Spectrum Sensing Methodologies for Cognitive Radio Systems: A Review <s> a) Optimal LRT-Based Sensing <s> Preface I. Introduction II. Elements of Hypothesis Testing III. Signal Detection in Discrete Time IV. Elements of Parameter Estimation V. Elements of Signal Estimation VI. Signal Detection in Continuous Time VII. Signal Estimation in Continuous Time References Index <s> BIB001 </s> Spectrum Sensing Methodologies for Cognitive Radio Systems: A Review <s> a) Optimal LRT-Based Sensing <s> It has recently been shown that multiple-input multiple-output (MIMO) antenna systems have the potential to improve dramatically the performance of communication systems over single antenna systems. Unlike beamforming, which presumes a high correlation between signals either transmitted or received by an array, the MIMO concept exploits the independence between signals at the array elements. In conventional radar, target scintillations are regarded as a nuisance parameter that degrades radar performance. The novelty of MIMO radar is that it takes the opposite view; namely, it capitalizes on target scintillations to improve the radar's performance. We introduce the MIMO concept for radar. The MIMO radar system under consideration consists of a transmit array with widely-spaced elements such that each views a different aspect of the target. The array at the receiver is a conventional array used for direction finding (DF). The system performance analysis is carried out in terms of the Cramer-Rao bound of the mean-square error in estimating the target direction. It is shown that MIMO radar leads to significant performance improvement in DF accuracy. <s> BIB002 </s> Spectrum Sensing Methodologies for Cognitive Radio Systems: A Review <s> a) Optimal LRT-Based Sensing <s> Superiority of multiple-input multiple-output (MIMO) radars over conventional phased array radars has been recently shown in many aspects. MIMO radars have better detection performance and can extract target information more precisely than phased array radars. In this paper, the problem of target detection using temporal coherent pulse terrain is considered for MIMO radars. We have introduced a structure that is suitable for coherent processing. Based on this structure, we have formulated this problem as a hypothesis test. One decision rule is developed for detection of target with unknown amplitude embedded in additive white gaussian noise (AWGN) with known power using the generalized likelihood ratio test (GLRT). Another adaptive scheme for the case of unknown power is presented. We have also proposed an ad-hoc constant false alarm rate (CFAR) detector against AWGN with unknown power. Performance of these detectors have been evaluated analytically. The results show the superiority of the MIMO radars with temporal coherent processing over the conventional phased arrays. <s> BIB003
The Neyman-Pearson states that for a given probability of false alarm, the test statistics that maximizes the probability of detection is the likelihood ratio test (LRT)  BIB001  which is defined as: where P (.) denotes the probability density function (PDF) and (x) denotes the received signal vector that is the aggregation of x(n), n = 0.1……., N -1. Such likelihood ratio test decides ℋR 1 when LRT ( )R exceeds a threshold , R otherwise it usesℋ 0 . The main challenge in in implementing the LRT is the requirement on the distribution give in equation  . The distribution of random vector x less than ℋR 1 is related to the source signal distribution, the wireless channels and the noise distribution. The distribution of x under ℋR 0 is related to the noise distribution  . In order to implement the LRT, a prior knowledge of the channel as a well as the signal and noise distribution is of paramount importance. This is practically difficult to realize. Assuming that the channels are flat-fading and the received source signal sample ( ) is independent over, the PDF in LRT is decoupled as:- Furthermore assuming that noise and signal samples are both Gaussian distributed, such that (n) ∼ Ν (0, 2 ) and s (n) ∼ N (0, Rs ) , the LRT becomes the estimator correlator (EC) detector as shown in  for which test statistic is given as: From equation (3) it is shown that R s (R s +2 P 2 I) -1 x (n) is the minimum mean squared error (MMSE) estimation of the source signal s (n). Thus, T EC (x) in (3) can be seen as the correlation of the signal x (n) with the MMSE estimation of s (n). EC detector needs to know the source signal covariance matrix R s and noise power 2 .Hence when the signal presence is unknown it becomes unrealistic to require signal covariance matrix for detection. It should be noted that if we assume that the noise is Gaussian distributed and the signals source is deterministic and known to the receiver, which in this case is the radar signal processing BIB002 BIB003  , it would then it can be easy to show that LRT becomes the matched filter based detector and its test statics is  .