A review on spectrum sensing for cognitive radio: challenges and solutions <s> Detection Threshold and Test Statistic Distribution <s> We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble. <s> BIB001 </s> A review on spectrum sensing for cognitive radio: challenges and solutions <s> Detection Threshold and Test Statistic Distribution <s> The focus of this survey is on the distribution function F Nβ (t) for the largest eigenvalue in the finite N Gaussian Orthogonal Ensemble (GOE, β = 1), the Gaussian Unitary Ensemble (GUE, β = 2), and the Gaussian Symplectic Ensemble (GSE, β = 4) in the edge scaling limit of N → ∝. These limiting distribution functions are expressible in terms of a particular Painleve II function. Comparisons are made with finite N simulations and of the universality of these distribution functions is discussed. <s> BIB002 </s> A review on spectrum sensing for cognitive radio: challenges and solutions <s> Detection Threshold and Test Statistic Distribution <s> Sensing (signal detection) is a fundamental problem in cognitive radio. In this paper, a new method is proposed based on the eigenvalues of the covariance matrix of the received signal. It is shown that the ratio of the maximum eigenvalue to the minimum eigenvalue can be used to detect the signal existence. Based on some latest random matrix theories (RMT), we can quantize the ratio and find the threshold. The probability of false alarm is also found by using the RMT. The proposed method overcomes the noise uncertainty difficulty while keeps the advantages of the energy detection. The method can be used for various sensing applications without knowledge of the signal, the channel and noise power. Simulations based on randomly generated signals and captured ATSC DTV signals are presented to verify the methods. <s> BIB003 </s> A review on spectrum sensing for cognitive radio: challenges and solutions <s> Detection Threshold and Test Statistic Distribution <s> In this paper, we give a brief review of the theory of spectral analysis of large dimensional random matrices. Most of the existing work in the literature has been stated for real matrices but the corresponding results for the complex case are also of interest, especially for researchers in Electrical and Electronic Engineering. Thus, we convert almost all results to the complex case, whenever possible. Only the latest results, including some new ones, are stated as theorems here. The main purpose of the paper is to show how important methodologies, or mathematical tools, have helped to develop the theory. Some unsolved problems are also stated. <s> BIB004 </s> A review on spectrum sensing for cognitive radio: challenges and solutions <s> Detection Threshold and Test Statistic Distribution <s> Random matrices play a crucial role in the design and analysis of multiple-input multiple-output (MIMO) systems. In particular, performance of MIMO systems depends on the statistical properties of a subclass of random matrices known as Wishart when the propagation environment is characterized by Rayleigh or Rician fading. This paper focuses on the stochastic analysis of this class of matrices and proposes a general methodology to evaluate some multiple nested integrals of interest. With this methodology we obtain a closed-form expression for the joint probability density function of k consecutive ordered eigenvalues and, as a special case, the PDF of the lscrth ordered eigenvalue of Wishart matrices. The distribution of the largest eigenvalue can be used to analyze the performance of MIMO maximal ratio combining systems. The PDF of the smallest eigenvalue can be used for MIMO antenna selection techniques. Finally, the PDF the kth largest eigenvalue finds applications in the performance analysis of MIMO singular value decomposition systems. <s> BIB005 </s> A review on spectrum sensing for cognitive radio: challenges and solutions <s> Detection Threshold and Test Statistic Distribution <s> Spectrum sensing is a fundamental component in a cognitive radio. In this paper, we propose new sensing methods based on the eigenvalues of the covariance matrix of signals received at the secondary users. In particular, two sensing algorithms are suggested, one is based on the ratio of the maximum eigenvalue to minimum eigenvalue; the other is based on the ratio of the average eigenvalue to minimum eigenvalue. Using some latest random matrix theories (RMT), we quantify the distributions of these ratios and derive the probabilities of false alarm and probabilities of detection for the proposed algorithms. We also find the thresholds of the methods for a given probability of false alarm. The proposed methods overcome the noise uncertainty problem, and can even perform better than the ideal energy detection when the signals to be detected are highly correlated. The methods can be used for various signal detection applications without requiring the knowledge of signal, channel and noise power. Simulations based on randomly generated signals, wireless microphone signals and captured ATSC DTV signals are presented to verify the effectiveness of the proposed methods. <s> BIB006
To make a decision on whether signal is present, we need to set a threshold γ for each proposed test statistic, such that certain P d and/or P f a can be achieved. For a fixed sample size N, we cannot set the threshold to meet the targets for arbitrarily high P d and low P f a at the same time, as they are conflicting to each other. Since we have little or no prior information on the signal (actually we even do not know whether there is a signal or not), it is difficult to set the threshold based on P d . Hence, a common practice is to choose the threshold based on P f a under hypothesis H 0 . Without loss of generality, the test threshold can be decomposed into the following form: γ = γ 1 T 0 (x), where γ 1 is related to the sample size N and the target P f a , and T 0 (x) is a statistic related to the noise distribution under H 0 . For example, for the energy detection with known noise power, we have For the matched-filtering detection with known noise power, we have For the EME/MME detection with no knowledge on the noise power, we have where λ min (N) is the minimum eigenvalue of the sample covariance matrix. For the CAV detection, we can set In practice, the parameter γ 1 can be set either empirically based on the observations over a period of time when the signal is known to be absent, or analytically based on the distribution of the test statistic under H 0 . In general, such distributions are difficult to find, while some known results are given as follows. For energy detection defined in (8) , it can be shown that for a sufficiently large values of N, its test statistic can be well approximated by the Gaussian distribution, that is, Accordingly, for given P f a and N, the corresponding γ 1 can be found as where For the matched-filtering detection defined in (9), for a sufficiently large N, we have Thereby, for given P f a and N, it can be shown that For the GLRT-based detection, it can be shown that the asymptotic (as N → ∞) log-likelihood ratio is central chisquare distributed  . More precisely, where r is the number of independent scalar unknowns under H 0 and H 1 . For instance, if σ 2 η is known while R s is not, r will be equal to the number of independent real-valued scalar variables in R s . However, there is no explicit expression for γ 1 in this case. Random matrix theory (RMT) is useful for determining the test statistic distribution and the parameter γ 1 for the class of eigenvalue-based detection methods. In the following, we provide an example for the BCED detection method with known noise power, that is, T 0 (x) = σ 2 η . For this method, we actually compare the ratio of the maximum eigenvalue of the sample covariance matrix R x (N) to the noise power σ 2 η with a threshold γ 1 . To set the value for γ 1 , we need to know the distribution of λ max (N)/σ 2 η for any finite N. With a finite N, R x (N) may be very different from the actual covariance matrix R x due to the noise. In fact, characterizing the eigenvalue distributions for R x (N) is a very complicated problem   BIB001 BIB004 , which also makes the choice of γ 1 difficult in general. When there is no signal, R x (N) reduces to R η (N), which is the sample covariance matrix of the noise only. It is known that R η (N) is a Wishart random matrix  . The study of the eigenvalue distributions for random matrices is a very hot research topic over recent years in mathematics, communications engineering, and physics. The joint PDF of the ordered eigenvalues of a Wishart random matrix has been known for many years  . However, since the expression of the joint PDF is very complicated, no simple closed-form expressions have been found for the marginal PDFs of the ordered eigenvalues, although some computable expressions have been found in BIB005 . Recently, Johnstone and Johansson have found the distribution of the largest eigenvalue  BIB001 of a Wishart random matrix as described in the following theorem. and ν = ( /ν converges (with probability one) to the Tracy-Widom distribution of order 1  BIB002 . The Tracy-Widom distribution provides the limiting law for the largest eigenvalue of certain random matrices  BIB002 . Let F 1 be the cumulative distribution function (CDF) of the Tracy-Widom distribution of order 1. We have where q(u) is the solution of the nonlinear Painlevé II differential equation given by Accordingly, numerical solutions can be found for function F 1 (t) at different values of t. Also, there have been tables for values of F 1 (t)  and Matlab codes to compute them  . Based on the above results, the probability of false alarm for the BCED detection can be obtained as which leads to or equivalently, From the definitions of μ and ν in Theorem 1, we finally obtain the value for γ 1 as Note that γ 1 depends only on N and P f a . A similar approach like the above can be used for the case of MME detection, as shown in BIB003 BIB006 . Figure 3 shows the expected (theoretical) and actual (by simulation) probability of false alarm values based on the theoretical threshold in  for N = 5000, M = 8, and K = 1. It is observed that the differences between these two sets of values are reasonably small, suggesting that the choice of the theoretical threshold is quite accurate.