Some Results in the Theory Modulated Filter Banks and Modulated Wavelet Tight Frames

Perfect reconstruction (PR) FIR filter banks, obtained by modulation of a linear-phase, lowpass, prototype filter and of length 2Mm are well known. Recently, PR modulated filter banks (MFBs) with the analysis and synthesis banks obtained from different prototypes have been reported. This paper describes a general form of modulation that includes modulations used in the literature. This modulation depends on an integer parameter, the modulation phase. The PR property is characterized for MFBs with finite and infinite impulse response filters. The MFB PR problem reduces to roughly M/2 two-channel PR problems. A natural dichotomy in the PR conditions leads us to the concepts of Type 1 and Type 2 MFBs. Unitary MFBs are characterized by the M/2 two-channel PR filter banks also being unitary (for FIR filters of length N = 2Mm, these results are given in (Malvar, Electr Lett. 26, June 1990, 906–907; Koilpillai and Vaidyanathan, IEEE Trans. SP40, No. 4, Apr. 1992, 770–783)). We also give a necessary and sufficient condition for a large class (including FIR) unitary MFB prototypes to have symmetric (even or odd) prototype filters, and exhibit unitary MFBs without symmetric prototypes. A parameterization of all FIR unitary MFBs is also given. An efficient design procedure for FIR unitary MFBs is developed. It turns out that MFBs can be implemented efficiently using Type III and Type IV DCTs. Compactly supported modulated wavelet tight frames are shown to exist and completely parameterized. K-regular modulated WTFs are designed numerically and analytically by solving a set of non-linear equations over the parameters. Design of optimal modulated WTFs for the representation of any given signal is described with examples, and this is used to design smooth modulated WTFs.     

New Synthesis of 2,2‐Bis(hydroxymethyl) methylenecyclopropane,a Key Intermediate for Antiviral Cyclopropavir

Synthesis of 2,2‐bis(hydroxymethyl)methylenecyclopropane (2) is described. Cyclopropane dicarboxylate 3 was reduced to diol 4, which was acetylated to give diacetate 5. Acetoxy‐bromo elimination afforded methylenecyclopropane 6. Debenzylation gave the title compound 2.     

Analysis of open dielectric waveguides using the finite-element penalty method

This paper analyzes open dielectric waveguides using the vector finite-element method and boundary integral equations derived from the second Green's theorem. This finite-element formulation, together with the boundary operator, is solved using a penalty function method. Comparison with previously published results shows good agreement for the analysis of the rectangular dielectric waveguide.     

Analytical computation of the Compton continuum in gamma-ray spectrometry

In gamma-ray spectrometry with solid state detectors if the photons have only one collision in the detector, the peak to total ratio should be equal to the ratio of photoelectric to total cross sections. However, it is well known that it is higher and increases with the detector size for all energies due to multiple collisions in the detector. With the data on the difference between the two ratios we estimate the higher order collisions and use this estimate to compute the counts distribution in the Compton continuum. The method is simple and essentially physics oriented without recourse to any empirical fitting of the spectral data. Results compare remarkably well with the experimental observations of Compton continuum.