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Wideband signal detection for cognitive radio applications with limited resources
EURASIP Journal on Advances in Signal Processingvolume 2019, Article number: 2 (2019)
Abstract
Wideband signals are expected to be used to achieve the required quality of service (QoS) in the next generation of wireless communications, civil and military radar, and many wireless sensor network (WSN) scenarios. Wideband signal detection has been identified as one of the most challenging problems in the proliferation of the cognitive radio technology. Moreover in many applications, spectrum sensing in cognitive radio (CR) is expected to be performed with limited resources in terms of time, computation, and complexity. This paper is dedicated to the detection of a wideband signal with small sample size. Aiming at using small sample size, a statistical model of samples is given based on Student’s t distribution. However, the limited number of channel observations brings a reduction of confidence in the decision. A set of new basic probability assignments associated with the hypothesis of the occupied or vacant channel are then proposed to perform the DempsterShafer (DS) decision process. Simulation results show that the proposed method has much higher sensitivity to sense an occupied channel than the traditional energy detection method (ED) and the decision fusion method when small sample size is used.
Introduction
With the evolution and development of various wireless technologies, spectrum resources are becoming scarce due to the increasing need for spectral bandwidth and number of users. Cognitive radio (CR) technology has attracted a lot of interest, especially for the next generation of wireless communications, many types of radar systems and wireless sensor network (WSN) [1–4]. In all those systems, wideband signals are expected to be used to achieve the required quality of service (QoS). Therefore, wideband signal detection plays an important role in a wide range of wireless communication systems and has been identified as one of the most challenging problems in the CR technology applications [5–7].
Although, there are numerous current research works focusing on wideband signal detection, many severe challenges still exist [8, 9]. First of all, in realistic scenarios, it is very difficult to know the number of antennas, the coding scheme, and the structure of the detected signal. Therefore, an accurate blind spectrum sensing method without any prior information is of great interest. Moreover, in order to avoid unexpected harmful interference, CR user must be able to quickly vacate the frequency band when the licensed user starts transmitting. Thus, the sensing time must be limited to an acceptable level, while still guarantee a sufficiently low detection error probability.
For this purpose, a number of spectrum sensing methods have been proposed and investigated in [10–17]. Under no prior knowledge about the wideband signal, energy detection (ED) has been shown to be the most popular technique in cooperative sensing thanks to its low computational power requirements on wireless devices [3, 13]. However, energy detection is limited by the signaltonoise ratio (SNR) wall and has high probability of false alarm [14]. In order to overcome these shortcomings, eigenvaluebased spectrum sensing methods have been proposed [14, 15], which are mainly based on the asymptotic or limiting distribution of extreme eigenvalues in order to overcome the noise uncertainty problem. Unfortunately, they cannot be extended to a more general dimensional setting due to their daunting computational cost. Moreover, these techniques require large number of samples, which is often not suitable for real application scenarios [16]. Thus [18–20], study the spectrum sensing method using goodnessoffit (GoF) test for small sample size, relying on the AndersonDarling (AD) statistic. In that case, the GoF test is only performed to assess the rejection (or not) of the null hypothesis (i.e., the absence of PU signal). Different from the GoF test mentioned above, both hypotheses of the presence and absence of the wideband signal would be considered in the proposed method in order to make full use of the statical information of the binary hypotheses and improve the detection performance.
Considering the challenges mentioned above, a robust spectrum sensing method with small sample size is proposed in this work. On the one hand, this means short time in realtime data processing. Especially when the detecting devices have only a singleradio architecture, the time of sampling and observing the channel is expected to be as short as possible. On the other hand, we consider that only less steady state reception can be obtained in some complex information environment. Firstly, considering that the Gaussian approximation in ED is good only when the sample size is sufficiently large [21], we reformulate the spectrum sensing into Student’s t distribution test problem as in [19], which is popular in situations where the sample size is small. Besides, based on the characteristics of Student’s tdistribution, new basic probability assignment (BPA) functions are proposed for estimating the presence or not of a wideband signal emitter. However, due to the small number of samples, the estimation performance inevitably suffers from lack of reliability. In order to improve the reliability, DempsterShafer (DS) theory of evidence [22–30] is used to make a final decision. As in [31], in the proposed method, Student’s t distribution of a reduced number of samples is used. The main contribution stands in the proposition of two new BPA functions to evaluate the credibility of the collected small samples from a wideband signal and the combination of BPA functions in order to make a more reliable decision. Specifically, the novelty is that the cumulative distribution function (CDF) of Student’s t distribution is utilized to define the BPA functions, which is different from the BPA functions based on the CDF of normal distribution and ED in [23–27]. The new BPAs based on Student’s t distribution are more appropriate for dealing with the situation of small samples than the BPAs based on normal distribution. Moreover, in the proposed scheme, in order to fully exploit the collected samples, both hypotheses of presence or absence of wideband signal are used. Simulations show that the proposed method has much higher sensitivity to detect the presence of a signal than ED and GoFbased methods.
The rest of the paper is organized as follows. In Section 2, some spectrum sensing preliminaries are presented. The proposed spectrum sensing scheme including the statistical model of the received small samples, basic probability assignment functions and DS fusion, is described in Section 3. Simulation results and conclusions are given in Sections 4 and 5, respectively.
Spectrum sensing preliminaries
In this paper, we assume that a wideband signal needs to be detected. According to [32], a signal having a fractional bandwidth greater than 0.01 and smaller than 0.25 is categorized as wideband. Consider that the observed bandwidth is subdivided into K subbands with equal bandwidth B_{sub}. Then the full observed bandwidth is B_{tot}=KB_{sub}. In each subband, signals are bandpassed and downconverted to the baseband. In order to provide a detection in a very short time, a limited number of realvalued samples Q are collected in each subband. The Nyquist rate in each subband is 2B_{sub}, and by setting the oversampling factor to N, the sampling period is \(T_{s}=\frac {1}{2B_{\text {sub}}N}\). In each subband, the signal is oversampled with a factor N, which means that the sampling frequency is very much larger than the subband width. Actually, when N is large enough while maintaining a small Q (number of samples), the scheme in each subband can be seen as a narrowband signal sampling process. As the over sampling factor N is increased, the observation duration QT_{s} is reduced. Let \(x_{q}^{(k)}\) be the qth sample in the kth subband. Wideband signal detection can be formulated as a binary hypothesis problem as follows
where H_{0} and H_{1} respectively represent the hypotheses of absence and presence of a signal in the kth subband. \(w_{q}^{(k)}\) is the noise contribution in the sample. Without loss of generality, we assume that the noise is an additive white Gaussian noise (AWGN) with zero mean and variance σ^{2}. \(s_{q}^{(k)}\) is the signal component in the kth subband when it is present. According to the oversampling assumption and small number of samples (short observation), it can be assumed that the signal is constant during its observation so that \(s_{q}^{(k)}=s^{(k)}\). In each of the K subbands, Q samples are collected with oversampling factor N. The more K is large, the more the narrowband signal hypothesis in each subband is true. The more K and N are large, the more the constant signal assumption over Q samples tends to be true. Since the number of samples Q is small, the observation duration is very short and during this short period, the narrowband signal can be approximated as constant. Moreover, in practice, the distribution of the power spectral density of the signal is unknown, we assume that the signal uniformly occupies the full bandwidth which is the most reasonable, fair, and neutral assumption. As an example, this assumption holds in many multicarrier signals schemes. It allows to model the signal as a constant in both time and frequency domains. Let us consider in this study that s^{(k)}=1, without loss of generality. In the simulations section below, the values of K, N, and Q have been selected arbitrarily as a matter of example, and some simulations are provided in Subsection 4.4 to support the signal assumptions.
In this case, the spectrum sensing problem is equivalent to a standard scenario with Gaussian distributions having equal variance and different means under each hypothesis.
Methods
The proposed spectrum sensing method relies on a fusion processing using DS theory and a new set of BPA functions. BPA definition and evaluation are the key points of the DS fusion. In most applications, it is generally assumed that the number of available samples is sufficiently large in order to correctly estimate the BPAs and perform a reliable fusion. But in this work, we consider that the CR device is very limited in terms of sample size. Hence, we propose to define some new BPAs relying on Student’s t distribution.
Statistical model of the received small samples
Considering the small number of samples and the special sensing scenario about detecting a wideband signal in a zero mean Gaussian noise, it is shown that the optimal test in signal detection is Student’s t test [31]. In order to construct the test statistic in accordance with Student’s t distribution, we denote, respectively, X_{k} and \(S_{k}^{2}\) as the mean and variance of the samples in the kth subband,
and
where k=1,2,⋯,K. Hence, K variables (one from each subband) are obtained as
Under H_{0} hypothesis, there exists only noise, \(x_{q}^{(k)} \thicksim \mathcal {N}\left (0,\sigma ^{2}\right)\), then Y_{k} follows Student’s t distribution with degree v=Q−1 degrees of freedom. Otherwise, under H_{1} hypothesis, the received signal samples include the wideband signal and noise, then \(x_{q}^{(k)} \thicksim \mathcal {N}\left (\mu,\sigma ^{2}\right)\), with μ=1 in our case. It comes that Y_{k} has a noncentral Student’s t distribution with v = Q − 1 degrees of freedom and noncentrality parameter \(\delta = \sqrt {Q\mu ^{2}/\sigma ^{2}}\), where μ^{2}/σ^{2} is the SNR [19].
The probability density function (PDF) of Student’s t distribution has the similar bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. The larger Q is, the more Student’s t distribution approaches the standard normal distribution [33]. Conversely, when Q is small, the tails of Student’s t distribution are much heavier than those of the normal distribution, as shown in Fig. 1. Moreover, the PDFs of the noncentral t distribution with different Q are also given in Fig. 1. As we can see, there is a certain overlap between the tails of the t distribution and the noncentral t distribution with the same degrees of freedom v = Q − 1, and the overlapping region decreases with the increase of Q from Fig. 1a to d. This also validates that Y_{k} in Eq. (4) for small Q is prone to taking values that fall far from their statistical mean and leads to an unreliable BPA estimation. Therefore, we propose to calculate K variables Y_{k} and combine them by DS theory of evidence for a reliable decision. In addition, in order to estimate the belief of the observed signal in each subband, the cumulative distribution functions (CDF) of Y_{k} under H_{0} and H_{1} denoted by F_{0}(y) and F_{1}(y) are applied, which are given in [33]. For DS fusion, we assume that Y_{k} has at least two values, that is K≥2.
Note that in this statistical model we reformulate the received samples \(x_{q}^{(k)}\) into a new variable Y_{k}, which has Student’s t distribution and noncentral t distribution under H_{0} and H_{1} hypotheses, respectively. The CDF F_{0}(y) of Y_{k} under H_{0} hypothesis only depends on the degrees of freedom v, while F_{1}(y) is related to parameter \(\delta = \sqrt {Q\cdot \text {SNR}}\). In this work, we assume that the noise variance σ^{2} is known, as in EDbased methods. Moreover, for the proposed method, as explained in the next section, the wideband signal detection is done by evaluating the reliabilities of both H_{0} and H_{1} hypotheses, which is a beneficial feature that is not used in the conventional GoF test based methods.
Basic probability assignment estimation
According to the DS theory of evidence [22], Ω denotes the universal set, and let 2^{Ω} be its power set. A function m:2^{Ω}↦[0,1] named basic probability assignment (BPA) is defined to quantify the candidate proposition as follows:
where for any set A⊂2^{Ω}, m(A)>0 which provides the degree of confidence that proposition A is true. Then, in our framework, 2^{Ω}={∅,{H_{0}},{H_{1}},Ω}.
In order to evaluate the credibility of the collected samples in the kth subband, we propose two new BPA functions m_{k}(H_{0}) and m_{k}(H_{1}) for H_{0} and H_{1} hypotheses in Eqs. (6) and (7), respectively
where m_{k}(H_{0}) and m_{k}(H_{1}) are related to the CDF F_{0}(y) and F_{1}(y) of Y_{k}, respectively. Importantly, these BPA functions indicate the credibilities for hypotheses H_{0} and H_{1} to be true, respectively. For example, a larger value of Y_{k} results in a larger m_{k}(H_{1}) and a smaller m_{k}(H_{0}), and vice versa, as shown in Fig. 2. Thus, we can make a decision on the presence or not of the wideband signal by comparing m_{k}(H_{0}) and m_{k}(H_{1}). If m_{k}(H_{1})>m_{k}(H_{0}), the signal exists; otherwise, the signal does not exist. However, since the number of samples Q is small, Y_{k} has been obtained with a small number. This will cause a big uncertainty and increase the conflict between m_{k}(H_{0}) and m_{k}(H_{1}). Then a third BPA function is defined as follows:
where Ω={H_{1},H_{0}} denotes either hypothesis could be true and m_{k}(Ω) indicates the total uncertainty of the kth subband of samples. m_{k}(Ω) is finally equal to F_{0}−F_{1} by combining Eqs. (6), (7), and (8). We can show that the value of F_{0}−F_{1} is nonnegative with the CDFs of Student’s t distribution and the noncentral t distribution. As shown in Fig. 3, the CDF of Student’s t distribution F_{0} (the solid line) is on the left side of the noncentral t distribution F_{1} (the dashed line). Then, for the same value Y_{k}, F_{0} must be greater than or equal to F_{1}. In order to improve the probability of detection and reduce the influence of the conflict evidence, we make a final reliable decision by fusing all BPA functions obtained from the K groups of samples.
DS fusion and final decision
In order to improve the reliability of detection, we need to combine the K BPA functions and make a final decision. Then, according to the basic DS theory of evidence and Eqs. (6), (7), and (8), two new combined BPA functions can be obtained as [23]
where A_{k}⊂2^{Ω} for k∈{1,…,K} and κ is a measure of the amount of conflict among the mass sets:
From Eqs. (9), (10), and (11), we can see that the two new BPA functions m(H_{0}) and m(H_{1}) are obtained by using the orthogonal sum for K BPA functions, which indicate the credibilities for hypotheses H_{0} and H_{1} to be true, respectively. The notation m=m_{1}⊕m_{2} is called the orthogonal sum of m_{1} and m_{2}, which is commutative and associative. If and only if there exist at least two subsets A_{1} and A_{2} of 2^{Ω} with A_{1}∩A_{2}≠∅ such that m_{1}(A_{1})≠0 and m_{2}(A_{2})≠0, then m is calculated as shown in Eq. (12) [23].
Finally, based on all K subband observations, the decision is made by comparing the ratio between m(H_{1}) and m(H_{0}) as follows:
where η is the decision threshold. In fact, it is difficult to derive the closedform expression for the threshold η and the probability of detection for the proposed method. Therefore, we have developed a simulation model. According to the constant false alarm rate (CFAR) definition, the threshold η corresponding to a given probability of false alarm is determined in advance by a Monte Carlo simulation with 10,000 independent runs. With the changing of η, the corresponding detection probability is determined, which is used to draw the receiver operating characteristics curves (ROC). The simulation settings and some examples are given in Section 4.
Consequently, the pseudo code of the proposed spectrum sensing method with small sample size is given in Algorithm 1. Note that the computational complexity of the proposed method mainly comes from DS fusion (step 10 in Algorithm 1. Generally, it increases rapidly with the number of elements in the frame of discernment (Ω) and the number of the subbands K, as shown in Eqs. (9) and (10). However, since the frame of discernment consists of only two elements {H_{0},H_{1}} for the spectrum sensing, the combination of two mass functions requires the computation of 2×2 intersections [28]. Moreover, in the proposed scheme, due to the division of the observed bandwidth into K subbands, a large K increases the number of BPA functions then the computational complexity of the DS fusion.
Simulation results and analysis
In this section, the performance of the proposed method is evaluated with simulations. At first, we compare the proposed method with ED under different sampling numbers. Secondly, we evaluate the detection performance of the proposed method with the same total sampling numbers and different subband and sampling numbers. Finally, we compare with the methods in [19, 20, 34].
In the following simulations, we consider a baseband signal with B_{tot} bandwidth and decompose it into K subbands, which means that the bandwidth of each subband is B_{sub}=B_{tot}/K. With an oversampling factor N, the sampling rate in each subband is then 2B_{sub}N. The parameters are summarized in Table 1. Note that the simulation settings in Table 1 are only chosen to illustrate our purpose and some other values could be chosen.
Performance comparison with ED
In the first simulation, we examine the performance of the proposed method by comparison with the basic ED method, which calculates the total energy in the full bandwidth as
The noise power σ^{2} is assumed to be known in ED method and in the proposed method. As performed in many practical schemes, the noise power can be estimated periodically when no signal is expected in the frequency band of interest. Note that in the comparison, the test statistic in ED method is approximated as Gaussian distributed thanks to the number of the total samples (e.g., KQ > 150) and by using the central limit theorem.
Figure 4 presents the probability of detection (P_{d}) of the proposed method with different sampling numbers Q = 16,8, where the probability of false alarm (P_{fa}) is set to 0.05 for different SNR. As can be seen, with the increase of SNR and for a given value of Q, the probability of detection of the proposed method rises up quickly which is better than the trend of the curves of ED method. For the proposed method, when Q=8, the probability of detection is 0.9796 at − 10 dB. In order to clearly reveal the performance of the proposed method, the receiver operating characteristic (ROC) curves with different sampling numbers are shown in Fig. 5 when SNR = − 15 dB. It is obvious that the performance of the proposed method and ED is improved with the increase of the number of samples. When the probability of false alarm P_{fa} is 0.1 with Q = 16,8, the corresponding detection probabilities of the proposed method P_{d} reach 0.9632 and 0.8155, respectively.
Performance analysis with different Q and K
In the second simulation, the detection performance of the proposed method with different Q and K is assessed. Figure 6 shows that P_{d} of the proposed method goes up quickly with the increase of SNR, where P_{fa} is 0.05 for different SNRs. As shown in Fig. 6, when the total number of samples is KQ=600, the detection performance of the proposed method is better than when KQ=360 and KQ=120. Moreover, for the same total number of samples, when the wideband is divided into more groups, a better performance can be obtained. As en example, for SNR = −15 dB and KQ=360, the magnified part of Fig. 6 shows that the detection probabilities are respectively 0.9501, 0.9461, and 0.9455 for (Q = 9, K = 40), (Q = 12, K = 30), and (Q = 18, K = 20), respectively. This also verifies that a large K results in a more reliable BPA estimation and finally gives a high detection probability. Moreover, the corresponding ROC curves are shown in Fig. 7 for SNR = − 15 dB. It also verifies that a larger K gives a better detection performance with the same total number of samples.
Performance comparison with the methods in [19, 20, 34] and ED method
In the third simulation, we compare the performance of the proposed method with the GoF methods, such as the AD testbased method in [19], the Z_{c} testbased method in [20], the decision fusion method in [34] and ED method with Q=8,K=20 for SNR = −15 dB. The GoF method is nonparametric which utilizes the distance between the empirical CDF of the received samples and the CDF of the noise distribution, to detect the presence of a signal. In [19], a blind spectrum sensing method based on the AD statistic and Student’s t distribution is proposed to outperform the energy detection. In [20], a nonparametric sensing scheme is proposed for the nonGaussian environment modelled by Middleton class A noise, where the PDF of the test statistics is approximated as lognormal and an expression of P_{d} is derived. In [34], a semisoft fusion scheme is proposed to achieve a tradeoff between sensing performance and bandwidth cost. However, these methods are not designed for the small number of samples situation, nor wideband signal. As we can see in Fig. 8, the proposed method, the AD testbased method in [19], the Z_{c} testbased method in [20], and the decision fusion method in [34] are better than ED when Q = 8 and K = 20 are used. When the probability of false alarm is equal to 0.1, the detection probability of the proposed method is 0.8191. This is about 21% better than the detection probability of its counterparts (the GoF methods in [19, 20]). Even comparing with the decision method proposed in [34], the detection probability of the proposed method has also about 8.1% increase.
In addition, in order to facilitate the comparisons with other methods, the number of samples required by the methods in [19, 20, 34] and ED method to achieve a similar performance is determined. In Table 2, for the targeted performance (P_{fa},P_{d})=(0.1,0.8191), it can be observed that the required number of samples for the methods in [19, 20, 34] and ED method is 260, 260, 200, and 9900. However, the proposed method only requires 160 samples to achieve the same performance. In fact, reducing the number of samples without sacrificing the detection performance is a very attractive feature in practice, because it brings an economy in terms of computational burden, sensing time, and energy consumption.
Discussion
Considering that no prior knowledge about the channel is available, a simplified assumption of the constant amplitude in each subband is made in this paper. In order to support this assumption, some simulations are proposed to show that the method still works well even if the whole bandwidth is not perfectly uniformly occupied. Let us consider amplitude s^{(k)} in each subband as a Gaussian distribution with mean \(\bar {s}~=~1\) and variance \(\sigma _{s}^{2}\). Given that \(\sigma _{s}^{2}~=~0\) corresponds to the constant amplitude assumption, we present the results with \(\sigma _{s}^{2}~=~0.3\) and \(\sigma _{s}^{2}~=~0.6\) in the following simulations, where Q = 16 and K = 20. As shown in Fig. 9, when \(\sigma _{s}^{2}\) is increased from 0 to 0.3 and 0.6, the proposed method still performs well compared to the classical ED technique. We can also see that the proposed method is not sensitive to the variation of power spectral density of the signal.
Conclusion
In this paper, we propose a wideband signal detection method for CR applications with limited resource. The advantage of the proposed technique compared to the traditional detection methods is that only small number of samples is required. In this work, the samples are fully exploited with Student’s t test which is more suitable to the small sample size case. New BPA functions based on the CDF of Student’s t distribution are constructed and used in the DS fusion process, which improves the detection performance. Simulation results show that the proposed method can achieve a higher probability of detection than other compared methods with small sample size. The proposed method uses a small number of samples without sacrificing the detection performance, therefore brings an important economy in terms of computational burden, sensing time, and energy consumption. In the future work, multipath frequency selective channels between the PUs and the CR devices could be considered. As a result of such harsh environment (multipath with no line of sight and strong propagation loss), cooperative spectrum sensing techniques with small sample size will be developed. In addition, due to an imperfect knowledge of the bandwidth especially at the edge of the spectrum or strong frequency attenuations through the channel, it may occur in practice that sometimes the band is not fully occupied when the signal is present. Although this situation is not considered in this study, a rejection of misleading subbands could be elaborated according to an evaluation of a similarity criteria among the subbands. This problem deserves to be taken into account in some future work.
Abbreviations
 AWGN:

Additive white Gaussian noise
 BPA:

Basic probability assignment
 CDF:

Cumulative distribution function
 CR:

Cognitive radio
 DS:

Dempstershafer
 ED:

Energy detection
 GoF:

Goodness of fit
 PDF:

Probability density function
 SNR:

Signaltonoise ratio
 SS:

Spectrum sensing
 WSN:

Wireless sensor network
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Funding
This work is supported by the Medical Science and Technology Research Foundation of Guangdong under Grant No. B2018108, the Youth Creative Talent Project (Natural Science) of Guangdong under Grant No. 2017KQNCX041 and the National Natural Science Foundation of China (NSFC) under Grant No. 61673260.
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The work presented here was carried out in collaboration between all authors. The general idea was proposed by SM and PC. SM, YW, and JL designed and performed the simulations, and analyzed the results. SM wrote the paper. PC, YW, and JL revised the manuscript and provided many valuable suggestions. All authors read and approved the final manuscript.
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Correspondence to Jianzhong Li.
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Keywords
 Wideband signal detection
 Cognitive radio
 Small sample size
 DempsterShafer theory of evidence