Product High-order Ambiguity Function Research Articles

Imaging of target with complicated motion using ISAR system based on IPHAF-TVA

The technique of imaging a target with a complicated motion using an Inverse Synthetic Aperture Radar (ISAR) system is an effective tool in the field of radar signal processing. After the translational compensation, the received signal reflected from the target can take the form of a multi-component Polynomial Phase Signal (m-PPS), and the high quality ISAR image can be provided via the combination between the estimated parameters of the m-PPS and the Range Instantaneous-Doppler technique (RID). For a target with a high maneuvrability, the occurrence of scatterers Migration Through Resolution Cell (MTRC), caused by the rotational movement could be appearing. That is why the variation in the amplitude of the echo during the time of observation cannot be neglected. The purpose of this study is the parameters estimation of the m-PPS signal with order three in the case of the Time Varying Amplitude (TVA). The Improved-version of the Product High-order Ambiguity Function (IPHAF) with TVA is proposed to improve the quality of the ISAR image compared with traditional techniques based on a constant amplitude; the experimental outcomes confirm that the new IPHAF-TVA method presented in this study is an effective technique to make the ISAR image very clear.

Retrieval of Three-Dimensional Surface Deformation Using an Improved Differential SAR Tomography System

Conventional differential synthetic aperture radar tomography (D-TomoSAR) can only capture the scatterers’ one-dimensional (1-D) deformation information along the line of sight (LOS) of the synthetic aperture radar (SAR), which means that it cannot retrieve the three-dimensional (3-D) movements of the ground surface. To retrieve the 3-D deformation displacements, several methods have been proposed; the performance is limited due to the insufficient sensitivity for retrieving the North-South motion component. In this paper, an improved D-TomoSAR model is established by introducing the scatterers’ 3-D deformation parameters in slant range, azimuth, and elevation directions into the traditional D-TomoSAR model. The improved D-TomoSAR can be regarded as a multi-component two-dimensional (2-D) polynomial phase signal (PPS). Then, an effective algorithm is proposed to retrieve the 3-D deformation parameters of the ground surface by the 2-D product high-order ambiguity function (PHAF) with the relax (RELAX) algorithm. The estimation performance is investigated and compared with the traditional algorithm. Simulations and experimental results with semi-real data verify the effectiveness of the proposed signal model and algorithm.

Imaging of high‐speed manoeuvering target via improved version of product high‐order ambiguity function

This study presents a novel algorithm for radar imaging of high-speed manoeuvering target. It is based on the fact that the azimuth part of the echo is modelled as the superposition of multiple quadratic frequency modulated (QFM) signals, and the range-Doppler algorithm is inappropriate to generate a focused radar image in this case. In this study, the improved version of product high-order ambiguity function (IPHAF) is proposed to estimate the parameters of multiple QFM signals. This estimator requires only one-dimensional (1D) maximisations without the scaling operation compared with the traditional product high-order ambiguity function. The radar image quality can be improved by the IPHAF technique, and the experimental results validate the effectiveness of the method presented in this study.

Parameter Estimation of Multicomponent 2D Polynomial-Phase Signals Using the 2D PHAF-Based Approach

This paper considers parameter estimation of multicomponent 2D polynomial-phase signals (mc-2D PPSs). The main contributions included are: i) Derivation of the Cramer-Rao lower bound (CRLB) for mc-2D PPSs; ii) Two dimensional product high-order ambiguity function (2D PHAF) based estimation procedure, recently proposed by Barbarossa, for the estimation of third-order mc-2D PPSs is extended to signals of arbitrary orders; iii) Procedure for optimal lag set selection in the proposed 2D PHAF has been proposed; iv) Since the 2D PHAF is characterized by large mean squared error (MSE) for higher-order signals, refinement procedure is introduced that is able to improve the estimation accuracy and to reduce MSE up to the CRLB. The proposed estimator has been used for target focusing in radar images.

Parameter Estimation of 2D Multi-Component Polynomial Phase Signals: An Application to SAR Imaging of Moving Targets

Polynomial-phase signals (PPS) appear in a variety of applications and several algorithms are available to estimate their parameters in the presence of noise. Among the available tools, the product high order ambiguity function (PHAF) has the merit of performing well in the presence of a superposition of PPS's. In this work, we generalize the PHAF to handle two-dimensional PPS's. Then we show an example of application motivating such an extension: the high resolution imaging of moving targets from synthetic aperture radars (SAR). Using the 2D-PHAF, we will propose an algorithm that compensates jointly for the range cell migration and the phase modulation induced by the relative radar-target motion in order to produce a focused image of the moving target. Numerical results illustrate the advantages of the proposed method when compared to existing auto-focusing algorithms.

Quasi‐maximum‐likelihood estimator of polynomial phase signals

A novel approach for the parameter estimation of the polynomial phase signals (PPS) based on the short-time Fourier transform (STFT) is proposed. Estimator accuracy is significantly higher compared to the high-order ambiguity function (HAF), product HAF, and similar the non-linear transforms based strategies. The proposed approach is more efficient than the maximum likelihood (ML) estimator for the high-order PPS. One-dimensional search is performed over a set of window widths in the STFT, whereas in the case of ML estimators, it has been done over the space of phase parameters. The proposed estimator is implemented in several steps: the instantaneous frequency (IF) estimation using the STFT for various window widths; a polynomial regression from the IF estimate producing a coarse signal coefficients estimate; refinement procedure, producing fine coefficients estimates; determination of the optimal window width in the STFT. The proposed technique is extended for the non-parametric estimation with a 2D search over a set of polynomial orders and window widths. Good estimation results are achieved up to the SNR threshold of about SNR=0 dB for the parametric case and SNR=2 dB for the non-parametric case.

A Hybrid CPF-HAF Estimation of Polynomial-Phase Signals: Detailed Statistical Analysis

In this paper, we consider parameter estimation of high-order polynomial-phase signals (PPSs). We propose an approach that combines the cubic phase function (CPF) and the high-order ambiguity function (HAF), and is referred to as the hybrid CPF-HAF method. In the proposed method, the phase differentiation is first applied on the observed PPS to produce a cubic phase signal, whose parameters are, in turn, estimated by the CPF. The performance analysis, carried out in the paper, considers up to the tenth-order PPSs, and is supported by numerical examples revealing that the proposed approach outperforms the HAF in terms of the accuracy and signal-to-noise-ratio threshold. Extensions to multicomponent and multidimensional PPSs are also considered, all supported by numerical examples. Specifically, when multicomponent PPSs are considered, the product version of the CPF-HAF outperforms the product HAF (PHAF) that fails to estimate parameters of components whose PPS order exceeds three.

A Parametric Method for Multicomponent Interference Suppression in Noise Radars

A method for interference suppression in noise radars is proposed. The phase of interference is locally approximated by a polynomial, whose coefficients are estimated using the product high-order ambiguity function (HAF), which successfully resolves the components of multicomponent interference and improves noise rejection. The phase approximation is used to concentrate and suppress the interference. For nonpolynomial phases, an algorithm that performs a piecewise polynomial approximation is developed. The algorithm is extended to deal with multicomponent interferences.

Are genetic algorithms useful for the parameter estimation of FM signals?

The estimation of polynomial-phase signals (PPSs) is a multiparameter problem, and the maximum likelihood (ML) optimization functions have numerous local optima, making the application of gradient techniques impossible. The common solution to this problem is based on the phase differentiation (PD) techniques that reduce the number of dimensions but, at the same time, reduce the accuracy and generate additional difficulties such as spurious components and error propagation. Here we show that genetic algorithms (GAs) can serve as a powerful alternative to the PD techniques. We investigate the limits of accuracy of the ML technique, and of some alternatives such as the high-order cubic phase function (HO-CPF) and high-order Wigner distribution (HO-WD). The ML approach combined with the proposed GA setup is limited up to the fifth-order PPS, which is not sufficient in many applications. However, the HO-CPF and HO-WD techniques coupled with the GA are able to accurately estimate phase parameters up to the tenth-order PPS. They significantly improve the accuracy with respect to the high-order ambiguity function (HAF) and product HAF (PHAF) and, for higher-order PPSs, they are much simpler and more efficient than the integrated generalized ambiguity function (IGAF).

Estimation of multiple phases from a single fringe pattern in digital holographic interferometry

Simultaneous measurement of multidimensional displacements using digital holographic interferometry involves multi-directional illumination of the deformed object and requires the reliable estimation of the resulting multiple interference phase distributions. The paper introduces an elegant method to simultaneously estimate the desired multiple phases from a single fringe pattern. The proposed method relies on modeling the reconstructed interference field as a piecewise multicomponent polynomial phase signal. Effectively, in a given region or segment, the reconstructed interference field is represented as the sum of different components i.e. complex signals with polynomial phases. The corresponding polynomial coefficients are estimated using the product high-order ambiguity function. To ensure proper matching of the estimated coefficients with the corresponding components, an amplitude based discrimination criterion is used. The main advantage of the proposed method is direct retrieval of multiple phases without the application of spatial carrier based filtering operations.

A parametric method for non-stationary interference suppression in direct sequence spread-spectrum systems

The problem of non-stationary interference suppression in direct sequence spread-spectrum (DS-SS) systems is considered. The phase of interference is approximated by a polynomial within the considered interval. According to the local polynomial Fourier transform (LPFT) principle, the received signal is dechirped by using the obtained phase approximation and the interference is, in turn, suppressed by excising the corrupted low-pass frequency band. For the estimation of polynomial coefficients, we use the product high-order ambiguity function (PHAF), known for its capability to successfully resolve components of a multicomponent polynomial-phase signal (PPS). The proposed method can suppress interferences with both polynomial and non-polynomial phase. In addition, it can suppress both monocomponent and multicomponent interferences. The simulations show that the proposed method outperforms time-frequency (TF) methods, that successfully deal with multicomponent interferences, in terms of the error probability and computational complexity.

Autofocusing of SAR images based on parameters estimated from the PHAF

The local polynomial Fourier transform (LPFT) based algorithm for auto-focusing SAR images has recently been proposed by the authors. It produces a well focused image of moving targets, without defocusing stationary targets or inducing undesired cross-terms. The drawback of this algorithm is its high computational burden caused by the large number of elements in the set of used chirp-rates. We propose an algorithm with decreased number of elements used for the LPFT-based SAR imaging. The product high-order ambiguity function (PHAF) is applied to estimate parameters of a radar signal. The estimated chirp-rate is used as an initial value for forming the set of chirp-rates. The proposed algorithm has significantly smaller set of chirp-rate values (tens comparing to several hundreds or thousands used in the previous algorithm version). In this manner, the calculation complexity is significantly reduced. The proposed procedure is fully automated, meaning that it follows the change of motion parameters. In addition, our procedure considers the third-order phase compensation.

Extraction of mixed-order multicomponent ship target signals from broadened sea clutter in bistatic shipborne surface wave radar

A dynamic geometry model was built to describe the Doppler-broadening characteristics of the first-order Bragg lines for bistatic shipborne surface wave radar. They are time-varying because the Doppler frequency shifts of sea echoes are simultaneously modulated by the velocity components projected from the unavoidably different motions of both platforms, which is more complex than its counterpart in the monostatic mode. The geometrical relation is used to obtain the received signal model: strong first-order sea clutter interferences and submerged ship targets with constant and non-constant speeds are considered as mixed-order multicomponent polynomial-phase signals (mc-PPSs) and both second-order sea clutter continuum and atmospheric noise as additive noise. Then, a scheme based on the product high-order ambiguity function is proposed to extract the targets; it is a recursive procedure in which the first-order sea clutter is removed by an existing time–space cascaded filtering method. Monte Carlo simulations show the validity and efficiency of the proposed scheme.

Time-Frequency Analysis Using Warped-Based High-Order Phase Modeling

The high-order ambiguity function (HAF) was introduced for the estimation of polynomial-phase signals (PPS) embedded in noise. Since the HAF is a nonlinear operator, it suffers from noise-masking effects and from the appearance of undesired cross-terms when multicomponents PPS are analyzed. In order to improve the performances of the HAF, the multi-lag HAF concept was proposed. Based on this approach, several advanced methods (e.g., product high-order ambiguity function (PHAF)) have been recently proposed. Nevertheless, performances of these new methods are affected by the error propagation effect which drastically limits the order of the polynomial approximation. This phenomenon acts especially when a high-order polynomial modeling is needed: representation of the digital modulation signals or the acoustic transient signals. This effect is caused by the technique used for polynomial order reduction, common for existing approaches: signal multiplication with the complex conjugated exponentials formed with the estimated coefficients. In this paper, we introduce an alternative method to reduce the polynomial order, based on the successive unitary signal transformation, according to each polynomial order. We will prove that this method reduces considerably the effect of error propagation. Namely, with this order reduction method, the estimation error at a given order will depend only on the performances of the estimation method.

Statistical analysis of the product high-order ambiguity function

The high-order ambiguity function (HAF) was introduced for the estimation of polynomial-phase signals (PPS) embedded in noise. Since the HAF is a nonlinear operator, it suffers from noise-masking effects and from the appearance of undesired cross terms and, possibly, spurious harmonics in the presence of multicomponent (mc) signals. The product HAF (PHAF) was then proposed as a way to improve the performance of the HAF in the presence of noise and to solve the ambiguity problem. In this correspondence we derive a statistical analysis of the PHAF in the presence of additive white Gaussian noise (AWGN) valid for high signal-to-noise ratio (SNR) and a finite number of data samples. The analysis is carried out in detail for single-component PPS but the multicomponent case is also discussed. Error propagation phenomena implicit in the recursive structure of the PHAF-based estimator are explicitly taken into account. The analysis is validated by simulation results for both single- and multicomponent PPSs.

Product high-order ambiguity function for multicomponent polynomial-phase signal modeling

Parameter estimation and performance analysis issues are studied for multicomponent polynomial-phase signals (PPSs) embedded in white Gaussian noise. Identifiability issues arising with existing approaches are described first when dealing with multicomponent PPS having the same highest order phase coefficients. This situation is encountered in applications such as synthetic aperture radar imaging or propagation of polynomial phase signals through channels affected by multipath and is thus worthy of a careful analysis. A new approach is proposed based on a transformation called product high-order ambiguity function (PHAF). The use of the PHAF offers a number of advantages with respect to the high-order ambiguity function (HAF). More specifically, it removes the identifiability problem and improves noise rejection capabilities. Performance analysis is carried out using the perturbation method and verified by simulation results.

Autofocusing of SAR images based on the product high-order ambiguity function

An autofocusing technique is proposed, based on the modelling of the instantaneous phase, induced by the relative radar/target motion, as a generic order polynomial and on the estimation of the polynomial coefficients by the so-called product high-order ambiguity function. The proposed estimation method is able to deal with multicomponent signals and is robust against model order mismatching, provided that the selected order is greater than the true order. The proposed method is compared with the phase gradient autofocusing method, and applications are presented of focusing of extended scenes and detection and imaging of moving targets.