# Spectrum Sensing under Correlated Antenna Array Using Generalized Detector in Cognitive Radio Systems

Authors: Modar Safir Shbat; Vyacheslav Tuzlukov
Journal: International Journal of Antennas and Propagation
 (2013)

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## Abstract

We derive the probability of false alarm and detection threshold under employment of the generalized detector (GD) in cognitive radio (CR) systems for two scenarios: firstly, the independent antenna array elements; the secondly, the correlated antenna array elements. The energy detector (ED) and GD spectrum sensing performances are compared under the same initial conditions. The simulation results show that implementation of the GD improves the spectrum sensing performance in CR systems both for independent and correlated antenna array elements.


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## Body

## 1. Introduction

Simple random access protocols such as the carrier sense multiple access (CSMA) are widely used in many network applications. Using these protocols, the users and nodes have to define an availability of the radio channel or possibility to use a definite spectrum in order to start the transmission process after an arbitrary delay.  The cognitive radio (CR) concept depends on the spectrum sharing and opportunistic spectrum access when there is a secondary network additionally to the primary network that has priority in access to spectral resources. The CR is an effective approach to improve spectrum utilization or radio resources by introducing an opportunistic use of frequency bands unused by the primary or licensed users. The CR systems have ability to measure, sense, learn, and define the radio channel parameters, the spectrum availability, and the radio operating conditions.Two types of users are considered in the CR systems, namely, the primary user and the secondary user. The primary users (the licensed users) have a priority to use the available designated spectrum. The secondary users are allowed to temporally use idle spectrum unused by the primary users. The secondary user should take down the radio resources if the primary user needs to use the same radio resources. Thus, the secondary user should try to find another idle radio resources or frequency bands.In general, most of the existed spectrum sensing approaches are based on the energy detector [1, 2], matched filter, [3, 4], and cyclostationary detector [5, 6]. The matched filter requires a complete knowledge about the signal received from the primary user and signaling features. The cyclostationary detector exploits features of signal received from the primary user caused by periodicity. Advantage of the energy detector (ED) is an absence of any required information about the signal sent by the primary user. The ED is considered to be optimal in the case of independent antenna array elements [1], but it is not true in the opposite case, that is, the correlated antenna array elements.  In general, the ED is sensitive to noise when variations in the noise power can cause a serious decline in the detection performance.The employment of ED with dynamic threshold in CR systems is investigated in [7] when the detection performance is defined under the fast fluctuated average noise power. The ED dynamic threshold is proposed to solve the problem of degradation in detection performance and sensitivity under the fast fluctuated average noise power, especially, at low values of the signal-to-noise ratio (SNR). The spectrum sensing performance under implementation of ED in CR systems is investigated in [8] using the dynamic threshold.Several spectrum sensing approaches based on the generalized likelihood ratio test (GLRT) are investigated in [8] with the purpose to be implemented by CR systems. The techniques proposed in [9] use the eigenvalues of the sample covariance matrix of the received signal vector by treating unknown parameters of the probability density function (pdf) of observed data independently of the presence of the signal from the primary user. Another signal detection scheme based on the eigenvalues of the covariance matrix of the signal vector received from the primary user is proposed in [10]. This scheme generates a decision statistics for the detection of signal sent by the primary user based on the ratio between the largest and the smallest eigenvalues of the covariance matrix of the received signal vector. In this case, the probability of miss is defined as a function of the number of cooperative receivers, number of samples, and SNR [11].Detection performance improving can be achieved by cooperative spectrum sensing using the two-step threshold ED [12]. The two-step thresholds are used for local detection allowing us to make a reliable local decision at each sensing node. The final decision is defined by combining the results of local decisions using the data fusion center. Under the spectrum sensing based on known signal pattern (waveform) of the primary user [13], the preamble (a known data sequence transmitted before each data burst) and the midamble (a known data sequence transmitted in the middle of the data burst) are used. Thus, if the signal pattern is known, the sensing process is performed by correlating the received signal with a known copy of itself (the coherent sensing). Some parameters extracted from the received signal, for example, the signal energy, and power spectral density are employed by radio identification sensing approach [14, 15]. More information about other spectrum sensing techniques such as the multitaper spectral estimation, wavelet transform estimation, and time-frequency analysis can be found in [13].There are many problems under spectrum sensing in CR systems, namely, the detection of the signal received from the primary user under correlation of the antenna array elements, interference cancellation, hidden primary user, and sensing efficiency when the data transmission is not allowed for the CR users during the observation period. The last problem decreases the transmission opportunities [16].Because of the low computation costs and implementation complexity, the ED is widely used in the spectrum sensing. Additionally, it does not need any knowledge about the signal sent by the primary user. The ED detects the signal by comparing the decision statistics with the detection threshold depended on the noise power (variance) [17]. The ED has some problems related to spectrum sensing including the threshold selection, interference cancellation, noise differentiation, noise power estimation, and detection performance degradation under the correlated antenna array elements and at the low SNR. The noise variance estimation problem is solved by distinguishing the noise and signal subspaces using the multiple signal classification (MUSIC) algorithm [18].The idea to employ the generalized detector (GD) for spectrum sensing in CR is proposed with the purpose to improve the sensing performance under the correlated antenna array elements because the GD has the same advantage as the ED; that is, no knowledge about parameters of the signal sent by the primary user is required. The GD is a combination of the Neyman-Pearson (NP) detector and ED based on the generalized approach to signal processing in noise [19]. As well known, the NP detector is optimal for the detection of signals with known parameters and the ED is optimal for the detection of signals with unknown parameters. The GD allows us to formulate a decision-making rule about the presence or absence of the signal based on definition of the jointly sufficient statistics of the mean and variance of likelihood function [20]. The GD implementation in wireless communication systems and GD detection performance are discussed in [21]. How we can improve the detection performance employing GD in radar sensor systems is investigated in [22, 23].In this paper, the spectrum sensing in CR systems based on employment of the GD is evaluated. We define the detection threshold and the probability of false alarm under GD employment in CR systems. The sensing performance of the ED and GD is compared under the same conditions for two scenarios: firstly, the independent antenna array elements; secondly, the correlated antenna array elements. The simulation results demonstrate the better sensing performance of the GD in comparison with the ED one both for independent and correlated antenna array elements.The remainder of this paper is organized as follows. The system model is presented in Section2. The GD main structure and the decision statistics are introduced in Section 3. Section 4 describes a definition of the GD threshold and a derivation of the probability of false alarm. The threshold and the probability of false alarm for ED are discussed in Section 5. The simulation results are presented in Section 6. The conclusion remarks are made in Section 7.


## 2. Spectrum Sensing in Correlated Antenna Array Elements

Assume that the spectrum sensing is carried out by the secondary user and/or secondary sensing node with the number of antennas equal toM (M antenna array elements). At the specific kth time instant and for the ith antenna array element, the binary hypothesis test for the spectrum sensing can be presented in the following form:
(1)xi[k]={wi[k],i=1,…,M;k=0,…,N-1⟹ℋ0;hi[k]a[k]+wi[k],i=1,…,M;k=0,…,N-1⟹ℋ1,
where xi[k] is the discrete-time received signal at the input of secondary user or secondary sensing node; a[k] is the discrete-time shift phase keying (PSK) modulated transmitted signal with the equal likely probability of transmission for all symbols; hi[k] is the discrete-time channel coefficient; and wi[k] is the discrete-time additive white Gaussian noise (AWGN) with zero mean and variance equal to σn2, that is, wi[k]~𝒞𝒩(0,σn2), where 𝒞 denotes that wi[k] is the complex random variable. The PSK modulated signal a[k] is transmitted over a Rayleigh fading channel with coefficients obeying the complex Gaussian distribution with zero mean and variance equal to σh2, that is, hi[k]~𝒞𝒩(0,σh2). The channel coefficients hi[k] corresponding to the ith antenna array element, i=1,…,M, are correlated between each other and independent of the time. The PSK modulated signal a[k], the channel coefficients hi[k], and the AWGN wi[k] are independent between each other.The exponential correlation model of antenna array elements is widely used owing to its simplicity and complete description of the spatial correlation [24]. The components of the M×M correlation matrix Cor are presented in the following form:
(2)Corij={ρi-j,i≤j,Corji*,i>j,i,j=1,…,M,
where ρ is the coefficient of correlation between two adjacent antenna array elements, 0≤ρ≤1 (real values), and * denotes the complex conjugate. Using the approximated cross-correlation function defined in [25], the correlation coefficient ρ can be given as
(3)ρ=exp(-23Λ2(dλc)2),
where Λ is the angular spread, λc is the wavelength, and d is the distance between adjacent antenna array elements (antenna spacing). Thus, under these conditions, the correlation matrix Cor is the symmetric Toeplitz matrix [26].The signals are received byM antenna array elements. If the sample size of received signals is N, the MN×1 received signal vector can be defined in the following form
(4)X=[x1(0),…,xM(0),…,x1(N-1),…,xM(N-1)]T,
where T denotes a transpose. The covariance matrices of the received signal vector X under the hypotheses ℋ0 and ℋ1  can be written in the following form:
(5)ℋ0⟹Cov0=E[XXH∣ℋ0]=σn2I,ℋ1⟹Cov1=E[XXH∣ℋ1]=Eaσh2A+σn2I,
where H denotes the Hermitian conjugate (conjugate transpose), I is the MN×MN identity matrix, Ea is the received signal energy at the input of the secondary user or sensing node, and A is the MN×MN matrix defined based on the correlation matrix Cor [26]
(6)A=[Cor0M⋯0M0M⋱⋱⋮⋮⋱⋱0M0M⋯0MCor]MN×MN,
where 0M is an M×M zero matrix.


## 3. GD Structure and Decision Statistics

The GD structure is presented in Figure1. Here MSG is the model signal generator (local oscillator), AF is the additional filter, and PF is the preliminary filter. The threshold apparatus (THRA) device defines the GD threshold and the signal model generator switching apparatus (SGSA) is used to switch on the MSG with the purpose to define the unknown parameters of the detected signal. The noise power estimator evaluates σn2 that is the variance of the noise at the GD input.Figure 1
GD structure.PF and AF are two linear systems at the GD front end that can be presented, for example, as the band-pass filters with the impulse responseshPF(τ) and hAF(τ). For simplicity of analysis, we think that these filters have the same amplitude-frequency responses and bandwidths. Moreover, a resonant or centered frequency of the AF is detuned relative to a resonant frequency of the PF on such a value that the information signal cannot pass through the AF.  Thus, the information signal and noise can appear at the PF output and only the noise appear at the AF output. If a value of detuning between the AF and PF resonant frequencies is more than 4÷5Δfa, where Δfa is the information signal bandwidth, the processes forming at the AF and PF outputs can be considered as independent and uncorrelated processes. In practice, under this condition the coefficient of correlation is not more than 0.05 [20, Chapter 3]. When the Gaussian noise w(t) comes in at the AF and PF inputs (the GD linear system front end), the noise forming at the AF and PF outputs is Gaussian, too, because the AF and PF are the linear systems. We may think that the AF and PF do not change the statistical parameters of input process since they are the linear GD front end systems. For this reason, the AF can be considered as a generator of reference sample with a priori information a “no” signal. A detailed discussion of the AF and PF can be found in [27, Chapter 5].The signal at the PF output can be defined asyi[k]=ai[k]+ζi[k], where ζi[k] is the observed noise at the PF output and ai[k]=hi[k]a[k]. Under the hypothesis ℋ0 and for all ith and kth, yi[k] is subjected to the complex Gaussian distribution with zero mean and variance σn2 and is considered as the independent and identically distributed (i.i.d) process. The AF output signal is the reference noise ηi[k]. The model signal is defined as
(7)aim[k]=βai[k],
where aim[k] is the generated model signal and β is the coefficient of the proportionality. The main functioning condition of GD is an equality over the whole range of parameters between the model signal forming at the GD MSG output and the detected signal forming at the GD input linear system (the PF) output [19]. How we can do it in practice is discussed in [20, Chapter 7].The decision statistics at the GD output can be presented in the following form [20, Chapter 3]:
(8)TGD(X)=∑k=0N-1∑i=1M2yi[k]aim[k]-∑k=0N-1∑i=1Myi2[k]+∑k=0N-1∑i=1Mηi2[k]≷ℋ0ℋ1THRGD,
where THRGD is the GD threshold. The first term in (8) corresponds to the NP detector with twice the gain and is considered as the sufficient statistics of the likelihood function mean. The second term in (8) corresponds to the ED and is considered as the sufficient statistics of the likelihood function variance. The third term in (8) presents the reference noise power generated according to the main functioning principles of the GD [19, Chapter 3]. Under the hypothesis ℋ1 corresponding to yi[k]=ai[k]+ζi[k] and the condition aim[k]=ai[k], the GD decision statistics takes the form
(9)TGD(X)=∑k=0N-1∑i=1Mai2[k]-∑k=0N-1∑i=1Mηi2[k]-∑k=0N-1∑i=1Mζi2[k],
where the second and third terms in (9) present the background noise at the GD output. The background noise is a difference between the noise power forming at the PF and AF outputs. In the opposite case (the hypothesis ℋ0) corresponding to yi[k]=ζi[k], the right side of (9), is the background noise only.


## 4. Spectrum Sensing Performance of GD

### 4.1. Correlated Antenna Array Elements

According to the GD decision statistics at the hypothesisℋ1 given by (9) if the energy of signal received by each of M antenna elements is combined with equal gain and the condition ai[k]=aim[k] is satisfied, the GD defines the total received signal energy within the limits of the observation interval and compares this energy with the GD threshold THRGD to make a decision of a “yes” or a “no” signal sent by the primary user. The probability of false alarm PFAGD and the probability of miss PmissGD are defined using the following forms [20, Chapter 6]:
(10)PFAGD=P(TGD(X)≥THRGD∣ℋ0)=1-Φ(THRGD-mℋ0GDVarℋ0GD),PmissGD=P(TGD(X)<THRGD∣ℋ1)=Φ(THRGD-mℋ1GDVarℋ1GD),
where
(11)Φ(x)=12+12erf(x2)
is the integral of probability,
(12)erf(x)=∫0xexp(-t2)dt
is the error function which is identical to Φ(x),mℋ0GD is the mean of the decision statistics TGD(X) under the hypothesis ℋ0, Varℋ0GD is the variance of the decision statistics TGD(X) under the hypothesis ℋ0, and mℋ1GD and Varℋ1GD are the mean and variance of the decision statistics under the hypothesis ℋ1, respectively. The decision statistics TGD(X) is a sum of M×N i.i.d. random variables.  Using a relationship between the probability of detection PDGD and the probability of miss PmissGD(13)PDGD=1-PmissGD
and taking into consideration a definition of the Gaussian Q-function
(14)Q(x)=12-12erf(x2),
based on (10) the probability of false alarm PFAGD and the probability of detection PDGD can be defined in the following forms:
(15)PFAGD=P(TGD(X)≥THRGD∣ℋ0)=Q(THRGD-mℋ0GDVarℋ0GD),(16)PDGD=P(TGD(X)>THRGD∣ℋ1)=Q(THRGD-mℋ1GDVarℋ1GD),
where the Gaussian Q-function can be presented in another form:
(17)Q(x)=12π∫x∞exp(-t22)dt.
The moment generating function (MGF) of the partial decision statistics given by
(18)TGD(Xk)=∑i=1Mai2[k]+∑i=1Mηi2[k]-∑i=1Mζi2[k],
where ∑i=1Mai2[k] is the sum of correlated random variables, can be delivered using the procedure discussed in [28]. As follows from (18), the MGF is defined as
(19)ℳTGD(Xk)(s)=∏i=1M(1-s(Eaσh2λi))-1×∏i=1M(1-s(2σn2))-1∏i=1M(1+s(2σn2))-1,
where λi is the eigenvalue of the ith spatial channel of the correlation matrix Cor. The mean and the variance of the partial decision statistics under the hypothesis ℋ1 can be presented in the following form:
(20)mℋ1GD=E[TGD(Xk)∣ℋ1]=M(Eaσh2),Varℋ1GD=Var[TGD(Xk)∣ℋ1]=∑i=1M(Eaσh2λi)2+4Mσn4.
Similarly, the variance of the partial decision statistics TGD(Xk) under the hypothesis ℋ0 takes the form
(21)Varℋ0GD=Var[TGD(Xk)∣ℋ0]=4Mσn4.
Under the hypothesis ℋ0, the mean of the partial decision statistics TGD(Xk) is equal to zero, mℋ0GD=0 [20, Chapter 3]. For large values of N, the central limit theorem can be applied to obtain the pdf of the GD decision statistics.With the purpose of avoiding the interference for the primary user in the CR systems, we define a lower bound of the probability of detectionPD. Thus, the sensing performance is evaluated by the probability of false alarm PFA while the probability of detection PD is maintained in accordance with the determined lower bound. In this case, there is a need to define the GD threshold THRGD as a function of the probability of detection PD applying the required lower bound. In practice, in the case of GD, a knowledge of the GD input noise variance is sufficient to define the detection threshold. In other words, the noise variance at the GD input can be estimated.We assume that the probability of detection is lower bounded, that is,PDGD≥α, where α is the constraint. Based on (16), the GD threshold THRGD can be presented in the following form:
(22)THRGD=mℋ1GD+Varℋ1GDQ-1(α).
As follows from (20), the GD threshold THRGD can be rewritten in the following form:
(23)THRGD=NMEaσh2+Q-1(α)N{∑i=1M(Eaσh2λi)2+4Mσn4}.
The SNR at the secondary sensing node input is defined as
(24)γ=Eaσh2σn2.
Taking into account (24), the THRGD can be presented in the following form:
(25)THRGD=NMγσn2+Q-1(α)N{∑i=1M(γσn2λi)2+4Mσn4}.
Based on (15), (21), and (23), the probability of false alarm PFAGD under correlated antenna array elements is defined in the following form:
(26)PFAGDcor=Q(NMEaσh2+Q-1(α)N{∑i=1M(Eaσh2λi)2+4Mσn4}4NMσn4).
After some elementary mathematical transformations and using (24), we can rewrite the PFAGDcor as follows:
(27)PFAGDcor=Q(γNM+Q-1(α)(1/M)(∑i=1M(γλi)2+4M)2).


### 4.2. Independent Antenna Array Elements

Under conditions that the value ofd/λc is high and the angular spread Λ value is close to π, there is no correlation between the adjacent antenna array elements in the GD; that is, the correlation coefficient is equal to zero (ρ=0). Then, taking into consideration that the correlation matrix becomes M×M identity matrix, the probability of false alarm PFA can be presented as a limiting case [26]:
(28)PFAuncor=limρ→0PFAcor.
Thus, based on (27), the probability of false alarm PFAGD under uncorrelated antenna array elements can be presented in the following form:
(29)PFAGDuncor=limρ→0PFAGDcor=Q(Q-1(α)(γ2+4)+γNM2).
Equation (29) presents the lower bound of the probability of false alarm PFAGD.


## 4.1. Correlated Antenna Array Elements

According to the GD decision statistics at the hypothesisℋ1 given by (9) if the energy of signal received by each of M antenna elements is combined with equal gain and the condition ai[k]=aim[k] is satisfied, the GD defines the total received signal energy within the limits of the observation interval and compares this energy with the GD threshold THRGD to make a decision of a “yes” or a “no” signal sent by the primary user. The probability of false alarm PFAGD and the probability of miss PmissGD are defined using the following forms [20, Chapter 6]:
(10)PFAGD=P(TGD(X)≥THRGD∣ℋ0)=1-Φ(THRGD-mℋ0GDVarℋ0GD),PmissGD=P(TGD(X)<THRGD∣ℋ1)=Φ(THRGD-mℋ1GDVarℋ1GD),
where
(11)Φ(x)=12+12erf(x2)
is the integral of probability,
(12)erf(x)=∫0xexp(-t2)dt
is the error function which is identical to Φ(x),mℋ0GD is the mean of the decision statistics TGD(X) under the hypothesis ℋ0, Varℋ0GD is the variance of the decision statistics TGD(X) under the hypothesis ℋ0, and mℋ1GD and Varℋ1GD are the mean and variance of the decision statistics under the hypothesis ℋ1, respectively. The decision statistics TGD(X) is a sum of M×N i.i.d. random variables.  Using a relationship between the probability of detection PDGD and the probability of miss PmissGD(13)PDGD=1-PmissGD
and taking into consideration a definition of the Gaussian Q-function
(14)Q(x)=12-12erf(x2),
based on (10) the probability of false alarm PFAGD and the probability of detection PDGD can be defined in the following forms:
(15)PFAGD=P(TGD(X)≥THRGD∣ℋ0)=Q(THRGD-mℋ0GDVarℋ0GD),(16)PDGD=P(TGD(X)>THRGD∣ℋ1)=Q(THRGD-mℋ1GDVarℋ1GD),
where the Gaussian Q-function can be presented in another form:
(17)Q(x)=12π∫x∞exp(-t22)dt.
The moment generating function (MGF) of the partial decision statistics given by
(18)TGD(Xk)=∑i=1Mai2[k]+∑i=1Mηi2[k]-∑i=1Mζi2[k],
where ∑i=1Mai2[k] is the sum of correlated random variables, can be delivered using the procedure discussed in [28]. As follows from (18), the MGF is defined as
(19)ℳTGD(Xk)(s)=∏i=1M(1-s(Eaσh2λi))-1×∏i=1M(1-s(2σn2))-1∏i=1M(1+s(2σn2))-1,
where λi is the eigenvalue of the ith spatial channel of the correlation matrix Cor. The mean and the variance of the partial decision statistics under the hypothesis ℋ1 can be presented in the following form:
(20)mℋ1GD=E[TGD(Xk)∣ℋ1]=M(Eaσh2),Varℋ1GD=Var[TGD(Xk)∣ℋ1]=∑i=1M(Eaσh2λi)2+4Mσn4.
Similarly, the variance of the partial decision statistics TGD(Xk) under the hypothesis ℋ0 takes the form
(21)Varℋ0GD=Var[TGD(Xk)∣ℋ0]=4Mσn4.
Under the hypothesis ℋ0, the mean of the partial decision statistics TGD(Xk) is equal to zero, mℋ0GD=0 [20, Chapter 3]. For large values of N, the central limit theorem can be applied to obtain the pdf of the GD decision statistics.With the purpose of avoiding the interference for the primary user in the CR systems, we define a lower bound of the probability of detectionPD. Thus, the sensing performance is evaluated by the probability of false alarm PFA while the probability of detection PD is maintained in accordance with the determined lower bound. In this case, there is a need to define the GD threshold THRGD as a function of the probability of detection PD applying the required lower bound. In practice, in the case of GD, a knowledge of the GD input noise variance is sufficient to define the detection threshold. In other words, the noise variance at the GD input can be estimated.We assume that the probability of detection is lower bounded, that is,PDGD≥α, where α is the constraint. Based on (16), the GD threshold THRGD can be presented in the following form:
(22)THRGD=mℋ1GD+Varℋ1GDQ-1(α).
As follows from (20), the GD threshold THRGD can be rewritten in the following form:
(23)THRGD=NMEaσh2+Q-1(α)N{∑i=1M(Eaσh2λi)2+4Mσn4}.
The SNR at the secondary sensing node input is defined as
(24)γ=Eaσh2σn2.
Taking into account (24), the THRGD can be presented in the following form:
(25)THRGD=NMγσn2+Q-1(α)N{∑i=1M(γσn2λi)2+4Mσn4}.
Based on (15), (21), and (23), the probability of false alarm PFAGD under correlated antenna array elements is defined in the following form:
(26)PFAGDcor=Q(NMEaσh2+Q-1(α)N{∑i=1M(Eaσh2λi)2+4Mσn4}4NMσn4).
After some elementary mathematical transformations and using (24), we can rewrite the PFAGDcor as follows:
(27)PFAGDcor=Q(γNM+Q-1(α)(1/M)(∑i=1M(γλi)2+4M)2).


## 4.2. Independent Antenna Array Elements

Under conditions that the value ofd/λc is high and the angular spread Λ value is close to π, there is no correlation between the adjacent antenna array elements in the GD; that is, the correlation coefficient is equal to zero (ρ=0). Then, taking into consideration that the correlation matrix becomes M×M identity matrix, the probability of false alarm PFA can be presented as a limiting case [26]:
(28)PFAuncor=limρ→0PFAcor.
Thus, based on (27), the probability of false alarm PFAGD under uncorrelated antenna array elements can be presented in the following form:
(29)PFAGDuncor=limρ→0PFAGDcor=Q(Q-1(α)(γ2+4)+γNM2).
Equation (29) presents the lower bound of the probability of false alarm PFAGD.


## 5. ED Spectrum Sensing Performance

The ED flowchart is presented in Figure2 where we use the following notations: A/D is the analog-to-digital converter, FFT is the fast Fourier transform, and (⋯)2 denotes the square law function. The spectrum sensing performance of ED was discussed in [26] for two cases: there is correlation between the antenna array elements and there is no correlation between the antenna array elements. Under the initial conditions discussed in Section 2, the decision statistics at the ED output can be defined as
(30)TED(X)=∑k=0N-1∑i=1Mxi2[k].
The ED decision statistics under the hypotheses ℋ0 and ℋ1 obeys the following distributions [26]:
(31)ℋ0⟹TED(X)~𝒩(NMσn2,NMσn4),ℋ1⟹TED(X)~𝒩(NM(Eaσh2+σn2),N∑i=1M(Eaσh2λi+σn2)2).
In the case of correlated antenna array elements, the probability of false alarm PFAED can be derived also based on the detection threshold and the lower bounded probability of detection PDED, that is, PDED≥α. According to [26], the probability of false alarm PFAED can be written as
(32)PFAEDcor=Q[Q-1(α)1M∑i=1M(γλi+1)2+γNM].
In the case when the antenna array elements are uncorrelated, the probability of false alarm PFAED takes the following form [26]:
(33)PFAEDuncor=limρ→0PFAEDcor=Q[Q-1(α)(γ+1)+γNM].Figure 2
ED structure.


## 6. Simulation Results

The ED and GD sensing performances in CR systems are compared under the same initial conditions for two cases, namely, the independent antenna array elements and the correlated antenna array elements. We verify the spectrum sensing performance analysis for both detectors using MATLAB where the simulation conditions and parameters are established according to IEEE 802.22 [29]. The main simulation parameters are presented in Table 1.Table 1
Simulation parameters.
Parameter
Value
The angular spread (Λ), the correlated antenna array elements
0.5
°
The angular spread (Λ), the uncorrelated antenna array elements
180
°
Distance between antenna elements (d), the correlated antenna array elements
d
=
λ
c
/
8
Distance between antenna elements (d), the uncorrelated antenna array elements
d
=
λ
c
/
2
Number of antenna array elements (M)
2
÷
10
SNR
-
20
÷
0 (dB)
P
D constraint  (α)
0.99
Coefficient  of correlation (ρ)
0; 0.1; 0.25; 0.5; 0.75; 1
Channel parameter (σh2)
1
N
20In Figure3, the probability of false alarm PFA of ED and GD is presented as a function of SNR when the antenna array elements are independent, and when the antenna array elements are correlated with the coefficient of correlation ρ=1, the number of antenna array elements is M=6.  The GD demonstrates better performance in comparison with the ED for all cases. For example, in the case of independent antenna array elements, at the probability of false alarm PFA equal to 0.5 the SNR gain in favor of GD is approximately 4 dB in comparison with the ED. Under the correlated antenna array elements and at the same probability of false alarm PFA=0.5, the SNR gain is about 2 dB in favor of GD comparing with the ED. In general, as shown in Figure 3, the probability of false alarm PFA for the correlated antenna array elements both for the ED and GD is higher in comparison with the case when the correlation between antenna array elements is absent.Figure 3
Comparison between the ED and GD sensing performance.Figure4 presents the receiver operation characteristic (ROC) curves for the GD and ED when the antenna array elements are independent; the number of antenna array elements is M=6 and SNR= −5 dB and −10 dB. The GD demonstrates superiority in sensing performance. For example, at the probability of false alarm PFA being equal to 0.1 and SNR = −10 dB, the probability of detection PD in the case of ED is equal approximately to 0.45, while the GD achieves the probability of detection PD equal to 0.8 under the same conditions. At the SNR = −5 dB and if the probability of false alarm PFA is equal to 0.1, both ED and GD achieve a probability of detection PD of more than 0.9.Figure 4
ROC of ED and GD.In Figure5, we illustrate the GD performance in terms of the probability of false alarm PFAGD; when the number of antenna array elements is variable 2≤M≤10, the coefficient of correlation ρ is changed as a parameter within the limits of 0.1≤ρ≤1, and the SNR = −5 dB. As we can see from Figure 5, the probability of false alarm PFAGD increases monotonically with the increasing in the coefficient of correlation ρ between antenna array elements. The use of large number of antenna array elements M allows us to reduce the negative action of the coefficient of correlation ρ on the probability of false alarm PFAGD.Figure 5
The probability of false alarm for GD versus the number of antenna array elements at various values of the coefficient of correlation.


## 7. Conclusions

Comparison of the spectrum sensing performance between the ED and GD is performed under the independent and correlated antenna array elements in CR systems at the low SNR. The GD overcomes the ED in the sensing performance when the antenna array elements are either independent or correlated. The simulation results show a validity to use the GD for spectrum sensing in CR systems and confirm a superiority of GD implementation in comparison with ED.  GD and ED performance analysis allows us to conclude that the probability of false alarm is lower bounded when the antenna array elements are independent. The GD sensing performance is a function of the coefficient of correlation between the antenna array elements. It follows from the fact that the probability of false alarm increases with the increasing in the coefficient of correlation between the antenna array elements. The use of large number of antenna array elements allows us to reduce a negative influence of correlation between the antenna array elements and, consequently, a degradation of the GD sensing performance.



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*Source: 853746-2013-06-19.xml*