Polynomial Phase Signals Research Articles

Three-dimensional displacement measurement from phase signals embedded in a frame in digital holographic interferometry.

A novel technique is proposed for the simultaneous measurement of all three components of displacement from a single recording of the interference field in digital holographic interferometry. The interference field is divided into a number of rectangular segments, and in each of these segments, the interference field is represented as a multicomponent low-order two-dimensional (2D) polynomial phase signal. 2D-product high-order ambiguity function based analysis is applied to obtain an accurate estimation of the 2D polynomial coefficients. These coefficients are further used to compute the interference phases. The simulation and experimental results show the robustness of the proposed method to noise and its effectiveness in multiple phase estimation.

Adaptive Viterbi-Based Range-Instantaneous Doppler Algorithm for ISAR Imaging of Ship Target at Sea

The range-instantaneous Doppler (RID) algorithm has been successfully used in high-resolution inverse synthetic aperture radar (ISAR) imaging of maneuvering ship targets. Parametric technique of RID is an important method to produce the well-focused RID ISAR images by estimating the parameters of received signal modeled as multicomponent polynomial-phase signals in each range bin. However, the cross terms of signals and false maxima of time–frequency distributions cause image blurring and the loss of fine detail. In this paper, we first propose a new time–frequency transform, named multiproduct high-order matched-phase transform (MPHMT), to suppress the cross terms and amplify the auto terms of signals. Then, an adaptive Viterbi search is proposed to select the optimal model parameters by an adaptive penalization function. Finally, associating MPHMT and adaptive Viterbi search with RID, we further propose the adaptive Viterbi-based RID algorithm for ISAR ship imaging. Simulation and real-data-based results demonstrate that our proposed algorithm is valid.

Parameter Estimation of Polynomial Phase Signal by Unscented Kalman Filtering

The parameter estimation of the Polynomial Phase Signals (PPS) is one of the core issues. In this paper, UKF-based algorithm is proposed to estimate the parameter of PPS embedded in Gaussian noise. The algorithm constructs an adequate state-space model to represent the PPS and the model can also be implied in real radar signal. Unscented Kalman filtering is applied to estimate the signal parameters. The method achieves the lower SNR threshold, the faster convergence speed, the higher accuracy and more stable estimation performance compared with the existing methods. Simulation also verifies the efficiency of the proposed method.

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.

A New Maneuvering Target Detection Algorithm in OTHR

In the three-order polynomial phase signal, HAF can be used to reduce phase order to achieve the phase parameter estimation. The loss of signal to noise ratio in the order reduction processing is serious. In order to reduce the SNR loss, this paper uses CPF to reduce phase order. It requiers two one-dimensional searches to estimate the rate of acceleration and acceleration. The estimated motion parameters are used to compensate the target echo doppler and doppler spectrum expansion can be suppressed. This paper proposes a new maneuvering target detection algorithm with low non-linearity. The algorithm makes use of the cubic phase function (CPF) to estimate target motion parameters and reduce the number of phase order. Simulation demonstrates the effectiveness of the proposed algorithm and the adaptability to low signal noise ratio.

Parameter estimation of non-uniform sampled polynomial-phase signals using the HOCPF-WD

Parameter estimation of non-uniformly sampled polynomial-phase signals (PPSs) is addressed in this paper. It is assumed that signals are sampled below the rate required by the sampling theorem. The recent advance in the field, based on combining the high-order cubic phase function (HO-CPF) and high-order Wigner distribution (HO-WD), has been applied for parameter estimation of such signals. Two cases of the sampling scheme are considered: (i) random sampling with symmetrically distributed samples around the middle point in the time interval and (ii) random sampling without the symmetry. A performance analysis is conducted showing high estimation accuracy even for large percentage of missing samples with the later case giving, as expected, slightly worse results than the former one.

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.

Modified Cramér-Rao Bounds for Continuous-Phase Modulated Signals

We consider the estimation of signal amplitude, time offset, and the parameters of a polynomial-phase signal from a continuous-phase modulated (CPM) signal that contains both unknown data symbols and known pilot symbols. Transformation of the polynomial-phase signal into an orthogonal basis allows us to derive the modified Cramér-Rao bounds (MCRB) for the problem of vector-parameter estimation in closed form. Numerical results demonstrate that the achievable estimation accuracy significantly depends on the burst structure and highlight the need for properly designed pilot sequences. Since our bounds are easy to evaluate, they can aid the design of pilot and burst configuration for CPM systems.

Polynomial Phase Estimation by Least Squares Phase Unwrapping

Estimating the coefficients of a noisy polynomial phase signal is important in fields including radar, biology and radio communications. One approach attempts to perform polynomial regression on the phase of the signal. This is complicated by the fact that the phase is wrapped modulo 2π and must be unwrapped before regression can be performed. In this paper, we consider an estimator that performs phase unwrapping in a least squares manner. We call this the least squares unwrapping (LSU) estimator. The LSU estimator can be computed in a reasonable amount of time for data sets of moderate size using existing general purpose algorithms from algebraic number theory. Under mild conditions on the distribution of the noise we describe the asymptotic properties of this estimator, showing that it is strongly consistent and asymptotically normally distributed. A key feature is that the LSU estimator is accurate over a far wider range of parameters than many popular existing estimators. Monte-Carlo simulations support our theoretical results and demonstrate the excellent statistical performance of the LSU estimator when compared with existing state-of-the-art estimators.

Non-uniform sampled cubic phase function

Parameter estimation of polynomial phase signals (PPSs) based on the cubic phase function (CPF) and its extensions cannot be performed by using the fast Fourier transform (FT) algorithm. Therefore, in order to express the CPF by means of the FT, in this paper we propose a scheme for the CPF evaluation based on non-uniform sampling. Calculation complexity of the estimation procedure is significantly reduced, whereas the accuracy is the same or better compared to the original algorithm.

Quasi maximum likelihood estimator of polynomial phase signals for compressed sensed data

Several papers in the literature cover parameter estimation of frequency modulated (FM) signals under reduced number of signal samples with respect to the Nyquist/Shannon criterion, i.e., within the compressive sensing (CS) framework. However, scope of these papers is mainly limited to sinusoids or sum of sinusoids. In this paper, the CS framework is extended to parameter estimation of higher order polynomial phase signals (PPSs) using the quasi-maximum likelihood (QML) estimator and robust short-time Fourier transform (STFT). The considered signal is assumed to be non-uniformly sampled PPS with smaller number of samples with respect to the Nyquist/Shannon criterion. However, the proposed technique can also be generalized to uniformly sampled signals with missing or unreliable samples.

Support vector regression ‐based robust frequency estimation algorithm by instantaneous phase

There are two important factors which impact the performance of phase-based frequency estimation algorithms remarkably: the approximations of noise phase model and imperfections in phase unwrapping process. Support vector regression (SVR) exits excellent capabilities for learning unknown data and forecasting the future ones, especially under the small sample condition. Therefore, the authors introduce it to predict the variation trend of instantaneous phase and unwrap phases efficiently. Even though with that being the case, errors still exist in phase unwrapping process because of its ambiguous phase characteristic. Furthermore, a SVR-based frequency estimation algorithm is proposed and makes it immune to these error phases by means of setting the SVR's parameters properly. The results show that, compared with other phased-based algorithms, not only does the proposed one maintain a wide estimation range and quality capabilities at low frequencies, but also improves accuracy at high frequencies and decreases the impact with the initial phase. The proposed algorithm is fit for not only linear phase signal but also polynomial phase signal, under both the Gaussian and non-Gaussian condition.

A recursive method for compensating ionospheric phase contamination based on multistage Taylor expansion

A recursive approach for the compensation of the ionospheric phase contamination is proposed based on multistage Taylor expansion and the extended Kalman filter. The proposed algorithm can dynamically compensate the ionospheric phase contamination. We show by numerical examples that the proposed algorithm outperformances the conventional methods such as the polynomial phase signal (PPS) and the singular value decomposition (SVD).

ISAR Imaging of Non-Uniformly Rotating Target via Range-Instantaneous-Doppler-Derivatives Algorithm

Inverse synthetic aperture radar (ISAR) imaging of non-uniformly rotating target is very significant in many applications. In this case, the Doppler frequency for a scatterer is time-varying, and the traditional Range-Doppler (RD) algorithm is not appropriate for the azimuth focusing. In this paper, the received signal in a range bin is assumed to be multi-component polynomial phase signal (PPS), and a novel approach for ISAR imaging of non-uniformly rotating target via Range-Instantaneous-Doppler-Derivatives (RIDD) algorithm is proposed. For the RIDD algorithm, the cross-range resolution is derived by the arbitrary order of Doppler derivatives, and it is the generalization and extension form of the traditional Range-Instantaneous-Doppler (RID) and Range-Instantaneous-Chirp-Rate (RIC) algorithms. Results of simulated and raw data demonstrate the effectiveness of the new method proposed.

Combined HO-CPF and HO-WD PPS estimator

Recently, the high-order cubic phase function (HO-CPF) and high-order Wigner distribution (HO-WD) have been proposed for parameter estimation of polynomial-phase signals (PPSs). To estimate PPS parameters, both the HO-CPF and HO-WD are evaluated at two points. One point is usually the center of the considered time interval, whereas the other one is shifted from the center. Shifting shortens the signal sequence and reduces the performance of the considered technique. In this paper, we propose an estimation procedure that combines the HO-CPF and HO-WD. The procedure evaluates both functions at the origin only, implying no signal shortening. Simulation results and statistical study have shown a significant performance improvement over the HO-CPF- and the HO-WD- based estimators.

An Improved Algorithm for Estimating the Parameters of Polynomial-Phase Signals

This paper presents an algorithm for estimating the parameters of polynomial-phase signals (PPSs). This algorithm combines the CPF-HAF method and Radon transform, and can be referred to as the Radon-CPF-HAF method. In the proposed algorithm, the HAF is first applied on the original PPS to produce a cubic phase signal, whose parameters are then estimated by the Radon transform of the cubic phase function for the derived cubic phase signal. As involving in lower order nonlinearity, the proposed algorithm outperforms the HAF in terms of the accuracy and signal-to-noise-ratio (SNR) threshold. When multicomponent PPSs are considered, the product version of the proposed algorithm removes the identifiability problem which the product version of the CPF-HAF (PCPF-HAF) suffers from. Computer simulations have been carried out to support the theoretical results.

On the Cramér–Rao bound for polynomial phase signals

Polynomial-phase signals have applications including radar, sonar, biology, and radio communication. Of practical importance is the estimation of the parameters of a polynomial phase signal from a sequence of noisy observations. Assuming that the noise is additive and Gaussian, the direct evaluation of the Cramér–Rao lower bound for this estimation problem involves evaluating the inverse of a matrix. Computing this inverse is numerically difficult for polynomial phase signals of large order. By making use of a family of orthogonal polynomials, we derive formulae for the Cramér–Rao bounds that are numerically stable and easy to compute.

A novel refinement technique for 2-D PPS parameter estimation

This paper considers refinement in parameter estimation of 2-D polynomial phase signals (PPSs). For this purpose, a method based on phase unwrapping, filtering and polynomial fitting is developed. In the method, 2-D parameter estimation is reduced to 1-D refinement procedure, thus significantly gaining in calculation efficiency. As initial estimates, the proposed method takes the estimates provided by any 2-D parameter estimator. In this paper, we use initial estimates provided by the standard Francos–Friedlander algorithm. Numerical simulations performed on third-order 2-D PPSs show that mean squared error of the proposed estimator reaches the Cramér–Rao lower bound above the signal-to-noise ratio threshold.

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.

An Algorithm for the Fine Estimation of Polynomial-Phase Signals

To precisely estimate the parameters of polynomial-phase signals (PPSs), we adopt a recently-proposed algorithm for sinusoid frequency estimation. The algorithm is combined with the Wigner distribution to estimate the instantaneous frequency of linear FM signals and with the high-order ambiguity function to estimate parameters of higher order PPSs. Both Gaussian and non-Gaussian noises are considered in simulations, which show that the algorithm converges within two iterations and approach achievable limits for the considered techniques.