Robust Spectrum Sensing in Cognitive RadioYonghong Zeng and Ying-Chang Liang

Institute for Infocomm Research, A*STAR, SingaporeEmails: {yhzeng, ycliang}@i2r.a-star.edu.sg

Abstract—Spectrum sensing is a critical step in cognitive radioto learn the radio environment. Despite its long history, inrecent years spectrum sensing has attracted substantial interestsfrom the cognitive radio community as well as other areas.Although there have been many methods, most of them needsome idealistic assumptions and are hardly applicable in realcognitive radio practice. In this paper, we first discuss thepossible hostile environment facing the spectrum sensing incognitive radio. We then investigate some methods which couldsurvive under such hostile environment. These methods includeeigenvalue/covariance based detections, cooperative sensing, andcyclostationary detections. The robustness of these methods arediscussed in detail through theoretic analysis and simulations.

Index Terms—Cognitive radio, spectrum sensing, sensing al-gorithm, signal detection, robust, robustness

I. CHALLENGES OF SPECTRUM SENSING IN COGNITIVE

RADIO

Spectrum sensing in cognitive radio have a few specialchallenges, among others, as follows. (1) A cognitive radiomay need to sense the primary signal at very low signalto noise ratio (SNR). This is to overcome the hidden nodeproblem: a sensor hears very weak signal from the primarytransmitter but can strongly interfere the primary receiver if ittransmits (here the primary receiver likes a hidden node). Toavoid the interference, one solution is to require the sensor tohave the capability of sensing at very low SNR. For example,in the 802.22 standard, the sensing sensitivity requirementis about -20dB. (2) Propagation channel uncertainty makesthe spectrum sensing difficult. In wireless communications,it is common that the channel is multipath fading and timedispersive. The unknown time dispersive channel turns mostcoherent detections unreliable. (3) It is hard to synchronize thereceived signal with the primary signal in time and frequency.This will cause some methods like preamble/pilot based detec-tions less effective. (4) The noise level may change with timeand location, which yields the noise power uncertainty issuefor detection [1], [2], [3], [4]. This makes methods relyingon accurate noise power unreliable. (5) The noise may notbe white, which will affect many methods with white noiseassumption. (6) There could be interferences from intentionalor unintentional transmitters. This requires the detector hasthe capability to suppress the interference while identify thesignal.

Although there have been many methods (see survey papers[5], [6], [7], [8] and the references therein), many methodsbased on ideal assumptions may not work well in such a hostileradio environment. We need the spectrum sensing to be robustto the unknown and maybe time-varying channel, noise and

interference. While there are various blind channel estimationand synchronization methods, most of them do not performwell at very low SNR. Furthermore, channel estimation andsynchronization are only valid when the signal is determinedto exist. Hence, in this paper we will discuss methods whichdo not rely heavily on the knowledge of propagation channeland synchronization. Therefore the coherent detections, likethe matched filtering, which heavily depends on the channeland synchronization, will not be our focus. There have beensome methods, which are proved to be robust in some sense.We will discuss the robustness of these methods including theeigenvalue/covariance based detections [9], [10], [11], [12],[13], [14], [15], [16], [17], [18], [19], [20], [21], cooperativesensing [22], [23], [24], [25], [26], [27], [28], [29], [30],[31], and cyclostationary detection [32], [33], [34]. All thesemethods require little information on the signal, channel andhave some immunity to synchronization error, fading andmultipath, noise uncertainty, and unknown interference.

II. SYSTEM MODEL

We consider a receiver/detector with 𝑀 ≥ 1 antennas. It isassumed that a centralized unit is available for processing thesignals from all the antennas. A similar scenario is the multi-node cooperative sensing, if all nodes are able to send theirobserved signals to a central node. There are two hypotheses:ℋ0, signal absent; and ℋ1, signal present. The received signalat antenna/receiver 𝑖 is given by

ℋ0 : 𝑥𝑖(𝑛) = 𝜂𝑖(𝑛) + 𝜌𝑖(𝑛)

ℋ1 : 𝑥𝑖(𝑛) = 𝑠𝑖(𝑛) + 𝜂𝑖(𝑛) + 𝜌𝑖(𝑛) (1)

𝑖 = 1, . . . ,𝑀,

where 𝜂𝑖(𝑛) is the noise and 𝜌𝑖(𝑛) is the possible interference.At hypothesis ℋ1, 𝑠𝑖(𝑛) is the received source signal atantenna/receiver 𝑖. Note that 𝑠𝑖(𝑛) is the transmitted primarysignal after going through the fading and multipath propaga-tion channel. That is, 𝑠𝑖(𝑛) can be written as

𝑠𝑖(𝑛) =

𝑁𝑝∑𝑘=1

𝑞𝑖𝑘∑𝑙=0

ℎ𝑖𝑘(𝑙)𝑠𝑘(𝑛− 𝑙) (2)

where 𝑁𝑝 is the number of primary signals, 𝑠𝑘(𝑛) stands forthe transmitted primary signal from primary user or antenna𝑘, ℎ𝑖𝑘(𝑙) denotes the propagation channel coefficient from the𝑘th primary user or antenna to the 𝑖th receiver/antenna, and𝑞𝑖𝑘 is the channel order for ℎ𝑖𝑘. For simplicity, it is assumedthat the signal, noise, interference, and channel coefficients areall real numbers.

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

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Spectrum sensing is to choose one of the two hypotheses(ℋ0 or ℋ1) based on the received signal. If ℋ1 is chosen, thesensor may further give information on the signal waveformand modulation schemes in some applications. The probabilityof detection, 𝑃𝑑, and probability of false alarm, 𝑃𝑓𝑎, aredefined as follows:

𝑃𝑑 = 𝑃 (ℋ1∣ℋ1) (3)

𝑃𝑓𝑎 = 𝑃 (ℋ1∣ℋ0) (4)

In general a sensing algorithm is said to be “optimal” if itachieves the highest 𝑃𝑑 for a given 𝑃𝑓𝑎 with a fixed numberof samples, though there could be other criteria to evaluate theperformance of a sensing algorithm.

In order to apply space and time processing, we stack thesignals from the 𝑀 antennas/receivers and 𝐿 time samples toyield the following 𝑀𝐿× 1 vectors:

x(𝑛) = [𝑥1(𝑛) ⋅ ⋅ ⋅𝑥𝑀 (𝑛) 𝑥1(𝑛− 1) ⋅ ⋅ ⋅𝑥𝑀 (𝑛− 1)

⋅ ⋅ ⋅𝑥1(𝑛− 𝐿+ 1) ⋅ ⋅ ⋅𝑥𝑀 (𝑛− 𝐿+ 1)]𝑇 (5)

s(𝑛) = [𝑠1(𝑛) ⋅ ⋅ ⋅ 𝑠𝑀 (𝑛) 𝑠1(𝑛− 1) ⋅ ⋅ ⋅ 𝑠𝑀 (𝑛− 1)

⋅ ⋅ ⋅ 𝑠1(𝑛− 𝐿+ 1) ⋅ ⋅ ⋅ 𝑠𝑀 (𝑛− 𝐿+ 1)]𝑇 (6)

𝜼(𝑛) = [𝜂1(𝑛) ⋅ ⋅ ⋅ 𝜂𝑀 (𝑛) 𝜂1(𝑛− 1) ⋅ ⋅ ⋅ 𝜂𝑀 (𝑛− 1)

⋅ ⋅ ⋅ 𝜂1(𝑛− 𝐿+ 1) ⋅ ⋅ ⋅ 𝜂𝑀 (𝑛− 𝐿+ 1)]𝑇 (7)

Based on the vector form, the hypothesis testing problemcan be reformulated as

ℋ0 : x(𝑛) = 𝜼(𝑛)

ℋ1 : x(𝑛) = s(𝑛) + 𝜼(𝑛), 𝑛 = 0, . . . , 𝑁 − 1. (8)

III. ROBUSTNESS OF COVARIANCE/EIGENVALUE BASED

DETECTIONS

In this section, it is assumed that the noise samples 𝜂𝑖(𝑛)’sand interference samples (if any) 𝜌𝑖(𝑛)’s are independent andidentically distributed (i.i.d) over both 𝑛 and 𝑖.

For 𝑀 > 1, the signal components at different anten-nas/receivers are spatially correlated, because all 𝑠𝑖(𝑛)’s aregenerated from the same source signal 𝑠𝑘(𝑛)’s. The signalcomponent at any particular receiver/antenna may also becorrelated in time. The correlation in space and time of thesignal components is the key for the covariance/eigenvaluebased detections.

To use this space and time correlations, the eigen-value/covariance based detections calculate the sample covari-ance matrix of the received signal, which is defined as

R̂𝑥(𝑁) =1

𝑁

𝑁−1∑𝑛=0

x(𝑛)x𝑇 (𝑛). (9)

Based on the generalized likelihood ratio test (GLRT) orinformation/signal processing theory, there have been a fewmethods proposed based on the sample covariance matrix.These methods can be classified into two classes: the eigen-value based detections (EBD) [9], [10], [11], [12], [13], [17],[19], [20], [21] and the covariance based detections (CBD)

[14], [15], [16], [18]. Here we summarize the methods in thefollowing .

Let 𝜆1 ≥ 𝜆2 ≥ ⋅ ⋅ ⋅ ≥ 𝜆𝑀𝐿 be the eigenvalues of the samplecovariance matrix.

Algorithm 1: Eigenvalue based detectionsStep 1. Compute the sample covariance matrix as defined

in (9).Step 2. Calculate the eigenvalues of the sample covariance

matrix.Step 3. Compute a test statistic from the eigenvalues. There

are different approaches to construct the test statistic. A fewsimple but effective method are as follows:

1) Maximum eigenvalue to trace detection (MET). Thetest statistic is

𝑇𝑀𝐸𝑇 = 𝜆1/tr(R̂𝑥(𝑁)) (10)

where tr(⋅) is the trace of a matrix, tr(R̂𝑥(𝑁)) =∑𝑀𝐿𝑖=1 𝜆𝑖. This method is also called blindly combined

energy detection (BCED) in [13].2) Maximum to minimum eigenvalue detection (MME).

The test statistic is

𝑇𝑀𝑀𝐸 = 𝜆1/𝜆𝑀𝐿 (11)

3) Arithmetic to geometric mean (AGM). The test statis-tic is

𝑇𝐴𝐺𝑀 =1

𝑀𝐿

𝑀𝐿∑𝑖=1

𝜆𝑖/(𝑀𝐿∏𝑖=1

𝜆𝑖)1/𝑀𝐿 (12)

Step 4. Compare the test statistic with a threshold to makea decision.

Other than using the eigenvalues, the covariance baseddetections directly use the elements of the covariance matrix toconstruct detection methods, which can reduce computationalcomplexity. The methods are summarized in the following.

Let the entries of the matrix R̂𝑥(𝑁) be 𝑟𝑚𝑛 (𝑚,𝑛 =1, 2, ⋅ ⋅ ⋅ ,𝑀𝐿).

Algorithm 2: Covariance based detectionsStep 1. Compute the sample covariance matrix as defined

in (9).Step 2. Construct a test statistic directly from the entries of

the sample covariance matrix. In general, the test statistic ofthe CBD is

𝑇𝐶𝐵𝐷 = F1(𝑟𝑚𝑛)/F2(𝑟𝑚𝑚) (13)

where F1 and F2 are two functions. There are many ways tochoose the two functions. Some special cases are shown in thefollowing.

1) Covariance absolute value detection (CAV). The teststatistic is

𝑇𝐶𝐴𝑉 =

𝑀𝐿∑𝑚=1

𝑀𝐿∑𝑛=1

∣𝑟𝑚𝑛∣/𝑀𝐿∑𝑚=1

∣𝑟𝑚𝑚∣ (14)

2) Maximum auto-correlation detection (MAC). The teststatistic is

𝑇𝑀𝐴𝐶 = max𝑚 ∕=𝑛

∣𝑟𝑚𝑛∣/𝑀𝐿∑𝑚=1

∣𝑟𝑚𝑚∣ (15)

2

3) Fixed auto-correlation detection (FAC): The teststatistic is

𝑇𝐹𝐴𝐶 = ∣𝑟𝑚0𝑛0∣/

𝑀𝐿∑𝑚=1

∣𝑟𝑚𝑚∣ (16)

where 𝑚0 and 𝑛0 are fixed numbers between 1 and𝑀𝐿. This detection is especially useful when we havesome prior information on the source signal correlationand knows the lag that produces the maximum auto-correlation.

Step 3. Compare the test statistic with a threshold to makea decision.

All these methods do not use the information of the signal,channel and noise power as well. The methods are robust tosynchronization error, channel impairment, and noise uncer-tainty.

The test statistic is compared with a threshold 𝛾 to makea decision. The threshold 𝛾 is determined based on the given𝑃𝑓𝑎. To find a formula for the thresholds is mathematicallyinvolved. In general it needs to find the theoretical distributionof some combination of the eigenvalues of a random matrix.There have been some exciting works on this by using therandom matrix theory [11], [16], [19], [20], [21], [35], [36],[37].

To show the performance and the robustness of the methods,here we give some simulation results for the EBDs (the resultsfor CBDs are similar). Comparison with the energy detection(ED) is also included. We consider two cases here: the signalis time uncorrelated and the signal is time correlated. TheReceiver Operating Characteristics (ROC) curves (𝑃𝑑 versus𝑃𝑓𝑎) at SNR = −15dB, 𝑁 = 5000, and 𝑀 = 4 are plottedat the two cases. The performance at first case in shown inFigures 1 with 𝐿 = 1 and that at the second case is shown inFigure 2 with 𝐿 = 6, where “ED-𝑢dB” means energy detectionwith 𝑢 dB noise uncertainty. In Figure 2, the source signalis the wireless microphone signal [38], [15] and a multipathfading channel is assumed. For both cases, MET, MME andAGM perform better than ED. MET, MME and AGM aretotally immune to noise uncertainty. However, the ED is veryvulnerable to noise power uncertainty.

IV. ROBUSTNESS OF COOPERATIVE SENSING

If interference exists and it is correlated in time, the EBDand CBD may claim the interference as primary signal. Thismay cause high probability of false alarm. At this case, singlesensor EDB and CBD may not be reliable [39]. However, ifwe assume that interferences at different sensors are differentand independently distributed, it is possible that interferencescan be canceled through data fusion. A fusion scheme basedon EBD/CBD is proposed in [31].

In this section, we assume that there are 𝐾 sensors dis-tributed in different locations. For simplicity, we assume thateach sensor only has one antenna. The system model for eachsensor is the same as the model in (1) with 𝑀 = 1. Choosethe same smooth factor 𝐿 for all sensors. Let the samplecovariance matrix (based on 𝑁 samples) of the received signal

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Fig. 1. ROC curve: i.i.d source signal.

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Fig. 2. ROC curve: wireless microphone source signal.

at sensor 𝑖 be R̂𝑥,𝑖(𝑁). Each sensor calculates its samplecovariance matrix and sends it to a fusion center (could beone of the sensor). The fusion sensor sums all the samplecovariance matrices as

R̂𝑥(𝑁) =1

𝐾

𝐾∑𝑖=1

R̂𝑥,𝑖(𝑁) (17)

Based on the analysis in [31], at hypothesis ℋ0, R̂𝑥(𝑁) isapproximately an diagonal matrix, while at hypothesis ℋ1,R̂𝑥(𝑁) is far from diagonal if primary signal samples are timecorrelated. After average, the inferences from different sensorsare canceled. Hence the fusion center can use the EBD or CBDdescribed above for the detection. The method is summarizedas follows.

Algorithm 3: Cooperative Covariance Based Detection(CCBD)

Step 1. Each sensor computes its sample covariance matrix.Step 2. Every sensor sends its sample covariance matrix to

the fusion center.

3

Step 3. The fusion center averages the sample covariancematrices.

Step 4. The CBD for R̂𝑥(𝑁) is used for detection.Algorithm 4: Cooperative Eigenvalue Based Detection

(CCED)Step 1 to Step 3. Same as those in Algorithm 3.Step 4. The EBD for R̂𝑥(𝑁) is used for detection.Here we give some simulation results to show how robust to

the interference the methods are. We consider the case whenboth noise uncertainty and time correlated interference existat each sensor. In more specific, the simulation are basedon: (1) there is noise uncertainty at each sensor and thenoise uncertainty bound is 𝜇dB; (2) there are interferencesat each sensor and the number of interferences and theirpowers are unpredictable. Aggregate interference power ateach sensor is uncertain and interference uncertainty bound(defined similar to noise uncertainty bound) is 2dB; (3) thenumber of interferences at each sensor is uncertain but lessthan four. (4) interferences at different sensors are different.

In the simulation, two types of signals are used as interfer-ences: harmonics and correlated moving average (MA) noisedefined as follows:

𝜌(𝑛) = 𝜗(𝑛) + 𝜗(𝑛− 1) (18)

where 𝜗(𝑛) is the complex Gaussian noise. Comparison withthe cooperative energy detection (CED) is also included. TheCED is a widely used approach which averages the receivedsignal energies from all the sensors.

We consider the sensing at low SNR. So we just show theresults for the average SNR of all sensors being -22dB and theaverage interference to noise ratio (INR) being -22dB as well.That is, the interference strength is similar to that of signal.The parameters are: 𝐾 = 50, 𝑁 = 1000 and 𝐿 = 10. FMmodulated wireless microphone signal [38], [15] is used asthe primary signal. The sampling rate at receiver is 6 MHz.The channel is assumed to be invariant within the sensingperiod.

Although CEBD is usually slightly better than CCBD, wechoose CCBD here due to its low computational complexity.

In Figure 3 the real 𝑃𝑓𝑎 versus the expected 𝑃𝑓𝑎 is shownwhen only harmonic interference presents (no signal). It is nowonder that the real 𝑃𝑓𝑎 of CED-𝜇dB is much higher than theexpected 𝑃𝑓𝑎, because CED takes the unexpected amount ofnoise and interference as signal. To reduce the 𝑃𝑓𝑎, the onlyapproach is to set a more conservative threshold. For instance,if we require the real 𝑃𝑓𝑎 smaller than 0.1, we should set thethreshold based on expected 𝑃𝑓𝑎 at about 0.0025 for the CED,and much much lower expected 𝑃𝑓𝑎 for CED-𝜇dB (𝜇 ≥ 0.5).On the other hand, the real 𝑃𝑓𝑎 of CCBD or CCBD-𝜇dB isonly slightly higher than the expected 𝑃𝑓𝑎. Furthermore, thedifference between CCBD and CCBD-𝜇dB is not large, whichmeans that the impact of noise uncertainty is not significantfor CCBD.

In Figure 4 we see the ROC curve when both primarysignal and harmonic interference presents. Obviously CCBD

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CCBDCCBD−0.5dBCCBD−1dBCCBD−1.5dBCCBD−2dBCEDCED−0.5dBCED−1dBCED−1.5dBCED−2dB

Fig. 3. Real 𝑃𝑓𝑎 versus expected 𝑃𝑓𝑎 with harmonic interference

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Fig. 4. ROC curve at hypothesis ℋ1 with harmonic interference

and CCBD-𝜇dB have higher 𝑃𝑑 than CED and their dif-ferences are small. The CCBD guarantees certain detectionperformance if a suitable threshold is set no matter whatnoise uncertainties and interference are. That means that robustsensing is achieved. For instance, if the required 𝑃𝑓𝑎 ≤ 0.1,the threshold can be set based on the expected 𝑃𝑓𝑎 = 0.04(see Figure 3) and the achieved real 𝑃𝑑 is thus higher than 0.9at all cases. In contrary, CED and CED-𝜇dB cannot guaranteecertain detection performance by setting any threshold. Thismeans robust sensing is impossible. For instance, as shownabove, in order to get real 𝑃𝑓𝑎 ≤ 0.1 it is required to set thethreshold based on expected 𝑃𝑓𝑎 ≤ 0.0025 for CED. Howeverat expected 𝑃𝑓𝑎 ≤ 0.0025, the real 𝑃𝑑 ≤ 0.525. For CED-𝜇dB(𝜇 > 0) the situation is even much worse.

Figure 5 and 6 show the simulation results for the casewith MA inferences. Basically the results are similar to thosein Figure 3 and 4.

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Fig. 5. Real 𝑃𝑓𝑎 versus expected 𝑃𝑓𝑎 with MA interference

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Fig. 6. ROC curve at hypothesis ℋ1 with MA interference

V. ROBUSTNESS OF CYCLOSTATIONARY DETECTIONS

In many practical communication systems, the signals areusually manipulated in some special ways. This may introducespecial statistical features. The cyclostationarity is a wellstudied feature in digital modulated signals. Cyclostationarityrefers to some periodical property in the statistics of thesignal. For example, the second order statistics of many digitalmodulated signals have shown to be periodical in time. Onthe other hand, noise is usually purely stationary. Therefore itis possible to distinguish signal from noise by checking thecyclostationary feature. This cyclostationarity can be extractedby the cyclic auto-correlation (CAC) or the spectral-correlationdensity (SCD) [32], [33], [34]. For a cyclostationary signal,its CAC/SCD takes non-zero values at some non-zero cyclicfrequencies. On the other hand, noise does not have any cyclo-stationarity at all, i.e., its SCD function has zero values at allnon-zero cyclic frequencies. Hence, the signal is distinguished

from noise by analyzing the CAC/SCD function. Furthermore,it is possible to distinguish the signal type because differentsignal may have different non-zero cyclic frequencies.

For simplicity, in this section we consider single an-tenna/receiver case, that is, 𝑀 = 1. For notation simplicity,we omit the subscript for antenna/receiver. The discrete CACof the received signal 𝑥(𝑛) is defined as

𝑅𝛼𝑥 (𝑘) =1

𝑁

𝑁−1∑𝑛=0

𝑥(𝑛+ 𝑘)𝑥∗(𝑛)𝑒−𝑗2𝜋𝛼𝑛𝑇𝑠 (19)

where the lag 𝑘 = 0, 1, , ⋅ ⋅ ⋅ , 𝐼 − 1 with 𝐼 << 𝑁 , and 𝑇𝑠 isthe sampling period.

Based on the CAC or its frequency domain version, theSCD, we can construct a test statistic for detection. There havebeen many different test statistics including the generalizedLRT (GLRT) [40]. Basically different tests have differentperformances and complexities.

When interference exists, the CSD may still work well aslong as the interference does not have the same cyclostationaryfeature as the primary signal. In general, the chance that theprimary signal and the interference have the same cyclostation-ary feature is slim. That means CSD is robust to interferenceand noise uncertainty. The time dispersive channel may havea major impact on the strength of the cyclostationary feature.The impact of this is studied in [41].

If we look at 𝑅𝛼𝑥 (𝑘) at the case of infinite number ofsamples (𝑁 → ∞) [32], [33], [34], it heaves like a deltafunction respect to variable 𝛼. At limited number of samples,expressions for 𝑅𝛼𝑥 (𝑘) of some signals are shown in [42].From the expressions, we see that the function also changesvery sharply with respect to variable 𝛼. This means thatin the detection we need to know the exact 𝛼 which givesthe cyclostationary feature. Any small error in 𝛼 will causecatastrophic result. However, even if the detector knows thetheoretic cyclic frequency, the true cyclic frequency may beunavailable due to the timing or frequency error. Detaileddescription of this is included in [42]. Hence the detectormay calculate the CAC/SCD at the cyclic frequency �̂� thatis different from the true 𝛼. Let 𝜖 = �̂� − 𝛼 be the mismatcherror and 𝛿 = ∣𝜖∣/∣𝛼∣ be the relative error. The error maycause the CSD to be unreliable. As an example, we show howthe detection performance is affected for a test statistic in thefollowing. To save space, we omit the detailed derivations.

We consider a CSD detection with the test statistic:

𝑇𝐶𝑆𝐷 =

𝐼−1∑𝑘=0

∣𝑅𝛼𝑥 (𝑘)∣2 (20)

where 𝛼 is the cyclic frequency where the signal has thecyclostationary feature. The test statistic is compared withan estimated noise power to make a decision. The maximumlikelihood (ML) estimation of the noise power is

�̂�2𝜂 =1

𝑁

𝑁−1∑𝑛=0

∣𝑥(𝑛)∣2. (21)

5

The threshold is thus chosen as 𝛽�̂�4𝜂 [43], where 𝛽 is a scalarto meet the pre-defined probability of false alarm.

Obviously the performance of the CSD is affected by therelative difference between the test statistics at ℋ0 and ℋ1,that is, the ratio

𝜑 = (𝑇𝐶𝑆𝐷∣ℋ1)/(𝑇𝐶𝑆𝐷∣ℋ0). (22)

The larger the ratio, the better the detection performance willbe.

We consider two types of primary signals: single carrier(SC) signal 𝑠(𝑡) = cos(2𝜋𝑓0𝑡) and the digitally modulated(DM) signal: 𝑠(𝑡) =

∑+∞𝑛=−∞ 𝑎(𝑛)𝑝(𝑡−𝑛𝑇𝑏), where 𝑇𝑏 is the

symbol duration, 𝑎(𝑛) is the quantized and modulated signalat time 𝑛𝑇𝑏, and 𝑝(𝑡) is the pulse shaping filter supported ininterval [0, 𝑇𝑏). For the SC signal, at cyclic frequency ±2𝑓0the CAC/SCD is non-zero. The digitally modulated signalis the basic signal model in digital communications, whosecyclostationary features occur at cyclic frequency 𝑘/𝑇𝑏 forany integer 𝑘.

Our interest is to see how the CSD performance degradesif the receiver chooses the cyclic frequency different to thetrue one. We can prove the following results on the detec-tion performance with respect to the mismatch error 𝜖. Inthe following, some notations are used: E(∣𝜂(𝑛)∣2) = 𝜛2,E(∣𝜂(𝑛)∣4) = 𝜛4, E(∣𝑎(𝑛)∣2) = 𝛿2, E(∣𝑎(𝑛)∣4) = 𝛿4, and𝐿𝑠 = 𝑇𝑏/𝑇𝑠.

Theorem 1: For the SC signal, the performance ratio isapproximated by

𝜑 ≈ 1 +𝐼

16𝑁((𝐼 − 2)𝜛22 +𝜛4)

(sin(𝜋𝜖𝑁𝑇𝑠)

sin(𝜋𝜖𝑇𝑠)

)2

, 𝜖 ∕= 0

(23)

and

𝜑 ≈ 1 +𝑁𝐼

16((𝐼 − 2)𝜛22 +𝜛4)

, 𝜖 = 0. (24)

Theorem 2: For the DM signal, the performance ratio isapproximated by

𝜑 ≈ 1 +𝐼(𝑑+ 𝑐)𝛿22 + 𝐿𝑠𝑑(𝛿4 − 2𝛿22)

(𝐼 − 2)𝜛22 +𝜛4

+𝐿2𝑠𝑑𝛿

22

𝑁((𝐼 − 2)𝜛22 +𝜛4)

(sin(𝜋𝜖𝑁𝑇𝑏/𝐿𝑠)

sin(𝜋𝜖𝑇𝑏)

)2

, 𝜖 ∕= 0

(25)

and

𝜑 ≈ 1 +𝐼(𝑑+ 𝑐)𝛿22 + 𝐿𝑠𝑑(𝛿4 − 2𝛿22)

(𝐼 − 2)𝜛22 +𝜛4

+𝑁𝑑𝛿22

(𝐼 − 2)𝜛22 +𝜛4

, 𝜖 = 0, (26)

where and in the following, 𝑑 and 𝑐 are some constants relatedto the pulse shaping filter 𝑝(𝑡).

Obviously the cyclic frequency mismatch always degradesthe detection performance, because 𝜑∣𝜖 = 0 > 𝜑∣𝜖 ∕= 0.

For many detectors, increasing the sample size (sensingtime) usually lifts the detection performance. If there is no

cyclic frequency mismatch, it is true for CSD as well. Actuallyfor 𝜖 = 0, it is easy to verify that

lim𝑁→∞

𝜑 = ∞. (27)

Unfortunately, if there exists cyclic frequency mismatch, thestatement may be untrue. In fact, if 𝜖 ∕= 0, it can be verifiedthat, for the SC signal,

lim𝑁→∞

𝜑 = 1, (28)

and for the DM signal,

lim𝑁→∞

𝜑 = 1 +𝐼(𝑑+ 𝑐)𝛿22 + 𝐿𝑠𝑑(𝛿4 − 2𝛿22)

(𝐼 − 2)𝜛22 +𝜛4

. (29)

So even if we increase the sensing time, the performance maybe not improved or even degraded.

Here we give some simulations to show how the detectionperformance degrades with 𝜖. Two scenarios are consideredhere. Scenario one: the primary signal is a SC signal withfrequency 1.5 MHz. At the receiver, the signal is sampled withsampling frequency 10 MHz. We compare the performancewhen the CSD choose the correct cyclic frequency 𝛼 = 3MHz (without mismatch) and the wrong cyclic frequency𝛼 = 3× (1+2×10−5) MHz (with mismatch). Scenario two:the primary signal is a digitally modulated BPSK signal withsymbol duration 𝑇𝑏 = 0.4𝜇𝑠 (micro second) and pulse shapingfilter 𝑝(𝑡) = 1 for 𝑡 ∈ [0, 𝑇𝑏). At the receiver, the signalis sampled with sampling frequency 10 MHz. We comparethe performance when the CSD choose the correct cyclicfrequency 𝛼 = 2.5 MHz (without mismatch) and wrong cyclicfrequency 𝛼 = 2.5 × (1 + 2 × 10−5) MHz (with mismatch).We choose the number of lags 𝐼 = 256.

Since 𝜑 does not change much at different Monte-Carlo testsfor relatively large 𝑁 , we just show it at one randomly chosentest. In Figure 7 we see the ratio 𝜑 when no cyclic frequencymismatch (𝛿 = 0) at scenario one, where the ratio increasesalmost linearly with the sample size. Figure 8 gives the ratiofor 𝛿 = 2× 10−5. The ratio decreases substantially comparedto that without mismatch at any sample sizes. Furthermore, theratio decreases with oscillation when the sample size increases.Figure 9 gives the ratio at scenario two when 𝛿 = 0 and Figure10 shows that ratio when 𝛿 = 2×10−5. Here we see the similarphenomenon as that for scenario one.

Figure 11 gives the probability of detection at scenario one,where the threshold is set for 𝑃𝑓𝑎 = 0.1 and the SNR is -25dB. The performance for scenario two is given in Figure 12 atthe same settings except that the SNR is -15 dB. In both cases,the relative mismatch error is very small at 𝛿 = 2 × 10−5.Unfortunately even with such a very small mismatch errorthe detection performance decreases dramatically. Anothervery bad thing is that increasing the sample size does notimprove the detection performance when mismatch exists. Thismeans that we cannot rely on sensing longer time for betterperformance, which is contrast to our common knowledge.

6

1 2 3 4 5 6 7

x 105

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x 104 Ratio of the test statistics: without cyclic frequency mismatch

Sample size

Rat

io o

f the

test

sta

tistic

s

By simulationTheoretic

Fig. 7. The ratio without mismatch (SC signal)

1 2 3 4 5 6 7

x 105

101

102

103

Ratio of the test statistics: with cyclic frequency mismatch

Sample size

Rat

io o

f the

test

sta

tistic

s

By simulationTheoretic

Fig. 8. The ratio with mismatch error 2× 10−5 (SC signal)

1 2 3 4 5 6 7

x 105

100

200

300

400

500

600

700

Ratio of the test statistics: without cyclic frequency mismatch

Sample size

Rat

io o

f the

test

sta

tistic

s

By simulationTheoretic

Fig. 9. The ratio without mismatch (digitally modulated signal)

1 2 3 4 5 6 7

x 105

101

Ratio of the test statistics: with cyclic frequency mismatch

Sample size

Rat

io o

f the

test

sta

tistic

s

By simulationTheoretic

Fig. 10. The ratio with mismatch error 2×10−5 (digitally modulated signal)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

x 105

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sample size

Pro

babi

lity

of d

etec

tion

without mismatchwith mismatch

Fig. 11. Probability of detection without and with mismatch (SC signal)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

x 105

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sample size

Pro

babi

lity

of d

etec

tion

without mismatchwith mismatch

Fig. 12. Probability of detection without and with mismatch (digitallymodulated signal)

7

VI. CONCLUSION

Lack of synchronization, fading and multipath, noise uncer-tainty, unknown interference, and low SNR are some majorchallenges for spectrum sensing in cognitive radio. Manyknown methods do not perform well in such a hostile en-vironment. We have shown that some methods like eigen-value/covariance based detections, cooperative sensing, andcyclostationary detections have certain degree of robustnessto the impairments. We have also shown that sometimes amethod is robust in one aspect but is not in another aspect,like the cyclostationary detection. To achieve reliable spectrumsensing, we should have a full picture of a possible methodand choose the right solution.

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