Nonlinear dynamics are basic to the characterization of many physical phenomena encountered in practice. Typically, we are given a time series of some observable(s), and the requirement is to uncover the underlying dynamics responsible for generating the time series. This problem becomes particularly challenging when the process and measurement equations of the dynamics are both nonlinear and noisy. Such a problem is exemplified by the case study of sea clutter, which refers to radar backscatter from an ocean surface. After setting the stage for this case study, the paper presents tutorial reviews of (1) the classical models of sea clutter based on the compound K-distribution and (2) the application of chaos theory to sea clutter. Experimental results are presented that cast doubts on chaos as a possible nonlinear dynamical mechanism for the generation of sea clutter. Most importantly, experimental results show that on timescales smaller than a few seconds, sea clutter is very well described as a complex autoregressive process of order 4 or 5. On larger timescales, gravity or swell waves cause this process to be modulated in both amplitude and frequency. It is shown that the amount of frequency modulation is correlated with the nonlinearity of the clutter signal. The dynamical model is an important step forward from the classical statistical approaches, but it is in its early stages of development.
In this paper we consider modelling and estimating the texture in high resolution non-
Gaussian sea clutter. Cyclostationarity of sea clutter is investigated and validated by
processing measured high resolution data. The clutter is modelled as a compound Gaussian
process and the texture as the superposition of real cosines with unknown frequencies,
amplitudes, and phases. We propose a method for estimating the model parameters and
retrieving the texture component from the intensity data in the presence of multiplicative noise (the speckle) with unknown power spectral density. The method exploits the clutter
cyclostationarity and is based on a RELAXation optimisation approach. The ability of the
proposed method to retrieve texture information is investigated by processing simulated and
measured sea clutter data.
The performance of two adaptive detection schemes developed in the literature, Kelly's generalized likelihood ratio test (GLRT) and the adaptive linear-quadratic (ALQ) detector, are tested on real sea clutter data recorded by the IPIX experimental radar. The results of first- and second-order statistical analyses performed on two data sets are reported. Amplitude analysis has been carried out by checking the fitting to Weibull, log-normal, K, and generalized K models. The results show good agreement between performance prediction based on the generalized K model, with texture strongly correlated among primary and secondary data, and the performance obtained by processing the real sea clutter data.
The contribution of this paper is the derivation of the joint maximum likelihood (ML) estimator of complex amplitude and Doppler frequency of a radar target signal embedded in correlated non-Gaussian clutter modelled as a compound-Gaussian process. The estimation accuracy of the ML frequency estimator is investigated and compared with that of the well-known periodogram and ESPRIT estimators under various operational scenarios. The hybrid Cramér-Rao lower bound (HCRLB) and a large sample closed-form expression for the mean square estimation error are also derived for Swerling I target signal. Finally, numerical results obtained by Monte Carlo simulation are checked by means of measured sea clutter data for the general case of fluctuating target amplitude.
In this work we present a thorough performance analysis of two algorithms for estimating Toeplitz covariance matrices, the structured sample covariance matrix estimator (SCME) and the structured normalised SCME (NSCME), which are employed by adaptive radar detectors against Gaussian and compound-Gaussian clutter. Performance predictions are checked with real-life sea clutter data.
A discussion is presented on the chaotic dynamics of sea clutter in the light of current knowledge about chaos theory. The discussion builds on experimental results reported by Haykin and Puthusserypady (see Chaos, vol.7, p.777-802, 1997, IEE 1997 Radar Conf., Edinburgh, Scotland, p.14-16, 1997, and Chaotic dynamics of sea clutters, Wiley, 1999) on the chaotic dynamics of sea clutter under varying environmental conditions and on the dynamic reconstruction of sea clutter using a regularised radial-basis function network. The experimental results presented are for a marine radar operating at low grazing angles. The implications of the chaotic dynamics of sea clutter are discussed in the context of physical aspects of sea clutter, simulation of sea clutter, the detection of a target signal embedded in sea clutter, and the stability of the predictive model used to perform the dynamic reconstruction of sea clutter.
This paper addresses the problem of signal detection in correlated non-Gaussian clutter modeled as a spherically invariant random process. The optimum strategy to detect a constant signal, with either known or unknown complex amplitude, embedded in correlated Gaussian clutter is given by comparing the whitening-matched filter output with a fixed threshold. When the clutter is non-Gaussian, the performance of the matched filter sensibly degrades. The optimum strategy is the classical whitening-matched filter output compared with a data-dependent threshold. This interpretation provides a deeper insight into the structure of the optimum detector and allows us to single out a family of suboptimum detectors based on a polynomial approximation of the data-dependent threshold. They are easy to implement and have performance that is really close to the optimal. The adaptive implementation of the polynomial detectors is also investigated, and their performance is analyzed by means of Monte Carlo simulation for various clutter scenarios.
The authors present a thorough performance analysis of two covariance matrix estimators, the sample covariance matrix estimator (SCME) and the normalised SCME (NSCME), which are employed by adaptive radar detectors in Gaussian and compound-Gaussian clutter. Theoretical performance predictions are derived, compared with the modified Cramer-Rao lower bound and checked with real-life sea clutter data. The results of the analysis show that the NSCME has superior performance in compound-Gaussian clutter and its performance is insensitive to the clutter multivariate distribution within the range cell under test and to the shape of the clutter correlation among different range cells. Conversely, the performance of the SCME heavily depends on the clutter distribution and has a dramatic worsening in spiky non-Gaussian clutter.
Concerns the improved detection of a nonstationary target signal in a nonstationary background. Ways to deal with the issue of nonstationarity are discussed, starting with Loeve's probabilistic theory of stationarity processes (1946, 1963). Three important tools emerge: the dynamic spectrum, the Wigner-Ville distribution as an instantaneous estimate of the dynamic spectrum and the Loeve spectrum. Procedures for the estimation of these spectra are described, and their applications are demonstrated using real-life radar data. Time, an essential dimension of learning, appears explicitly in the dynamic spectrum and Wigner-Ville distribution and implicitly in the Loeve spectrum. In each case, the 1D time series is transformed into a 2D image where the presence of nonstationarity is displayed in a more visible manner than in the original time series. This sets the stage for reformulating the signal detection problem as an adaptive pattern classification whereby we can exploit the learning property of neural nets. Hence, we describe a novel learning strategy for distinguishing between the different classes of received signals, such as 1) there is no target signal present in the received signal; 2) the target signal is weak; and 3) the target signal is strong. We present a case study based on real-life radar data. The case study demonstrates that the adaptive approach described is superior to the classical approach.
The notion that a deterministic nonlinear dynamical system (with relatively few degrees of freedom) can display aperiodic behavior has a strong bearing on sea clutter characterization: random-looking sea clutter may be the outcome of a chaotic process. This new approach envisages deterministic rules for the underlying sea clutter dynamics, in contrast to the stochastic approach where sea clutter is viewed as a random process with a large number of degrees of freedom. In this paper, we demonstrate, convincingly for the first time, the chaotic dynamics of sea clutter. We say so on the basis of results obtained using radar data collected from a series of extensive and thorough experiments, which have been carried out with ground-truthed sea clutter data sets at three different sites. The study includes correlation dimension analysis (based on the maximum likelihood principle) and Lyapunov spectrum analysis. The Lyapunov (Kaplan-Yorke) dimension, which is a byproduct of Lyapunov spectrum analysis, shows that it is indeed a good estimator of the correlation
dimension. The Lyapunov spectrum also reveals that sea clutter is produced by a coupled system of nonlinear differential equations of order five or six.
We describe a novel modular learning strategy for the detection of a target signal of interest in a nonstationary environment, which is motivated by the information preservation rule. The strategy makes no assumptions on the environment. It incorporates three functional blocks: (1) time-frequency analysis, (2) feature extraction, and (3) pattern classification, the delineations of which are guided by the information preservation rule. The time-frequency analysis, which is implemented using the Wigner-Ville distribution (WVD), transforms the incoming received signal into a time-frequency image that accounts for the time-varying nature of the received signal's spectral content. This image provides a common input to a pair of channels, one of which is adaptively matched to the interference acting alone, and the other is adaptively matched to the target signal plus interference. Each channel of the receiver consists of a principal components analyzer (for feature extraction) followed by a multilayer perceptron (for feature classification), which are implemented using self-organized and supervised forms of learning in feedforward neural networks, respectively. Experimental results based on real-life radar data are presented to demonstrate the superior performance of the new detection strategy over a conventional detector using constant false-alarm rate (CFAR) processing. The data used in the experiment pertain to an ocean environment, representing radar returns from small ice targets buried in sea clutter; they were collected with an instrument quality coherent radar and properly ground truthed.
Novel detection schemes are developed using a coherent X-band radar for the detection of small pieces of icebergs. The methods use Wigner-Ville (WV) distribution to perform detection in a joint time-frequency space. Two separate methodologies are presented. The first method extracts classification features from the ambiguity function of the received signal and a neural network is used to perform detection based on these features. The second method uses the method of Principal Components Analysis (PCA) to extract essential information from the time-frequency space for classification. Using real radar data, results are presented and the developed methods are also compared to a conventional Doppler constant false-alarm rate (CFAR) processor.
The paper is devoted to a detailed analysis of experimental data, collected at Osborne Head Gunnery Range with McMaster University IPIX radar, to test theoretical models developed in the literature. The validity of the compound model has been proven for VV polarisation both for amplitude and correlation properties. Cross-polarised data also exhibit a compound behaviour but require an additional Gaussian component due to thermal noise. HH data deviate from the K model and seem to better approach a log-normal distribution. Previous results have been obtained by a correlation test that allows separation of the short and long correlation terms, a modified Kolmogoroff Smirnoff test to verify the fitting and a cumulants domain analysis to quantify the Gaussian component. The interest of the work lies in its application for successful radar design.
The detection of targets from a coherent radar scan is examined, as well as how sophisticated signal processing techniques may be incorporated into such tasks. Techniques of coherent constant-false-alarm-rate (CFAR) detection are discussed for this scanning-mode problem. In an attempt to better utilise the information present in the scan plane for detection, the use of higher-dimensional analysis techniques is discussed. The fundamental advantage is their ability to provide spatial separation (in these higher dimensions) of signal features, which it is hoped will enhance the detection performance. A method, employing such techniques, is presented, and contrasted with traditional CFAR approaches. Some examples are presented using real data, collected with McMaster University's X-band coherent radar.
We describe a computationally efficient scheme for the nonlinear adaptive prediction of nonstationary signals whose generation is governed by a nonlinear dynamical mechanism. The complete predictor consists of two subsections. One performs a nonlinear mapping from the input space to an intermediate space with the aim of linearizing the input signal, and the other performs a linear mapping from the new space to the output space. The nonlinear subsection consists of a pipelined recurrent neural network (PRNN), and the linear section consists of a conventional tapped-delay-line (TDL) filter. The nonlinear adaptive predictor described is of general application. The dynamic behavior of the predictor is demonstrated for the case of a speech signal; for this application, it is shown that the nonlinear adaptive predictor outperforms the traditional linear adaptive scheme in a significant way.
Ship navigation through growler-infested waters is a problem of deep concern to the Canadian oil exploration industry. Growlers are small pieces of glacial ice weighing up to 100 tonnes. Conventional noncoherent marine radars do not perform well in detecting these small but dangerous objects. A coherent detector based on a Gaussian spectral width (GSW) parameterisation of received radar echoes from growlers and sea clutter is considered. Using real X-band radar measurements, the new GSW detector is shown to offer significant improvement in the detectability of growlers in the sea. Nevertheless, the performance achieved indicates that further integration is still required to satisfy practical performance requirements.
Fractal theory is applied to the analysis of real radar signals which are scattered from rough sea surfaces. The databases formed by sampling the radar signals include the two general cases, i.e. both forward-scattered and backscattered signals. The signals for the two cases were recorded using two entirely different radar systems and at two entirely differently geographic locations. The box counting method is used to estimate the fractal dimension of the scattered signals. To corroborate this result, a computation of the fractal dimension is based on the index alpha in the power spectrum relation, P(f) varies as f/sup - alpha /. The estimates derived from both methods are consistent. It is observed that the forward-scattered and back-scattered radar signals have very similar fractal dimensions, i.e. 1746+or-0.033 for the 9.6 GHz forward-scattered signals, 1.753+or-0.024 for the 8.6 GHz forward-scattered signals, and 1.758+or-0.015 for the 9.39 GHz back-scattered signals. Finally, it is shown that there is a detectable variation in the fractal dimension when a target is present. Based on this variation, it is therefore possible to detect the presence of a target by observing the fractal dimension of the radar returns.
Ship navigation through ice-infested waters is a problem of deep concern to the Canadian shipping and exploration industry. Conventional marine radars have difficulty detecting small pieces of glacial ice called growlers which are very hazardous to vessels if struck. In an effort to improve detection performance, X-band radar measurements were collected and analyzed to determine the characteristics of clutter and growler returns that could lead to their separability. These analyses suggested that coherent medium dwell-time processing (i.e., integration times of a fraction of a second) could provide improvement In growler detectability over conventional methods; and long dwell-time processing (i.e., integration times on the order of seconds) could provide even further improvement. We report on the performance of two new coherent, medium dwell-time detectors. A third detector which is representative of conventional marine radar serves as a basis for comparison Although significant improvement in growler detectability is achieved, the two coherent detectors still fall short of operational requirements. This leads to the development of a long dwell-time detector which provides considerably better performance. Empirical results indicate that this new detector could well satisfy stringent operational requirements.
A technique that performs better than conventional, noncoherent marine radars in detecting small pieces of glacial ice (growlers) in the sea is reported. A coherent detector based on an autoregressive (AR) parameterization of received radar echoes that distinguishes growler returns from the background sea clutter returns is considered. A conventional noncoherent detector is also evaluated as a basis for comparison. The AR-based detector is shown to offer significant improvements in detecting growlers. The results reported are based on real X-band radar data.
The use of the K-distribution to describe the amplitude statistics of sea clutter collected with an instrument-quality X-band radar off the East Coast of Canada is investigated. It is shown that the K-distribution is suitable for modelling the amplitude statistics of both likepolarised and crosspolarised radar configurations. The amplitude statistics of small ice targets (growlers) in spiky clutter are also examined; the results obtained indicate that using only long-term amplitude statistics is not sufficient for the reliable detection of such targets in the presence of sea clutter. Furthermore, it is shown that the crosspolarised channel does not offer any added benefit to small target detection when considering only long-term amplitude statistics. The authors also consider the phase statistics of sea clutter and show them to be uniformly distributed on 2 pi .
Using experimental sea clutter data obtained with an instrument-quality radar research system, we show the existence of a chaotic attractor in the clutter. This new and exciting result is in sharp contrast with the conventional stochastic modeling of clutter as some kind of a random process with certain distribution.
We consider modeling and estimating the texture in high-resolution non-Gaussian sea clutter. The cyclostationarity of sea clutter is investigated and validated by processing measured high-resolution data. The clutter is modeled as a compound Gaussian process and the texture as the superposition of real cosines with unknown frequencies, amplitudes, and phases. We propose a method for estimating the model parameters and retrieving the texture component from the intensity data in the presence of multiplicative noise (the speckle) with unknown power spectral density. The method exploits the clutter cyclostationarity and is based on a relaxation optimization approach. The ability of the proposed method to retrieve texture information is investigated by processing simulated and measured sea clutter data.
Detecting signals hidden beneath the noise floor is a challenging task. As the signal-to-noise ratio (S/N) dips below 0, false alarms and detection misses become a serious problem. Furthermore, to satisfy the real-time or near real-time requirement, detection schemes that are computationally intensive do not enjoy wide-spread adoption. In this paper, we present a new detection algorithm consisting of phase-space reconstruction technique and principal components analysis. The goal is to achieve the detection of weak signals in noisy environments. With the new algorithm, our study shows that in addition to detection, the frequency of the signal can be extracted even when the S/N reaches negative value and the FFT power spectrum shows no trace of its spectral characteristics. The signal detection scheme is insensitive to the nature of the background noise, making it viable to achieve good performance in various signal application domains. In this paper, we chose to report on the results pertaining to the analysis of time series from IPIX radar. The new detection algorithm is also computationally lean, thus enabling its use in real-time applications.
We perform a detailed analysis of experimental data, collected at Osborne Gunnery Range with MacMaster University IPIX radar, to test the theoretical models developed in the literature. A comparison of the amplitude histograms with the K and lognormal distribution shows that VV data exhibit a K behaviour, while HH data are better fitted by a lognormal model and cross-polarised data by a K plus thermal noise model. The results have been confirmed by comparing the first four moments and through a modified version of the Kolmogoroff-Smirnoff test. The validity of the compound model, that identifies two components of clutter fluctuations, has been proved for the VV polarisation. The texture component, deriving from the spatially varying mean level has been isolated, since the speckle component was decorrelated by the frequency agility. These results are of great importance for design of optimum target detection schemes.
An overview on a number of new theoretical findings is presented for the optimum and suboptimum radar detection of fluctuating targets against a composite disturbance, which is modeled as a mixture of coherent K-distributed and Gaussian distributed disturbance. The optimum coherent detector, which derives from the likelihood ratio test, is used as the performance bound. Starting from some of its possible different analytical formulations, the optimum detector is approximated to give a family of suboptimum detectors, with performance close to optimal. The adaptive implementation of these detectors is discussed and the adaptivity loss is evaluated. The detection schemes are then fed with recorded live clutter data, and their performance are analyzed in such environment. The validity of the theoretical analysis is thus assessed as well as the practicality of the suboptimum detection schemes presented.