Nonparametric if and DOA estimation

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 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.

An analytic signal s(t) is modeled over a T second duration by a pole-zero model by considering its periodic extensions. This type of representation is analogous to that used in discrete-time systems theory, where the periodic frequency response of a system is characterized by a finite number of poles and zeros in the z-plane. Except, in this case, the poles and zeros are located in the complex-time plane. Using this signal model, expressions are derived for the envelope, phase, and the instantaneous frequency of the signal s(t). In the special case of an analytic signal having poles and zeros in reciprocal complex conjugate locations about the unit circle in the complex-time plane, it is shown that their instantaneous frequency (IF) is always positive. This result paves the way for representing signals by positive envelopes and positive IF (PIF). An algorithm is proposed for decomposing an analytic signal into two analytic signals, one completely characterized by its envelope and the other having a positive IF. This algorithm is new and does not have a counterpart in the cepstral literature. It consists of two steps. In the first step, the envelope of the signal is approximated to desired accuracy using a minimum-phase approximation by using the dual of the autocorrelation method of linear prediction, well known in spectral analysis. The criterion that is optimized is a waveform flatness measure as opposed to the spectral flatness measure used in spectral analysis. This method is called linear prediction in spectral domain (LPSD). The resulting residual error signal is an all-phase or phase-only analytic signal. In the second step, the derivative of the error signal, which is the PIF, is computed. The two steps together provide a unique AM-FM or minimum-phase/all-phase decomposition of a signal. This method is then applied to synthetic signals and filtered speech signals.

The estimation of the instantaneous frequency (IF) of a harmonic complex-valued signal with an additive noise using the Wigner distribution is considered. If the IF is a nonlinear function of time, the bias of the estimate depends on the window length. The optimal choice of the window length, based on the asymptotic formulae for the variance and bias, can be used in order to resolve the bias-variance tradeoff. However, the practical value of this solution is not significant because the optimal window length depends on the unknown smoothness of the IF. The goal of this paper is to develop an adaptive IF estimator with a time-varying and data-driven window length, which is able to provide quality close to what could be achieved if the smoothness of the IF were known in advance. The algorithm uses the asymptotic formula for the variance of the estimator only. Its value may be easily obtained in the case of white noise and relatively high sampling rate. Simulation shows good accuracy for the proposed adaptive algorithm.

In this study, the problem of estimating multiple frequency-hopping (FH) signal parameters in situations in which random observations are missing is addressed. A uniform linear array (ULA) is utilized to receive FH signals, and the missing observations are considered to be equivalent to randomly missing elements in the space-time matrix obtained from the ULA. Bayesian compressed sensing (BCS) estimates the hopping time of received FH signals by the spatial information from the space-time matrix and restores the missing observations. The estimated hopping time is implemented as the boundary to divide the signals such that each segment contains a superposition of time-invariant multiple components. Then, the instantaneous frequency (IF) of multiple components can be estimated precisely by atomic norm soft thresholding (AST) within each segment. After estimating the hopping time and IF, the direction of arrival (DOA) of multiple signals is directly calculated using these two parameters. The simulation results show that the proposed method is superior to existing approaches. Provided that sufficient array elements are available, multi-FH signal parameters can be estimated with satisfactory accuracy even when a large portion of observations is missing.

Chapter 10 - Instantaneous Frequency Estimation and Localization

Tremor can be a severely disabling symptom. It occurs in patients with many movement disorders including Parkinson's disease (PD) and essential tremor (ET). Physical tremor can be measured with a variety of instruments including electromyograms (EMG) and accelerometers. Neural tremor can be measured by microelectrodes placed near nerve cells or obtained from brain cells during stereotactic neurosurgery. Recent studies have demonstrated that tremor activity measured from two distinct signals, possibly from different types of sensors, may be uncoupled even when the dominant frequency of the tremor is the same in both signals. The coupling between signals may also change over time. Several measures of phase coupling have been proposed in previous studies. We describe the results of a simulation study to assess the reliability and inverse variability of three popular phase-coupling indices and an index based on the correlation coefficient of the instantaneous frequency (IF) signals. The results of our study demonstrate that the common-mean correlation coefficient of the IF signals is the most accurate measure of phase coupling and has many desirable properties not shared by the other indices

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.

An Instantaneous Frequency Estimation Approach in a High Noise Environment

Uniform linear array (ULA) geometry does not perform well for direction of arrival (DOA) estimation at directions close to the array endfires. Shirvani and Akbari solved this problem by displacing two elements from both ends of the ULA to the top and/or bottom of the array axis. Shirvani-Akbari array (SAA) presents a considerable improvement in the DOA estimation of narrowband sources arriving at endfire directions in terms of DOA estimation accuracy and angular resolution. In this paper, all new proposed SAA configurations are modelled and also examined, numerically. In this paper, two well-known DOA estimation algorithms, multiple signal classification (MUSIC) and minimum variance distortionless response (MVDR), are used to evaluate the effectiveness of proposed arrays using total root mean square error (RMSE) criterion. In addition, two new scenarios are proposed which divide angular search to two parts, directions close to array endfires as well as middle angles. For middle angles, which belong to (−70∘≤𝜃≤70∘), ULA is considered, and for endfire angles, the angles which belong to (−90∘≤𝜃≤−70∘) and (70∘≤𝜃≤90∘), SAA is considered. Simulation results of new proposed scenarios for DOA estimation of narrowband signals show the better performance with lower computational load.

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.

Currently, mostly used method for Instantaneous Frequency (IF) feature extraction of radar signal is two-dimensional graph, which is easy for people to distinguish, but not for machines. This article proposes a feature extraction algorithm based on instantaneous frequency of radiation source. Using the statistical change difference of different IF signals, we formulate the classification feature vectors describing the various types of signal pulse modulation laws, according which we do some classification experiments. The experiments prove that instantaneous frequency characteristics of radiation sources can characterize the change difference of different modulation signals and have some anti-noise performance. We are sure that this method can be used in signal sorting, and can be realized in practice.

An adaptive Generalized S-transform for instantaneous frequency estimation

Time-scale-chirp_rate operator for recovery of non-stationary signal components with crossover instantaneous frequency curves

Direction of arrival (DOA) estimation of narrowband wavefronts impinging on an array of sensors has long been of great research interest. In this paper, it is shown that for a uniform linear array (ULA), discrete Fourier transform (DFT) domain-based DOA estimation rely on knowledge of the power spectral density (PSD). We present a novel approach for signal tracking, which updates the signal PSD for each DFT coefficient in the presence of both signal and noise. This method is based on the eigenvalue decomposition of correlation matrices that are constructed from snapshots of time series of array sensors. The presented method is very well capable of estimating more than five signals from different spatial directions. In comparison to the well-known multiple signal classification (MUSIC) algorithms the proposed method reduces the estimation error between the estimated and the true signal direction.