Ali Ghobadzadeh

Separating Function Estimation Tests: A New Perspective on Binary Composite Hypothesis Testing

In this paper, we study some relationships between the detection and estimation theories for a binary composite hypothesis test H0 against H1 and a related estimation problem. We start with a one-dimensional (1D) space for the unknown parameter space and one-sided hypothesis problems and then extend out results into more general cases. For one-sided tests, we show that the uniformly most powerful (UMP) test is achieved by comparing the minimum variance and unbiased estimator (MVUE) of the unknown parameter with a threshold. Thus for the case where the UMP test does not exist, the MVUE of the unknown parameter does not exist either. Therefore for such cases, a good estimator of the unknown parameter is deemed as a good decision statistic for the test. For a more general class of composite testing with multiple unknown parameters, we prove that the MVUE of a separating function (SF) can serve as the optimal decision statistic for the UMP unbiased test where the SF is continuous, differentiable, positive for all parameters under H1 and is negative for the parameters under H0. We then prove that the UMP unbiased statistic is equal to the MVUE of an SF. In many problems with multiple unknown parameters, the UMP test does not exist. For such cases, we show that if one detector between two detectors has a better receiver operating characteristic (ROC) curve, then using its decision statistic we can estimate the SF more ε-accurately, in probability. For example, the SF is the signal-to-noise ratio (SNR) in some problems. These results motivate us to introduce new suboptimal SF-estimator tests (SFETs) which are easy to derive for many problems. Finally, we provide some practical examples to study the relationship between the decision statistic of a test and the estimator of its corresponding SF.

Invariance and Optimality of CFAR Detectors in Binary Composite Hypothesis Tests

We investigate the relationship between constant false alarm rate (CFAR) and invariant tests. We introduce the minimal invariant group (MIG). We show that for a family of distributions, the unknown parameters are eliminated from the distribution of the maximal invariant statistic under the MIG while the maximum information of the observed signal is preserved. We prove that any invariant test with respect to MIG is CFAR and conversely, for any CFAR test an invariant statistic exists with respect to an MIG under some mild conditions. Moreover for a given CFAR test, we propose a systematic method for deriving an enhanced test, i.e., a function of observations exists such that the likelihood ratio (LR) of the maximal invariant of its MIG gives an enhanced test. Furthermore, we introduce the uniformly most powerful-CFAR (UMP-CFAR) test as the optimal CFAR bound among all CFAR tests. We then prove that the UMP-CFAR test for the minimally invariant hypothesis testing problem is given by the LR of the maximal invariant under MIG. For some problems, this test (MP-CFAR) depends on the unknown parameters of the alternative hypothesis, however, provides an upper-performance bound for all suboptimal CFAR tests. We also propose three suboptimal novel CFAR tests among which one is asymptotically optimal.

Asymptotically Optimal CFAR Detectors

This paper investigates the asymptotic optimality of the Constant False Alarm Rate (CFAR) tests obtained using the Minimal Invariant Group (MIG) reduction. We show that the CFAR tests obtained after MIG reduction using the Wald test is a Separating Function Estimation Test (SFET) and that the Generalized Likelihood Ratio Test (GLRT) and the Rao test are asymptotically SFET using Maximum Likelihood Estimation (MLE) under some mild conditions. Thus, they are asymptotically optimal. In order to find an improved test and motivated by the invariance property of the MLE of induced maximal invariant, we maximize the asymptotic Probability of Detection of the SFET using the MLE after reduction. We propose a systematic method allowing to derive the asymptotically optimal Separating Function (AOSF). This AOSF is obtained as the Euclidean distance of the transformed parameters under two hypotheses such that the gradient of the transformed parameters is the Cholesky decomposition of the Fisher Information Matrix (FIM), i.e., the FIM is transformed into an identity matrix. Interestingly, the AOSF Estimation Test (AOSFET) using MLE simplifies to the Wald-CFAR wherever the FIM does not depend on the unknown parameters. The simulation results show that the proposed AOSFET usually outperforms the GLRT, Wald test and Rao test.

GLR approach for MIMO radar signal sampling in unknown clutter parameter

Multiple-input multiple-output (MIMO) radar has been shown to provide enhanced performance in theory and practice. The MIMO radar is a type of multi-static radar, that uses multiple antennas at both the transmitter and receiver to improve the detection performance. In this paper, we propose a generalized likelihood ratio test (GLRT) for target detection by MIMO radar in unknown Gaussian clutter. We focus on MIMO radar with widely separated antennas. Mathematical results indicate that the number of receiver antennas should be more than the number of received signal samples in a pulse repetition interval (PRI); i.e, in each processing round, the number of secondary data set (from different antennas), should not be less than the number of the received signal samples in each antenna receiver.

Generalized Wald Test for Binary Composite Hypothesis Test

This letter provides a generalization of the well-known Wald test. The proposed generalized Wald test (GWT) is a Separating Function Estimation Test (SFET) which is a type of detector recently introduced for a wide class of composite problems. The test statistics of an SFET is an estimate of a real-valued Separating Function (SF). It is already proved that a Minimum Variance Unbiased Estimator of any SF leads to the optimal Uniformly Most Powerful unbiased detector. In many practical cases, such an optimal detector does not exist; hence, suboptimal ones are used, instead. Selecting an SF with a guaranteed performance is still an open problem which is investigated in this letter. First, we derive a lower bound for the detection probability of the SFET in terms of corresponding SF and the Fisher Information Matrix. Then we optimize the proposed bound with respect to the SF. The solution of the optimization problem leads to a generalization of the Wald test which is asymptotically optimal and reduces to the Wald detector in some special cases. Simulation results show the superiority of the proposed GWT over its counterparts, namely, the Wald test and GLR detector in some examples.

Transformation effects on invariant property of invariant hypothesis test and UMPI detector

The Uniformly Most Powerful (UMP) test, which is one with the highest probability of detection over all unknown parameters in a composite hypothesis test, does not exist in the most practical problems. It is common to derive the suboptimal Generalize Likelihood Ratio (GLR) detector, that is shown to perform close to optimal invariant tests in some problems. The Uniformly Most Powerful Invariant (UMPI) test is the one that is used for the performance evaluation of the GLR detector. However, the GLR or UMPI detectors are not simply derivable in many problems. In this paper, we suggest that the problem be transformed to a new space. We derive the conditions of the transformation, such that the detectors for the original problem and the transformed ones become equivalent. In addition, using this method, we discuss two practical problems (one in radar and other in communication) and derive the GLR detectors. We also find the UMPI detectors in these problems to verify the performance of the derived GLR detector.

The role of MVU estimator and CRB in binary composite hypothesis test

This work presents a new perspective to the relationship between the composite binary hypothesis test and the estimation of its unknown parameters, i.e. the Uniformly Most Powerful (UMP) test and the Minimum Variance and Unbiased Estimator(MVUE). We show that for the one-sided binary composite hypothesis test, if the UMP test exists, it is nothing but comparing the MVUE for the unknown parameter with a threshold. The paper tries to make a link between the Cramer Rao Bound (CRB) in estimation theory and the UMP performance bound in detection theory. In addition to the intrinsic theoretical interest of such relationship discussed in the paper, it leads us to proposing a novel detection method. For such problems in which the UMP test does not exist, we suggest using a good estimator of the unknown parameter as the decision statistic. The simulation results confirm the idea that the closer we get to the CRB in estimating the unknown parameter, the more we get near to the UMP performance bound in detection.

The separating function estimation test and the UMPI test for target detection problem using optimal signal design in MIMO radar

In this paper, we study the MIMO signal detection problem using widely separated antennas in Gaussian interference. The interference is assumed to be colored with unknown N × N covariance matrix. We derive the Uniformly Most Powerful Invariant (UMPI) test for this detection problem as the upper performance bound for invariant tests. Also the Separating Function Estimation Test (SFET) is derived for this problem using the Signal and Scatter to Interference Ratio (SIR). Then, based on the eigenvalues expansion of SIR, we propose a set of orthogonal signals in transmitters which maximizes the detection probability of UMPI and SFET. Simulation results show that 1) the performance of our proposed detector is close to the UMPI bound and 2) the performance of the optimal invariant bound improves when the transmitters use the proposed set of signals instead of the orthogonal complex exponential signals.

Invariant target detection of MIMO radar with unknown parameters

In this paper, three target detectors Uniformly Most Powerful Invariant (UMPI), Generalized Likelihood Ratio Test (GLRT) and a Separating Function Estimation Test (SFET) based on the scale group of transformations are proposed and applied to Widely Separated Antennas Multiple-Input Multiple-Output (WSA MIMO) radars. It is shown that for this problem the UMPI test depends on the scatter to noise ratio, hence the UMPI test provides the upper performance bound for all invariant tests. To derive the asymptotically optimal SFET the Maximum Likelihood Estimation (MLE) of unknown parameters are replaced into the induced maximal invariant, which is equal to the scatter to noise ratio. The MLE of the scatter to noise ratio does not have a closed form, hence we propose an iterative estimator to calculate the SFET statistic. Similarly an iterative GLRT is also proposed for this problem. The simulation results show that the performance of SFET tends to the optimal invariant bound by increasing the scatter to noise ratio.

Asymptotically optimal narrowband signal detection using uniform linear array antenna

This paper addresses the detection of a narrowband signal in Gaussian noise with unknown parameters. Assuming unknown direction-of-arrival (DoA), amplitude, frequency, phase and noise variance, two Separating Function Estimation Tests (SFETs) and a Generalized Likelihood Ratio Test (GLRT) are proposed to detect the signal. These SFETs are estimates of a proposed Separating Function (SF). This proposed SF provides asymptotically optimal detectors using Maximum Likelihood Estimation (MLE) and is derived by the decomposition of Fisher information function of the induced maximal invariant. We propose two estimators MLE and Outlier Processed MLE (OPMLE) for estimation of the SF. It is shown that, the MLE of frequency and DoA are obtained by an exhaustive search to maximize the absolute of the two-dimensional discrete Fourier transform (DFT) of the received signals. We propose OP-MLE as an MLE based estimator by first eliminating the outliers from the DFT of the received signal using a pre-estimation of DoA and frequency. The simulation results show that the omission of outliers results in considerable improvement. Similarly, the resulting SFET using OP-MLE provides a higher probability of detection comparing with SFET using MLE and GLRT.