A novel technique is presented for designing finite impulse response orthonormal wavelet filters. The filters are obtained from the spectral factorization of an appropriately designed parametric Bernstein polynomial. We show that by strategically "pinning" some of the zeros of the polynomial, the nonnegativity requirement on the polynomial, which is mandatory for orthonormal filter design, can be easily achieved. Filters with a high number of vanishing moments and sharper frequency response (but lower vanishing moments) than the maximally flat Daubechies filters can be easily designed. The technique is simple as it only involves solving linear equations yet is versatile as filters with different characteristics can be obtained with ease
One hundred years after the introduction of the Bernstein polynomial basis, we survey the historical development and current state of theory, algorithms, and applications associated with this remarkable method of representing polynomials over finite domains. Originally introduced by Sergei Natanovich Bernstein to facilitate a constructive proof of the Weierstrass approximation theorem, the leisurely convergence rate of Bernstein polynomial approximations to continuous functions caused them to languish in obscurity, pending the advent of digital computers. With the desire to exploit the power of computers for geometric design applications, however, the Bernstein form began to enjoy widespread use as a versatile means of intuitively constructing and manipulating geometric shapes, spurring further development of basic theory, simple and efficient recursive algorithms, recognition of its excellent numerical stability properties, and an increasing diversification of its repertoire of applications. This survey provides a brief historical perspective on the evolution of the Bernstein polynomial basis, and a synopsis of the current state of associated algorithms and applications. [All rights reserved Elsevier].
For a smooth strictly plurisubharmonic function u on an open set OmegasubC n and F a C 1 nondecreasing function on R +*, we investigate the complex partial differential equations Delta g log det(u ij) = F(det(u ij))||nablag log det(u ij)||g 2, where Delta g, ||.|| g and nabla g are the Laplacian, tensor norm and the Levi-Civita connexion, respectively, with respect to the Kahler metric g = partpartu. We show that the above PDE's has a Bernstein property, i.e. det (u ij) is constant on Omega, provided that g is complete, the Ricci curvature of g is bounded below and F satisfies inf tisinR+(2tF'(t)+F(t) 2/n) > 14 and F(max B(R) det u ij) = o (R). [All rights reserved Elsevier].
Perpendicularly propagating electron Bernstein modes in a uniformly magnetized plasma having an isotropic kappa velocity distribution are investigated within the framework of a fully electromagnetic plasma model but one which ignores particle relativistic effects. The dispersion relations for the Bernstein mode waves are found to be significantly dependent on the spectral index, kappa, of the electron kappa distribution. In particular, waves with frequencies exceeding the upper hybrid frequency are seen to occupy a diminishing range of frequencies above the nearest cyclotron harmonic as kappa is reduced. The Bernstein mode wave whose frequency lies closest to the upper hybrid frequency is found to couple to the cold plasma, electromagnetic Z mode, as it does in a Maxwellian plasma. For waves whose frequencies lie below the upper hybrid frequency, diminishing kappa gives rise to an increasingly weak dependence of frequency on wave number and a slower frequency fall off with this parameter, but the frequency occupies the entire intraharmonic band here. All Bernstein modes are observed to become significantly electromagnetic at very long wavelengths, or small wave numbers, having in general elliptical polarization whose elliptical eccentricity depends on kappa and wave number. At smaller wavelengths the modes are found to be electrostatic to a very good approximation, irrespective of kappa value. The significance of the results to the interpretation of banded emissions in planetary magnetospheres is briefly discussed
R. I. Pinsker
M. D. Carter
C. B. Forest
V. A. Svidzinski
P. K. Chattopadhyay
Conventional electron cyclotron heating using the O- and X-modes to carry energy from the plasma edge to the cyclotron resonance layer is not possible for high density, low magnetic field devices (RFPs and STs, for example), since these modes are evanescent in all but the very edge of the plasma. As an alternative, we consider coupling to the electron Bernstein wave (EBW) with a phased waveguide array, the mouth of which is inserted to the vicinity of the upper hybrid resonance at the edge of the plasma. The calculation of the waveguide reflection coefficient is similar to the lower hybrid coupling problem solved by Brambilla, but the character of the plasma surface admittance is quite different for the EBW from that for the lower hybrid wave. Two models for the surface admittance are compared. In the first, the lowest-order EBW is included in the calculation, while in the second, the cold plasma model (which does not have any mode corresponding to the EBW) with weak collisions is used. The surface admittances obtained in those two models for parameters relevant to coupling experiments performed in the Madison symmetric torus (MST) are compared, and found to agree closely, despite the very different physics in the models. A significant asymmetry with respect to the direction mutually perpendicular to the static magnetic field and the radial direction (the poloidal direction in tokamak geometry, the toroidal direction in the edge of the RFP) is found. This phenomenon is related to the well-known up/down asymmetry in fast wave launch in the ion cyclotron radio frequency and to the up/down asymmetry previously reported for the fast wave in the lower hybrid radio frequency. The effect is very strong in this situation due to the density gradient scale length being much shorter than the local wavelength (violation of WKB condition) in the coupling region
R I Pinsker
M D Carter
C B Forest
V A Svidzinski
P K Chattopadhyay
Conventional electron cyclotron heating using the O- and X-modes to carry energy from the plasma edge to the cyclotron resonance layer is not possible for high density, low magnetic field devices (RFPs and STs, for example), since these modes are evanescent in all but the very edge of the plasma. As an alternative, we consider coupling to the electron Bernstein wave (EBW) with a phased waveguide array, the mouth of which is inserted to the vicinity of the upper hybrid resonance at the edge of the plasma. The calculation of the waveguide reflection coefficient is similar to the lower hybrid coupling problem solved by Brambilla, but the character of the plasma surface admittance is quite different for the EBW from that for the lower hybrid wave. Two models for the surface admittance are compared. In the first, the lowest-order EBW is included in the calculation, while in the second, the cold plasma model (which does not have any mode corresponding to the EBW) with weak collisions is used. The surface admittances obtained in those two models for parameters relevant to coupling experiments performed in the Madison symmetric torus (MST) are compared, and found to agree closely, despite the very different physics in the models. A significant asymmetry with respect to the direction mutually perpendicular to the static magnetic field and the radial direction (the poloidal direction in tokamak geometry, the toroidal direction in the edge of the RFP) is found. This phenomenon is related to the well-known up/down asymmetry in fast wave launch in the ion cyclotron radio frequency and to the up/down asymmetry previously reported for the fast wave in the lower hybrid radio frequency. The effect is very strong in this situation due to the density gradient scale length being much shorter than the local wavelength (violation of WKB condition) in the coupling region.