Cooperative Spectrum Sensing based on the Limiting Eigenvalue Ratio Distribution in Wishart Matrices

Recent advances in random matrix theory have spurred the adoption of eigenvalue-based detection techniques for cooperative spectrum sensing in cognitive radio. Most of such techniques use the ratio between the largest and the smallest eigenvalues of the received signal covariance matrix to infer the presence or absence of the primary signal. The results derived so far in this field are based on asymptotical assumptions, due to the difficulties in characterizing the exact distribution of the eigenvalues ratio. By exploiting a recent result on the limiting distribution of the smallest eigenvalue in complex Wishart matrices, in this paper we derive an expression for the limiting eigenvalue ratio distribution, which turns out to be much more accurate than the previous approximations also in the non-asymptotical region. This result is then straightforwardly applied to calculate the decision threshold as a function of a target probability of false alarm. Numerical simulations show that the proposed detection rule provides a substantial performance improvement compared to the other eigenvalue-based algorithms.Comment: 7 pages, 2 figures, submitted to IEEE Communications Letter

On the Performance of Spectrum Sensing Algorithms using Multiple Antennas

In recent years, some spectrum sensing algorithms using multiple antennas, such as the eigenvalue based detection (EBD), have attracted a lot of attention. In this paper, we are interested in deriving the asymptotic distributions of the test statistics of the EBD algorithms. Two EBD algorithms using sample covariance matrices are considered: maximum eigenvalue detection (MED) and condition number detection (CND). The earlier studies usually assume that the number of antennas (K) and the number of samples (N) are both large, thus random matrix theory (RMT) can be used to derive the asymptotic distributions of the maximum and minimum eigenvalues of the sample covariance matrices. While assuming the number of antennas being large simplifies the derivations, in practice, the number of antennas equipped at a single secondary user is usually small, say 2 or 3, and once designed, this antenna number is fixed. Thus in this paper, our objective is to derive the asymptotic distributions of the eigenvalues and condition numbers of the sample covariance matrices for any fixed K but large N, from which the probability of detection and probability of false alarm can be obtained. The proposed methodology can also be used to analyze the performance of other EBD algorithms. Finally, computer simulations are presented to validate the accuracy of the derived results.Comment: IEEE GlobeCom 201

Eigenvalue Ratio Detection Based on Exact Moments of Smallest and Largest Eigenvalues

Detection based on eigenvalues of received signal covariance matrix is currently one of the most effective solution for spectrum sensing problem in cognitive radios. However, the results of these schemes always depend on asymptotic assumptions since the close-formed expression of exact eigenvalues ratio distribution is exceptionally complex to compute in practice. In this paper, non-asymptotic spectrum sensing approach to approximate the extreme eigenvalues is introduced. In this context, the Gaussian approximation approach based on exact analytical moments of extreme eigenvalues is presented. In this approach, the extreme eigenvalues are considered as dependent Gaussian random variables such that the joint probability density function (PDF) is approximated by bivariate Gaussian distribution function for any number of cooperating secondary users and received samples. In this context, the definition of Copula is cited to analyze the extent of the dependency between the extreme eigenvalues. Later, the decision threshold based on the ratio of dependent Gaussian extreme eigenvalues is derived. The performance analysis of our newly proposed approach is compared with the already published asymptotic Tracy-Widom approximation approach

SPECTRUM SENSING USING CYCLIC PREFIX IN COGNITIVE RADIO WIRELESS SYSTEM

The rapid growth of wireless communications has made the problem of spectrum utilization ever more critical. The increasing diversity (voice, short message, Web & multimedia) and demand of high quality-of-service (QoS) applications have resulted in overcrowding of the allocated (officially sanctioned) spectrum bands, leading to significantly reduced levels of user satisfaction. The concepts of GLRT algorithm and substantial improvement over the U-GLRT algorithm are explained. This paper presents a model which uses efficient CP method for CR in Wireless Systems. Primary signal has been detected in the OFDM transmission with both the CPCC and MP–based C-GLRT algorithms greatly outperform energy detection in multi path environment has been implemented using software design. The signal model in our analysis is to efficiently exploit the correlation among the transmitted signals due to the presence of CP. Proposed method of cognitive radio takes two steps of implementation .first named as MP based is to detect the noise and de-noise the signal and the second is cp based in which the signals are identified based on the cyclic prefix

Performance Analysis of Multi-Antenna Hybrid Detectors and Optimization with Noise Variance Estimation

In this paper, a performance analysis of multi-antenna spectrum sensing techniques is carried out. Both well known algorithms, such as Energy Detector (ED) and eigenvalue based detectors, and an eigenvector based algorithm, are considered. With the idea of auxiliary noise variance estimation, the performance analysis is extended to the hybrid approaches of the considered detectors. Moreover, optimization for Hybrid ED under constant estimation plus detection time is performed. Performance results are evaluated in terms of Receiver Operating Characteristic (ROC) curves and performance curves, i.e., detection probability as a function of the Signal-to-Noise Ratio (SNR). It is concluded that the eigenvector based detector and its hybrid approach are able to approach the optimal Neyman-Pearson performance

Phase transitions in the condition number distribution of Gaussian random matrices

We study the statistics of the condition number $\kappa=\lambda_{\mathrm{max}}/\lambda_{\mathrm{min}}$ (the ratio between largest and smallest squared singular values) of $N\times M$ Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large $N$ the cumulative $\mathcal{P}[\kappa<x]$ and tail-cumulative $\mathcal{P}[\kappa>x]$ distributions of $\kappa$. We find that these distributions decay as $\mathcal{P}[\kappa<x]\approx\exp\left(-\beta N^2 \Phi_{-}(x)\right)$ and $\mathcal{P}[\kappa>x]\approx\exp\left(-\beta N \Phi_{+}(x)\right)$, where $\beta$ is the Dyson index of the ensemble. The left and right rate functions $\Phi_{\pm}(x)$ are independent of $\beta$ and calculated exactly for any choice of the rectangularity parameter $\alpha=M/N-1>0$. Interestingly, they show a weak non-analytic behavior at their minimum $\langle\kappa\rangle$ (corresponding to the average condition number), a direct consequence of a phase transition in the associated Coulomb fluid problem. Matching the behavior of the rate functions around $\langle\kappa\rangle$, we determine exactly the scale of typical fluctuations $\sim\mathcal{O}(N^{-2/3})$ and the tails of the limiting distribution of $\kappa$. The analytical results are in excellent agreement with numerical simulations.Comment: 5 pag. + 7 pag. Suppl. Material. 3 Figure

On the probability that all eigenvalues of Gaussian, Wishart, and double Wishart random matrices lie within an interval

We derive the probability that all eigenvalues of a random matrix $\bf M$ lie within an arbitrary interval $[a,b]$, $\psi(a,b)\triangleq\Pr\{a\leq\lambda_{\min}({\bf M}), \lambda_{\max}({\bf M})\leq b\}$, when $\bf M$ is a real or complex finite dimensional Wishart, double Wishart, or Gaussian symmetric/Hermitian matrix. We give efficient recursive formulas allowing the exact evaluation of $\psi(a,b)$ for Wishart matrices, even with large number of variates and degrees of freedom. We also prove that the probability that all eigenvalues are within the limiting spectral support (given by the Mar{\v{c}}enko-Pastur or the semicircle laws) tends for large dimensions to the universal values $0.6921$ and $0.9397$ for the real and complex cases, respectively. Applications include improved bounds for the probability that a Gaussian measurement matrix has a given restricted isometry constant in compressed sensing.Comment: IEEE Transactions on Information Theory, 201