Multiple-Antenna Cooperative Spectrum Sensing Based on the Wavelet Transform and Gaussian Mixture Model

Spectrum sensing is a core technology in cognitive radio (CR) systems. In this paper, a multiple-antenna cooperative spectrum sensor based on the wavelet transform and Gaussian mixture model (MAWG) is proposed. Compared with traditional methods, the MAWG method avoids the derivation of the threshold and improves the performance of single secondary user (SU) spectrum sensing in cases of channel loss and hidden terminal. The MAWG method reduces the noise of the signal which collected by the multiple-antenna SUs through the wavelet transform. Then, the fusion center (FC) extracts the statistical features from the signals that are pre-processed by the wavelet transform. To extract the statistical features, an sensing data fusion method is proposed. The MAWG method divides all SUs that are involved in the cooperative spectrum sensing into two clusters and extracts a two-dimensional feature vector. In order to avoid complicated decision threshold derivation, the Gaussian mixture model (GMM) is used to train a classifier for spectrum sensing according to these two-dimensional feature vectors. Simulation experiments are performed in the κ−μ channel model. The simulation shows that the MAWG can effectively improve spectrum sensing performance under the κ−μ channel model.


Introduction
Spectrum sensing technology is used to detect and judge whether the primary user (PU) signal is present and find the spectrum holes for secondary users (SUs) to access [1][2][3]. The single SU spectrum sensing methods are susceptible to channel fadding, hidden terminal and other issues. To solve these shortcomings of classical spectrum sensing, random matrix theory (RMT) is applied to cooperative spectrum sensing (CSS), which has become a research hotspot [4]. In these methods, the covariance matrix should be calculated based on the signal matrix from SUs. Furthermore, the corresponding eigenvalue is calculated as a statistical feature of the covariance matrix for spectrum sensing. There are many CSS methods based on RMT have been proposed [5][6][7][8], such as the ratio of the maximum and minimum eigenvalue (MME), the ratio of the maximum eigenvalue to the trace (RMET), the difference between the maximum eigenvalue and the average eigenvalues (DMEAE), and the difference between the maximum and the minimum eigenvalue (DMM). These methods need to derive threshold based on

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A new spectrum sensing method is developed to reduce the noise associated with the signal by using the wavelet transform. In this method, each SU collects spectrum sensing data from environment, performs wavelet transform to reduce noise and send the pre-processed sensing data to fusion center (FC).

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A spectrum sensing method based on GMM is proposed to avoid threshold derivation. The GMM is used to train the CSS classifier. After training, the FC uses the classifier to make the final decision about the PU state.

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In the experimental simulation section, we compare and analyze the performance of MAWG and single antenna CSS method [21,23]. These methods are simulated using the κ − µ channel. The simulation results show that the MAWG can effectively improve the spectrum sensing performance.
This paper is organized as follows. Section 2 introduces the system model of multiple-antenna CSS. Section 3 proposes a CSS method based on the GMM, which is called MAWG. Section 4 simulates the MAWG method. Results indicate that the MAWG can effectively improve the spectrum sensing performance. Section 5 summarizes the full text and outlines a simple plan for future research work.

Basic Multiple-Antenna CSS and Eigenvalues in Random Matrix
There are some problems in cognitive radio networks (CRN), which are path losses and shadows. It is difficult for a single SU to accurately determine and judge whether the PU is using the licensed spectrum [24][25][26][27]. Therefore, in order to combat and reduce the impact of fading channels on spectrum sensing performance, this paper study cooperative SUs with multiple-antenna for spectrum sensing. The basic multiple-antenna CSS diagram is shown in Figure 1. According to Figure 1, it is assumed that there is only one PU, M cooperative SUs and a FC in the cognitive radio network. Each SU participating in the CSS has Aantennas. The PU and the FC only have one antenna, respectively. In Figure 1, the task of each SU with multiple-antenna is collecting sensing data, using wavelet transform to reduce noise and upload the pre-processed sensing data to the FC. The FC clusters the received signals from SUs and extracts feature vectors according to different clusters. Based on these feature vectors, a classifier is trained by using GMM. After training, the classifier is used for spectrum sensing in the FC.
According to the signal status received by each antenna of the SUs, a binary hypothesis model can be expressed by [28] x l i (n) = (1) n = 1, 2, ..., N, where x l i (n) represents the signal that is received by the lth antenna of the ith SU; h l i (n) represents the channel gain between PU and lth antenna of the ith SU; s l i (n) represents the signal that is transmitted by the PU; w l i (n) represents Gaussian white noise (GWN); H 1 and H 0 represent the presence and absence of the PU signal, respectively. N represents the number of sampling points. In the multiple-antenna system, the received signals from different antennas exist correlation. The correlation [29] between the ath, a ∈ 1, 2, ..., A and the bth, b ∈ 1, 2, ..., A antenna can be defined as , d ab represents the distance between the ath and the bth antenna, ν represents the wavelength and θ indicates the propagation direction of antenna. If ν = 2d ab and θ → 0 rad, the correlation C ab is the largest because of the Λ 2 ( d ab ν ) → 0. Thus, in this condition, s a i (n) = s b i (n) = s i (n). In this paper, a sample condition is considered, furthermore Equations (1) is rewritten as Based on the above assumption, the definition x l i = [x l i (1), x l i (2), . . . , x l i (N)] represents the signal that is received by the lth antenna of the ith SU. Thus, a signal matrix can be obtained where X i ∈ R A×N . For the convenience of the representation, the covariance matrix of the SU received signal is . Define a signal matrix S i of PU received by all antennas of ith SU after channel losses. The covariance matrix of the S i is [9,12] can be calculated by where I represents the identity matrix. The eigenvalue of R X i can be expressed by where α j represents the jth eigenvalue of R S i . For H 0 , the PU signal does not exist, and only the GWN exists in the signal matrix X i , which means that R X i = σ 2 I. At this time α j = 0 and λ max = λ 2 = λ 3 = · · · = λ min = σ 2 .
When H 1 is established, R X i = R S i + σ 2 I. Since the PU signal itself has a correlation, α j makes λ j no longer be equal. Therefore, we can get λ max > λ 2 > λ 3 > · · · > λ min .
In the following, the difference between the maximum eigenvalue and the minimum eigenvalue T DMM is calculated by The ratio of the maximum eigenvalue to the matrix trace T RMET is calculated by where tr(·) represents the trace of the matrix. For the T DMM feature, when H 0 is established, For H 1 , Equations (9) and (10) indicate that the values of T DMM are significantly different at H 0 and H 1 , respectively. Therefore, the T DMM can be used for spectrum sensing.
Similarly, for the T RMET feature, when H 0 is established, For H 1 , λ max = α max + σ 2 , and where α represents the average eigenvalue of R S i . We can see that the values of T RMET are different under the conditions of H 0 and H 1 from analyzing Equations (11) and (12), respectively. Therefore, these T RMET can be used for spectrum sensing.
In the experimental simulation analysis section, to effectively evaluate the performance of the MAWG algorithm, we use the detection probability P d and false alarm probability P f as the performance evaluation indicators. The specific form is as follows: whereĤ 1 is the measured status of PU being exist while H 1 is the actual status of the PU being exist.
where H 0 is the actual status of the PU being absent. For convenient reference, the symbols that are used in the paper are summarized as shown in Table 1.
Two clusters composed of different SUs X, P Matrix corresponding to C 1 and C 2

Symbol Notations
R X ,R P Covariance matrix of to X and P T DMM DMM eigenvalue T RMET RMET eigenvalue T X,z , T P,z Feature from R X and R P . T z A feature vector composed of T X,z and T P,z , where z ∈ {DMME, RMET} S Training feature set Threshold for controlling P f and P d

Spectrum Sensing Based on GMM
The GMM is a widely used clustering algorithm which uses multiple Gaussian distributions as parameter models according to the number of clusters. The expected maximum (EM) algorithm obtains the most optimal Gaussian distribution parameters by using samples. In spectrum sensing, the presence of the PU signal and the absence of the PU signal can be considered as two different Gaussian distributions according to Equation (3). Therefore, the GMM can be used for training. It is noted that the samples are two-dimensional feature vectors extracted from the sensing signals of SUs.

Spectrum Sensing System Model Based on GMM
In this section, the GMM is used for spectrum sensing. The whole process is divided into two parts, that is, the training part and the spectrum sensing part. As shown in Figure 2, the blue dotted box indicates the training part, and the yellow dotted box indicates the spectrum sensing part. Each SU previews the authorized spectrum, collects enough sensing data and pre-processes these data by Wavelet transform. Assume that these sensing data contain both states of PU. Then, the two-dimensional feature vectors T z are extracted from these sensing data. Finally, the classifier is trained on the FC. After the training, the classifier is used for spectrum sensing.

Signal Preprocessing Based on Wavelet Transform
Before calculating the two-dimensional feature vector of the SUs signals, in order to reduce the impacts of noise on the feature and improve the spectrum sesing performance under a low signal-noise ratio (SNR), the wavelet transform is used to conduct denoising in each SU. For the specific case in this paper, it is assumed that the signal collected by the lth antenna of the ith SU is . , x l i (N)] and the specific algorithm steps are as follows [30].
Step 1: The wavelet transform signal x l i is used to obtain the wavelet coefficient W.
Step 2: The wavelet coefficient W is the threshold that is used to obtain the estimated coefficient W.
Step 3: Perform wavelet reconstruction using W and obtain denoised signal. This paper uses a soft threshold function that is as follows: where β is the VisuShrink threshold [31,32], which satisfies where σ is the standard deviation of the noise. After the signals of SUs received are reduced noise by using wavelet noise, a new signal matrix can be obtained After sensing data is pre-processed by SUs, these sensing data is uploaded to the FC. In the CSS, i > 2. To extract signal feature, a sensing data fusion method is used to obtain a two-dimensional feature vector. It is noted that the fusion method can fuse the sensing data from SUs which equip different number antennas. Specifically, The FC divides the SUs into two clusters C 1 and C 2 . When i ≥ 2 and M is an odd number, let J 1 , J 3 , . . . , J M ∈ C 1 and J 2 , J 4 , . . . , J M−1 ∈ C 2 . When i ≥ 2 and M is even, let J 1 , J 3 , . . . , J M−1 ∈ C 1 and J 2 , J 4 , . . . , J M ∈ C 2 . Then, the matrices in C 1 and C 2 are recombined to obtain matrices X and P.
When i ≥ 2 and M is an odd number, the signal data that is collected by the SUs in the recombination cluster C 1 can obtain X, which is a By reorganizing the matrix in cluster C 2 , P can be obtained as a Similarly, when i ≥ 2 and M is even, X and P are also obtained, and both are MA 2 × N matrices. According to the obtained X and P matrices, the corresponding covariance matrices R X = E[XX T ] and R P = E[PP T ] are respectively calculated.
When i ≥ 2 and M is an odd number, it is assumed that the eigenvalues of matrices R X and R P are λ 1 (λ max ) > λ 2 > λ 3 > · · · > λ (M+1)A/2 (λ min ) or λ (M−1)A)/2 (λ min ) from the maximum to the minimum. When i ≥ 2 and M is even, then the eigenvalues of matrices R X and R P are λ 1 (λ max ) > λ 2 > λ 3 > · · · > λ MA/2 (λ min ) from the maximum to the minimum. According to this, T DMM can be obtained by T RMET can be calculated by Based on the covariance matrix, the corresponding statistical feature T DMM or T RMET is calculated. Let T R X ,z and T R P ,z , where z ∈ {DMM, RMET}, denote the features corresponding to R X and R P , respectively. Thus, a feature vector is obtained In the next section, the classifier is trained by using a sufficient number of T z feature vectors and GMM, which is used to achieve spectrum sensing.

Offline Training Based on GMM
Before training begins, we need to prepare a training feature vectors set [33], where B is the number of training feature vectors, T b z , b = 1, 2, . . . , B is the bth feature vector that is extracted according to the method that is proposed in this paper. The distributions of GMM can be expressed by [19] p where K represents the number of mixed components, π k is the mixing coefficient that satisfies ∑ K k π k = 1, N (x|µ k , Σ k ) is a Gaussian distribution with a mean of µ k and a variance of Σ k , According to the situation of spectrum sensing, spectrum sensing can be considered as a two-class problem whether the PU is using the licensed spectrum, which means K = 2. Thus, Equation (24) can be rewritten as p(x) = π 1 N (x|µ 1 , Σ 1 ) + π 2 N (x|µ 2 , Σ 2 ).
The process is described by Algorithm 1.
Step 3: Check whether the parameters converge. If they do not converge, return to Step 1 and continue executing the algorithm. End

Online Spectrum Sensing Based on GMM
After the training is completed, the optimal parameters π * k , µ * k , and Σ * k can be obtained. According to these optimal parameters, a classifier for spectrum sensing can be constructed where, the parameter ξ is used to control P f in the spectrum sensing system. If ξ is smaller, the PU is more likely not to use the authorized channel, which means the channel is available. Then, the probability of miss detection and P f are increased. Conversely, if the ξ is larger, the PU is more likely to use the authorized channel, which means the channel is unavailable. Hence, the P d and spectrum utilization are reduced. When performing online sensing, the two-dimensional feature vector T z are extracted from the channel which needs to be perceived. Finally, we use Equation (28) for spectrum sensing.

Experimental Simulation Analysis
In this section, the MAWG method is verified in κ − µ channel fading model. The κ − µ channel model is a widely accepted model because it can generate many known wireless channel models by adjusting the parameters κ and µ. By setting κ and µ in the κ − µ fading channel to some specific parameter values, it can be converted into known models, such as the Rayleigh fading channel (µ = 1, κ → 0), the Rician channel (µ = 1) and the Nakagami-m channel (κ → 0).
To demonstrate the performance of the MAWG method, in the simulation experiment, DMM or RMET is selected as the characteristic of the signal. The PU signal is a multiple component signal [22] in this experiment. According to the spectrum sensing statistical feature extraction method that is described above, 2000 feature vectors are extracted. Firstly, 1000 feature vectors are used to train the GMM framework. After the training is completed, the classifiers which are used for spectrum sensing are obtained. Then, the other 1000 feature vectors are used for testing.

Clustering Performance Analysis
In this section, we analyze the clustering effect of the GMM clustering algorithm under different characteristics and different channel conditions. The channel conditions are the Rayleigh fading channel (µ = 3, κ → 0) and Rician fading channel (µ = 1, κ = 3). Figures 3 and 4 show the clustering effects of the different features under Rayleigh channel with an SNR = −10 dB.  In Figures 3-6, the yellow circles represent the feature vectors that are classified as noise. The blue circles represent the feature vectors that are classified as the PU signal existence class. The star represents the mean µ * 2 of the PU signal existence classes. The square represents the mean µ * 1 of the noise class.
By observing and analyzing Figures 3-6, it can be found that the DMM feature can contain higher characteristic information for the reaction signal. Therefore, in the following simulation analysis, the simulation experiments are mainly carried out for DMM features in the MAWG method, which is called the MAWGDMM method.

Experimental Results and Performance Analysis with Different SNR
The IQDMM and IQRMET methods in Figures 7 and 8 are proposed in Reference [21]. DARDMM and DARRMET are proposed in Reference [23]. These methods use the IQ and DAR decomposition to increase the logic SUs. In the feature extraction, the DMM and RMET were chosen to construct feature vector. For achieve CSS, the K-means is used in [21] and K-medoids is used in Reference [23]. The simulation parameters are set as follows in the MAWGDMM algorithm: the number of SUs is M = 2, the number of sampling points is N = 1000 and the number of antennas is A = 3. Figure 7 shows the simulation results when the SNR = −14 dB with Rayleigh and Rician fading channels. Figure 8 shows the simulation results when the SNR = −16 dB with Rayleigh and Rician fading channels. Tables 2 and 3 show the detection probabilities of the different algorithms under different fading channels at the same false alarm probability. The receiver operating characteristics (ROC) have been drawn for a comparative performance analysis.  Multiple-antenna spectrum sensing can make full use of multiplexing and spatial diversity, which can reduces the effects of path losses and shadows on spectrum sensing. Thus, according to Figures 7 and 8, Tables 2 and 3, it can conclude that the MAWGDMM has better spectrum sensing.

Performance Analysis with Different Values of A and M
This section analyzes the impacts of different number of cooperative SUs and the number of different antennas on the spectrum sensing performance. In Figure 9, the simulation parameters are set as follows-SNR = −16 dB, N = 1000, A = 2, and M is 2, 3, 4, 5, respectively.
As the number of SUs increases, more comprehensive information for the PU signal is collected, which can overcome the problem hidden terminal and improve the spatial diversity. The final decision of the FC is more reliable. Thus, from Figure 9 and Table 4, we can see that the spectrum sensing performance is further improved as the number of SUs increases.  When the SNR = −16 dB, a Rayleigh channel is used, and P f = 0.1; the spectrum perceptual performance when M = 5 is improved by 15.29%, 44.12%, and 145.00%, respectively, compares to conditions when M = 4, M = 3, and M = 2. When P f = 0.2, the perceived performance is increased by 5.32%, 22.22%, and 47.76%, respectively. Under a Rician channel, the performance is increased by 12.94%, 41.18%, and 134.15%, respectively, when P f = 0.1; the performance is increased by 4.26%, 16.67%, and 36.11%, respectively, when P f = 0.2.
As the number of SU antennas increases, the spatial diversity and spatial multiplexing gain are improved. The MAWGDMM can work well when many antennas observe the authorized spectrum together. By analyzing Figure 10 and Table 5, we can conclude that the spectrum sensing performance is improved.
When the SNR = −16 dB, a Rayleigh channel is used and P f = 0.1, the spectrum sensing performance when A = 6 is improved by 22.22%, 65.00%, and 153.85%, respectively, compares to conditions when A = 4, A = 3, and A = 2. When P f = 0.2, the performance is increased by 2.04%, 26.59%, and 75.44%, respectively. Under a Rician channel, the performance is increased by 15.29%, 48.48%, and 145.00% respectively, when P f = 0.1, and the performance is increased by 4.21%, 19.28%, and 65.00%, respectively, when P f = 0.2. Table 5. Detection probabilities for the different numbers of antennas when the SNR = −16 dB.

Performance Analysis with Different Numbers of Sample Points
This section will analyze the effect of different numbers of samples on the spectrum sensing performance. In Figure 11, the simulation parameters are set as follows-the SNR = −16 dB, M = 2, A = 2, N is 1000, 1200, 1600, and 2000, respectively. As the number of sampling points increases, a more complete PU signal can be collected by the SUs. The feature vector is more representative of the status of the PU. By analyzing Figure 11 and Table 6, it can be concluded that as the number of sample points increases, the spectrum sensing performance is improved. When the SNR = −16 dB, a Rayleigh channel is used and P f = 0.1; the spectrum sensing performance when N = 2000 is improved by 56.67%, 70.9%, and 135.00%, respectively, compares to conditions when N = 1600, N = 1200, and N = 1000. In the case when P f = 0.2, the performance is increased by 4.35%, 41.18%, and 68.42%, respectively. Under a Rician channel, the performance is increased by 17.14%, 78.26%, and 100.00%, respectively, when P f = 0.1, and the performance is increased by 6.67%, 45.45%, and 60.00%, respectively, when P f = 0.2.

Conclusions
This paper aims to improve spectrum sensing performance, especially the spectrum sensing performance of the fading channel. Based on this aim, this paper proposes the multiple-antenna CSS based on the wavelet transform and GMM. This method adopts cooperative SUs and the multiple antenna spectrum sensing method, which can effectively overcome the problems that are encountered by single SU spectrum sensing, such as path losses and shadows. Specifically, this paper proposes a new signal feature extraction method and combines the GMM to achieve spectrum sensing. In the experimental simulation section, the simulation with the κ − µ channel is performed and the simulation results are analyzed. The results show that the MAWG method can improve the spectrum sensing performance to some extent. In this paper, the analysis of the overall cost is ignored; in future research, we will further analyze the overall cost and improve the applicability of the algorithm. Acknowledgments: The author wants to thank the author's organization because they have provided us with many conveniences.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: