Estimation Of Polynomial-phase Signals Research Articles

A Novel Parameter Estimation for Polynomial Phase Signals Using the Spectrum Phase

A novel approach for the parameter estimation of polynomial phase signals (PPSs) is proposed. Unlike the existing methods, the proposed method uses the spectrum phase (SP) rather than the traditional instantaneous phase (IP) for parameter estimation. First, we present an unwrapping procedure for the SP. Second, we derive a method that can convert the unwrapped SP into the IP and then perform polynomial regression on the IP to obtain the estimates of the parameters. Finally, we use the O'Shea refinement strategy to refine the estimates to achieve the Cramér-Rao lower bound. The simulation results show that the proposed technique has a lower signal-to-noise ratio threshold and is less complex than the existing methods for high-order PPSs.

A method for efficient maximization of PPS estimation functions

Parameter estimation of polynomial phase signals is usually performed by maximizing the cost function. To reduce the calculation complexity, the maximization consists of coarse and fine estimation stages. To refine coarse estimate, in this paper, we propose a method based on filtering and phase unwrapping of transform function. Simulation experiments have shown excellent results compared to classical fine estimation algorithms such as Aboutanios–Mulgrew and dichotomous search.

Estimation of Higher Order Polynomial Phase Signals in an Impulsive Noise

Parametric estimation of polynomial phase signals (PPSs) corrupted by an impulsive noise has been considered. The interpolation in joint-variable domain has been combined with the hybrid cubic phase function—high-order ambiguity function-based estimator to address this issue. Technique for estimating the percentage of impulses and threshold selection is also proposed. In addition, we propose the refinement procedure to further decrease the mean squared error of PPS parameters estimates.

Compound time‐frequency domain method for estimating parameters of uniform‐sampling polynomial‐phase signals on the entire identifiable region

Parameter estimation of polynomial-phase signals (PPSs) observed in additive white Gaussian noise (AWGN) is addressed. Most of the existing estimators cannot work on a fully identifiable region. Using the algebraic number theory, McKilliam et al. proposed a least squares unwrapping (LSU) estimator, which can operate on the entire identifiable region. However, its computational load may be large, especially when the number of samples is large. In this study, the authors first extend the amplitude-weighted phase-based estimator (AWPE) for sinusoidal and chirp signals to PPSs and derive a time domain maximum likelihood estimator. The performance is analysed and compared with the Cramer–Rao lower bound (CRLB). Then, the authors propose an iterative compound time-frequency domain parameter estimation method, which includes a coarse estimation step and a fine estimation step conducted by the discrete polynomial phase transform and AWPE estimator, respectively. Monte–Carlo simulations show that the proposed method can work on the entire identifiable region and that it outperforms the existing state-of-the-art estimators. Its computational complexity is considerably lower than that of the LSU estimator, while its threshold signal-to-noise ratio is a few decibels higher than that of the LSU estimator.

Resolving aliasing effect in the QML estimation of PPSs

The quasimaximum likelihood (QML) estimator of polynomial-phase signals (PPSs) is based on the maximization of the short-time Fourier transform and suffers from aliasing when signals are sampled below the Nyquist sampling rate. In this paper, a phase unwrapping procedure has been proposed as an additional step in the QML to estimate parameters of such signals. Statistical study has shown excellent performance of the proposed approach.

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.

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 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.

Improving Polynomial Phase Parameter Estimation by Using Nonuniformly Spaced Signal Sample Methods

This paper investigates the computationally efficient parameter estimation of polynomial phase signals embedded in noise. Many authors have previously proposed multilinear analysis methods which operate on uniformly spaced samples of the signal. Such methods include the higher-order ambiguity functions (HAFs), the Polynomial Wigner-Ville distributions (PWVDs) and the higher-order phase (HP) functions. This paper investigates the use of multilinear methods which operate on nonuniformly spaced signal samples. It is seen that the relaxation of the requirement to use uniformly spaced samples in the analysis can lead to significant performance improvements. A theoretical analysis and simulations are presented in support of these claims.

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).

New method for parameter estimation of polynomial phase signals

In this correspondence, a linear transform called polynomial to Hermite polynomial transform (PHPT) is proposed, and the PHPT-based method for parameter estimation of polynomial phase signals (PPS) embedded in additive white Gaussian noise is addressed. The first advantage is the low computational complexity. If the number of samples is N, the computational complexity is O(3N log2N) regardless of the order or the component number of PPS. The second advantage is that there is no non-linear operation in PHPT algorithm; therefore there is no cross-term interference in between the various components. In addition, the proposed method can well process the PPS with unknown order phase. Simulations illustrate that the proposed algorithm is more effective as compared with the existing ones.

Estimation of FM signal parameters in impulse noise environments

A simple algorithm based on discrete chirp Fourier transform (DCFT) is used for the chirp signal parameters estimation in impulse noise environments. A modification of the DCFT is introduced in order to produce accurate estimates in this case. This modification, called the robust DCFT, produces highly accurate results in impulse noise environments, while for the Gaussian noise it is only slightly worse than the standard one. Generalization to the parametric estimation of polynomial phase signals is given. It is based on the robust form of the integrated generalized ambiguity function (IGAF). We applied the IGAF-based procedure on the signal filtered by using the robust filter designed in the frequency domain.

On the performance of cyclic moments-based parameter estimators of amplitude modulated polynomial phase signals

Parameter estimation of amplitude-modulated polynomial phase signals embedded in additive white Gaussian noise is considered. The amplitude modulation is modeled as the sum of a real-valued deterministic function and a zero mean correlated stationary random process. It is shown that cyclic moments-based estimators, previously proposed for parameter estimation of polynomial phase signals modulated by stationary random processes, can be adapted to the more general signal model considered here. The covariance matrices of the cyclic moments-based amplitude and phase parameter estimators are derived for large sample lengths. Using this result, it is shown how the lags can be chosen to minimize the large-sample variances of the cyclic moments-based phase parameter estimators. Comparisons with the Cramer-Rao bounds are performed under the assumption of a Gaussian modulating process. The theoretical derivations are confirmed by simulation results.

Statistical analysis of polynomial phase signals affected by multiplicative and additive noise

In this paper we are interested in the problem of the estimation of polynomial phase signals affected by a multiplicative noise and an additive noise using the high-order ambiguity function. We distinguish for the multiplicative noise the non-zero mean case and the zero mean case. We analyse the performances of the proposed estimators and compare their respective variances with the corresponding Cramer–Rao bounds. The theoretical results are illustrated by computer simulations.

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.