The fractional Fourier transform (FRT) is an extension of the ordinary Fourier transform (FT). Applying the language of the unified FT, we develop FRT expressions for discrete and continuous signals, introducing a particular form of periodicity: chirp-periodicity. The FRT sampling theorem is derived as an extension of its ordinary counterpart

The paper investigates the possibility for giving a general definition of the fractional Fourier transform (FRT) for all signal classes [one-dimensional (1-D) and multidimensional, continuous and discrete, periodic and aperiodic]. Since the definition is based on the eigenfunctions of the ordinary Fourier transform (FT), the preliminary conditions is that the signal domain/periodicity be the same as the FT domain/periodicity. Within these classes, a general FRT definition is formulated, and the FRT properties are established. In addition, the multiplicity (which is intrinsic in a fractional operator) is clearly developed. The general definition is checked in the case in which the FRT is presently available and, moreover, to establish the FRT in new classes of signals

The multiplicity of the fractional Fourier transform (FRT), which is intrinsic in any fractional operator, has been claimed by several authors, but never systematically developed. The paper starts with a general FRT definition, based on eigenfunctions and eigenvalues of the ordinary Fourier transform, which allows us to generate all possible definitions. The multiplicity is due to different choices of both the eigenfunction and the eigenvalue classes. A main result, obtained by a generalized form of the sampling theorem, gives explicit relationships between the different FRT

The extension of the Fourier transform operator to a fractional power has received much attention in signal theory and is finding attractive applications. The paper introduces and develops the fractional discrete cosine transform (DCT) on the same lines, discussing multiplicity and computational aspects. Similarities and differences with respect to the fractional Fourier transform are pointed ou

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The paper establishes the commutativity conditions of upsampling and downsampling for multidimensional signals defined on discrete Abelian groups (lattices). The general condition includes results available in the literature as particular cases. Complete generality is achieved by giving an abstract definition of up/downsampling and by working in the signal domain instead of the traditional approach based on z and Fourier transforms. Examples of applications are outlined 1) for the one-dimensional (1-D) case verifying existing results, 2) for television scanning, 3) for degenerate (reduced-dimensionality) lattices, and 4) for multiplicative groups

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