| We consider the issue of wavelet design based on Gröbner bases. Such design requirements as perfect reconstruction and orthogonality describing the underlying filters result in nonlinear equations. This in turn makes the Gröbner approach ideal. All the wavelets designed in this thesis are based on FIR filters.; The design flexibility made possible by the Gröbner bases approach allows finding wavelets with various properties. The thesis begins by investigating the approximate orthogonality of 2-band symmetric filters, in addition to 2-band orthogonal filters approximating symmetry. In both cases the method allows for arbitrary degrees of approximation. The lack of symmetry and orthogonality is addressed in 4-channel filterbank allowing both properties, in addition to interpolation. Furthermore, multiwavelets are revisited, and scaling functions with interpolation, symmetry, and orthogonality are investigated.; Properties of tight frame wavelets are exploited in the design of 4-channel over-sampled symmetric filters enjoying a high degree of smoothness when compared with orthogonal counterparts. The thesis also presents a means of designing the filterbank through spectral factorization. Lastly, 4-channel tight frames are taken one step further, this time in the design of a Hilbert-like pair of tight frame 4-channel filterbanks with a symmetric wavelet envelope taking on the form . |
