − 1.

(53)



Lemma 2: When there is a signal, we have

E (λ(0)) =σ2s + σ2

η (54)

var (λ(0)) = var(

1Ns

ST0 S0

)+

2σ2η

Ns

(2σ2

s + σ2η

)(55)

E (λ(l)) =αlσ2s (56)

var (λ(l)) = var(

1Ns

ST0 Sl

)

+σ2

η

Ns

(σ2

η + 2σ2s +

2(Ns − l)α2l

Nsσ2

s

),

l = 1, . . . , L − 1. (57)

Note that var((1/Ns)ST0 Sl), l = 0, 1, . . . , L − 1 depends on

the signal properties.For simplicity, we denote E(λ(l)) by Θl and var(λ(l)) by

Δl. Note that, usually, Ns is very large. Based on the centrallimit theorem, λ(l) can be approximated by the Gaussiandistribution.

Lemma 3: When the signal is not present, we have

E (|λ(l)|) =√

2πNs

σ2η, l = 1, 2, . . . , L − 1. (58)

When the signal is present, we have

E (|λ(l)|) =

√2Δl

π

(2 − e

−Θ2

l2Δl

)

+ |Θl|

⎛⎜⎝1 −

√2π

+∞∫|Θl|/

√Δl

e−u22 du

⎞⎟⎠

l = 1, 2, . . . , L − 1. (59)

For a large Ns and a low SNR

E (|λ(l)|)≈√

2πNs

(σ2

s + σ2η

) (2−e

−τ2l

2

)

+ |Θl|

⎛⎝1−

√2π

+∞∫τl

e−u22 du

⎞⎠ , l=1, 2, . . . , L − 1 (60)

where

τl =|αl|SNR

√Ns

1 + SNR, SNR =

σ2s

σ2η

. (61)

Proof: Based on the central limit theorem, we have

E (|λ(l)|) =1√

2πΔl

+∞∫−∞

|u|e−(u−Θl)

2

2Δl du

=1√2π

+∞∫−∞

|√

Δlu + Θl|e−u22 du

=1√2π

−Θl/√

Δl∫−∞

(−√

Δlu − Θl)e−u22 du

+1√2π

+∞∫−Θl/

√Δl

(√

Δlu + Θl)e−u22 du

=

√2Δl

π

+∞∫0

ue−u22 du +

√2Δl

π

0∫−Θl/

√Δl

ue−u22du

+2Θl√2π

0∫−Θl/

√Δl

e−u22 du

=

√2Δl

π

(2 − e

−Θ2

l2Δl

)+

2Θl√2π

0∫−Θl/

√Δl

e−u22 du

=

√2Δl

π

(2 − e

−Θ2

l2Δl

)

+ |Θl|

⎛⎜⎝1 −

√2π

+∞∫|Θl|/

√Δl

e−u22 du

⎞⎟⎠ . (62)

For a large Ns and a low SNR

Δl ≈(σ2

s + σ2η)2

Ns

|Θl|√Δl

≈ τl.

Thus, we obtain (60).When there is no signal, Θl = 0, and Δl = (1/Ns)σ4

η.Hence, we obtain (58). �

Theorem 1: When there is no signal, we have

E (T1(Ns)) =(

1 + (L − 1)√

2πNs

)σ2

η (63)

E (T2(Ns)) = σ2η (64)

var (T2(Ns)) =2

Nsσ4

η. (65)

When there is a signal and for a large Ns, we have

E (T1(Ns)) ≈σ2s + σ2

η

+2σ2

s

L

L−1∑l=1

(L − l)|αl|

⎛⎝1−

√2π

+∞∫τl

e−u22 du

⎞⎠

+2(σ2

s +σ2η)

L

L−1∑l=1

(L−l)√

2πNs

(2−e

−τ2l

2

)(66)

E (T2(Ns)) = σ2s +σ2

η (67)

var (T2(Ns)) = var(

1Ns

ST0 S0

)+

2σ2η

Ns

(2σ2

s + σ2η

). (68)

For a large Ns, T1(Ns) and T2(Ns) approach Gaussiandistributions.



Proof: Equations (63)–(68) are direct results fromLemmas 1–3. Noting that λ(l) is a summation of Ns randomvariables, when Ns is large, based on the central limit theorem,it can be approximated by Gaussian distributions. From thedefinition of T1(Ns) and T2(Ns), we know that they alsoapproach Gaussian distributions. �

B. Detection Probability and the Associated Threshold

From the preceding theorem, we have

limNs→∞

E (T1(Ns)) = σ2s + σ2

η +2σ2

s

L

L−1∑l=1

(L − l)|αl|. (69)

For simplicity, we denote

ΥLΔ=

2L

L−1∑l=1

(L − l)|αl| (70)

which is the overall correlation strength among the consecutiveL samples. When there is no signal, we have

T1(Ns)/T2(Ns) ≈E (T1(Ns)) /E (T2(Ns))

= 1 + (L − 1)√

2πNs

. (71)

Note that this ratio approaches 1 as Ns approaches infinity. Inaddition, note that the ratio is not related to the noise power(variance). On the other hand, when there is a signal (the signal-plus-noise case), we have

T1(Ns)/T2(Ns) ≈E (T1(Ns)) /E (T2(Ns))

≈ 1 +σ2

s

σ2s + σ2

η

ΥL (72)

= 1 +SNR

SNR + 1ΥL. (73)

Here, the ratio approaches a number that is larger than 1as Ns approaches infinity. The number is determined by thecorrelation strength among the signal samples and the SNR.Hence, for any fixed SNR, if there are a sufficiently largenumber of samples, we can always differentiate if there is asignal or not based on the ratio.

However, in practice, we have only a limited number ofsamples. Thus, we need to evaluate the performance at fixedNs. First, we analyze the Pfa at hypothesis H0. The probabilityof false alarm for the CAV algorithm is

Pfa =P (T1(Ns) > γ1T2(Ns))

=P

(T2(Ns) <

1γ1

T1(Ns))

≈P

(T2(Ns) <

1γ1

(1 + (L − 1)

√2

Nsπ

)σ2

η

)

= P

⎛⎝T2(Ns) − σ2

η√2

Nsσ2

η

<

1γ1

(1 + (L − 1)

√2

Nsπ

)− 1√

2/Ns

⎞⎠

≈ 1 − Q

⎛⎝ 1

γ1

(1 + (L − 1)

√2

Nsπ

)− 1√

2/Ns

⎞⎠

where

Q(t) =1√2π

+∞∫t

e−u2/2du. (74)

For a given Pfa, the associated threshold should be chosensuch that

1γ1

(1 + (L − 1)

√2

Nsπ

)− 1√

2/Ns

= −Q−1(Pfa). (75)

That is

γ1 =1 + (L − 1)

√2

Nsπ

1 − Q−1(Pfa)√

2Ns

. (76)

Note that, here, the threshold is not related to noise power andSNR. After the threshold is set, we now calculate the proba-bility of detection at various SNRs. For the given threshold γ1,when the signal is present

Pd =P (T1(Ns) > γ1T2(Ns))

=P

(T2(Ns) <

1γ1

T1(Ns))

≈P

(T2(Ns) <

1γ1

E (T1(Ns)))

=P

(T2(Ns) − σ2

s − σ2η√

var (T2(Ns))<

1γ1

E (T1(Ns)) − σ2s − σ2

η√var (T2(Ns))

)

= 1 − Q

(1γ1

E (T1(Ns)) − σ2s − σ2

η√var (T2(Ns))

). (77)

For a very large Ns and a low SNR

var (T2(Ns)) ≈2σ2

η

Ns

(2σ2

s + σ2η

)≈

2(σ2

s + σ2η

)2

Ns

E (T1(Ns)) ≈σ2s + σ2

η + σ2sΥL.

Hence, we have a further approximation, i.e.,

Pd ≈ 1 − Q

⎛⎜⎝

1γ1

+ ΥLσ2s

γ1(σ2s+σ2

η)− 1√

2/Ns

⎞⎟⎠

= 1 − Q

⎛⎝ 1

γ1+ ΥLSNR

γ1(SNR+1)− 1√

2/Ns

⎞⎠ . (78)



TABLE IPROBABILITIES OF FALSE ALARM

Obviously, Pd increases with the number of samples Ns, theSNR, and the correlation strength among the signal samples.Note that γ1 is also related to Ns, as previously shown, andlimNs→∞ γ1 = 1. Hence, for fixed SNR, Pd approaches 1 whenNs approaches infinity.

For a target pair of Pd and Pfa, based on (78) and (76), wecan find the required number of samples as

Nc ≈2(Q−1(Pfa) − Q−1(Pd) + (L − 1)/

√π)2

(ΥLSNR)2. (79)

For fixed Pd and Pfa, Nc is only related to the smoothingfactor L and the overall correlation strength ΥL. Hence, thebest smoothing factor is

Lbest = minL

{Nc} (80)

which is related to the correlation strength among the signalsamples.

C. Comparison With Energy Detection

Energy detection is the basic sensing method, which wasfirst proposed in [14] and further studied in [10]–[12] and[15]. It does not need any information of the signal to bedetected and is robust to unknown dispersive channels. Energydetection compares the average power of the received signalwith the noise power to make a decision. To guarantee a reliabledetection, the threshold must be set according to the noisepower and the number of samples [10]–[12]. On the other hand,the proposed methods do not rely on the noise power to setthe threshold [see (76)] while keeping the other advantages ofenergy detection.

Accurate knowledge on the noise power is then the key of theenergy detection. Unfortunately, in practice, noise uncertaintyis always present. Due to noise uncertainty [10]–[12], theestimated (or assumed) noise power may be different from thereal noise power. Let the estimated noise power be σ2

η = ασ2η.

We define the noise uncertainty factor (in decibels) as

B = sup{10 log10 α}. (81)

It is assumed that α (in decibels) is evenly distributed in aninterval [−B,B] [11]. In practice, the noise uncertainty factorof a receiving device normally ranges from 1 to 2 dB [11], [20].Environment/interference noise uncertainty can be much higher[11]. When there is noise uncertainty, the energy detection isnot effective [10]–[12], [20]. The simulation presented in thenext section also shows that the proposed method is much betterthan the energy detection when noise uncertainty is present.Hence, here, we only compare the proposed method with idealenergy detection (without noise uncertainty).

To compare the performances of the two methods, we firstneed a criterion. By properly choosing the thresholds, bothmethods can achieve any given Pd and Pfa > 0 if a sufficiently

large number of samples are available. The key point is howmany samples are needed to achieve the given Pd and Pfa > 0.Hence, we choose this as the criterion for comparing thetwo algorithms. For energy detection, the required number ofsamples is approximately [11]

Ne =2(Q−1(Pfa) − Q−1(Pd)

)2

SNR2 . (82)

Comparing (79) and (82), if we want Nc < Ne, we need

ΥL > 1 +L − 1√

π (Q−1(Pfa) − Q−1(Pd)). (83)

For example, if Pd = 0.9 and Pfa = 0.1, we need ΥL > 1 +(L − 1/4.54). In conclusion, if the signal samples are highlycorrelated such that (83) holds, CAV detection is better thanideal energy detection; otherwise, ideal energy detection isbetter.

In terms of the computational complexity, the energy detec-tion needs about Ns multiplications and additions. Hence, thecomputational complexity of the proposed methods is about Ltimes that of the energy detection.

IV. SIMULATION AND DISCUSSION

In this section, we will give some simulation results for threesituations: 1) narrow-band signals; 2) captured DTV signals[21]; and 3) multiple antenna received signals.

First, we simulate the probabilities of false alarm Pfa, be-cause Pfa is not related to the signal. (At H0, there is no signalat all.) We set the target Pfa = 0.1 and choose L = 10 andNs = 50 000. We then obtain the thresholds based on Pfa, L,and Ns. The threshold for energy detection is given in [11].Table I gives the simulation results for various cases, where, andin the following, “EG-x dB” means the energy detection withx-dB noise uncertainty. The Pfa’s for the proposed method andenergy detection without noise uncertainty meet the target, butthe Pfa for the energy detection with noise uncertainty (even aslow as 0.5 dB) far exceeds the limit. This means that the energydetection is very unreliable in practical situations with noiseuncertainty.

Second, we fix the thresholds based on Pfa and simulate theprobability of detection Pd for various cases. We consider twosignal types.

1) Narrow-band signals: A frequency-modulated wirelessmicrophone signal is used here (soft speaker) [22]. Thecentral frequency is fc = 100 MHz. The sampling rateat the receiver is 6 MHz (the same as the TV band-width in the USA). Fig. 1 shows the simulation results.(The corresponding Pfa is shown in Table I.) Notethat “CAV-theo” means the theoretical results given inSection III-B. Due to some approximations, the theoret-ical results do not exactly match the simulated results.



Fig. 1. Probability of detection for a wireless microphone signal:Ns = 50 000.

Fig. 2. Pd versus Pfa for a wireless microphone signal: Ns = 50 000, andSNR = −20 dB.

CAV detection is better than ideal energy detection (with-out noise uncertainty), which verifies our assertion inSection III-C. The reason is that, as we pointed out inSection II-B, the source signal is a narrow-band signal;therefore, their samples are highly correlated. As shownin the figure, if there is noise uncertainty, the Pd of theenergy detection is much worse than that of the proposedmethod. Fig. 2 shows the receiver operating characteristiccurve (Pd versus Pfa) at fixed SNR = −20 dB. Theperformances of the methods at different sample sizes(sensing times) are given in Fig. 3. It is clear that CAVdetection is always better than ideal energy detection.

To test the impact of the smoothing factor, we fixSNR = −20 dB, Pfa = 0.01, and Ns = 50 000 and varythe smoothing factor L from 4 to 14. Fig. 4 shows theresults for Pd. We see that Pd is not very sensitive tothe smoothing factor for L ≥ 8. Noting that a smaller Lmeans lower complexity, in practice, we can choose

Fig. 3. Pd versus sample size Ns for a wireless microphone signal:Pfa = 0.01, and SNR = −20 dB.

Fig. 4. Pd versus the smoothing factor for a wireless microphone signal:Pfa = 0.01, Ns = 50 000, and SNR = −20 dB.

a relatively small L. However, it is very difficult tochoose the best L, because it is related to the signalproperty (unknown). Note that energy detection is notaffected by L.

2) Captured DTV signals. The real DTV signals (fieldmeasurements) are collected at Washington, DC. Thedata rate of the vestigial sideband DTV signal is10.762 Msample/s. The recorded DTV signals were sam-pled at 21.524476 Msample/s and downconverted to alow central intermediate frequency of 5.381119 MHz(one fourth of the sampling rate). The A/D conversionof the radio-frequency signal used a 10-bit or a 12-bitA/D converter. Each sample was encoded into a 2-B word(signed int16 with two’s complement format). The mul-tipath channel and SNR of the received signal are un-known. To use the signals for simulating the algorithmsat a very low SNR, we need to add white noise to obtainvarious SNR levels [23].



Fig. 5. Probability of detection for the DTV signal WAS-051/35/01:Ns = 50 000.

Fig. 5 shows the simulation results based on the DTV signalfile WAS-051/35/01. (The receiving antenna is outside andlocated 20.29 mi from the DTV station; the antenna height is30 ft.) [21]. The corresponding Pfa is shown in Table I. If thenoise variance is exactly known (B = 0), the energy detectionis better than the proposed method. However, as discussed in[10]–[12], noise uncertainty is always present. Even if the noiseuncertainty is only 0.5 dB, the Pd of the energy detection ismuch worse than that of the proposed method.

In summary, all the preceding simulations show that the pro-posed method works well without using the information aboutthe signal, channel, and noise power. The energy detection isnot reliable (i.e., with low probability of detection and highprobability of false alarm) when there is noise uncertainty.

Third, we simulate the proposed algorithms with multipleantennas/receivers. We consider a system with four receivingantennas. Assume that the antennas are well separated (withthe separation larger than a half-wavelength) such that theirchannels are independent. This assumption is only for sim-plicity. In fact, the proposed algorithms perform better if thechannels are correlated. Assume that each multipath channelhi(k) has five taps (Ni = 4) and that all the channel taps areindependent with equal power. The channel taps are generatedas Gaussian random numbers and different for different MonteCarlo realizations. Source signal s0(n) is i.i.d. and binaryphase-shift keying modulated. The received signal at antennai is defined in (29). The smoothing factor is L = 8, and thenumber of samples at each antenna is Ns = 25 000. We fix thePfa = 0.1 at all cases. Fig. 6 shows the Pd for three cases. Fromthe figure, we see that, when only one antenna’s signal is used(M = 1), the method still works. This verifies our assertion inSection II-B that the method is valid, even if the inputs are i.i.d.,but the channel is dispersive. When the signals of two (M = 2)or four (M = 4) antennas are combined based on the methodin Section II-D, Pd is much better. The more the antennasused, the better the Pd. The optimal LRT [13] detection forM = 4 is also included as an upper bound for any detectionmethods.

Fig. 6. Probability of detection using multiple antennas: Pfa = 0.1, andNs = 25 000, with one source signal.

Fig. 7. Probability of detection for time variant channels: M = 2, Pfa = 0.1,and Ns = 25 000.

To check the performance of the methods for time-variantchannels, we give a simulation result here. The time-variantchannel is generated based on the simplified Jake’s model. Letfd be the normalized maximum Doppler frequency (DF). Thetime-variant channel for simulation is defined as

hi(n, k)=exp(

j2πn6−k

5fd

)hi(k), k=1, . . . , 5 (84)

where hi(k) is the time-invariant channel previously defined.For different DFs ranging from 0 to 10−2, the simulation resultis shown in Fig. 7 for M = 2. For fast time-variant channels,the performance of the proposed methods will degrade.

We also simulate the situation when there are multiple sourcesignals. Fig. 8 shows the results for the case of three sourcesignals. Compared with Fig. 6, here, the results do not changemuch. Hence, the proposed method is valid when there aremultiple source signals.



Fig. 8. Probability of detection using multiple antennas: Pfa = 0.1, andNs = 25 000, with three source signals.

V. CONCLUSION

In this paper, sensing algorithms based on the sample co-variance matrix of the received signal have been proposed.Statistical theories have been used to set the thresholds andobtain the probabilities of detection. The methods can be usedfor various signal detection applications without knowledge ofthe signal, channel, and noise power. Simulations based onthe narrow-band signals, captured DTV signals, and multipleantenna signals have been carried out to evaluate the perfor-mance of the proposed methods. It is shown that the proposedmethods are, in general, better than the energy detector whennoise uncertainty is present. Furthermore, when the receivedsignals are highly correlated, the proposed method is betterthan the energy detector, even if the noise power is perfectlyknown.

APPENDIX

At the receiving end, the received signal is sometimes filteredby a narrow-band filter. Therefore, the noise embedded in thereceived signal is also filtered. Let η(n) be the noise samplesbefore the filter, which are assumed to be i.i.d. Let f(k),k = 0, 1, . . . , K be the filter. After filtering, the noise samplesturn to

η(n) =K∑

k=0

f(k)η(n − k), n = 0, 1, . . . . (85)

Consider L consecutive outputs, and define

η(n) = [η(n), . . . , η(n − L + 1)]T . (86)

The statistical covariance matrix of the filtered noise becomes

Rη = E(η(n)η(n)T

)= σ2

ηFFT (87)

where F is an L × (L + K) matrix defined as

F=

⎡⎢⎣

f(0) · · · f(K − 1) f(K) · · · 0. . .

. . .0 · · · f(0) · · · · · · f(K)

⎤⎥⎦ . (88)

Let G = FFT . If an analog filter or both analog and digitalfilters are used, matrix G should be defined based on those filterproperties. Note that G is a positive-definite symmetric matrix.It can be decomposed to

G = Q2 (89)

where Q is also a positive-definite symmetric matrix. Hence,we can transform the statistical covariance matrix into

Q−1RηQ−1 = σ2ηIL. (90)

Note that Q is only related to the filter. This means that wecan always transform the statistical covariance matrix Rx in(6) (by using a matrix obtained from the filter) such that (8)holds when the noise has passed through a narrow-band filter.Furthermore, since Q is not related to signal and noise, we canprecompute its inverse Q−1 and store it for later usage.

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Yonghong Zeng (A’01–M’01–SM’05) received theB.S. degree from Peking University, Beijing, China,and the M.S. and Ph.D. degrees from the NationalUniversity of Defense Technology, Changsha, China.

He was an Assistant Professor and AssociateProfessor with the National University of DefenseTechnology until July 1999. From August 1999 toOctober 2004, he was a Research Fellow withNanyang Technological University, Singapore, andthen with University of Hong Kong, Hong Kong.Since November 2004, he has been with the Institute

for Infocomm Research, A*STAR, Singapore, as a Senior Research Fellow andthen as a Research Scientist. He has coauthored six books, including Trans-forms and Fast Algorithms for Signal Analysis and Representation (Boston,MA: Springer-Birkhäuser, 2003), and more than 60 refereed journal papers.He is the holder of four granted patents. His current research interests includesignal processing and wireless communication, particularly cognitive radioand software-defined radio, channel estimation, equalization, detection, andsynchronization.

Dr. Zeng was the recipient of the ministry-level Scientific and TechnologicalDevelopment Awards in China four times and the Institute of EngineersSingapore Prestigious Engineering Achievement Award in 2007. He servedas a technical program committee or an organizing committee member formany prestigious international conferences, such as the IEEE InternationalConference on Communications (2008), the IEEE Wireless Communicationsand Networking Conference (2007 and 2008), the IEEE Vehicular TechnologyConference (2008), and the Third International Conference on Cognitive RadioOriented Wireless Networks and Communications (CrownCom 2008).

Ying-Chang Liang (SM’00) received the Ph.D. de-gree in electrical engineering in 1993.

He is currently a Senior Scientist with the In-stitute for Infocomm Research (I2R), A*STAR,Singapore, where he has been leading research ac-tivities in the area of cognitive radio and cooperativecommunications and the standardization activities inIEEE 802.22 wireless regional networks, for whichhis team has made fundamental contributions inthe physical layer, medium-access-control layer, andspectrum-sensing solutions. He is also an Adjunct

Associate Professor with Nanyang Technological University, Singapore, andthe National University of Singapore (NUS) and an adjunct Professor with theUniversity of Electronic Science and Technology of China, Chengdu, China.Since 2004, he has been teaching graduate courses with NUS. From December2002 to December 2003, he was a Visiting Scholar with the Departmentof Electrical Engineering, Stanford University, Stanford, CA. He is a GuestEditor for the Computer Networks Journal Special Issue on Cognitive WirelessNetworks (Elsevier). He is the holder of six granted patents and more than15 pending patents. His research interests include cognitive radio, dynamicspectrum access, reconfigurable signal processing for broadband communica-tions, space-time wireless communications, wireless networking, informationtheory, and statistical signal processing.

Dr. Liang is an Associate Editor for the IEEE TRANSACTIONS ON

VEHICULAR TECHNOLOGY. He was an Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS from 2002 to 2005 andthe Lead Guest Editor of the IEEE JOURNAL ON SELECTED AREAS IN

COMMUNICATIONS Special Issue on Cognitive Radio: Theory and Applica-tions. He has served for various IEEE conferences as a technical programcommittee (TPC) member. He was the Publication Chair of the 2001 IEEEWorkshop on Statistical Signal Processing, a TPC Co-Chair of the 2006 IEEEInternational Conference on Communication Systems, a Panel Co-Chair ofthe 2008 IEEE Vehicular Technology Conference Spring (IEEE VTC Spring2008), a TPC Co-Chair of the Third International Conference on CognitiveRadio Oriented Wireless Networks and Communications (CrownCom 2008),the Deputy Chair of the 2008 IEEE Symposium on New Frontiers in DynamicSpectrum Access Networks (DySPAN 2008), and a Co-Chair of the ThematicProgram on Random Matrix Theory and its Applications in Statistics andWireless Communications of the Institute for Mathematical Sciences, NationalUniversity of Singapore, in 2006. He was the recipient of the Best Paper Awardsfrom the IEEE VTC in Fall 1999, the 2005 IEEE International Symposium onPersonal, Indoor, and Mobile Radio Communications, and the 2007 Institute ofEngineers Singapore Prestigious Engineering Achievement Award.


spectrum-sensing algorithms for cognitive radio based on statistical covariances

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