Richard Scheper

We use an asymptotic integral approximation of a wavelet transform as a model for the estimation of instantaneous frequency (IF). Our approach allows the calculation of the Cramer-Rao bound for the IF variance at each time directly, without the need for explicit phase parameterization. This is in contrast to other approaches where the Cramer-Rao bounds rely on a preliminary decomposition of the IF with respect to a (usually polynomial) basis. Attention is confined to the Morlet wavelet… Show more

We use an asymptotic integral approximation of a wavelet transform as a model for the estimation of instantaneous frequency (IF). Our approach allows the calculation of the Cramer-Rao bound for the IF variance at each time directly, without the need for explicit phase parameterization. This is in contrast to other approaches where the Cramer-Rao bounds rely on a preliminary decomposition of the IF with respect to a (usually polynomial) basis. Attention is confined to the Morlet wavelet transform of single-component signals corrupted with additive Gaussian noise. Potential computationally and statistically efficient IF extraction algorithms suggested by the analysis are also discussed. Show less

Presented here are general limits on bandpass filter performance as a denoising step for parameter estimation. Monocomponent analytic FM signals in white Gaussian noise are assumed. Analytic and numerical evidence is presented to support the argument that the Gaussian filter can be used as a canonical bandpass filter for the purpose of such an analysis. Simple, approximate Cramer-Rao bound formulas for amplitude, phase and IF estimates are presented for limiting cases.

Wavelet filter banks are potentially useful tools for analyzing and extracting information from frequency modulated (FM) signals in noise. Chief among the advantages of such filter banks is the tendency of wavelet transforms to concentrate signal energy while simultaneously dispersing noise energy over the time-frequency plane, thus raising the effective signal to noise ratio of filtered signals. Over the past decade, much effort has gone into devising new algorithms to extract the relevant… Show more

Wavelet filter banks are potentially useful tools for analyzing and extracting information from frequency modulated (FM) signals in noise. Chief among the advantages of such filter banks is the tendency of wavelet transforms to concentrate signal energy while simultaneously dispersing noise energy over the time-frequency plane, thus raising the effective signal to noise ratio of filtered signals. Over the past decade, much effort has gone into devising new algorithms to extract the relevant information from transformed signals while identifying and discarding the transformed noise. Therefore, estimates of the ultimate performance bounds on such algorithms would serve as valuable benchmarks in the process of choosing optimal algorithms for given signal classes. Discussed here is the specific case of FM signals analyzed by Morlet wavelet filter banks. By making use of the stationary phase approximation of the Morlet transform, and assuming that the measured signals are well resolved digitally, the asymptotic form of the Fisher Information Matrix is derived. From this, Cramer-Rao bounds are analytically derived for simple cases. Show less

A transform--model based approach for suppressing noise in frequency modulated (FM) signals is presented. This approach is based on a model of the expected output of a wavelet filter bank in response to a noise-free FM signal. Resolving the discrepancy between filter bank output in response to a noisy FM signal and the expected output given by the model provides the mechanism for noise suppression. Specifically, a stationary phase approximation to the Morlet wavelet transform is used to form… Show more

A transform--model based approach for suppressing noise in frequency modulated (FM) signals is presented. This approach is based on a model of the expected output of a wavelet filter bank in response to a noise-free FM signal. Resolving the discrepancy between filter bank output in response to a noisy FM signal and the expected output given by the model provides the mechanism for noise suppression. Specifically, a stationary phase approximation to the Morlet wavelet transform is used to form the model. The approach is shown to perform favorably on a numerical example when compared to both simple lowpass filtering (linear) and wavelet thresholding (non-linear) denoising techniques. Show less

We present a framework for the use of stationary phase approximations to a Morlet wavelet transform as a device to generate computationally efficient algorithms for extracting modulation information in frequency modulated (FM) signals. Presented here are two specific FM estimators generated from this approach that may be implemented in terms of filter banks with very few filters.

The "reduced equations" of MHD are based on the ordering that variations transverse to the magnetic field are sharper than longitudinal variations. We show that this ordering breaks down near conducting surfaces satisfying "line-tied" boundary conditions. A boundary layer is shown to exist there. Reduced equations are derived for the layer and matched asymptotically to the usual equations away from the surface. It is shown that the boundary layer does not affect the usual procedures to solve… Show more

The "reduced equations" of MHD are based on the ordering that variations transverse to the magnetic field are sharper than longitudinal variations. We show that this ordering breaks down near conducting surfaces satisfying "line-tied" boundary conditions. A boundary layer is shown to exist there. Reduced equations are derived for the layer and matched asymptotically to the usual equations away from the surface. It is shown that the boundary layer does not affect the usual procedures to solve for the flux surfaces and the flow streamlines. Show less

We present two examples of current sheets that form in a magnetic configuration when it is subjected to quasi-static motions at the footpoints. The entire system is two-dimensional. There are no preexisting X-points and the footpoint motions are continuous. The calculations are motivated by the hypothesis of Parker that quasi-static deformations of MHD equilibria are generally accompanied by the formation of current sheets. The results demonstrate that three dimensions are not a necessary… Show more

We present two examples of current sheets that form in a magnetic configuration when it is subjected to quasi-static motions at the footpoints. The entire system is two-dimensional. There are no preexisting X-points and the footpoint motions are continuous. The calculations are motivated by the hypothesis of Parker that quasi-static deformations of MHD equilibria are generally accompanied by the formation of current sheets. The results demonstrate that three dimensions are not a necessary condition for current sheet formation. In addition, the calculation of Van Ballegooijen is not contradicted, because the initial magnetic field in our case is not dominantly collinear. Possible applications to the solar corona are discussed. Show less