Statistical Resolution Limit Research Articles

Refinement and derivation of statistical resolution limits for circular or rectilinear correlated sources in CES data models

This paper analyzes the deterministic (DCRB) and the stochastic (SCRB) Cramér-Rao bound on direction-of-arrival (DOA) estimation for two equi-powered correlated complex circular or rectilinear sources affected by complex circular white noise in different complex elliptically symmetric (CES) data models. Beginning by decomposing these CRBs, into factors depending on signal and noise parameters, and on geometric parameters of the array, some new interpretable closed-form expressions are derived in particular scenarios. These expressions provide useful insight into the behavior of these CRB’s dependence on the correlation factor. Approximate closed-form expressions of these CRBs for small DOA separation are also derived. These results lead to new formulas for statistical resolution limit (SRL) based on the Smith criterion at which an unbiased DOA estimation algorithm can resolve two closely-spaced circular or rectilinear sources. These SCRB-derived formulas are much less optimistic than those which have so far been deduced only from the DCRB under the assumption of known sources.

Colocated MIMO radar waveform design for angular statistical resolution limit in the context of hypothesis testing

Colocated MIMO radar waveform design for angular statistical resolution limit in the context of hypothesis testing

Angular Statistical Resolution Limit of Two Closely-Spaced Point Targets: A GLRT-Based Study

The study of statistical resolution limit (SRL) of two closely spaced targets has attracted considerable interests in the last decade. Two definitions for SRL have been proposed: one is based on the Cramer–Rao bound (CRB) and the other is based on a decision-based process. In this paper, we focus on the latter one. Different from the existing study, which assumes that the center parameter of interests (POIs) is known a priori , we use a more general model where all the POIs are unknown. We exploit the first-order Taylor expansion of the signals to get an approximate linear model with respect to the tested parameter, namely, the separation of directions of arrival of two sources. Then, we apply the general likelihood rate test to get a closed-form expression of SRL. We consider both the cases with known and unknown noise variance. Moreover, we analyze the impact of some parameters (including the resolution rate, the false-alarm rate, and the waveforms) on the SRL. For comparison, we also derive the CRB-based SRL, which is essentially different with our decision-based counterpart. Numerical simulation results demonstrate the validity of our theoretical results.

On the Statistical Resolution Limit (SRL) for Time-Reversal based MIMO radar

In the single-input multiple-output radar, the system transmits scaled (coherent) versions of a single waveform. The multiple-input multiple-output (MIMO) radar uses multiple antennas to simultaneously transmit several non-coherent waveforms and exploits multiple antennas to receive the reflected signals (echoes). This diversity in term of waveform coding allows to transmit orthogonal waveforms which enables the MIMO radar superiority in several fundamental aspects, including: improved parameter identifiability and estimation and much enhanced flexibility for transmit beam-pattern design. The context of this work is the co-located MIMO radar where the transmit and the receive arrays are close in space. In this paper, we provide a theoretical performance analysis to compare two configurations of MIMO radar: Conventional configuration and Time Reversal (TR) configuration in term of Statistical Resolution Limit (SRL). This study provides new insights on the performance gain of the TR scheme which is discussed and illustrated by appropriate simulation results depending on the receive noise level.

Statistical Resolution Limit Analysis of Two Closely Spaced Signal Sources Using Rao Test

High computational complexity and difficulty in taking an analytical expression of detection statistics are some of the problems encountered when using the statistical resolution limit (SRL) analysis in the generalized likelihood ratio test (GLRT). A Rao test-based method is used in this paper to analyze the 1-D SRL of two closely spaced signals, which can simplify the derivation of detection statistics, reduce the computational complexity, and improve the detection performance in some scenarios. In the proposed approach, two cases are considered based on the interference sources that are either involved or not in the received signals. Moreover, in each case, the detection statistic is derived by assuming the known or unknown variance of the added noise, respectively. For the performance comparison, the GLRT detection statistic is also identified. The relationship between the SRL and the required minimum signal-to-noise ratio/signal-to-interference-plus-noise ratio is also obtained to resolve the two adjacent signal sources. Then, the detection performance and computational complexity are evaluated. The simulation results are used to verify the effectiveness and superiority of the Rao test.

Cramér–Rao bound and statistical resolution limit investigation for near-field source localization

In this paper, we investigate the performance analysis for near-field source localization in terms of the mean square error and resolvability. We first derive and analyze non-matrix, closed-form expressions of the deterministic Cramér–Rao bound for two closely spaced, time-varying near-field sources in the context of linear arrays. Numerical simulations confirm the validity of the obtained expressions. Using these expressions and based on Smith's criterion, we discuss the behavior of the statistical resolution limit with respect to some features of interest, namely, the correlation factor, the central frequency, the minimum resolution limit boundary and the array geometry. Finally, to avoid the complexity of the optimal geometry design procedure, we propose a fast, nearly optimal, array design scheme to enhance the capacity of resolvability under given constraints (e.g., number of sensors and array aperture).

On the Accuracy and Resolvability of Vector Parameter Estimates

In this paper we address the problem of fundamental limitations on resolution in deterministic parameters estimation. We introduce a definition of resolvability based on probability and incorporating a requirement for accuracy unlike most existing definitions. Indeed in many application the key problem is to obtain distributions of estimates that are not only distinguishable but also accurate and compliant with a required precision. We exemplify the proposed definition with estimators that produce normal estimates, as in the conditional model for which the Gaussianity and efficiency of maximum likelihood estimators (MLEs) in the asymptotic region of operation (in terms of signal-to-noise ratio and/or in large number of snapshots) is well established, even for a single snapshot. In order to measure the convergence in distribution, we derive a simple test allowing to check whether the conditional MLEs operate in the asymptotic region of operation. Last, we discuss the resolution of two complex exponentials with closely spaced frequencies and compare the results obtained with the ones provided by the various statistical resolution limit released in the open literature.

Three-Dimensional Extensions of Resolvability of Two Point Targets in MIMO Radar Using a Hypothesis Test Formulation

Multiple-input multiple-output (MIMO) radar has been extensively studied in recent years, due to its outstanding performance in various aspects. Resolution and estimation of the targets’ parameters are central capabilities of a MIMO radar system. In this paper, the resolution performance of MIMO radar in terms of the three-dimensional statistical resolution limit (3D SRL) is investigated, where three-dimension refers to direction of arrival (DOA), direction of departure (DOD), and Doppler frequency of a target. The scenario of two closely spaced targets corrupted by broadband noise with/without clutter sources is considered and the closed-form expression of the resolution threshold, which is the minimum signal-to-noise ratio (SNR) required for a given resolution performance, is derived using a generalized likelihood ratio test (GLRT) formulation. Finally, both theoretical and numerical analyses are provided for some insights regarding the resolution limit.

Analysis on the Resolution of Polarimetric Radar and Performance Evaluation of the Polarimetric Bandwidth Extrapolation Method

Polarimetric radar provides useful information on scattering mechanisms, and advances in polarimetric synthetic aperture radar (POLSAR) technology have shown the effect of polarization information on various applications. The disadvantage of POLSAR is low azimuth resolution capability. Since the effective pulse repetition frequency (PRF) in each polarimetric channel of POLSAR is half the total PRF, the resolution is twice as low as the single-polarization SAR. This drawback may be mitigated by employing a superresolution method designed for polarimetric radar such as polarimetric bandwidth extrapolation (PBWE), previously proposed by the authors. Thus far, the authors only empirically showed that polarization information helps in improving resolution via numerical simulations. In this paper, theoretical analysis on the resolution of a polarimetric radar is provided. Along with the resolution, estimation accuracy of the polarization property of the target is studied. In the analysis, we employ the recently proposed metric called the statistical resolution limit (SRL) that provides the highest resolution achievable by any unbiased parametric spectral estimator. The SRL of polarimetric radar is derived and compared with that of single-polarization radar. The Cramér–Rao bound (CRB) of the polarization estimation error of the polarimetric radar is also derived. The complexified full Fisher information matrix and the CRBs of signal parameters are derived first for the general multichannel signal in nonwhite Gaussian noise. Then, the SRL and the polarization estimation error are derived using the CRBs of signal parameters for the cases where two point targets are closely located. It is shown analytically and numerically that the polarization information helps in improving the SRL. It is also shown that the polarimetric processing significantly improves the accuracy of the polarization estimation when two targets are located closer than the Fourier resolution cell. Finally, the performance of the PBWE is evaluated via simulations and is compared to the SRL and the CRB for the polarization estimation.

On the asymptotic resolvability of two point sources in known subspace interference using a GLRT-based framework

The asymptotic statistical resolution limit (SRL), denoted by δ, characterizing the minimal separation to resolve two closely spaced far-field narrowband sources for a large number of observations, among a total number of M≥2, impinging on a linear array is derived. The two sources of interest (SOI) are corrupted by (1) the interference resulting from the M−2 remaining sources and by (2) a broadband noise. Toward this end, a hypothesis test formulation is conducted. Depending on the a priori knowledge on the SOI, on the interfering sources and on the noise variance, the (constrained) maximum likelihood estimators (MLEs) of the SRL subject to δ∈R and/or in the context of the matched subspace detector theory are derived. Finally, we show that the SRL which is the minimum separation that allows a correct resolvability for given probabilities of false alarm and of detection can always be linked to a particular form of the Cramér–Rao bound (CRB), called the interference CRB (I-CRB), which takes into account the M−2 interfering sources. As a by product, we give the theoretical expression of the minimum signal-to-interference-plus-noise ratio (SINR) required to resolve two closely spaced sources for several typical scenarios.

Statistical Resolution Limit for Source Localization With Clutter Interference in a MIMO Radar Context

During the last decade, multiple-input multiple-ouput (MIMO) radar has received an increasing interest. One can find several estimation schemes in the literature related to the direction of arrivals and/or direction of departures, but their ultimate performance in terms of the statistical resolution limit (SRL) have not been fully investigated. In this correspondence, we fill this lack. Particulary, we derive the SRL to resolve two closely spaced targets in clutter interference using a MIMO radar with widely separated antennas. Toward this end, we use a hypothesis test formulation based on the generalized likelihood ratio test (GLRT). Furthermore, we investigate the link between the SRL and the minimum signal-to-noise ratio (SNR) required to resolve two closely spaced targets for a given probability of false alarm and for a given probability of detection. Finally, theoretical and numerical analysis of the SRL are given for several scenarios (with/without clutter interference, known/unknown parameters of interest and known/unknown noise variance).

Statistical analysis of achievable resolution limit in the near field source localization context

In this fast communication, we derive the statistical resolution limit (SRL), characterizing the minimal parameter separation, to resolve two closely spaced known near-field sources impinging on a linear array. Toward this goal, we conduct on the first-order Taylor expansion of the observation model a Generalized Likelihood Ratio Test (GLRT) based on a Constrained Maximum Likelihood Estimator (CMLE) of the SRL. More precisely, the minimum separation between two near-field sources, that is detectable for a given probability of false alarm and a given probability of detection, is derived herein. Finally, numerical simulations are done to quantify the impact of the array geometry of the signal sources power distribution and of the array aperture on the statistical resolution limit.

Statistical resolution limit for the multidimensional harmonic retrieval model: hypothesis test and Cramér-Rao Bound approaches

The statistical resolution limit (SRL), which is defined as the minimal separation between parameters to allow a correct resolvability, is an important statistical tool to quantify the ultimate performance for parametric estimation problems. In this article, we generalize the concept of the SRL to the multidimensional SRL (MSRL) applied to the multidimensional harmonic retrieval model. In this article, we derive the SRL for the so-called multidimensional harmonic retrieval model using a generalization of the previously introduced SRL concepts that we call multidimensional SRL (MSRL). We first derive the MSRL using an hypothesis test approach. This statistical test is shown to be asymptotically an uniformly most powerful test which is the strongest optimality statement that one could expect to obtain. Second, we link the proposed asymptotic MSRL based on the hypothesis test approach to a new extension of the SRL based on the Cramér-Rao Bound approach. Thus, a closed-form expression of the asymptotic MSRL is given and analyzed in the framework of the multidimensional harmonic retrieval model. Particularly, it is proved that the optimal MSRL is obtained for equi-powered sources and/or an equi-distributed number of sensors on each multi-way array.

Statistical Resolution Limit of the Uniform Linear Cocentered Orthogonal Loop and Dipole Array

Among the family of polarization sensitive arrays, we can find the so-called cocentered orthogonal loop and dipole uniform linear array (COLD-ULA). The COLD-ULA exhibits some interesting properties, e.g., the insensibility of the polarization vector with respect to the source localization in the plan of the array. In this correspondence, we derive the statistical resolution limit (SRL) characterizing the minimal separation, in terms of direction-of-arrivals, to resolve two closely spaced known polarized sources impinging on a COLD-ULA. Toward this end, nonmatrix closed form expressions of the deterministic Cramer-Rao bound (CRB) are derived and thus, the SRL is deduced. A comparison between the SRL of the COLD-ULA and the classical ULA are given. Particularly, it is shown that, in the case of orthogonal known signal sources, the SRL of the COLD-ULA is equal to the SRL of the ULA, meaning that it is not a function of polarization parameters. Furthermore, due to the derived SRL, it is shown that, under some general conditions, the SRL of the COLD-ULA is smaller than the one of the ULA.

Statistical Angular Resolution Limit for Point Sources

We define a statistical angular resolution limit (ARL) on resolving two closely spaced point sources in a 3-D reference frame, using constraints on the probabilities of false alarm and detection for a hypothesis test. The ARL can be used as a performance measure for sensor arrays in localizing remote sources and is applicable to different measurement models and applications (e.g., radar, sonar, or astronomy). By considering the asymptotic performance of the generalized likelihood ratio test (GLRT), we derive the analytical expression of the ARL and show that it is proportional to the square root of the Cramer-Rao bound (CRB) on the angular source separation, or asymptotically the lower bound on the mean-square angular error (MSAECRB) . Numerical examples illustrate that the proposed ARL is practically computable and achievable with large data samples. Our analytical result can replace the commonly used ad hoc resolution limits in existing literature.

Statistical resolution limits and the complexified Crame/spl acute/r-Rao bound

Array resolution limits and accuracy bounds on the multitude of signal parameters (e.g., azimuth, elevation, Doppler, range, cross-range, depth, frequency, chirp, polarization, amplitude, phase, etc.) estimated by array processing algorithms are essential tools in the evaluation of system performance. The case in which the complex amplitudes of the signals are unknown is of particular practical interest. A computationally efficient formulation of these bounds (from the perspective of derivations and analysis) is presented for the case of deterministic and unknown signal amplitudes. A new derivation is given using the unknown complex signal parameters and their complex conjugates. The new formula is readily applicable to obtaining either symbolic or numerical solutions to estimation bounds for a very wide class of problems encountered in adaptive sensor array processing. This formula is shown to yield several of the standard Crame/spl acute/r-Rao results for array processing, along with new results of fundamental interest. Specifically, a new closed-form expression for the statistical resolution limit of an aperture for any asymptotically unbiased superresolution algorithm (e.g., MUSIC, ESPRIT) is provided. The statistical resolution limit is defined as the source separation that equals its own Crame/spl acute/r-Rao bound, providing an algorithm-independent bound on the resolution of any high-resolution method. It is shown that the statistical resolution limit of an array or coherent integration window is about 1.2/spl middot/SNR/sup -1/4/ relative to the Fourier resolution limit of 2/spl pi//N radians (large number N of array elements). That is, the highest achievable resolution is proportional to the reciprocal of the fourth root of the signal-to-noise ratio (SNR), in contrast to the square-root (SNR/sup -1/2/) dependence of standard accuracy bounds. These theoretical results are consistent with previously published bounds for specific superresolution algorithms derived by other methods. It is also shown that the potential resolution improvement obtained by separating two collinear arrays (synthetic ultra-wideband), each with a fixed aperture B wavelengths by M wavelengths (assumed large), is approximately (M/B)/sup 1/2/, in contrast to the resolution improvement of M/B for a full aperture. Exact closed-form results for these problems with their asymptotic approximations are presented.

Position-sensitive single carrier CdZnTe detectors

Single polarity charge sensing on room temperature semiconductor gamma-ray detectors can be achieved by using the coplanar electrode read-out technique. This mehod can eliminate the hole-trapping problem of the wide band gap semiconductors which are currently available. Our previous results on 5 mm cube CZT detectors confirmed [6] that the energy resolution can be dramatically improved compared with that obtained using the conventional read-out method. This paper explores the application of this technique to CdZnTe detectors of larger volume, namely 1 cm 3. In our previous work, we suggested a method to obtain γ-ray interaction depth and further progress is reported here. This technique can be used to correct for the electron trapping as a function of distance from the anode. The intrinsic position resolution has been analyzed and energy resolutions of less than 2% FWHM at 662 keV were obtained on both detectors tested. Finally, the factors which inhibit attaining the statistical energy resolution limit of CdZnTe detectors have been explored. These results will be of interest in the design of higher performance, portable and imaging-related, room-temperature semiconductor γ-ray detectors.

Nonequilibrium dynamic conductivity of superconductors: An exploitable basis for high-energy resolution x-ray detectors

A new design for high-energy radiation/particle detectors is presented. The nonequilibrium response of a superconductor to the absorption of the incident quanta is sensed by electromagnetic measurements of the altered dynamic conductivity. Microwave absorption may be used to amplify the signal. Such a detector will provide better energy resolution than semiconducting charge-collection devices once the statistical resolution limit is reached.