Boualem Boashash

Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals

The concept of instantaneous frequency (IF), its definitions, and the correspondence between the various mathematical models formulated for representation of IF are discussed. The extent to which the IF corresponds to the intuitive expectation of reality is also considered. A historical review of the successive attempts to define the IF is presented. The relationships between the IF and the group-delay, analytic signal, and bandwidth-time (BT) product are explored, as well as the relationship with time-frequency distributions. The notions of monocomponent and multicomponent signals and instantaneous bandwidth are discussed. It is shown that these notions are well described in the context of the theory presented. >

A human identification technique using images of the iris and wavelet transform

A new approach for recognizing the iris of the human eye is presented. Zero-crossings of the wavelet transform at various resolution levels are calculated over concentric circles on the iris, and the resulting one-dimensional (1-D) signals are compared with model features using different dissimilarity functions.

Estimating and interpreting the instantaneous frequency of a signal. II. Algorithms and applications

For pt.I see ibid., vol.80, no.4, p.520-38 (1992). The concept of instantaneous frequency (IF) is extended to discrete-time signals. The specific problem explored is that of estimating the IF of frequency-modulated (FM) discrete-time signals embedded in Gaussian noise. Well-established methods for estimating the IF include differentiation of the phase and smoothing thereof, adaptive frequency estimation techniques such as the phase locked loop (PLL), and extraction of the peak from time-varying spectral representations. More recently, methods based on a modeling of the signal phase as a polynomial have been introduced. These methods are reviewed, and their performance compared on both simulated and real data. Guidelines are given as to which estimation method should be used for a given signal class and signal-to-noise ratio (SNR). >

The bootstrap and its application in signal processing

The bootstrap is an attractive tool for assessing the accuracy of estimators and testing hypothesis for parameters where conventional techniques are not valid, such as in small data-sample situations. We highlight the motivations for using the bootstrap in typical signal processing applications and give several practical examples. Bootstrap methods for testing statistical hypotheses are described and we provide an analysis of the accuracy of bootstrap tests. We also discuss how the bootstrap can be used to estimate a variance-stabilizing transformation to define a pivotal statistic, and we demonstrate the use of the bootstrap for constructing confidence intervals for flight parameters in a passive acoustic emission problem.

An efficient real-time implementation of the Wigner-Ville distribution

The Wigner-Ville distribution (WVD) is a valuable tool for time-frequency signal analysis. In order to implement the WVD in real time, an efficient algorithm and architecture have been developed which may be implemented with commercial components. This algorithm successively computes the analytic signal corresponding to the input signal, forms a weighted kernel function, and analyzes the kernel via a discrete Fourier transform (DFT). To evaluate the analytic signal required by the algorithm, it is shown that the time domain definition implemented as a finite impulse response (FIR) filter is practical and more efficient than the frequency domain definition of the analytic signal. The windowed resolution of the WVD in the frequency domain is shown to be similar to the resolution of a windowed Fourier transform. A real-time signal processor has been designed for evalution of the WVD analysis system. The system is easily paralleled and can be configured to meet a variety of frequency and time resolutions. The arithmetic unit is based on a pair of high-speed VLSI floating-point multiplier and adder chips.

Note on the use of the Wigner distribution for time-frequency signal analysis

It is shown that a correct use of the Wigner distribution (WD) for time-frequency signal analysis requires use of the analytic signal. This version, often referred to as the Wigner-Ville distribution (WVD), is straightforward to compute, does not exhibit any aliasing problem, and introduces no frequency artifacts. The problems introduced by the use of the Wigner distribution with a real signal are clarified. >

A high-resolution quadratic time-frequency distribution for multicomponent signals analysis

The paper introduces a new kernel for the design of a high resolution time-frequency distribution (TFD). We show that this distribution can solve problems that the Wigner-Ville distribution (WVD) or the spectrogram cannot. In particular, the proposed distribution can resolve two close signals in the time-frequency domain that the two other distributions cannot. Moreover, we show that the proposed distribution is more accurate than the WVD and the spectrogram in the estimation of the instantaneous frequency of a stepped FM signal embedded in additive Gaussian noise. Synthetic and real data collected from real-world applications are shown to validate the proposed distribution.

Separating more sources than sensors using time-frequency distributions

We examine the problem of blind separation of nonstationary sources in the underdetermined case, where there are more sources than sensors. Since time-frequency (TF) signal processing provides effective tools for dealing with nonstationary signals, we propose a new separation method that is based on time-frequency distributions (TFDs). The underlying assumption is that the original sources are disjoint in the time-frequency (TF) domain. The successful method recovers the sources by performing the following four main procedures. First, the spatial time-frequency distribution (STFD) matrices are computed from the observed mixtures. Next, the auto-source TF points are separated from cross-source TF points thanks to the special structure of these mixture STFD matrices. Then, the vectors that correspond to the selected auto-source points are clustered into different classes according to the spatial directions which differ among different sources; each class, now containing the auto-source points of only one source, gives an estimation of the TFD of this source. Finally, the source waveforms are recovered from their TFD estimates using TF synthesis. Simulated experiments indicate the success of the proposed algorithm in different scenarios. We also contribute with two other modified versions of the algorithm to better deal with auto-source point selection.

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.

Adaptive instantaneous frequency estimation of multicomponent FM signals using quadratic time-frequency distributions

An adaptive approach to the estimation of the instantaneous frequency (IF) of nonstationary mono- and multicomponent FM signals with additive Gaussian noise is presented. The IF estimation is based on the fact that quadratic time-frequency distributions (TFDs) have maxima around the IF law of the signal. It is shown that the bias and variance of the IF estimate are functions of the lag window length. If there is a bias-variance tradeoff, then the optimal window length for this tradeoff depends on the unknown IF law. Hence, an adaptive algorithm with a time-varying and data-driven window length is needed. The adaptive algorithm can utilize any quadratic TFD that satisfies the following three conditions: First, the IF estimation variance given by the chosen distribution should be a continuously decreasing function of the window length, whereas the bias should be continuously increasing so that the algorithm will converge at the optimal window length for the bias-variance tradeoff, second, the time-lag kernel filter of the chosen distribution should not perform narrowband filtering in the lag direction in order to not interfere with the adaptive window in that direction; third, the distribution should perform effective cross-terms reduction while keeping high resolution in order to be efficient for multicomponent signals. A quadratic distribution with high resolution, effective cross-terms reduction and no lag filtering is proposed. The algorithm estimates multiple IF laws by using a tracking algorithm for the signal components and utilizing the property that the proposed distribution enables nonparametric component amplitude estimation. An extension of the proposed TFD consisting of the use of time-only kernels for adaptive IF estimation is also proposed.