Spherically Invariant Random Process Research Articles

Compatibility Assessment of Multistatic/Polarimetric Clutter Data With the SIRP Model

This article deals with the statistical inference of simultaneously recorded co- and cross-polarized bistatic coherent sea-clutter returns at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$S$</tex-math></inline-formula> -band. This study is conducted employing appropriate statistical learning tools, involving the complex envelope of data, to assess the compliance of the available measurements with the spherically invariant random process (SIRP) representation, as well as to analyze possible texture correlations among the diverse polarimetric channels. Moreover, the spatial heterogeneity of the sea-clutter data is studied. The results highlight that the SIRP model is a good candidate for the representation of bistatic coherent clutter and usually the coherence time of the SIRP texture at the bistatic nodes is longer than that in the monostatic sensing. Notably, at bistatic angles in order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$60^\circ$</tex-math></inline-formula> , the quadrature components of the cross-polarized bistatic measurements substantially exhibit a Gaussian behavior. These achievements further shed light on the bistatic sea-clutter diversity from the geometric and polarimetric point of view.

A Novel Gamma Distributed Random Variable (RV) Generation Method for Clutter Simulation with Non-Integral Shape Parameters.

Sea clutter simulation is a well-known research endeavour in radar detector analysis and design, and many approaches to it have been proposed in recent years, among which zero memory non-linear (ZMNL) and spherically invariant random process (SIRP) are the most two widely used methods for compound Gaussian distribution. However, the shape parameter of the compound Gaussian clutter model cannot be a non-integer nor non-semi-integer in the ZMNL method, and the computational complexity of the SIRP method is very high because of the complex non-linear operation. Although some improved methods have been proposed to solve the problem, the fitting degree of these methods is not high because of the introduction of Beta distribution. To overcome these disadvantages, a novel Gamma distributed random variable (RV) generation method for clutter simulation is proposed in this paper. In our method, Gamma RV with non-integral or non-semi-integral shape parameters is generated directly by multiplying an integral-shape-parameter Gamma RV with a Beta RV whose parameters are larger than 0.5, thus avoiding the deviation of simulation of Beta RV. A large number of simulation experimental results show that the proposed method not only can be used in the clutter simulation with a non-integer or non-semi-integer shape parameter value, but also has higher fitting degree than the existing methods.

Tight Analytical and Asymptotic Upper Bound for the BER and FER of Linear Codes Over Exponentially Correlated Generalized-Fading Channels

In this paper, the upper bound and asymptotical approximation for the bit error rate (BER) as well as for the frame error rate of linear block codes under maximum-likelihood decoding operating over exponentially correlated (EC) generalized-fading channels are proposed. Particularly, our analysis takes the convolutional codes as an example and three different fading scenarios are considered, namely Rice, Nakagami- $m$ , and Weibull fading. Such scenarios have a common property that is spherically invariant random process (SIRP). Our analysis relies on the derivation of new lower bounds per each eigenvalue since the BER as well as the FER expressions are written in terms of the pairwise error probability, which by its turn depends on the eigenvalues of the channel covariance matrix. Based on that, the upper bounds and asymptotic expressions of the considered metrics subject to the three above fading models are derived, which show to be tighter than the previous results published elsewhere in the literature, especially for the high signal-to-noise ratio regions.

Iterative marginal maximum likelihood DOD and DOA estimation for MIMO radar in the presence of SIRP clutter

Abstract The spherically invariant random process (SIRP) clutter model is commonly used in scenarios where the radar clutter cannot be correctly modeled as a Gaussian process. In this short communication, we devise a novel Maximum-Likelihood (ML)-based iterative estimator for direction-of-departure and direction-of-arrival estimation in the Multiple-input multiple-output (MIMO) radar context in the presence of SIRP clutter. The proposed estimator employs a stepwise numerical concentration approach w.r.t. the objective function related to the marginal likelihood of the observation data. Our estimator leads to superior performance, as our simulations show, w.r.t. to the existing likelihood based methods, namely, the conventional, the conditional and the joint likelihood based estimators, and w.r.t. the robust subspace decomposition based methods. Finally, interconnections and comparison between the Iterative Marginal ML Estimator (IMMLE), Iterative Joint ML Estimator (IJMLE) and Iterative Conditional ML Estimator (ICdMLE) are provided.

MIMO radar target localization and performance evaluation under SIRP clutter

Multiple-input multiple-output (MIMO) radar has become a thriving subject of research during the past decades. In the MIMO radar context, it is sometimes more accurate to model the radar clutter as a non-Gaussian process, more specifically, by using the spherically invariant random process (SIRP) model. In this paper, we focus on the estimation and performance analysis of the angular spacing between two targets for the MIMO radar under the SIRP clutter. First, we propose an iterative maximum likelihood as well as an iterative maximum a posteriori estimator, for the target's spacing parameter estimation in the SIRP clutter context. Then we derive and compare various Cramér–Rao-like bounds (CRLBs) for performance assessment. Finally, we address the problem of target resolvability by using the concept of angular resolution limit (ARL), and derive an analytical, closed-form expression of the ARL based on Smith's criterion, between two closely spaced targets in a MIMO radar context under SIRP clutter. For this aim we also obtain the non-matrix, closed-form expressions for each of the CRLBs. Finally, we provide numerical simulations to assess the performance of the proposed algorithms, the validity of the derived ARL expression, and to reveal the ARL's insightful properties.

RANGE-SPREAD TARGET DETECTION IN COMPOUND GAUSSIAN CLUTTER WITH RECIPROCAL OF THE SQUARE ROOT OF GAMMA TEXTURE

In this paper, the range-spread target detection in compound Gaussian clutter with reciprocal of the square root of gamma (RSRG) texture is investigated. The RSRG distribution has been proved to be a good model for texture component of extremely heterogeneous radar clutter. Taking this compound Gaussian model as a spherically invariant random process (SIRP), the Neyman-Pearson optimal detector for the range-spread target detection with known target amplitude is derived flrstly. By replacing the ideal target amplitude with its maximum likelihood estimate (MLE), the generalized likelihood ratio test (GLRT) is then obtained. The statistical property of the texture component is taken into account in both of these two detectors, which makes the detectors computationally complicated. A suboptimal generalized likelihood ratio test based on order statistics (OS-GLRT) is flnally proposed by substituting the texture component with its MLE. The OS-GLRT makes use of some largest observations from the range cells occupied by the most likely target scatters. The performance assessment conducted by Monte Carlo simulation validates the efiectiveness of the proposed detectors.

Subspace detection for range and Doppler distributed targets with Rao and Wald tests

For the detection problem caused by the point target broken up into the distributed target in the high-range resolution radar system, adaptive detection algorithms based on the Rao and the Wald tests are proposed. The distributed target, which is modeled as a subspace random signal, may be distributed both in range and also in Doppler frequency axes. The clutter is modeled as a spherically invariant random process (SIRP). The unknown parameters are estimated by maximum likelihood estimation only under hypothesis H 0 in Rao test, while only under hypothesis H 1 in Wald test. Interestingly the proposed non-adaptive detectors from different deriving progresses have the same final form, which perform an incoherent processing over all the cells under test and ensure CFAR property with respect to the unknown statistics of the clutter texture component. In the end, performances of the proposed adaptive detectors are assessed through Monte Carlo simulations and are shown to have better detection performance compared with existing similar detectors.

GLRT subspace detection for range and Doppler distributed targets

A generalized likelihood ratio test (GLRT) is derived for adaptive detection of range and Doppler-distributed targets. The clutter is modeled as a spherically invariant random process (SIRP) and its texture component is range dependent (heterogeneous clutter). We suppose here that the speckle component covariance matrix is known or estimated thanks to a secondary data set. Thus, unknown parameters to be estimated are local texture values, the complex amplitudes and Doppler frequencies of all scattering centers. To do so, we use superresolution methods. The proposed detector assumes a priori knowledge on the spatial distribution of the target and has the precious property of having a constant false alarm rate (CFAR) with the assumption of a known speckle covariance matrix or by the use of frequency agility.

Adaptive subspace detection of range-distributed target in compound-Gaussian clutter

This paper deals with high-resolution radar (HRR) adaptive detection of range-distributed target embedded in compound-Gaussian clutter which is modeled as a spherically invariant random process (SIRP). Using multiple dominant scattering (MDS) model of range-distributed target, we justify that range-distributed target can be modeled as a subspace random signal. The unknown deterministic parameters are replaced by their ML estimates and then the nonadaptive detector is proposed. A closed-form expression for the probability of false alarm of the nonadaptive detector is derived and it ensures CFAR property with respect to the unknown statistics of the clutter texture component. Moreover, an adaptive detector is obtained relying on a two-step GLRT-based design procedure. Performances of these proposed detectors are assessed through Monte Carlo simulations and are shown to have better detection performance compared with existing similar detector.

Outage probability analysis of antenna array using maximum ratio combining in spherically invariant fading environment

AbstractIn this paper, the outage probability of maximum ratio combining (MRC) in the spherically invariant fading environment was analyzed. The study was done with and without consideration of the effect of noise. First, the probability density functions (pdf) of signal‐to‐interference plus noise ratio (SINR) and signal‐to‐interference ratio (SIR) were derived. In our derivation, the characteristic pdfs of spherically invariant random process (SIRP) corresponding to the interferences were assumed to be both different and identical. Moreover, both equal and different mean powers of interferences were considered. With the derived pdfs of SINR and SIR, the expressions of outage probability were computed. Numerical analysis was done using three numerical methods known as Gauss–Laguerre, Gauss–Hermite, and Gauss–Legendre integrations. It should be noted that SIRP, the generalization of Gaussian random process, is assumed to have K‐distribution, which is the generalization of Rayleigh. Our analysis was verified by comparing the outage probabilities of the considered environments with those of Rayleigh fading channel. From the numerical results, the derived outage probability converges to that of Rayleigh fading, when the shape parameter of characteristic pdf approaches infinity. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Statistical analysis of the nonhomogeneity detector for non-Gaussian interference backgrounds

We derive the nonhomogeneity detector (NHD) for non-Gaussian interference scenarios and present a statistical analysis of the method. The non-Gaussian interference scenario is assumed to be modeled by a spherically invariant random process (SIRP). We present a method for selecting representative (homogeneous) training data based on our statistical analysis of the NHD for finite sample support used in covariance estimation. In particular, an exact theoretical expression for the NHD test statistic probability density function (PDF) is derived. Performance analysis of the NHD is presented using both simulated data and measured data from the multichannel airborne radar measurement (MCARM) program. A performance comparison with existing NHD approaches is also included.

Sampling design for weak signal detection in SIRP noise

In this paper, we present a sampling design for weak signal detection in additive spherically invariant random process (SIRP) noise. Our approach for the design of sampling schemes is based on the class of Ali–Silvey distance measures. A design objective function for this problem is developed. This objective function is maximized to obtain the sampling scheme for the detection problem under consideration. Two examples are presented to illustrate our sampling design. Numerical results show that our sampling design outperforms the uniform sampling. It is also numerically shown that the sampling design is insensitive to the statistics (the characteristic probability density function) of the SIRP noise samples.

Expected number of maxima in the envelope of a spherically invariant random process

In many engineering applications, specially in communication engineering, one encounters a bandpass non-Gaussian random process, with a slowly varying envelope. Among the available models for non-Gaussian random processes, spherically invariant random processes (SIRPs) play an important role. These processes are of interest mainly due to the fact that they allow one to relax the assumption of Gaussianity, while keeping many of its useful characteristics. In this paper, we have derived a simple and closed-form formula for the expected number of maxima of a SIRP envelope. Since Gaussian random processes are special cases of SIRPs, this formula holds for Gaussian random processes as well. In contrast with the available complicated expression for the expected number of maxima in the envelope of a Gaussian random process, our simple result holds for an arbitrary power spectrum. The key idea in deriving this result is the application of the characteristic function, rather than the probability density function, for calculating the expected level crossing rate of a random process.

BORD: bayesian optimum radar detector

We derive the expression of an optimum non-Gaussian radar detector from the non-Gaussian spherically invariant random process (SIRP) clutter model and a bayesian estimator of the SIRP characteristic density. SIRP modelizes non-Gaussian process as a complex Gaussian process whose variance, the so-called texture, is itself a positive random variable (r.v.). After performing a bayesian estimation of the texture probability density function (PDF) from reference clutter cells we derive the so-called bayesian optimum radar detector (BORD) without any knowledge about the clutter statistics. We also derive the asymptotic expression of BORD (in law convergence), the so-called asymptotic BORD, as well as its theoretical performance (analytical threshold expression). BORD performance curves are shown for an unknown target signal embedded in correlated K-distributed and are compared with those of the optimum K-distributed detector. These results show that BORD reach optimal detector performances.

CFAR detection of distributed targets in non-Gaussian disturbance

The subject of detection of spatially distributed targets in non-Gaussian noise with unknown statistics is addressed. At the design stage, in order to cope with the a priori uncertainty, we model noise returns as Gaussian vectors with the same structure of the covariance matrix, but possibly different power levels (heterogeneous environment). We also assume that a set of secondary data, free of signal components, is available to estimate the correlation properties of the disturbance The proposed detector assumes no a priori knowledge about the spatial distribution of the target scatterers and ensures the constant false alarm rate (CFAR) property with respect to both the structure of the covariance matrix and the power levels. Finally, the performance assessment, conducted modeling the disturbance as a spherically invariant random process (SIRP), confirms its validity to operate in real radar scenarios.

Computer generation of correlated non-Gaussian radar clutter

We develop computer simulation procedures which enable us to generate any correlated non-Gaussian radar clutter that can be modeled as a spherically invariant random process (SIRP). In most cases, when the clutter is a correlated non-Gaussian random process, performance of the optimal radar signal processor cannot be evaluated analytically. Therefore, in order to evaluate such processors, there is a need for efficient computer simulation of the clutter. We present two canonical simulation procedures for the generation of correlated non-Gaussian clutter. A new approach for the goodness-of-fit test is proposed in order to assess the performance of the simulation procedure. >

Joint phase and envelope densities of a higher order

In dealing with narrowband noise processes the density p(R1,R2, Delta ) in which R1 and R2 are envelope samples and Delta is the phase difference can play an important role in determining system design features such as diversity performance, crossing rates, error probabilities, optimum processing and frequency or phase distributions. Standard texts on signal detection or radar usually include a discussion of p(R) and, less often, of p(R1,R2) but the number of instances in which p(R1,R2, Delta ) can be specified is limited, and here known results are touched upon while a general form for this joint density is developed. The work extends an earlier treatment of p(R1,R2, Delta ) and includes some important observations regarding the method used here and a SIRP (spherically invariant random process) approach which has been widely proposed for the analysis of correlated radar returns. Two differing families of non-Gaussian processes are used to illustrate the working and a number of densities for the jointly correlated pair R1 and R2, and the phase difference Delta are given, so extending the pool of such results available to the analyst. The approach used is heuristic and although the positivity of p( Delta ) is not proven outright, experience and the cases illustrated point to this requirement being met.