VOLUME-BASED SPECTRUM SENSING FOR COGNITIVE RADIO USING MULTIPLE ANTENNAS

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CR, whose main idea is to sense the spectrum over a wide range of frequency bands and use the temporally unengaged groups for capable wireless transmission. It is a promising epitome to increase the spectrum manipulation efficiency. In a CR network, when the vast cycle resources of a PU not occupied, an SU is legitimate to use them. That is to say, and the SU desires to perceive the occurrence of the PU scrupulously. The Generating as a binary hypothesis testing problem and is particularly challenging for small sample size and low signal-to-noise ratio (SNR) circumstances.
For the situation of signal absence, the noticed data only consist of noise and are usually given to be, Independent and identically distributed (IID). It is perceptible that the energy and correlation structure of the observations differ when the PU signal is present [4]. As a result, spectrum sensing can be concluded by making use of these dissimilarities. When the noise power is known, the energy detector(ED) [6,] has been shown to be optimal for the IID PU signals. The eigenvalues of the obtained signal sample covariance matrix (SCM) in the signal occurrence 2 situation are more spread out than those in the noise only case, which corresponds to a scaled identity matrix in the asymptotic sense. The spread-out Eigen-spectrum results from the connection contraction characteristic in the covariance matrix. As a consequence of many eigenvalues-based detectors which exploit the correlation for spectrum, sensing has been proposed in the literature [13]. Derived the framework of generalized likelihood ratio test (GLRT) [16], the arithmetic-to-geometric mean (AGM) method [17] can identify reliably the correlated signals embedded in IID noise. AGM algorithm has its root in the maximum likelihood (ML) theory which turns out to be inefficient when the temporal and spectacular amplitude are tiny, that is, the hypothetical verdict threshold cannot be precisely indomitable. On the other hand, the maximum-to-minimum eigenvalues (MME) approach is heuristically developed to test if the SCM corresponds to a particularity conditions or its applicator alternative with the use of its greatest extent and smallest amount eigenvalues. Since not all Eigen-values are occupied, its exposure enactment is incredibly perceptive to weak applicatory signals and tiny samples scales. Moreover, the reckoning of the hypothetical threshold for the MME algorithm relies on the dispersion of the limit and smallest eigenvalues in the structure of random matrix theory (RMT) [18]. In practice, the SU receivers are usually uncelebrated, making the noises at different antennas to be non-uniform.
Some approaches have suggested for robust SS in the literature, such as the GLRT test [18], independence test [17], Hadamard ratio test, locally most important invariant test(LMPIT) [17] and volume-based approaches [19]. The underlying idea is that the determinant of SCM or volume differs dramatically between the signal absence and signal-presence situations It is worth pointing that the volume-based detector developed [18] for real-valued observation, and its detection enactment has not yet been critiqued, which is the officer involvement of these work. By comparing first and second consequences of signal absence and signal-presence hypothesis simultaneously with using the Gamma distribution approximation, we originate accurate analytic formulae for the false alarm and detection probabilities for the situation of IID noise. It enables us to develop the theoretical decision threshold for possible primary signal detection as well as receiver operating characteristic (ROC) for performance evaluation.
The rest of the paper organized as follows. The problem formulation of spectrum sensing conferred in section II. Volume-based determined in section III. Simulation results presented in section IV. Conclusions presented in section V.
Problem Formulation:-Signal model:-Consider a multipath fading channel model and assume there 1 PU and (d-1) interference users with d >=1, and each of them equipped along with a single antenna in a CR network. The intervention users are now counted as PUs because they cover the same channel, that is, there are d PUs. To find the temporally unoccupied channel, an SU receiver with M antennas desires to scrutinize this channel. Denote the signal absence and signal-presence hypothesis by H 0 and H 1 , respectively. SU with M antennas tries to detect the signals emitted by d PUs with the single antenna. X t =H S t + n t (1) Here H ϵ C M X d denotes the channel matrix between the PUs and SU, which is stranger deterministic throughout the wise period, and the n t =[n 1 (t) …….,n M (t)] T (4) and the consideration, signal, and noise vectors, correspondingly, with (.) T being transposed operator. We assume that sounds are statically independent and satisfy Si (t) ~ N( 0,σ 2 Si )(i=1,….,d) with σ 2 Si being the ith unknown signal variance, and that Ni (t) ~ N(0, T i ) (i=1….M) with T i being the unknown noise variance. In signal -absence hypothesis H 0 , the population covariance matrix of the observation 3 Here Σ s =E[S t S t ].
When signal absence and signal presence SS issues cast as the binary hypothesis test, there four it can be written as Here H 0 is denoted as signal absence and H 1 is denoted as signal presence.
Sensing solution:-Σ is the hyper volume of the geometry determined by the row vectors of Σ. For example, let us consider the scenario of three receiving antennas where the observed data with zero mean and unity variance may be independent, correlated or coherent.
Here corresponding covariance matrices are the 3x3 identity matrix, full rank matrix, and list one arbitrary pattern. For namesake, it can be called as the cube, parallelepiped, and line. It formed by the row vectors of the matrices is drawn in Fig.1.
Here, the volume of the cube, parallelepiped, and line, are denoted by v 1, v 2, and v 3 , in the sense of cube represented as absence signal situation, whereas parallelepiped and line can represent a signal present case. For signal-absence situation, the covariance matrix is a 3x3 identity matrix, that is v 1 =1, and signal presence situation v 2 < v 1 , v 1 =0.
Here to utilize the correlation structure for practical spectrum sensing, we need to calculate the sample covariance matrix(SCM) rather than the population covariance matrix Σ; it represented as = 1 =1 And the volume of geometry with unity edge is |D - For the situation of signal absence, |D -(1/2) S D -(1/2) )| asymptotically get the identity matrix as the number samples tend to infinity, leading the volume of one. For the situation of signal presence, the correlation structure gets leads to the notable reduction of the total amount. So we compare the predetermined threshold and volume of the geometry, Here volume-based detector offers the same expression as the Hadamard ratio test but with a different diagonal matrix in the denominator. The Hadamard ratio rules given below In this configuration, ϒ is approximately equal to the ᶓℎ . On another hand, i PI is much larger than the iiRII due to the correlation, making much smaller than ᶓℎ under H 1 . 4 To consider the different behaviors of the volume-based and Hadamard ratio approaches, the empirical probability density functions (PDFs) of the test statistics for these two methods plotted in fig.2. In this configuration we are taking four antennas, three primary signals with powers [-2,-3, -4] Db are assumed to exist in the sensed channel, and the noise is IID with unity power. In fig.2(a), the number samples sufficiently large. Here N=5000, so that the SCM is the very accurate estimate of the population covariance matrix, and for absent signal situation, the covariance matrix is an identity matrix, it means that volume-based method is approximately equal to the Hadamard ratio, as drawn in fig. 2 (a). It seemed that the PDF under H 1, denoted as p 1 , is much further away from the PDF under H 0 , denoted as p 0 , volume based detector than that of the Hadamard ratio rule. When the number samples become small, like N=50, the SCM is the inaccurate estimate of the population one. These numerical results represented in fig.2(b).The thereby implies that the volume-based algorithm can detect the PUs correctly with higher probability than the Hadamard ratio approach especially in the small sample and low signal to noises ratio situations.
In theoretical analysis, the volume-based test statics is modified as Here ϒ is the decision threshold of the volume-based detector, the next section, derived the Gamma distribution approximation, accurate analytic formulae derived for false alarm probability, detection probability, the theoretical decision threshold as well as ROC. It enables us accurately to determine the theoretical decision threshold for the practical SS.

Performance analysis;-
In this one, the analytic formula for the false-alarm probability, detection probability, decision threshold as well as ROC is derived by assuming the additive noise is IID.
Here we are trying to first and second moments under the signal activation hypothesis and signal deactivation theory, and first, we derived the Detection probability of the volume-based method.
Detection Probability:-Let us considered the p 1 (y) be the PDF of the statistic variable ξ under presence signal hypothesis (H 1 ), which is determined the following proposition.
Provide that the population covariance matrix Σ gave, and F(ϒ;α 1 ,β 1 ) is the cumulative distribution function (CDF) of Gamma distribution. From the constant false-alarm rate(CFAR) perspective, the decision threshold ϒ usually is determined under the absence signal hypothesis, it must be resolute and independent of the noise variance. Here With μ 0 and ν 0 being the second order approximation to the mean of ξ and the first order approximation to the variance of ξ.
And the mapping between the false-alarm probability and detection probability yields in ROC. Hence the analytic ROC formula for the volume-based test is Numerical Results:- In this section, we are going to the discussion about the simulation results for analytic false alarm and detection probabilities. Moreover, the superiority of the volume-based approach over the representative methods for the spectrum sensing under the IID and non-IID noise conditions are demonstrated.

Analytic False-Alarm and Detection Probability:-
In this one, the accuracy of the analytic formulae for the false alarm and detection probabilities is numerically calculated. To comparison, exact false alarm and detection probabilities, which empirically determined by 10 5 Monte Carlo simulation trials, are presented as well.
In Let us try the accuracy of the detection probability for the proposed Gamma approximation. Similarly, the exact detection probability empirically determined by 10 5 Monte Carlo simulations is also presented for comparison. The Presence of primary signals, the Rayleigh-fading channel is adopted to evaluate the accuracy of the derived Gamma Approximation P d .
The Rayleigh-fading situation, the columns of the channel matrix H follow a complex Gaussian distribution with zero mean and covariance matrix ɸ due to the correlation between the signals at the receiving antennas which cannot sufficiently space for physical size constraints. The correlated Rayleigh-fading channel model is able precisely to characterize the behavior of the working channel. The (K, l) entry of ψ is determined as follows Where I 0 (.) denotes the zero-order modified Bessel function, k controls the width of the angles-of-arrival (AOAs) of the primary signal, impinging upon the receiving antennas of the SU. Which can vary from 0 up to 00 (extremely non-isotropic scattering), ψ Ɛ [-п,п) represents the mean direction of the AOAs, and d kl stands for the distance, which normalized on the wavelength λ, between the n-th and i-th antennas of the SU.
The figure. 4 demonstrate the numerical results of the single primary signal in the Rayleigh fading channel and under the situation of IID noise.
In this one M =4, N=100,200 to 300, and the power of the primary signal set as -5db. It simplified that the derived detection probability can accurately predict the detection performance for the volume-based approach. Fig4 (b) plots the Gamma approximate and exact detection likelihood of the volume-based detection method for IID noise. Here M=6, N=200,300 to 400, and the signal power are same as in Fig.4 (a). Here we can observe that the proposed Gamma relative detection probability is quite precise regarding fitting the exact one. In Fig.5 (b). The number of antennas and some samples set as M=6 and N=[200, 400, 800], respectively. Here in this one, we are observed that the derived P d is very accurate concerning the predicting the detection performance for the volume-based algorithm. Detection Performance:-Here we have derived robustness as well as the accuracy of the volume-based detection algorithm for complexvalued observed data by comparing its empirical ROC with those other representative methods. In particular, the decision threshold is varied to calculate the false-alarm probability and its corresponding detection probability, leading to the ROC curve, for the purpose of comparison, the numerical results of the John's, AGM (or ST ), Hadamard ratio as well as ED detectors are provided. All the numerical results obtained from 10 5 Monte Carlo trials.
The ROC of the volume-based, AGM, John"s, Hadamard ratio and Energy detection and IID noise drawn in Figure.6.
In fig. 6(a) where the number of antennas M=4, some samples N=10. The number of primary signals is one, and its power equals to 8 dB, the volume-based approach is superior to the AGM and Hadamard ratio methods but inferior to John's detector which is known to be the locally most powerful invariant test for sphere city. Moreover, all of them are inferior to the actual noise variance based ED approach.
Similarly, in Fig. 6 It indicates that the volume-based detector surpasses the AGM and Hadamard ratio methods but is not as accurate as John's approach in the scenario of IID noise. However, as the sample number increases to 50, the volume-based algorithm can provide the same accuracy as John's algorithm which has been proved to be the more potent for the small sample case. Similarly in Fig. 7(a) where the numbers of antennas M = 4 number of samples N = 10, primary signal d = 1, and its power equals 8db, and different noise models namely non-IID noise. Under such a sound condition, the volumebased detector is capable of offering the best detection performance among the blind methods. In turn, implies that the volume-based approach is superior to the Hadamard ratio method in accuracy and the AGM and John's schemes in robustness against the deviation of IID noise. In fig. 7(b) where the numbers of antennas M= 4 and number of samples N = 10, primary signals increases to 3 and its powers set as [ 1 2 , 2 2 , 3 2 ] = [5, 2,0]db, and different noise models namely non-IID noise. Under this condition, we plotted volume-based approach was superior to the Hadamard ratio method in accuracy and the AGM and John's schemes in robustness against the deviation of IID noise. In fig. 7(c) where the numbers of antennas M = 4 and number of samples N = 30, d = 3, and its powers are set as [ 1 2 , 2 2 , 3 2 ] = [0, -3 , -5] dB, and with non-IID noise. Under this condition, we plotted volume-based approach was superior to the Hadamard ratio method in accuracy and the AGM and John's schemes in robustness against the deviation of IID noise. Fig. 8(a) where the number of antennas M = 6, and some samples N = 10, primary signal d=1, and its power equals 8db, and noise models namely IID. Under such a sound condition, the volume-based method is superior to the AGM and Hadamard ratio schemes but inferior to John's system in detection performance. In fig.  8(b) where the number of antennas M = 6, number of samples N = 10, and primary signals increases to 3 and its powers as set as

Conclusion:-
The volume-based method has derived for the IID noise; it is providing accurate theoretical threshold calculation and evaluating its sensing performance. Here we are taking first and second moments of signal active, signal in a current hypothesis; we can employ the Moment-matching method accurately derive the Gamma distribution.
14 By computing the exact first and second moments of the test statistic under the current signal hypothesis, we are capable of approximating the Gamma distribution for the volume-based test static, ending up with an accurate analytic expression for the detection probability.