Luis F. Chaparro

Evolutionary periodogram for nonstationary signals

Presents a novel estimator for the time-dependent spectrum of a nonstationary signal. By modeling the signal, at any given frequency, as having a time-varying amplitude accurately represented by an orthonormal basis expansion, the authors are able to compute a minimum mean-squared error estimate of this time-varying amplitude. Repeating the process over all frequencies, they obtain a power distribution as a function of time and frequency that is consistent with the Wold-Cramer evolutionary spectrum. Based on the model assumptions, the authors develop the evolutionary periodogram (EP) for nonstationary signals, an estimator analogous to the periodogram used in the stationary case. They also derive the time-frequency resolution of the new estimator. The approach is free of some of the drawbacks of the bilinear distributions and of the short-time Fourier transform spectral estimates. It is guaranteed to produce nonnegative spectra without the cross-term behavior of the bilinear distributions, and it does not require windowing of data in the time domain. Examples illustrating the new estimator are given. >

Multi-window Gabor expansion for evolutionary spectral analysis

Abstract In this paper, we connect a new multi-window discrete Gabor expansion of finite extent, deterministic signals with their evolutionary spectra. In our method both the signal representation and its corresponding evolutionary spectrum are obtained simultaneously. Including a scale parameter in the discrete Gabor expansion, we develop a multi-window representation which can be related to the deterministic version of the Wold-Cramer decomposition of non-stationary signals. By choosing Gaussian windows and appropriate scales, the expansion can be used to represent the narrow- and wide-band components of a signal. The evolutionary spectrum is then easily calculated from the Gabor coefficients. The computation of the evolutionary spectrum using this approach can be efficiently done by means of the Fast Fourier Transform algorithm. As an application, we present an approximate implementation of time-frequency masking. This implementation only requires that the support of the Gabor coefficients be restricted in the time-frequency plane. Examples illustrating the evolutionary spectral analysis and the proposed time-frequency masking are given.

Discrete evolutionary transform for time–frequency signal analysis

Abstract In this paper, we introduce the discrete evolutionary transform (DET) capable of representing deterministic non-stationary signals. Besides the signal representation, the DET permits the computation of a kernel from which the evolutionary spectrum of the signal is obtained. The signal representation is modeled after the Wold–Cramer representation used for random non-stationary signals in Priestleys evolutionary spectral theory. The proposed transform generalizes the short-time Fourier transform and the spectrogram. To illustrate how to define the windows used in the DET we consider the Gabor and the Malvar cases. The Gabor-based window is time dependent and uses the bi-orthogonal analysis and synthesis windows of the expansion. The Malvar-based window is a function of time and of frequency, and depends on the orthogonal functions used in the expansion. Two types of transforms are shown: sinusoidal and chirp DETs. The sinusoidal DET represents well signals with narrow-band components, while the chirp transformation is capable of representing well signals with wide-band components provided that the instantaneous frequency information of the signal components is estimated. Examples are used to illustrate the implementation of the DET. The examples show the capability of the transform in providing excellent representation for a signal and a spectrum with very good time–frequency resolution.

Data-adaptive evolutionary spectral estimation

We present a novel data-adaptive estimator for the evolutionary spectrum of nonstationary signals. We model the signal at a frequency of interest as a sinusoid with a time-varying amplitude, which is accurately represented by an orthonormal basis expansion. We then compute a minimum mean-squared error estimate of this amplitude and use it to estimate the time-varying spectrum at that frequency, all while minimizing the interference from the signal components at other frequencies. Repeating the process over all frequencies, we obtain a power distribution that is consistent with the Wold-Cramer evolutionary spectrum and reduces to Capons (1969) method for the stationary case. Our estimator possesses desirable properties in terms of time-frequency resolution and positivity and is robust in the spectral estimation of noisy nonstationary data. We also propose a new estimator for the autocorrelation of nonstationary signals. This autocorrelation estimate is needed in the data-adaptive spectral estimation. We illustrate the performance of our estimator using simulation examples and compare it with the recently presented evolutionary periodogram and the bilinear time-frequency distribution with exponential kernels. >

Compressive sampling of swallowing accelerometry signals using time-frequency dictionaries based on modulated discrete prolate spheroidal sequences

Monitoring physiological functions such as swallowing often generates large volumes of samples to be stored and processed, which can introduce computational constraints especially if remote monitoring is desired. In this article, we propose a compressive sensing (CS) algorithm to alleviate some of these issues while acquiring dual-axis swallowing accelerometry signals. The proposed CS approach uses a time-frequency dictionary where the members are modulated discrete prolate spheroidal sequences (MDPSS). These waveforms are obtained by modulation and variation of discrete prolate spheroidal sequences (DPSS) in order to reflect the time-varying nature of swallowing acclerometry signals. While the modulated bases permit one to represent the signal behavior accurately, the matching pursuit algorithm is adopted to iteratively decompose the signals into an expansion of the dictionary bases. To test the accuracy of the proposed scheme, we carried out several numerical experiments with synthetic test signals and dual-axis swallowing accelerometry signals. In both cases, the proposed CS approach based on the MDPSS yields more accurate representations than the CS approach based on DPSS. Specifically, we show that dual-axis swallowing accelerometry signals can be accurately reconstructed even when the sampling rate is reduced to half of the Nyquist rate. The results clearly indicate that the MDPSS are suitable bases for swallowing accelerometry signals.

Reconstruction of nonuniformly sampled time-limited signals using prolate spheroidal wave functions

Shannons sampling theory is based on the reconstruction of bandlimited signals which requires infinite number of uniform time samples. Indeed, one can only have finite number of samples for numerical implementation. In this paper, as a dual of the bandlimited reconstruction, a solution for time-limited signal reconstruction from nonuniform samples is proposed. The system model we present is based on the idea that time-limited signals which are also nearly bandlimited can be well approximated by a low-dimensional subspace. This can be done by using prolate spheroidal wave functions as the basis. The order of the projection on this basis is obtained by means of the time-frequency dimension of the signal, especially in the case of non-stationary signals. The reconstruction requires the estimation of the nonuniform sampling times by means of an annihilating filter. We obtain the reconstruction parameters by solving a linear system of equations and show that our finite-dimensional model is not ill-conditioned. The practical aspects of our method including the dimensionality reduction are demonstrated by processing synthetic as well as real signals.

Evolutionary chirp representation of non-stationary signals via Gabor transform

Abstract In this paper, we propose a chirp time–frequency representation for non-stationary signals, and associate with it—via a multi-window Gabor expansion—the corresponding evolutionary spectra. Representations based on rectangular time–frequency plane tilings give poor time and frequency localization in the spectrum, especially when the signal is not modeled well by fixed bandwidth analysis. We propose a representation that uses scaled and translated windows modulated by chirps as bases. Considering a chirp-based Wold–Cramer model, the signal evolutionary spectrum with improved time and frequency resolutions is obtained from the kernel of the representation. The chirp representation optimally chooses scales and linear chirp slopes by maximizing a local energy concentration measure. Parsimonious signal representation and well-localized evolutionary spectrum are obtained simultaneously. As an application of our representation, we consider the excision of broad-band jammers in spread spectrum communications. Examples illustrating the improvement in the time and frequency resolution of the signal spectrum using our procedure are given.

Fuzzy morphological polynomial image representation

A novel signal representation using fuzzy mathematical morphology is developed. We take advantage of the optimum fuzzy fitting and the efficient implementation of morphological operators to extract geometric information from signals. The new representation provides results analogous to those given by the polynomial transform. Geometrical decomposition of a signal is achieved by windowing and applying sequentially fuzzy morphological opening with structuring functions. The resulting representation is made to resemble an orthogonal expansion by constraining the results of opening to equate adapted structuring functions. Properties of the geometric decomposition are considered and used to calculate the adaptation parameters. Our procedure provides an efficient and flexible representation which can be efficiently implemented in parallel. The application of the representation is illustrated in data compression and fractal dimension estimation temporal signals and images.

Evolutionary spectral analysis using a warped Gabor expansion

In this paper, we present a Gabor representation based on a nonrectangular tiling of the time-frequency plane and use it to improve the time and frequency resolutions of evolutionary spectra. In the traditional Gabor expansion, a signal is decomposed into a weighted combination of sinusoidally modulated windows resulting in a rectangular time-frequency plane tiling. Poor time and frequency localizations occur in the evolutionary spectrum when the corresponding signal is not modeled well by this fixed-window analysis. We are thus proposing the warped Gabor representation based on a linear chirp model for the signal. By means of a frequency transformation we are able to use the previous sinusoidal representation and choose the Gabor coefficients according to either a frequency masking or an energy concentration measure. Examples are given to illustrate our procedures.

Informative priors for minimum cross-entropy positive time-frequency distributions

A method for generating an informative prior when constructing a positive time-frequency distribution (TFD) by the method of the minimum cross-entropy (MCE) is developed. The prior is obtained from a combination of the Wigner distribution (WD) and the evolutionary periodogram, and results in a more informative MCE-TFD, as quantified via the mutual information of the distribution. The procedure allows any of the bilinear distributions to be used in the prior. Examples illustrate the performance of the new technique.