Theory of wavelet transform over finite fields

In this paper, we develop the theory of the wavelet transform over Galois elds. To avoid the limitations inherent in the number theoretic Fourier transform over nite elds, our wavelet transform relies on a basis decomposition in the time domain rather than in the frequency domain. First, we characterize the in nite dimensional vector spaces for which an orthonormal basis expansion of any sequence in the space can be obtained using a symmetric bilinear form. Then, by employing a symmetric, non-degenerate, canonical bilinear form we derive the necessary and su cient condition that basis functions over nite elds must satisfy in order to construct an orthogonal wavelet transform. Finally, we give a design methodology to generate the mother wavelet and scaling function over Galois elds by relating the wavelet transform to a two channel paraunitary lter bank. Online relevant information can be found at http://www.ee.gatech.edu/users/fekri.