Instantaneous frequency estimation using the Wigner distribution with varying and data-driven window length

Chapter 10 - Instantaneous Frequency Estimation and Localization

The Wigner distribution (WD) produces highly concentrated time-frequency (TF) representation of nonstationary signals. It may be used as an efficient signal analysis tool, including the cases of frequency modulated signals corrupted with the Gaussian noise. In some applications, a significant amount of impulse noise is present. Then, the WD fails to produce satisfactory results. The robust periodogram has been introduced for spectral estimation of this kind of noisy signals. It can produce good concentration for pure harmonic signals. However, it is not so efficient in the cases of signals with rapidly varying frequency. This is the motivation for introducing the robust WD. It is a reliable TF representation tool for wide class of nonstationary signals corrupted with impulse noise. This distribution produces good accuracy of the instantaneous frequency (IF) estimation. Using the Huber (1981) loss function, a generalization of the WD is presented. It includes both the standard and the robust WD as special cases. This distribution can be used for TF analysis of signals corrupted with a mixture of impulse and Gaussian noise. The presented theory is illustrated on examples, including applications on the IF estimation and time-varying filtering of signals corrupted with a mixture of the Gaussian and impulse noise. The case study analysis of the IF estimators' accuracy, based on the standard and the robust WD forms, is performed. In order to improve the IF estimation, a median filter is applied on the obtained IF estimate.

Estimation of the instantaneous frequency (IF) in a high noise environment, by using the Wigner distribution (WD) and the Viterbi algorithm, is considered. The proposed algorithm combines nonparametric IF estimation based on the WD maxima with minimization of the IF variations between consecutive points. Algorithm realization is performed recursively using the (modified) Viterbi algorithm. Performances are compared with IF estimation based on the WD maxima.

The use of linear time-frequency (TF) representa-tions as instantaneous frequency (IF) estimators arises in various fundamental disciplines, mainly thanks to their simplicity and immunity against interfering cross-terms. In this paper, we derive the optimal window width for IF estimation of noisy multicomponent signals based on a family of linear TF transforms that use Gaussian windowing functions and the Fourier oscillatory kernel. Closed-form formulas concerning the estimation bias and the variance are presented, which, thanks to their generality, describe the statistical performance of IF estimators based on transforms with fixed, time-adaptive, frequency-adaptive, or time-frequency adaptive windows. The optimal window is dependent on the unknown first derivative of the IF; therefore, we employ a low-complexity procedure to infer the derivative and optimize the width accordingly. Two adaptive and fully automated TF representations (TFRs) are developed; the first employs a time-adaptive window that minimizes the sum of the mean-squared errors (MSEs) of the IF estimates at each time instant, while in the second TFR, the window is adaptive over time and frequency, minimizing the estimation MSE at each location in the TF domain. Examples using synthetic and real-world signals demonstrate that the proposed algorithms may outperform many popular state-of-the-art techniques, including those that are signal-adaptive, in terms of IF estimation.

The paper presents a method for the estimation of instantaneous frequency (IF) of signals characterized by interfering IF trajectories. It would be a useful methodology for maintenance and troubleshooting of wireless networks sharing the same operating frequency interval or predictive maintenance of vibrating/rotating systems. Stemming from IF trajectories estimated through traditional or enhanced time-frequency representations, the method applies an agile digital signal processing approach in order to suitably merge different sections belonging to the same trajectory. Preliminary results show the efficacy of the method when signals characterized by multiple intersections in the time-frequency domain are experienced.

A nonparametric algorithm for estimation of instantaneous frequency (IF) and direction-of-arrival (DOA) of wideband FM signals, at the uniform linear array (ULA), is considered. The IF estimation of signals received at the sensor array is performed by using positions of the Wigner distribution (WD) maxima. Information of the impinging signal's IF is embedded in the IF estimates. It is extracted by linear interpolation of the obtained IF estimates. The L/sub 1/ norm is used for interpolation, instead of the standard L/sub 2/ norm, in order to improve performance in a high noise environment. Further improvement is achieved by applying a nonlinear filter, directly to the obtained IF estimates. The DOA is estimated by using a modified beamforming technique.

We address the problem of estimating instantaneous frequency (IF) of a real-valued constant amplitude time-varying sinusoid. Estimation of polynomial IF is formulated using the zero-crossings of the signal. We propose an algorithm to estimate nonpolynomial IF by local approximation using a low-order polynomial, over a short segment of the signal. This involves the choice of window length to minimize the mean square error (MSE). The optimal window length found by directly minimizing the MSE is a function of the higher-order derivatives of the IF which are not available a priori. However, an optimum solution is formulated using an adaptive window technique based on the concept of intersection of confidence intervals. The adaptive algorithm enables minimum MSE-IF (MMSE-IF) estimation without requiring a priori information about the IF. Simulation results show that the adaptive window zero-crossing-based IF estimation method is superior to fixed window methods and is also better than adaptive spectrogram and adaptive Wigner-Ville distribution (WVD)-based IF estimators for different signal-to-noise ratio (SNR).

An adaptive Generalized S-transform for instantaneous frequency estimation

Synchrosqueezing transform (SST) benefits from an instantaneous frequency (IF) estimator in the time-frequency domain, providing an energy-concentrated time-frequency representation to describe the time-varying frequency of seismic signals. To enhance the concentration performance of SST, this paper theoretically proposes a statistical SST (SSST) by constructing a spectrum-weighted IF estimator in the short-time Fourier transform (STFT) domain. In this SSST, a linear chirp signal is introduced to better capture its chirp rate, and then the quadratic STFT spectra are weighted by a window-related weighting function. On this basis, two equations related to the IF and chirp rate are constructed to derive the spectrum-weighted IF estimator. Finally, the STFT coefficients are squeezed to the estimated IF trajectories by a frequency fixed-point iterative algorithm, thereby providing a more concentrated time-frequency representation for multi-component signals than existing advanced methods, while enabling to retrieve each component. Two synthetic examples and one field seismic data on thin interbeds are utilized to demonstrate the effectiveness of the proposed SSST and show its ability to highlight the time-varying frequency features of seismic signals, which is a promising seismic data analysis tool, such as characterizing thin interbed thickness variations.

Instantaneous frequency (IF) estimation of multi-component signals with closely spaced and intersecting signal components of varying amplitudes is a challenging task. This paper presents a novel iterative time–frequency (TF) filtering approach to address this problem. The proposed algorithm first adopts a high-resolution time–frequency distribution to resolve close components in the TF domain. Then, IF of the strongest signal component is estimated by a new peak detection and tracking algorithm that takes into account both the amplitude and the direction of peaks in the TF domain. The estimated IF is used to remove the strongest component from the mixture, and this process is repeated till the IFs of all signal components are estimated. Experimental results show the superiority of the proposed method as compared to other state-of-the-art methods.

In this paper, a new concept for ultrasonic indoor positioning based on instantaneous frequency of ultrasonic signals is presented. Nonlinear phase characteristics of ultrasonic transducers introduce a frequency deviation in ultrasonic signals. By sweeping at very fast rates, a large spike in the deviation is introduced. The artefacts observable in instantaneous frequency estimations are highly localized and present an opportunity for accurate frequency detection. In order to be useful, the artefacts need to take place within the pulse and have sufficient magnitude for accurate processing. The system consists of a transducer transmitter and receiver pair, which have a center frequency of 40kHz and a bandwidth of 460Hz. In order to incorporate more transmitters, a time-division multiple access (TDMA) scheme is applied to ensure orthogonality of signals. The concept includes four ultrasonic transmitters and a single receiver, which can uniquely identify each transmitter by a distinct signal sweep. Linear chirp signals are used to form narrow pulses and ensure no interference in the TDMA scheme. The received signal is amplified and passed through a phase-locked loop (PLL) to detect the chirp signals. Accurate instantaneous frequency detection can be done on the voltage-controlled oscillator (VCO) of the PLL, which has a narrower bandwidth than the overall signal sweep. The instantaneous frequency estimation methods are largely explored in this work and consider two methods: the Hilbert transform and a zero-crossings method. This work highlights some of the advantages and disadvantages of both methods. Time of flight (ToF) in this system can ultimately be obtained by considering the instantaneous frequency estimations and the time for one particular frequency to be transmitted and received.

Objectives: Identification of cow lameness is important to farmers to improve and manage cattle health and welfare. No validated tools exist for automatic lameness detection. In this research, we aim to early detect the cow lameness by identifying the instantaneous fundamental gait harmonics from low frequency (16Hz) acceleration signals recorded using leg-worn sensors. Methods: A triaxial accelerometer has been worn on each cow leg. Synchrosqueezed wavelet transform (SSWT) has been applied to acceleration signals to generate the initial time-frequency spectrum related to the gait. This spectrum is given as an input to a designed deep neural network including time-frequency based long short-term memory (LSTM) to estimate instantaneous frequencies at each time point. An inverse SSWT (ISSWT) is then used to recover the gait harmonic and to estimate an enhanced spectrum. Results: Validation of instantaneous frequencies has been provided for each cow leg (combined signals from 23 cows) and the time-series cross validator across the three folds are provided. The average of mean squared errors in frequencies across 3 folds for each leg is obtained as 0.036, 0.033, 0.044 and 0.042 for left-front, right-front, right-back and left-back legs, respectively. Conclusion: Estimation of instantaneous gait frequencies is proved useful for identification of cow gait phases, lameness detection, accurate estimation of gait speed, coherency in movement among the legs and identification of non-gait episodes. Moreover, the proposed method can be used as a new frequency ridge estimation method exploiting SSWT for many other applications.

In this work, we derive general formulae for the instantaneous frequency (IF) and instantaneous bandwidth (IB) according to matching pursuit (MP) signal decomposition, irrespective of the kind of dictionary used. We show that via MP decomposition, the IF is now exactly the weighted average IF of the decomposed signal and that it is also always real-valued. In addition, the IB is always positive for all dictionaries with Gaussian envelopes and arbitrary polynomial phase

Traditional signal postprocessing methods suffer from the repeated assignment problem (RAP), which can result in inaccurate instantaneous frequency (IF) estimation and signal recovery. In this article, to solve this problem, a novel time-frequency (TF) analysis (TFA) method called the synchro-reassigning transform (SRT) is proposed. This method aims to obtain the ideal TF representation (TFR) by utilizing derivatives of the constructed amplitude function and a three-step selection rule to adaptively extract the TF coefficients on the IF trajectories in the TF plane, and reassigning these TF coefficients into a new TFR. In this way, SRT can eliminate the RAP, thus obtaining an approximately ideal TFR that helps to realize more accurate IF estimation and signal reconstruction. Furthermore, SRT shows satisfactory performance on signals with high nonlinear IFs or relatively close IFs, even under strong noise conditions. Two simulated signal and three real-life signals were used to demonstrate the performance of SRT.

This paper makes use of local polynomial Wigner-Ville distribution (LPWVD), originally designed for nonparametric instantaneous frequency (IF) estimation of transient signals, to propose a parametric IF estimation for polynomial phase signals (PPSs). Statistical performance such as asymptotic bias and variance of the LPWVD-based parametric IF estimator is derived in closed-form. Based on the analytical results, we extend the statistical efficiency of the Wigner-Ville distribution (WVD) for a second-order PPS only to that of the LP-WVD for an arbitrary order, when the IF is estimated at the middle of sample observations. Simulation results verify the analytical performance and comparisons with the polynomial Wigner-Ville distribution (PWVD) show that the LPWVD-based parametric IF estimator can provide better performance.