טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentGazit Amir
SubjectSampling Schemes of the Multiresolution Fourier Transform
DepartmentDepartment of Electrical Engineering
Supervisors Dr. Meir Zibulski
Professor Emeritus Yehoshua Zeevi


Abstract

The Multiresolution Fourier Transform (MFT) is a relatively new tool suitable for signal representation and processing. In this research, sampling schemes of the MFT are considered, in order to assess the benefits of its inherent redundancy. Sampling schemes are classified into three categories: 1) Regular periodic sampling, 2) Irregular periodic sampling, and 3) Irregular non-periodic sampling. This thesis is concerned with sampling schemes that belong to either one of the two first categories. A combination of a sampling scheme and a window function can be viewed as an inner vector product of the original function with a discrete set of functions.  When ever such a combination corresponds to a frame, reconstruction is possible. Theorems are proven for regular periodic sampling. It is shown that 3D sampling lattice, defined as being linearly spaced in time and frequency and logarithmically spaced in dilation, can not correspond to a frame. Several schemes of planar, i.e. planar 2D sampling lattice are devised and tested.  It is proven that none of these sampling schemes corresponds to frames.  Alternative definitions of lattices and planes may yield positive results, i.e. correspond to frames. Irregular periodic sampling is studied mainly by simulations.  Several finite subsets are formed, using various basic window functions.  Each subset is shifted in time/space, and is either shifted in frequency or dilation, to create larger function sets.  These function sets constitute frames only when they include frames consisting of non-dilated windows, or windows shifted only in time or frequency. The proven theorems, along with simulation results, suggest that any frames created by sampling the MFT are already spanned by Gabor frames, wavelet frames, multi-window Gabor frames or multi-window wavelet frames. However, this conclusion has yet to be proven.