Adaptive detection of transient signals

This paper presents a derivation of the Maximum Likelihood (ML) detector for acoustic signals that have propagated through a random or uncertain ocean environment. The derivation requires probability distribution functions (pdfs) of relevant signal and noise parameters belonging to the exponential class. (A companion paper describes how available knowledge of the environment and the Maximum Entropy method can be used to calculate exponential class pdfs for signal and noise parameters.) The resultant ML detector operates on the observations to compute parameter means, referred to as conditional mean estimates (CMEs), and then correlates the CME with the observations to obtain a detection statistic. For this reason, the detector is referred to as an Estimated Ocean Detector (EOD). For Gaussian signal and noise, the EOD reduces to the weighted sum of a correlation detector (CD) and an energy detector (ED), which agrees with Van Trees' result. When the signal variance is low the weights strongly favor the CD, while for high signal variance the weights strongly favor the ED. A closed-form solution is not available for the Gaussian EOD. However, results obtained using a Monte Carlo simulation show that when the signal variance is low, EOD performance equals that of the CD, while under high signal variance, performance equals the ED. For intermediate signal variance, the simulation shows that the EOD weights the CD and ED outputs so as to perform better than either detector by itself. Finally, EOD robustness to incorrect estimates of received signal and noise parameters pdfs is investigated using the Monte Carlo simulation. The EOD is shown to be a promising method for improving passive sonar performance by directly incorporating available information about the ocean environment without sacrificing robustness to errors in the signal parameter distributions

Likelihood ratio tests for variance components in linear mixed models

The authors propose a computationally efficient spectrum sensing solution for an orthogonal frequency division multiplexing (OFDM) signal in a frequency-selective fading channel with additive white Gaussian noise (AWGN). The authors' assumption is that the data symbols, channel coefficients and the noise variance are all unknown. The nature of the problem leads them to find an invariant detector. The optimum one is uniformly most powerful invariant (UMPI); their effort shows that this test does not exist, as the final decision statistic depends on some unknown parameters; although, they derive the most powerful invariant (MPI) detector, implanting these parameters, to provide an upper bound for invariant detectors. Instead, the authors develop the generalised likelihood ratio test (GLRT), substituting the unknown parameters by their maximum likelihood (ML) estimates in the Neyman–Pearson likelihood ratio. Furthermore, they propose a computationally efficient implementation of the resulting detector. Simulation results show a slight decrease in efficiency against gaining so much computational complexity (CC) improvement. For more comparisons they also derive the conventional energy detector, namely, generalised energy detector (GED) for a binary phase shift keying (BPSK) signal, to show the adequacy of the GLR detector performance which would be demonstrated by simulation results.

Abstract. This paper is devoted to an extension of a classical problem of statistics to the asymptotic distribution of likelihood ratios. Two main types of likelihood ratios are considered for Gaussian ARMA processes. It is assumed in both cases that the asymptotic Fisher information matrix of estimation is singular in the higher order models. It is proved that the asymptotic distributions of the log likelihood ratios are invariant with respect to the parameters generating the process. A simulation shows that the sample distribution of the log likelihood ratio approaches the asymptotic one. Finally, the likelihood ratio test is proposed for model order reduction.

: The report proposes a design of an adaptive receiver for the detection and estimation of nearly periodic signals in additive Gaussian noise. A nearly periodic signal is defined to be a sample function of a Gaussian random process which can be divided into equal length intervals, called periods, in such a manner that the correlation between periods decreases exponentially with their separation. The receiver computes a low signal-to-noise ratio conditional likelihood ratio from which the observer must make decisions. The likelihood ratio is conditional because the receiver estimates any unknown parameters necessary for the computation of the true likelihood ratio. Thus the receiver can only compute a likelihood ratio conditioned upon these estimates being the true values of the unknown parameters. The receiver consists of pre- emphasis filters followed by a comb filter, energy detector, and weighted summation. A theoretical evaluation of the receiver, in terms of ROC curves, is made for the special case of nearly periodic signals with statistically independent equal-strength harmonics in white noise of known power.

Bayesian and generalized likelihood ratio tests are derived for detection of a common unknown signal of known rank K in M > K independent channels of white gaussian noise. The cases of known and unknown noise variance are both treated. These derivations encompass the development of explicit expressions for an invariant measure on the grassmannian manifold of K-dimensional subspaces of complex N-dimensional space and parameterization of this manifold to enable the calculation of the necessary marginalization integrals. Performance of the detectors is compared by simulation.

Particle filters for probability hypothesis density filter with the presence of unknown measurement noise covariance

To improve the efficiency of bandwidth usage, the cognitive radios concept has emerged. Spectrum sensing is a pivotal function in cognitive radio systems so that the secondary users can be able to access the free spectrum holes without interfering with primary users. Energy detection is the most common and easiest spectrum sensing technique for cognitive radios. In this paper, the conventional energy detection receiver operating characteristic curve and performance metrics are simulated under additive white Gaussian noise (AWGN) environment by using Monte Carlo simulation. Also, the performance of energy detection is analyzed under different types of M-ary modulation. The analysis of performance will carry the assumption of noise uncertainty. This performance gain is compared with conventional energy detection without any noise effect. In addition, the relation between a number of samples and signal to noise ratio wall (SNR wall) is provided and simulated. The simulation tool used in the analysis is MATLAB software. Finally, the implementation of energy detection technique in the time domain and frequency domain is designed on Xilinx Spartan-3E (XC3S500E-FG320) Field Programmable Gate Array (FPGA) kit by using Verilog language and Xilinx ISE Simulator version 14.1.

Maximum likelihood (ML) amplitude and noise variance estimation without having to jointly estimate the frequency and the phase and based only on information from the noisy received signal magnitude, is studied for a single sinusoid in complex additive white Gaussian noise. This estimation problem is equivalent to the classic problem of parameter estimation for the Rician distribution. While solving the likelihood equation is impossible in general, we propose a new approach based on a large argument approximation. For the case with known noise variance, a closed-form ML amplitude estimator is obtained, which outperforms the conventional root-mean-square estimator. For the case with unknown noise variance, the closed-form joint amplitude and noise variance estimators obtained do not require prior knowledge of one another.

Likelihood ratio distributions and the (ir)relevance of error rates

Orthogonal frequency-division multiplexing (OFDM) is a popular transmission technology in cognitive radio (CR) networks, because the correlation of cyclic prefix (CP) in OFDM signals can be utilized to improve the reliability of spectrum sensing of secondary users (SUs). However, the optimal spectrum sensing over multipath fading channels remains an important and challenging issue. Therefore, this work proposes an optimal Neyman- Pearson (NP) detector for spectrum sensing using CP. To detect the OFDM signal of primary users (PUs), the log- likelihood ratio (LR) test is formulated by using the correlation characteristics of the redundancy of CP. Analytical results indicate that the LR of received samples is equivalent to their log- likelihood function (LF) plus LR of an energy detector (ED), subsequently allowing us to gain insights on the optimal NP detector. Since many unknown parameters need to be resolved, a practical generalized log-likelihood ratio test (GLRT) is presented. Moreover, to achieve a good performance over multipath fading channels, a channel- independent GLRT (CI-GLRT) is employed to derive an estimation of correlation coefficient independent of multipath channel profiles. Simulations confirm the advantages of the proposed detectors compared with state-of-the-art detectors.

The problem is target motion analysis (TMA) in situations where the variance (standard deviation) of additive white Gaussian measurement noise is unknown and time-varying. In particular, the paper examines a somewhat surprising result from the theoretical analysis based on the Cramer-Rao bound, which suggests that the best-achievable (second-order) error in target state estimation is unaffected by the lack of knowledge of the measurement noise variance. In order to examine this result, the paper develops three recursive Bayesian filters for TMA, which jointly estimate the target state and the measurement variance. The basis of all filters is the Cubature Kalman filter for bearings-only tracking, combined with (i) the variational Bayesian approach, (ii) the Rao-Blackwellised particle filter, and (iii) the interactive multiple-model (IMM), to deal with the unknown time-varying measurement variance. The paper presents extensive numerical simulation results and comparisons, which confirm that the lack of knowledge of the measurement noise variance is by no means a handicap for TMA.

In this paper, we investigate the detection of a band-limited signal with unknown central frequency, exact bandwidth and the spectrum in white Gaussian noise with unknown variance. We propose a practically implementable Generalized Likelihood Ratio (GLR) detector with these unknown parameters and verify its performance through some simulation examples. Furthermore, in order to evaluate the performance of this detector, we derive a Most Powerful Invariant (MPI) detector in the case of known partial Signal to Noise Ratios (SNRs), i.e., where the SNR of the different Discrete Fourier Transform components of the signal are known. Our results show the close performance of the GLR detector and this MPI detector. In addition, we propose a locally MPI detector in small SNRs, that performs slightly better than GLR in these SNRs.

We present a training-based approach for the classification of noisy unknown transient signals with arbitrary range and Doppler shift (time and frequency shifts). The method uses the magnitude-square of the signal ambiguity function to remove the unknown shifts. An ambiguity domain template is then generated from labeled training data (tens of observations), and classification is performed using an inner product. The method is tested on synthetic transient signals in Gaussian noise and performs as well as or better than another previously proposed time-frequency based method, and an energy detector.

In seismic exploration, the process of estimating the source produced signal common to a collection of sampled data recordings contaminated with additive noise is known as stacking. Stacking is used when a low energy acoustic source can be repeated in the same physical location (or nearly so) and a collection of data recordings are gathered at the same electromechanical transducer from the multiple source initiations. This stacking process is primarily aimed at reducing environmental noise which frequently dominates each individual recording when low energy sources are used. This is in contrast to other techniques used to reduce or eliminate so called source generated noise which behaves as signal in the stacking process. Specifically, the problem we address is the simultaneous maximum likelihood (ML) estimation of signal values and unknown noise distribution parameters from a collection of sampled data recordings, each containing the same signal sequence. In this paper we restrict our consideration to independent, zero mean, Gaussian noise variates with unknown variance which may differ between data records. This model is justified when a broadband analog noise source is subsequently low pass filtered along with the signal and the result sampled at the Nyquist rate. The linear filtering tends to produce a Gaussian noise process and the Nyquist rate sampling produces approximately uncorrelated samples. The ML estimates of signal values and noise variances are obtained numerically by steepest ascent.