A Second–Order Statistical Method for SpectrumSensing in Correlated Shadowing and Fading

EnvironmentsKhalid Qaraqe‡ and Serhan Yarkan‡‡

‡Department of Electrical and Computer Engineering, Texas A & M University at Qatar,Texas A & M Engineering Building, Education City, Doha, 23874, Qatar

E–mail: kqaraqe@tamu.edu

‡‡Electrical & Computer Engineering, Texas A & M University214 Zachry Engineering Center, College Station, TX, 77843–3128

E–mail: syarkan@ece.tamu.edu

Abstract—Spectrum sensing is one of the most importanttasks of cognitive radios (CRs) in future wireless systems andof user equipments (UEs) in next generation wireless networks(NGWNs). Therefore, deciding whether a specific portion of radiofrequency (RF) spectrum is occupied or not is of paramountimportance for all sorts of future wireless communicationssystems. In this study, a spectrum sensing method that employs asecond–order statistical approach is proposed for detecting fastfading signals in spatially correlated shadowing environments.Analysis and performance results are presented along with thediscussion related to the performance comparison of energydetection method.

Index Terms—fast fading, mobility, shadowing, spectrum sens-ing

I. INTRODUCTION

One of the most important resources for radio communica-tions is the radio frequency (RF) spectrum. With the increasingdemand for wireless applications and services, various wirelessdevices and technologies need to utilize the same portionof the RF spectrum. For instance, Bluetooth, Wi–Fi (IEEE802.11b/g/n), ZigBee (IEEE 802.15.4), cordless telephones,home microwave ovens, and baby monitoring devices operateon industrial, scientific, and medical (ISM) band (at 2.4GHz).It is not difficult to predict that the number of technologiescoexisting on the same portion of RF spectrum will increaseas new wireless applications and services emerge.

At first glance such a dense coexistence seems to cause ascarcity in the RF spectrum. However, measurements revealthat the RF spectrum is actually underutilized rather thanbeing scarce [1]. Recently, cognitive radio (CR) is proposedas a remedy to overcome this spectral underutilization concern[2]. Although there is no formal definition for CR, the termimplies a radio that can sense, be aware of, and learn about itssurrounding environment and adapt its parameters accordingly[3]. In this regard, one of the fundamental tasks of CR is to beaware of the RF spectrum by steadily sensing it in order to takeadvantage of available spectral opportunities for transmission.

Even though the notion “spectrum sensing” is coined forCR, it is extremely important for next generation wirelessnetworks (NGWNs) as well. In NGWNs, frequency reuseof one (FRO) is considered to be the prominent deploymentstrategy to avoid expensive network planning process. How-ever, FRO comes at the expense of significant co–channelinterference (CCI) levels especially for the user equipments(UEs) residing in the vicinity of cell borders. SignificantCCI levels cause very poor signal reception due to a lowcarrier–to–interference ratio (C/I), reduced system capacity,more frequent handoffs, and dropped calls. Because CCI isinevitable in FRO schemes, it needs to be managed by the

following strategies: interference avoidance, interference min-imization, or interference cancellation. Therefore, identifyingthe unused radio resources,1 which can be regarded as a specialcase of spectrum sensing, plays a crucial role in interferencemanagement for NGWNs.

Spectrum sensing methods can coarsely be categorized asfollows: (C1) matched filtering, (C2) waveform–based sensing,(C3) cyclostationary– and autocorrelation–based (or second–and higher–order statistical methods based) sensing, and(C4) energy detection. Although matched filtering provides anoptimum way of spectrum sensing [4], it requires the knowl-edge of various characteristics of the transmitted signal such assignaling bandwidth, operating frequency, modulation type andorder, pulse shape, and frame/burst format. Waveform–basedsensing rests upon searching for some of the known character-istics of the transmitted signal such as pilot sequences or mid–ambles throughout the received signal. Searching operationis carried out by specially constructed templates via pattern–matching techniques [5, 6]. As in matched filtering, majordrawback of the methods falling into (C2) is the necessity ofhaving the knowledge of some distinct characteristics of thetransmitted signal. Cyclostationarity–based approaches striveto exploit the periodicity in the statistics of a signal such as itsautocorrelation [7, 8]. Despite the fact that it is a very powerfulmethod, cyclostationarity–based methods might be relativelymore computationally complex. Similar to cyclostationarity–based methods, correlation–based approaches can also beapplied to spectrum sensing problems [9, 10]. However, spatialcorrelation of shadowing needs to be taken into account, sinceit changes the statistical characteristics of signals and affectsthe performance. Energy detection, in contrast to (C1)–(C3),adopts a very simplistic approach by comparing the energyof the received signal with a threshold [11, 12]. In spite ofits performance shortcomings, it is evident that (C4) requiresrelatively less computationally complex operations comparedto the methods classified in (C1)–(C3).2

In this study, an autocorrelation–based spectrum sensingmethod is proposed for a mobile receiver which strives todetect the absence/presence of an unknown transmitted signalin a general radio propagation environment. The methodproposed takes into account second–order statistics of thepropagation mechanisms such as spatially correlated shad-owing and fast fading in the received signal. Considering

1For instance, such resources correspond to resource blocks (time–frequency units) in Third Generation Long Term Evolution (3GLTE) down-link.

2A very comprehensive survey regarding this topic along with some otherprominent methods can be found in [13, and references therein].

2010 IEEE 21st International Symposium on Personal Indoor and Mobile Radio Communications

978-1-4244-8016-6/10/$26.00 ©2010 IEEE 780

the fact that shadowing process changes slowly comparedto the fast fading process, the method proposed separatesthe statistics of these processes from the received signal byemploying a low–pass filter followed by logarithmic detectorand investigating second–order statistics of the output. It isshown that in the absence of an unknown signal, the methodproposed causes second–order characteristics of noise processto converge a constant that depends on the filtering duration.When an unknown signal is present, on the other hand,shadowing correlation never allows the second–order statisticsto cross below that constant and identifies the presence of theunknown signal. Numerical results also show that the methodproposed outperforms the conventional energy detector for thesignals buried into noise. The rest of the paper is organizedas follows: Section II expresses statement of the problem andprovides signal model along with its statistical characteristics.Section III gives the details of the method proposed along withrelevant analysis. Section IV presents the numerical resultsincluding performance comparison and relevant discussions.Section V summarizes the observations and concludes thepaper.

II. STATEMENT OF THE PROBLEM, SIGNAL MODEL, ANDITS STATISTICAL CHARACTERISTICS

Statement of the Problem and Signal Model: Due to theambient noise present over the transmission medium, complexbaseband equivalent of the received signal can be representedas:

r(t) =

{n(t), H0,

x(t) + n(t), H1,(1)

where n(t) is complex additive white Gaussian noise (AWGN)with CN

(0, σ2

N

)in the form of n(t) = nI(t) + jnQ(t) as

both nI(t) and nQ(t) being N(0, σ2

N/2)

and j =√−1; x(t)

is the complex baseband equivalent of the unknown signal;H0 represents the hypothesis corresponding to absence of theunknown signal, whereas H1 refers to the hypothesis corre-sponding to presence of it. Then, statement of the problemcan be expressed as identifying the absence/presence of theunknown signal x(t) by looking at the statistical characteristicsof the received signal r(t) in the presence of ambient noisen(t).

Having said that, the unknown signal x(t) can be modeledby decomposing it into the following form under the narrow-band channel assumption:

x(t) = m(t)s(t)a(t), (2)

where m(t) = h(t)ejθ(t) represents complex fading channelprocess whose amplitude and phase are h(t) and θ(t), re-spectively; s(t) denotes the real–valued slow–fading processincluding the combined effects of both distance–dependentpath loss and shadowing; and a(t) is the unknown basebandsignal. In addition, all three processes in (2) are assumed tobe independent of each other and of the noise process n(t).

Statistical Characteristics of Fast–Fading Process [m(t)]:Transmitted signals arrive at the receiver through multiple rays(sometimes referred to as paths) in the wireless propagationenvironment. These rays are combined at the receiver antennaand form a complex fading channel process m(t) representinga change in both amplitude and phase of the received signal.Complex fading channel process manifests itself by causingrapid fluctuations in the power level of the received signal withrespect to very small displacements on the order of a coupleof wavelengths of the transmission. Assuming that there issufficiently large number of independent rays arriving at thereceiver antenna in the absence of a specular component suchas in line–of–sight (LOS) conditions, fading channel amplitudeh(t) = |m(t)| follows the Rayleigh distribution according to

the central limit theorem. Furthermore, due to mobility, fadingchannel process exhibits correlation in time (or depending onthe formalization, in space) [14]. As an idealized case, angle–of–arrival (AoA) of the rays at an omni–directional receiverantenna can be assumed to be uniformly distributed within[0, 2π) on a two–dimensional plane. This special case leadsto the very well–known Jakes’ Doppler spectrum implying acorrelation in time for the complex fading channel processwith Rh(τ) = J0 (2πfD |τ |), where J0 (·) is the zeroth–orderBessel function of the first kind; fD is the maximum Dopplerfrequency given by fcv/c, fc is the transmission frequency,v is the mobile speed, and c is the speed of transmittedwaves (in RF propagation c = 3 × 108m/s). Depending onthe fD |τ | value, one can conclude that fast–fading channelprocess decorrelates with itself over displacements on the orderof a couple of wavelengths of the transmission [14].

Statistical Characteristics of Slow–Fading Process [s(t)]:Fluctuations in the power level of the received signal is causednot only by fast–fading process but also by the transmitter–receiver separation and by the obstacles present in between.Transmitter–receiver separation causes a power loss in thereceived signal which is known as distance–dependent pathloss and decreases monotonically with respect to the distancebetween transmitter and receiver. On the other hand, the impactof obstacles in the propagation paths between transmitterand receiver manifests itself as drastic fluctuations in thepower level of the received signal and is called shadowing(or sometimes shadow fading). However, fluctuations in thereceived signal power caused by distance–dependent path lossand shadowing take place with larger displacements comparedto those caused by fast–fading process. In other words, forthe same speed, combined impact of path loss and shadowingexhibits a slower rate of change in the received signal powerlevel compared to that of fast–fading process; therefore, it isreferred to as “slow–fading” process. As frequently reportedand verified in most of the experimental studies present in theliterature, the first–order statistics of the slow–fading processcan be approximated by a log–normal distribution. Therefore,the combined impact of path loss and shadowing can becaptured by the following single process [15, 16]:

s(t) = exp

(1

2µ(t) +

σG

2g(t)

), (3)

where µ(t)/2 denotes mean, σG/2 is the standard deviation oflog–normal shadowing, and g(t) is a real–valued unit normalprocess N (0, 1) [16].3 From the practical point of view, themean of log–normal process represents the impact of distance–dependent path loss varying over longer periods of time.Measurements also show that g(·) exhibits correlation of anexponentially decaying form [17]:

Rg(τ) = E {g(t)g(t+ τ)} = exp

(−v |τ |

dρ

), (4)

where E {·} is the statistical expectation and dρ is thedecorrelation distance. In [17], dρ is calculated to be 5.75mand 350m for urban and suburban environments, respectively,which emphasizes the difference between correlation scalesof shadowing and fast–fading under practical scenarios. It isimportant to state that both (3) and (4) correspond to idealizedcases which are consistent with experimental results availablein the literature. Note also that there are some other studies inthe literature related to shadowing models such as static anddynamic shadowing [18]. Yet, the model defined by both (3)and (4) are adopted due to the following two reasons: (R1) It

3Since the statistics of the slow–fading process are given in terms of powerlevels, the constant 1/2 is introduced into (3) for notational convenience andeasier analysis implying ln

(s2(t)

)= µ(t) + σGg(t).

781

is clear that due to the mathematical tractability of both (3)and (4), the analysis will be simpler. (R2) Furthermore, sucha sharp (exponential) decay yields pessimistic results in termsof shadowing correlation, which provide some sort of upperbound for the problem considered. Having said this, as willbe shown subsequently, it is important to mention that theproposed method is independent of any sort of shadowingcorrelation model.

Finally, without loss of generality, it is assumed that receiverdisplacement within the duration of operation is negligiblysmall compared to the distance between the unknown signalsource and the receiver. Therefore, the impact of µ(t) can beneglected and s(t) is assumed solely to include the impact ofshadowing process.

III. PROPOSED METHOD

In Section II, it is stated that statistics of shadowing andfast–fading processes evolve in different scales on spatialdomain. This implies that shadowing process is not expected tovary within relatively short displacements such as in a coupleof wavelengths of the transmission. Bearing this in mind, firstconsider passing the received signal through a low–pass filterwhose (normalized) impulse response is given by:

w(t) =1

2TA

(sgn

(t+

TA

2

)− sgn

(t− TA

2

))(5)

where sgn(·) is the signum (or sign) function and TA denotesthe effective averaging duration. Now, let one focus on thehypothesis H1, since it includes both noise and the unknownsignal terms. If r(t) is passed through the low–pass filter underthe hypothesis H1, then, in the light of both (2) and (5):

z(t) =

∞∫−∞

w(t− τ) (x(τ) + n(τ)) dτ

=

t+TA/2∫t−TA/2

w(t− τ)m(τ)s(τ)a(τ) dτ

+

t+TA/2∫t−TA/2

w(t− τ)n(τ) dτ

︸ ︷︷ ︸NF (t)

.

(6)

is obtained. By choosing TA to be so short that shadowingdoes not change within,4 (6) can be rewritten as:

z(t) = s(t)

t+TA/2∫t−TA/2

w(t− τ)m(τ)a(τ) dτ

︸ ︷︷ ︸MF (t)

+NF (t)

= s(t)MF (t) +NF (t).

(7)

To reveal the impact of shadowing process, first absolutesquare of output of the low–pass filter is calculated:

z(t)z∗(t) = |z(t)|2 = Z(t) = |s(t)MF (t) +NF (t)|2

which can be rewritten as:

Z(t) = s2(t) |MF (t)|2 + |NF (t)|2 + 2s(t)ℜ{MF (t)N∗F (t)} ,

(8)

4A brief discussion regarding to what extent TA can be considered to beshort is given by Footnote 11 in Section IV for both practical scenarios andgeneral cases.

where (·)∗ and ℜ{·} denote the complex conjugate and thereal part of their input, respectively. Second, natural logarithmoperator is applied to (8) which yields after some algebraicmanipulations:

ln (Z(t)) = ln(s2(t) |MF (t)|2

)+ ln

(Z(t)

|s(t)MF (t)|2

). (9)

Plugging (3) into (9) by neglecting the impact of distance–dependent path loss reads:

ln (Z(t)) = σGg(t) + ln(|MF (t)|2

)+ ln

(Z(t)

|s(t)MF (t)|2

)︸ ︷︷ ︸

L(t)

.

(10)It is clear that autocorrelation of (10) will include the shad-owing correlation via g(t). Therefore, first let the unbiasedestimate of autocorrelation for any random process X (t) bedefined as:5

RX (τ) = limT→∞

1

T + τ

T/2+τ∫−T/2

X (t)X ∗(t− τ) dt. (11)

Thus, autocorrelation of (10) in terms of (11) becomes:

Rln (Z)(τ) = σ2Ge

−v|τ |/dρ +RL(τ)+RgL(τ)+RLg(τ). (12)

In (12), the impact of correlation characteristics of shadowingprocess can be seen within the first term at right hand side ofthe equation. However, (12) by itself is still not very conclusiveto make any decision regarding the absence/presence of theunknown signal due to the following reasons: (I) the impactof low–pass filtering operation still exists and affects espe-cially the delays (lags) τ around zero, (II) the autocorrelationestimates are biased with the mean of ln (Z(t)),6 and (III) inthe presence of finite support, autocorrelation estimates willfluctuate and make the decision process difficult. As will beshown in what follows, all these concerns can be alleviated byexamining the noise–only process through the steps (6)–(11).In the subsequent parts, (12) will be revisited and discussedfurther in the light of the results to be obtained for the noise–only case.

For the noise–only case, consider first the ideal scenariowhere T → ∞ and TA = 0. Note that this scenariocorresponds to infinite support with no low–pass filtering andimmediately degenerates to NF (t) = n(t). In that case, inputto the logarithm operator has a chi–square distribution withtwo degrees of freedom (χ2(2)). However, since input to thelogarithm operator is generated by the squaring device, one cantake advantage of the equality: ln

(|n(t)|2

)= 2 ln (|n(t)|).

This way, input–output relationship for the logarithm operatorcan be expressed in terms of Rayleigh distribution, which pro-vides analytical tractability in the subsequent steps. BecauseY(t) = |n(t)| is Rayleigh distributed, its probability densityfunction (PDF) is given by:

pY(y) =

{yαe

(− y2

2α

), 0 ≤ y

0, y < 0(13)

5Here, it must be stated that the reliability of autocorrelation estimates atlarger lags is very low; therefore, the estimates at larger delays should bedisregarded.

6As will be shown subsequently, mean of ln (Z(t)) actually plays a crucialrole in the decision step. Therefore, it is not removed from (12).

782

where α is the mode parameter of the distribution satisfyingµY = α

√π2 with µY being the mean of the PDF of Y(t)

and α =√

σ2N/2. Therefore, output of the logarithm, say

X(t), forms a time series that is composed of log-Rayleighdistributed values:

X(t) = ln (Y(t)) = ln (|n(t)|) , (14)

with the following PDF:

pX(x) =e2x

αexp

(−e2x

2α

)(15)

for all x ∈ R. Next, let:

r′(t) = limA→0

A cos (2πfAt+ ϕA) + n(t), (16)

where A, fA, and ϕA are some arbitrary amplitude, frequency,and phase values, respectively. Note that (16) is equivalent ofhypothesis H0 in the limiting sense. If r′(t) follows throughthe steps (6)–(11), then output of the correlator is given by[19]:

Ψ(τ) =∞∑i=1

(i + l even)

φiN (τ)

i∑l=1

((i+ l) /2− 1

(i− l) /2

)Υm

× 1F21 ((i+ l) /2; l + 1;−Υ) /l!l (i+ l)

+1

4

∞∑i=1

1F21 (i; 1;−Υ)φ2i

N (τ) +

(ln (A) +

1

2E1(Υ)

)2

,

(17)

where φN (·) is the normalized autocorrelation estimates of thequadrature components (i.e., nQ(·)) of n(t), Υ is the signal–to–noise ratio (SNR) and defined to be Υ , A2/

(2σ2

N

),

1F21 (·; ·; ·) is the confluent hypergeometric function, and E1(·)

is the exponential integral [20]. Since the purpose is to obtainthe characteristics of the noise–only process, one can consider(17) by expanding E1(·) into power series for A → 0 (orequivalently for Υ → 0). This allows one to see that (17) isdominated by the cross–noise terms as Υ diminishes and canbe expressed after some manipulations as:

RX(τ) , limΥ→0

Ψ(τ) ∼=1

4

∞∑i=1

φ2iN (τ)

i2+Φ(σN ), (18)

where Φ(σN ) represents a constant that depends on the noisevariance σ2

N . Since Φ(σN ) is a constant, one can readilycalculate the variance of X(t) by setting τ = 0 and ignoringΦ(σN ) as:

σ2X =

1

4

∞∑i=1

1

i2=

π2

24. (19)

Recalling that X(t) is a non–zero mean process (i.e., µX = 0)due to the non–linear transformation applied, one can concludethat:

Φ(σN ) = µ2X (20)

holds in (18), since RA(τ) = E {A(t)A∗(t+ τ)} andRA(0) = σ2

A + µ2A for any stationary stochastic process A(t)

with µA = 0.Then by assuming σN to be unity for the sake of simplicity,

µX can be calculated via (13) and (15)–(18) as:

µX = ln (α) +ln (2)− γ

2= ln

(1√2

)+

ln (2)− γ

2= −γ

2,

(21)

where γ is Euler–Mascheroni (or sometimes referred to solelyas Euler’s) constant and given by γ = −

∫∞0

ln (u)e−udu.7 In(18), it is clear that at larger delays (lags) τ , the autocorrela-tion estimates exhibit an asymptotic behavior and convergesΦ(σN ), which is a function of noise variance. However, itis desired that the method proposed is independent of noisevariance σ2

N . Thus, normalizing the autocorrelation estimateswith the signal power (i.e., with the value at τ = 0) will yieldthe following constant:8

Φ′ =Φ(σN )

RX(0)=

µ2X

µ2X + σ2

X

. (22)

Up until this point, TA is assumed to be zero correspondingto omitting the low–pass filtering operation for the noise–onlycase. It is evident that such a filtering operation prior to thelogarithmic detector will not change the convergent behaviorof Φ′; however, it will solely change the value of Φ′ to a newconstant, say Φ′′

TA. Due to the space limitations, derivations

regarding the value of Φ′′TA

is omitted and numerical resultsare presented in Section IV for different TA values.

As final step, one might want to obtain an approximationfor (18) in the presence of finite support, since it causesfluctuations around Φ′′

TAat larger τ values as stated earlier

in (III). Thus, the following unbiased estimator is applied tothe autocorrelation estimates in (12) prior to the decision step:

VU =1

U

TA+U∫TA

Rln (Z)(τ)dτ (23)

where U denotes the effective integration time. It is clear thatwhen TA = 0+ and U → ∞, then VU → Φ′.

Based on the discussion related to steps between (12) and(23), it is evident that when x(t) = 0, shadowing correlationwill still influence the autocorrelation estimates at larger delays(lags) τ and not give rise to a rapid decay. In other words, whenan unknown signal is abscent, a sharp decay is expected at theoutput of the correlator due to the aforementioned second–order statistical characteristics of noise. However, when anunknown is signal present, although there will still be a decayat the output of the correlator, that decay will never reach thelevel where noise–only case lies in spite of the exponentialdecaying assumed for the shadowing in (4).9 Therefore, onecan conclude that:

Φ′′TA

< VU (24)

always holds. Moreover, as shown earlier, Φ′′TA

implies thatno such measurement is required to determine a specificthreshold.

At this point, it is worth mentioning the practical implica-tions of the integration limits and of integration duration in(23). In parallel to the earlier dicussions, first and foremost,TA should be kept as low as possible in order not to violatethe assumption regarding the invariance of shadowing processin (8). In that sense, it is plausible to think that TA is selectedto be the lowest non–zero value possible at the receiver. Forinstance, considering the discrete sampling operation, TA canbe selected to be k∆t, where 2 ≤ k with ∆t = 1/fs denoting

7First five digit of γ in decimal form is γ ∼= 0.57721 . . .8When σN is unity: Φ′ = γ2

γ2+π2/6∼= 0.16843 . . . for the first five

significant digits in decimal.9This is very critical because of the reason (R2) stated in Section II.

Considering the fact that the proposed method is independent of any specificcorrelation model for shadowing, Φ′′

TAconstitutes the lower bound for the

problem considered here.

783

the sampling interval of the receiver.10 This way, it is ensuredthat (8) still holds. Second, in contrast to TA, U needs to bechosen as large as possible to obtain better estimates VU in(23). In this regard, U is actually bounded by the memorylimitations of the receiver while disregarding the values atlarger delays (lags) due to the reliability concerns stated inFootnote 5.

In the light of the aforementioned discussions, finally, thedecision step can be given as follows:

D =

{H0, 0 ≤ M,H1, otherwise, (25)

where the decision metric is given by M =(Φ′′

TA− VU

). The

block diagram of the method proposed is given in Figure 1along with the corresponding steps applied in the analysis.

IV. NUMERICAL RESULTS AND DISCUSSIONS

The receiver is assumed to be mobile with an average speedvalue of v = 10m/s and to operate on 2GHz. In parallel, thefast–fading channel is assumed to have a Rayleigh distributedamplitude with a Doppler spectrum of Jakes’ type. Spatiallycorrelated log–normal shadowing is imposed on the signalwith σG = {4.3, 7.5}dB and dρ = {5.75, 350}m for urbanand suburban environments, respectively, as reported in [17].These two particular environments are selected to investigatethe performance of the method proposed, because they exhibitvery different characteristics in terms of their decorrelationdistances implying different rate of change in their second–order statistics. The sampling frequency of the receiver isassumed to be fixed at 20KHz in order to satisfy the conditionfD ≪ fs as stated in Footnote 10. The impact of pathloss is neglected under hypothesis H1 by assuming a verysignificant spatial separation between the unknown sourceand the receiver with respect to the displacement that themobile traverses. The effective averaging time of the low–passintegrate–and–dump filter is set to be 0.05ms correspondingto the case k = 2, whereas the total sensing time is set to beU = 50ms.11

First, one might want to focus on the sampled version ofnormalized Rln (|n|)(τ) with Rln (|n|)(0), since it constitutesthe base of the method proposed and reveals the impact ofoperations applied between the steps (8)–(24). In Figure 2, twodistinct characteristics mentioned earlier can clearly be seen:the impact of filtering and convergence of autocorrelation ofthe noise passed through logarithmic detector. Note that theimpact of filtering reveals itself by forming a gradual decayingfrom the normalized autocorrelation value at τ = 0 to theconstant Φ′ and Φ′′

TAs. Here, different Φ′′

TAvalues can easily

be identified in Figure 2 as saturation points for correspondingk = {2, 3, 4}. For instance, Φ′′

TA= 0.495 for k = 2.

Next, the correlation behavior of the unknown signal con-taminated by noise can be examined. The results are givenin Figure 3 in comparison with the case where unknownsignal is absent (noise–only case) for urban environment.In accordance with (24), as the unknown signal dominates,the saturation level for the autocorrelation estimates divergesbecause shadowing correlation prevails (12) and maintainslarger correlation values as expected.

Urban environment settings are applied to both the methodproposed and the energy detection methods to obtain a com-parative performance investigation. Comparison is establishedby receiver operating characteristic (ROC), that is probability

10Of course, this conclusion depends on relatively larger sampling rates fsat the receiver which satisfy fD ≪ fs.

11Note that, TA is suggested to be 85ms in [21] as a general–purpose valueconsidering a plausible range of mobile velocities. Therefore, the TA valueselected here implies that shadowing process can safely be assumed to beinvariant under practical scenarios.

Low-pass

Filtering

( )

Logarithm

Operator

ln(•)

Squaring

Device

|•|2

Correlator

E{X(t)X*(t+τ)}

Estimator

Θ(τ;U)

Decision

Device

)(tz )(tZ

( ))(ln tZ

( ) )()(ln τtZ

RUV

)(tr

{ }10,HH

Fig. 1. Block diagram of the method proposed.

of detection (PD) with respect to probability of false alarm(PF ). In the literature, complementary ROC, which focuseson probability of miss–detection (PMD) with respect to PF , iswidely employed as well. In this study, complementary ROCwill be considered. The results are plotted in Figure 4 fordifferent SNR values. As seen from Figure 4, the proposedmethod performs better than the energy detection even for lowSNR values.

The impact of decorrelation distance dρ on the methodproposed can be investigated in a better way by looking atoutput of the correlator. The results are given in Figure 5for two environment instances under 0dB SNR. Figure 5indicates that the shorter the dρ, the more rapid decay in (12),since extremely large dρ values such as 350m in suburbanenvironments do not allow (12) to decay rapidly as expected.

V. CONCLUSION

In this study, a spectrum sensing method that is based on thesecond–order statistical characteristics of the received signalis proposed for mobile radio receivers. The method proposedtakes into account the impact of correlated shadowing andfast–fading process and exploits the spatial occurrence scalesof these processes. Upon low–pass filtering the received signalat the baseband, a logarithmic detector is applied right aftersquaring device and a decision is made by examining theautocorrelation characteristics of output.

The proposed method reveals that spatially correlated shad-owing is extremely important in spectrum sensing for mobilereceivers. It is clear that ignoring shadowing correlation mightlead to optimistic results for second–order statistics–basedmethods and must be avoided.

VI. ACKNOWLEDGMENT

This work is supported by Qatar National Research Fund(QNRF)12 grant through National Priority Research Programwith the project number 08–101–2–025.

REFERENCES

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12QNRF is an initiative of Qatar Foundation.

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0 2 4 6 8 10 12

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Autocorrelation sample lag index

Unb

iase

d A

utoc

orre

latio

n Es

timat

es (N

orm

aliz

ed)

k=4k=3k=2k=1 (i.e., T

A=0)

Phi’= 0.1628

Fig. 2. Correlator output of H0 for k = 1, 4, as k = 1 implies TA = 0(Φ′ is labeled as Phi′ and ≈ 0.1628).

5 10 15 20 25

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Autocorrelation sample lag index

Unb

iase

d A

utoc

orre

latio

n Es

timat

es (N

orm

aliz

ed)

H1

, SNR=0dB

H1

, SNR=−3dB

H1

, SNR=−6dB

H0

Phi’’ = 0.495

Fig. 3. Correlator output for x(t) = 0 with k = 2 (The plot belonging tonoise–only case is maintained for comparison purposes).

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10−4

10−3

10−2

10−1

100

10−4

10−3

10−2

10−1

100

Probabililty of False Alarm

Prob

abili

lty o

f Mis

s

ED, SNR=−12dBPM, SNR=−12dBED, SNR=−6dBPM, SNR=−6dB

Fig. 4. Complementary ROC curves of both energy detector and the methodproposed for different SNR levels in urban environment (k = 2).

2 4 6 8 10 12 14 16 18

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Autocorrelation sample lag index

Unb

iase

d A

utoc

orre

latio

n Es

timat

es (N

orm

aliz

ed)

H1

, drho

=350m

H1

, drho

=5.75m

H0

Phi’’ = 0.495

Fig. 5. The impact of different environments having different decorrelationdistances on output of the correlator block (k = 2).

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