Abstract:
The sampling theorem states that any frequency bandlimited signal can be exactly reconstructed from its sampled values. Different forms of this theorem have been reviewed since 1841 up to the present time. Kramer's generalized sampling theorem has been extended to two dimensions, and the form for higher dimensions can be derived by the same method. Upper bounds for the variations of sampled signals were investigated, and it was shown that tighter bounds can be established. Although the sampling theorem for sequency bandlimited signals was proved by the aid of Kramer's generalized sampling theorem we showed that this theorem can be proved directly or by the method of Shannon's sampling theorem in Fourier analysis. This is in agreement with the result of Campbell, that for some cases Kramer's generalized theorem does not enlarge the class of functions to which sampling theorems can be applied. Because of the recent interest in applications of sampling theorems in discrete and finite Walsh-Fourier analysis we introduced new forms of these theorems and proved them by simple techniques. These results were applied easily for the development of periodic and nonperiodic sampling theorems for two dimensions. Finally, a definition in Haar-Fourier analysis is created, which is analogous to the definition of M-sequency bandlimited signals. Based on this definition several useful theorems are established which show similarities between sampling theorems in Walsh and Haar-Fourier analysis.
