Optimal Likelihood Ratio Test Research Articles

Optimal Correlators for Detection and Estimation in Optical Receivers

Motivated by modern applications of light detection and ranging (LIDAR), we study the model of an optical receiver based on an avalanche photo-diode (APD), followed by electronic circuitry for detection of reflected optical signals and estimation of their delay. This model is known to be quite complicated as it consists of at least three different types of noise: thermal noise, shot noise, and multiplicative noise (excess noise) that stems from the random gain associated with the photo-multiplication of the APD. Consequently, the derivation of the optimal likelihood ratio test (LRT) associated with signal detection is a non-trivial task, which has no apparent exact closed-form solution. We consider instead a class of relatively simple detectors, that are based on correlating the noisy received signal with a given deterministic waveform, and our purpose is to characterize the optimal waveform in the sense of the best tradeoff between the false-alarm (FA) error exponent and the missed-detection (MD) error exponent. In the same spirit, we also study the problem of estimating the delay on the basis of maximizing the correlation between the received signal and a time-shifted waveform, as a function of this time shift. We characterize the optimal correlator waveform that minimizes the mean square error (MSE) in the regime of high signal-to-noise ratio (SNR). The optimal correlator waveforms for detection and for estimation turn out to be different, but their limiting behavior is the same: when the thermal Gaussian noise is dominant, the optimal correlator waveform becomes proportional to the clean signal, but when the thermal noise is negligible compared to the other noises, then it becomes a logarithmic function of the clean signal, as expected.

Extended GLRT Detection of Moving Targets for Multichannel SAR Based on Generalized Steering Vector.

A generalized likelihood ratio test (GLRT) with the constant false alarm rate (CFAR) property was recently developed for adaptive detection of moving targets in focusing synthetic aperture radar (SAR) images. However, in the multichannel SAR-ground moving-target indication (SAR-GMTI) system, image defocus is inevitable, which will remarkably degrade the performance of the GLRT detector, especially for the lower radar cross-section (RCS) and slower radial velocity moving targets. To address this issue, based on the generalized steering vector (GSV), an extended GLRT detector is proposed and its performance is evaluated by the optimum likelihood ratio test (LRT) in the Neyman-Pearson (NP) criterion. The joint data vector formulated by the current cell and its adjacent cells is used to obtain the GSV, and then the extended GLRT is derived, which coherently integrates signal and accomplishes moving-target detection and parameter estimation. Theoretical analysis and simulated SAR data demonstrate the effectiveness and robustness of the proposed detector in the defocusing SAR images.

A New Frame Synchronization Algorithm for Linear Periodic Channels With Memory

Identifying the start time of a sequence of symbols received at the receiver, commonly referred to as frame synchronization, is a critical task for achieving good performance in digital communications systems employing time-multiplexed transmission. In this work we focus on frame synchronization for linear channels with memory, in which the channel impulse response is periodic and the additive Gaussian noise is correlated and cyclostationary. Such channels appear in many communications scenarios, including narrowband power line communications and interference-limited wireless communications. We derive frame synchronization algorithms based on simplifications of the optimal likelihood-ratio test, assuming the channel impulse response is unknown at the receiver, which is applicable to many practical scenarios. The computational complexity of each of the derived algorithms is characterized, and a procedure for selecting nearly optimal synchronization sequences is proposed. The algorithms derived in this work achieve better performance than the noncoherent correlation detector, and, in fact, facilitate a controlled tradeoff between complexity and performance.

JUNIPR: a framework for unsupervised machine learning in particle physics

In applications of machine learning to particle physics, a persistent challenge is how to go beyond discrimination to learn about the underlying physics. To this end, a powerful tool would be a framework for unsupervised learning, where the machine learns the intricate high-dimensional contours of the data upon which it is trained, without reference to pre-established labels. In order to approach such a complex task, an unsupervised network must be structured intelligently, based on a qualitative understanding of the data. In this paper, we scaffold the neural network’s architecture around a leading-order model of the physics underlying the data. In addition to making unsupervised learning tractable, this design actually alleviates existing tensions between performance and interpretability. We call the framework Junipr: “Jets from UNsupervised Interpretable PRobabilistic models”. In this approach, the set of particle momenta composing a jet are clustered into a binary tree that the neural network examines sequentially. Training is unsupervised and unrestricted: the network could decide that the data bears little correspondence to the chosen tree structure. However, when there is a correspondence, the network’s output along the tree has a direct physical interpretation. Junipr models can perform discrimination tasks, through the statistically optimal likelihood-ratio test, and they permit visualizations of discrimination power at each branching in a jet’s tree. Additionally, Junipr models provide a probability distribution from which events can be drawn, providing a data-driven Monte Carlo generator. As a third application, Junipr models can reweight events from one (e.g. simulated) data set to agree with distributions from another (e.g. experimental) data set.

Optimal Detection and Error Exponents for Hidden Semi-Markov Models

We study detection of random signals corrupted by noise that over time switch their values (states) between a finite set of possible values, where the switchings occur at unknown points in time. We model such signals as hidden semi-Markov signals, which generalize classical Markov chains by introducing explicit (possibly nongeometric) distribution for the time spent in each state. Assuming two possible signal states and Gaussian noise, we derive optimal likelihood ratio test and show that it has a computationally tractable form of a matrix product, with the number of matrices involved in the product being the number of process observations. The product matrices are independent and identically distributed, constructed by a simple measurement modulation of the sparse semi-Markov model transition matrix that we define in the paper. Using this result, we show that the Neyman–Pearson error exponent is equal to the top Lyapunov exponent for the corresponding random matrices. Using theory of large deviations, we derive a lower bound on the error exponent. Finally, we show that this bound is tight by means of numerical simulations.

A density based empirical likelihood approach for testing bivariate normality

ABSTRACTSample entropy based tests, methods of sieves and Grenander estimation type procedures are known to be very efficient tools for assessing normality of underlying data distributions, in one-dimensional nonparametric settings. Recently, it has been shown that the density based empirical likelihood (EL) concept extends and standardizes these methods, presenting a powerful approach for approximating optimal parametric likelihood ratio test statistics, in a distribution-free manner. In this paper, we discuss difficulties related to constructing density based EL ratio techniques for testing bivariate normality and propose a solution regarding this problem. Toward this end, a novel bivariate sample entropy expression is derived and shown to satisfy the known concept related to bivariate histogram density estimations. Monte Carlo results show that the new density based EL ratio tests for bivariate normality behave very well for finite sample sizes. To exemplify the excellent applicability of the proposed approach, we demonstrate a real data example.

Identifying Individual Camera Device From RAW Images

This paper investigates the problem of identifying the individual imaging device source of the same model from a natural image in RAW format. We propose an enhanced Poissonian–Gaussian model describing the distribution of pixels from a RAW image. The parameters of this statistical noise model are considered as unique fingerprints and, hence, used to identify the source camera device. The source camera identification problem is cast within the framework of the hypothesis testing theory. In an ideal context, where all model parameters are perfectly known, the Likelihood Ratio Test (LRT) is presented, and its performance is theoretically established. The statistical performance of this optimal LRT serves as an upper bound for the detection power. For a practical use, when the nuisance parameters are unknown, a generalized LRT based on estimation of those parameters is designed to deal with unknown expectation of pixels (roughly, the image content). More importantly, by combining multiple sub classifiers together with the voting strategy, a multi-classifier is proposed to identify multiple individual devices. Numerical results on simulated data and on various large dataset of real natural images highlight the relevance of our proposed approach.

Linear Soft Combination for Cooperative Spectrum Sensing in Cognitive Radio Networks

In this letter, we study the cooperative energy-based combination for spectrum sensing in cognitive radio networks. Linear combination rule is investigated to minimize the probability of missed detection subject to an upper limit on the probability of false alarm. We propose a novel and practical linear combination rule, which only requires the mean and variance of the local test statistics. The performance of the proposed combination rule is verified in three typical fading scenarios: the slow fading, the block fading, and the fast fading. Simulations show that the proposed strategy has nearly the same performance as the optimal likelihood-ratio test approaches, and outperforms the conventional linear combination methods.

On Spectrum Sensing of OFDM Signals at Low SNR: New Detectors and Asymptotic Performance

This paper deals with near-optimal spectrum sensing of orthogonal frequency division multiplexing (OFDM) signals to achieve reliable detection at low signal-to-noise ratio (SNR). The likelihood ratio test (LRT), a simple hypothesis test , delivers the optimal performance, but it requires that the parameters involved in the test are known. Hence, the generalized likelihood ratio test (GLRT), a composite hypothesis test , has often been employed. However, GLRT-based detectors involve biased parameter estimation, which may lead to inferior performance. In this paper, with proper approximation, the LRT is reduced to a simpler form, which only requires the knowledge of noise power. Then a novel unbiased and consistent estimator of the noise power is developed. This estimator is combined with the approximate LRT, leading to an asymptotic simple hypothesis test (ASHT). Two ASHT-based detectors are presented for cases with and without time synchronization, and their theoretical performances are analyzed. Simulation results show that the ASHT-based detectors deliver performances very close to that of the optimal LRT, and significantly outperform the existing GLRT-based detectors. It is also shown that the ASHT-based detectors exhibit robustness against the influence of multipath channels.

Design of Binary Quantizers for Distributed Detection Under Secrecy Constraints

In this paper, we investigate the design of distributed detection networks in the presence of an eavesdropper (Eve). We consider the problem of designing binary sensor quantizers that maximize the Kullback-Leibler (KL) divergence at the fusion center (FC), when subject to a tolerable constraint on the KL divergence at Eve. We assume that the channels between the sensors and the FC (likewise the channels between the sensors and the Eve) are modeled as binary symmetric channels (BSCs). In the case of i.i.d. received symbols at both the FC and Eve, we prove that the structure of the optimal binary quantizers is a likelihood ratio test (LRT). We also present an algorithm to find the threshold of the optimal LRT, and illustrate it for the case of Additive white Gaussian noise (AWGN) observation models at the sensors. In the case of non-i.i.d. received symbols at both FC and Eve, we propose a dynamic-programming based algorithm to find efficient quantizers at the sensors. Numerical results are presented to illustrate the performance of the proposed network.

Study of spectrum sensing exploiting polarization: From optimal LRT to practical detectors

Spectrum sensing exploiting polarization can be leveraged to improve detection performance. In this paper, we present an optimal likelihood ratio test (LRT) which serves as the upper bound on the performance of all methods that exploit polarization under Gaussian assumption. However, the performance of LRT for realistic blind sensing is unclear since the practical implementation of polarization sensing has to face several challenges, which is the focus of this paper. Four practical sensing algorithms utilizing polarization are considered in this work, that is, a generalized likelihood ratio test (GLRT) for polarization sensing using maximum likelihood estimate (MLE) method when any prior knowledge is unknown, an optimal test when the noise of channel between primary user and secondary user is partially polarized, a realistic detector considering that the received polarization wave arrived with a specific direction rather than is almost perpendicular to the receiver dual-polarized antennas, and a more practical blind detector for polarization channel exhibiting polarization mode dispersion (PMD) phenomenon especially in wideband system. The simulation results show the performance gains possible with proposed detectors and how well the proposed detectors are expected to perform in practice.

On the Near-Optimality of Training-Based GLRT Spectrum Sensing

Spectrum sensing is of utmost importance in cognitive radio and dynamic spectrum access systems for achieving spectrum awareness. To provide reliable spectrum awareness, it is critical to develop (near-) optimal sensing techniques and understand achievable performance limits. In this paper, we study likelihood ratio test (LRT)-based methods for detection of a signal of interest (SOI) assuming multiple receive antennas in the presence of spatially correlated additive noise (colored noise). We show that with on/off status information of the SOI available at receiver, a training-based generalized likelihood ratio test (TB-GLRT) method can be designed which approximates the optimal LRT estimator-correlator (EC) detector. Using the inverse Laplace transform and complex Wishart distribution theory, we derive formulas for detection and false alarm probabilities of both LRT-EC and GLRT schemes. The convergence of the TB-GLRT to the LRT-EC is analyzed and proved based on convergence behaviors of the detectors and their decision statistics. Simulation results verify the analytical convergence properties. In addition, the results illustrate the effects of different system design parameters, and demonstrate that the TB-GLRT scheme can provide substantial performance improvement over several existing GLRT methods.

Decision Fusion with Channel Errors in Distributed Decode-Then-Fuse Sensor Networks.

Decision fusion for distributed detection in sensor networks under non-ideal channels is investigated in this paper. Usually, the local decisions are transmitted to the fusion center (FC) and decoded, and a fusion rule is then applied to achieve a global decision. We propose an optimal likelihood ratio test (LRT)-based fusion rule to take the uncertainty of the decoded binary data due to modulation, reception mode and communication channel into account. The average bit error rate (BER) is employed to characterize such an uncertainty. Further, the detection performance is analyzed under both non-identical and identical local detection performance indices. In addition, the performance of the proposed method is compared with the existing optimal and suboptimal LRT fusion rules. The results show that the proposed fusion rule is more robust compared to these existing ones.

Sequential Frame Synchronization Based on Hypothesis Testing With Unknown Channel State Information

In this paper, we propose a hypothesis testing based sequential frame synchronization technique that is robust against fluctuations and imperfect estimation of channel gain at the receiver. The proposed detector uses division in the decision rule to cancel the effect of channel gain, and therefore is called division based hypothesis testing (DHT). We show that the receiver operating characteristic (ROC) of DHT is close to the optimal likelihood ratio test (LRT), while the optimal LRT requires perfect knowledge of noise and channel gain at the receiver. Further, we use three performance metrics, namely, the probability of false synchronization P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fs</sub> , average reception duration l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rx</sub> , and probability of missed synchronization P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ms</sub> , to compare the performance of hypothesis testing against that of point estimation. We analytically show that hypothesis testing can have better reliability and lower energy consumption than point estimation if the probability of false alarm of the hypothesis testing is lower than the minimum of two thresholds. The cost of hypothesis testing is an arbitrarily small P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ms</sub> while point estimation can have zero P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ms</sub> . A modified implementation of hypothesis testing is also introduced to further reduce the energy consumption.

A review on polarization‐based spectrum sensing

AbstractIn cognitive radios, the biggest challenge is spectrum sensing. It usually makes use of the amplitude, frequency or space dimension of the received signal to differentiate signal from background noise. Polarization, also as an inherent characteristic of signals, provides an additional degree of freedom in signal space. Polarization‐based spectrum sensing has been identified as an effective method for improving detection performance. In this paper, various polarization‐based spectrum sensing methods including the optimal polarization likelihood ratio test using Neyman–Pearson theorem, virtual polarization detection utilizing maximum signal to noise ratio method and blind polarization detectors based on generalized likelihood ratio test method are reviewed. Then, a comprehensive comparison of various polarization‐based sensing methods in terms of detection performance and computational complexity is detailed. Finally, challenges and possible future studies on polarization‐based spectrum sensing are described. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Steganalysis of JSteg algorithm using hypothesis testing theory

This paper investigates the statistical detection of JSteg steganography. The approach is based on a statistical model of discrete cosine transformation (DCT) coefficients challenging the usual assumption that among a subband all the coefficients are independent and identically distributed (i. i. d.). The hidden information-detection problem is cast in the framework of hypothesis testing theory. In an ideal context where all model parameters are perfectly known, the likelihood ratio test (LRT) is presented, and its performances are theoretically established. The statistical performance of LRT serves as an upper bound for the detection power. For a practical use where the distribution parameters are unknown, by exploring a DCT channel selection, a detector based on estimation of those parameters is designed. The loss of power of the proposed detector compared with the optimal LRT is small, which shows the relevance of the proposed approach.

Efficient and Low Complexity Modulation Classification Algorithm for MIMO Systems

This study develops a feature-based Automatic Modulation Classification (AMC) algorithm for spatially multiplexed Multiple-Input Multiple-Output (MIMO) systems employing two Higher Order Cumulants (HOCs) of the estimated transmit signal streams as discriminating features and a multiclass Support Vector Machine (SVM) as a classification system. The algorithm under study has the capability to recognize a wide range of modulation schemes without any prior information about the channel state. The classification performance of the proposed algorithm was evaluated via extensive simulations under different operating conditions and was also compared with the one obtained with the optimal Hybrid Likelihood Ratio Test (HLRT) approach. The results show that the proposed algorithm is capable of classifying the considered modulation schemes with good classification accuracy and can achieve performance comparable to that of the HLRT approach while having a significantly lower computational complexity.

A local adaptive model of natural images for almost optimal detection of hidden data

This paper proposes a novel methodology to detect data hidden in the least significant bits of a natural image. The goal is twofold: first, the methodology aims at proposing a test specifically designed for natural images, to this end an original model of images is presented, and, second, the statistical properties of the designed test, probability of false alarm and power function, should be predictable.The problem of hidden data detection is set in the framework of hypothesis testing theory. When inspected image parameters are known, the Likelihood Ratio Test (LRT) is designed and its statistical performance is analytically established. In practice, unknown image parameters have to be estimated. The proposed model of natural images is used to estimate unknown parameters accurately and to design a Generalized Likelihood Ratio Test (GLRT). Finally, the statistical properties of the proposed GLRT are analytically established which permits us, first, to guarantee a prescribed false-alarm probability and, second, to show that the GLRT is almost as powerful as the optimal LRT. Numerical results on natural image databases and comparison with prior art steganalyzers show the relevance of theoretical findings.

Spectrum Sensing for OFDM Signals Using Pilot Induced Auto-Correlations

Orthogonal frequency division multiplex (OFDM) has been widely used in various wireless communications systems. Thus the detection of OFDM signals is of significant importance in cognitive radio and other spectrum sharing systems. A common feature of OFDM in many popular standards is that some pilot subcarriers repeat periodically after certain OFDM blocks. In this paper, sensing methods for OFDM signals are proposed by using such repetition structure of the pilots. Firstly, special properties for the auto-correlation (AC) of the received signals are identified, from which the optimal likelihood ratio test (LRT) is derived. However, this method requires the knowledge of channel information, carrier frequency offset (CFO) and noise power. To make the LRT method practical, we then propose an approximated LRT (ALRT) method that does not rely on the channel information and noise power, thus the CFO is the only remaining obstacle to the ALRT. To handle the problem, we propose a method to estimate the composite CFO and compensate its effect in the AC using multiple taps of ACs of the received signals. Computer simulations have shown that the proposed sensing methods are robust to frequency offset, noise power uncertainty, time delay uncertainty, and frequency selectiveness of the channel.

Characterization of escape times of Josephson junctions for signal detection

The measurement of the escape time of a Josephson junction might be used to detect the presence of a sinusoidal signal embedded in noise when use of standard signal processing tools can be prohibitive due to the extreme weakness of the source or to the huge amount of data. In this paper we show that the prescriptions for the experimental setup and some physical behaviors depend on the detection strategy. More specifically, by exploitation of the sample mean of escape times to perform detection, two resonant regions are identified. At low frequencies there is a stochastic resonance or activation phenomenon, while near the plasma frequency a geometric resonance appears. Furthermore, detection performance in the geometric resonance region is maximized at the prescribed value of the bias current. The naive sample mean detector is outperformed, in terms of error probability, by the optimal likelihood ratio test. The latter exhibits only geometric resonance, showing monotonically increasing performance as the bias current approaches the junction critical current. In this regime the escape times are vanishingly small and therefore performance is essentially limited by measurement electronics. The behavior of the likelihood ratio and sample mean detector for different values of incoming signal to noise ratio is discussed, and a relationship with the error probability is found. Detectors based on the likelihood ratio test could be employed also to estimate unknown parameters in the applied input signal. As a prototypical example we study the phase estimation problem of a sinusoidal current, which is accomplished by using the filter bank approach. Finally we show that for a physically feasible detector the performances are found to be very close to the Cramer-Rao theoretical bound. Applications might be found, for example, in some astronomical detection problems (where the all-sky gravitational and/or radio wave search for pulsars requires the analysis of nearly sinusoidal long-lived waveforms at very low signal-to-noise ratio) or to analyze weak signals in the subterahertz range (where the traditional electronics counterpart is difficult to implement).