Polynomial Wigner-Ville Distribution Research Articles

Motion Parameters Estimation of a Moving Target Based on Instantaneous Frequency Estimates Using a Passive Acoustic Sensor

This paper studies a method for estimating the motion parameters [Formula: see text] using the instantaneous frequency estimates of the acoustical signal emitted from a moving target in the air and received by an acoustic sensor placed on the ground. Here, [Formula: see text] is the target’s travel speed, [Formula: see text] is the closest point of approach slant range, [Formula: see text] is the source acoustical frequency, and [Formula: see text] is the time at the target get to the closest point of approach to the sensor. A novel relationship formula between the instantaneous frequency and motion parameters is derived and an algorithm for estimating the motion parameters based on the solution of quadratic nonlinear optimization from the instantaneous frequency estimates is proposed. The instantaneous frequency is estimated by searching the peak of a polynomial Wigner–Ville distribution. An approach for calculating the optimal window width in a polynomial Wigner–Ville distribution is proposed. Numerical examples are provided for validating our theoretic claims.

Using Polynomial Wigner–Ville Distribution for Velocity Estimation in Remote Toll Applications

This letter presents the use of polynomial Wigner-Ville distribution (PWVD) for accurate velocity estimation as used in remote road traffic management applications. In such applications based on inverse synthetic aperture radar, velocity estimation is central to obtain a suitable level of performance of the signal processing. Moreover, the precision of this velocity estimation is crucial in order to achieve the best detection and estimation of the gauge of the vehicles through the use of radar images. Hence, the PWVD is applied as an instantaneous frequency estimator used in this traffic surveillance application.

ISAR Imaging of Maneuvering Target Based on the L-Class of Fourth-Order Complex-Lag PWVD

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In inverse synthetic aperture radar (ISAR) imaging of a maneuvering target, the received signal in a range bin can be characterized as a multicomponent polynomial phase signal (PPS) after motion compensation. The traditional discrete Fourier transform algorithm is not appropriate in analyzing the PPS. In this paper, the L-class of a fourth-order complex-lag polynomial Wigner–Ville distribution (PWVD) is presented to generate a high-resolution time–frequency distribution (TFD) for the multicomponent PPS. For a signal with polynomial phase up to order four (this is accurate enough in ISAR imaging), the cross-terms between different components can be reduced by the convolution in the frequency domain of the L-class of the fourth-order complex-lag PWVD. The new TFD is used in the ISAR imaging of the maneuvering target, and high-quality instantaneous ISAR images are obtained. Results of simulated and real data demonstrate the effectiveness of the method above. </para>

Fourth-Order Complex-Lag PWVD for Multicomponent Signals with Application in ISAR Imaging of Maneuvering Targets

The fourth-order complex-lag polynomial Wigner–Ville distribution (PWVD) is extended to generate a high resolution time–frequency distribution for multicomponent signals in this paper. For signals with polynomial phase up to order four, the interferences between different components are reduced by the convolution in the frequency domain of the complex-lag PWVD. The complex-lag PWVD can achieve optimal energy concentration, and it is used in the inverse synthetic aperture radar (ISAR) imaging of maneuvering targets, where high quality instantaneous ISAR images are obtained. Simulated results demonstrate the effectiveness of the method.

New time–frequency distribution based on the polynomial Wigner–Ville distribution and L class of Wigner–Ville distribution

A new kind of time-frequency distribution based on the polynomial Wigner-Ville distribution (PWVD) and L class of Wigner-Ville distribution (LWVD) is presented in this study, and it is named as LPWVD. The LPWVD can be implemented based on the convolution of LWVD in frequency domain. Compared with the PWVD, the LPWVD is cross-term free for multi-component signals, and the time-frequency convergence is higher than the PWVD. The asymptotic statistical performance of LPWVD for estimating the instantaneous frequency (IF) of a polynomial phase signal (PPS) is studied. The simulated results demonstrate the validity of the LPWVD.

Polynomial Wigner–Ville distribution-based method for direct phase derivative estimation from optical fringes

This paper proposes a polynomial Wigner–Ville distribution-based method to directly estimate phase derivative from a single fringe pattern. In the proposed method, we evaluate the polynomial Wigner–Ville distribution along each row/column of the given fringe pattern. The peak of the polynomial Wigner–Ville distribution is used as the phase derivative estimator. To improve the robustness of the distribution against the artifacts or interference terms, a windowed form of the polynomial Wigner–Ville distribution is used. Simulation and experimental results are presented which validate the method's applicability for direct phase derivative estimation.

Generalized High-Order Phase Function for Parameter Estimation of Polynomial Phase Signal

The high-order phase function (HPF) has been introduced recently to estimate the parameters of a polynomial phase signal (PPS). In this correspondence, we generalize the standard HPF by introducing multiple time instants. Thus, the standard HPF can be treated as a special example of the generalized HPF with identical time instants. We propose a procedure for finding time instants minimizing the mean-square error (MSE). The proposed method achieves better performances than the high-order ambiguity function (HAF) and polynomial Wigner-Ville distribution (PWVD). The theoretical analysis as well as the Monte Carlo simulations verify the advantages such as lower MSE and lower SNR threshold for the PPS.

Effect of interpolation on PWVD computation and instantaneous frequency estimation

The peak of the polynomial Wigner–Ville distribution (PWVD) can be used for estimating the instantaneous frequency (IF) of monocomponent polynomial phase signals. However, the PWVD kernel, optimized to yield a time–frequency distribution (TFD) localized along the IF, comprises of fractional-time-sampled signals. When implemented in a discrete-time scenario, this calls for signal interpolation. We study three interpolation schemes—linear, cubic polynomial and sinc and derive expressions for the variance of the interpolated samples in the presence of noise. In representing nonstationary signals using the PWVD, the instantaneous energy content of noise auto-terms and signal-noise cross-terms is found to be the least for linear interpolation scheme. For polynomial IF estimation using the peak of the PWVD, it was found that linear interpolation is a computationally efficient way of obtaining reasonably good estimates at low signal-to-noise ratios (SNRs). For high SNRs, sinc interpolation outperforms the other two schemes. Similar results were found when the experiment was extended to sinusoidal IF signals also.

Analysis of polynomial FM signals corrupted by heavy-tailed noise

In this paper, we consider the analysis of polynomial FM signals corrupted by additive heavy-tailed noise. Standard time–frequency techniques fail to analyze such signals. For that, we propose here a new technique, named the robust polynomial Wigner-Ville distribution (r-PWVD) to handle this case. We show that this representation outperforms the robust Wigner-Ville distribution (r-WVD) and the robust spectrogram in terms of artifacts suppression and high time–frequency resolution for this class of signals. Also, we show that the peak of the r-PWVD is an accurate instantaneous frequency estimator. Examples and Monte-Carlo simulations are presented in order to validate and prove the performance of the proposed algorithm.

An architecture for the vlsi design of systems for time-frequency analysis and time-varying filtering

A flexible system for time-frequency signal analysis is presented. It is based on the S-method, which has a significant advantage in implementation since it can involve, as a key intermediate step, the Short-time Fourier transform or the Hartley transform, each widely studied and commonly used in practice. Signal invariant and signal dependent system forms are presented. Hardware design, for a fixed-point arithmetic, is well-structured and suitable for vlsi implementation. The same hardware, without additional time requirements, may be shared for the simultaneous realization of the fourth order L-Wigner distribution, as well as for the realization of the cross-terms free fourth order polynomial Wigner-Ville distribution. This possibility makes the designed hardware suitable for wide range of the applications. The proposed hardware is applied to the realization of time-varying filtering, as well. Finally, it has been implemented with fpga chips (Field Programmable Gate Array) in order to verify the results on real devices.

Instantaneous frequency estimation of nonlinear frequency-modulated signals in the presence of multiplicative and additive noise

This paper addresses the problem of estimating the instantaneous frequency (IF) of monocomponent nonlinear, not necessarily polynomial, frequency modulated (FM) signals affected by stationary multiplicative and additive noise. Both noise processes are assumed to be complex circular Gaussian and independent. The peak of the polynomial Wigner-Ville distribution (PWVD) is proposed here as an IF estimator. We derive analytical expressions for the bias and asymptotic variance of the estimator and propose an algorithm to select the optimal window length to resolve the bias-variance tradeoff in the IF estimation. Simulation results are presented to confirm the theoretical results.

Polynomial phase signal analysis based on the polynomial derivatives decompositions

An exact decomposition of the derivatives of any order of a polynomial /spl phi/(t) is proposed in terms of /spl phi/(t-t/sub 0/), ..., /spl phi/(t-t/sub n/). This result allows us to introduce generalized time-frequency distributions for studying signals having a polynomial phase and a constant amplitude in order to determine the degree and the coefficients of the corresponding phase. The relationships between these distributions and the already known polynomial distributions, i.e., the polynomial phase transform and the polynomial Wigner-Ville distribution, are discussed. Illustrations by example are proposed.

Instantaneous frequency estimation of polynomial FM signals using the peak of the PWVD: statistical performance in the presence of additive gaussian noise

The peak of the polynomial Wigner-Ville distribution (PWVD) has been previously proposed as an estimator of the instantaneous frequency (IF) for a monocomponent polynomial frequency modulated (FM) signal. In this paper, we evaluate the statistical performance of this estimator in the case of additive white Gaussian noise and provide an analytical expression for the variance. We show that for a given PWVD order, the estimator performance can be improved by a proper choice of the kernel coefficients in the PWVD. A performance comparison between the PWVD based IF estimator and another previously proposed one based on the high-order ambiguity function (HAF) is also provided, Simulation results show that for a signal-to-noise ratio larger than 3 dB, the proposed sixth-order PWVD outperforms the HAF in estimating the IF of a third- or fourth-order polynomial phase signal, evaluated at the central point of the observation interval.

Design of higher order polynomial Wigner-Ville distributions

The polynomial Wigner-Ville distribution (PWVD) is a time-frequency signal analysis tool for representing polynomial frequency modulated (FM) signals. We present an optimization procedure to design PWVDs and discuss the need for higher order PWVDs.

On the realization of the polynomial Wigner-Ville distribution for multicomponent signals

A method for the polynomial Wigner-Ville distributions realization, in the case of multicomponent signals, is presented. Using this method, one may theoretically get a sum of the polynomial Wigner-Ville distributions of each component separately. The method is illustrated by a numerical example.

Aircraft flight parameter estimation based on passive acoustic techniques using the polynomial Wigner–Ville distribution

The acoustic signal from an overflying aircraft, as heard by a stationary observer, is used to estimate an aircraft’s constant height, ground speed, range, and acoustic frequency. Central to the success of this flight parameter estimation scheme is the need for an accurate estimate of the instantaneous frequency of the observed acoustic signal. In this paper, the polynomial Wigner–Ville distribution is used in this application as the instantaneous frequency estimator. Its performance and the issue of the optimal time domain window length are addressed.

Instantaneous frequency estimation of quadratic and cubic FM signals using the cross polynomial Wigner-Ville distribution

The cross polynomial Wigner-Ville distribution (X-PWVD) is applied to instantaneous frequency (IF) estimation of polynomial FM signals. The X-PWVD is used in a simple iterative algorithm, similar to the one proposed by Boashash and O'Shea (see ibid., vol.41, no.3, p.1439, 1993) for the cross Wigner-Ville distribution (X-WVD). The advantage of the X-PWVD over the polynomial Wigner-Ville distribution is that the former is linear with respect to the observed signal and as a result it attains the Cramer-Rao bound at lower signal-to-noise ratios (SNRs).

Relationship between the polynomial and the higher order Wigner-Ville distribution

The paper establishes the relationship between the two methods of higher order time-frequency analysis: the polynomial Wigner-Ville distribution (WVD) and the higher order WVD. Using the projection-slice theorem, it is shown that the polynomial WVD represents a unique projection of the higher-order WVD from the time-multifrequency space to the time-frequency subspace. The implication of this relationship is investigated from the aspect of the analysis of multicomponent signals.