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This paper addresses the problem of multidimensional signal reconstruction from generalized samples in fractional Fourier domain including the deterministic case and the stochastic case. The generalized sampling expansion is investigated for the case where the fractional bandlimited input depends on N real variable, i.e., f ( t ) = f ( t 1 , ⋯ , t N ) and is used as a common input to a parallel bank of m independent N dimensional linear fractional Fourier filters Hα,k(u), k = 1 , ⋯ , m . For the deterministic input, the input is assumed to have its N dimensional fractional Fourier transform bandlimited to the frequency rang |ui|≤Ωi, for i = 1 , ⋯ , N . If m, the number of fractional Fourier filters, is written as a product of positive integers in the form m = m 1 m 2 ⋯ m N , and if the fractional bandlimited input f(t) is processed by fractional Fourier filter Hα,k(u)resulting m outputs gk(t), then f(t) can be reconstructed in terms of the samples gk(nT), each output being sampled at the identical rates of Ω 1 csc α / m 1 π , Ω 2 csc α / m 2 π , ⋯ , Ω N csc α / m N π samples/second in t 1 , ⋯ , t N respectively. This contrasts with the rates of Ω 1 csc α / π , Ω 2 csc α / π , ⋯ , Ω N csc α / π in t 1 , ⋯ , t N needed for reconstruction of the unfiltered input f(t). Input sampling expansions in terms of samples of the output filters are given for both deterministic and stochastic inputs, the generalized sampling expansion for random input having the same form as for the deterministic case but interpreted in the mean-square sense. Our formulation and results are general and include derivative sampling and periodic nonuniform sampling in the fractional Fourier domain for multidimensional signals as special case. Finally, the potensional application of the multidimensional generalized sampling is presented to show the advantage of the theory. Especially, the application of multidimensional generalized sampling in the context of the image scaling about image super-resolution is also discussed. The simulation results of image scaling are also presented.

The linear canonical transform (LCT) describes the effect of first-order quadratic phase optical system on a wave field. In this paper, we address the problem of signal reconstruction from multichannel samples in the LCT domain based on a new convolution theorem. Firstly, a new convolution structure is proposed for the LCT, which states that a modified ordinary convolution in the time domain is equivalent to a simple multiplication operation for LCT and Fourier transform (FT). Moreover, it is expressible by a one dimensional integral and easy to implement in the designing of filters. The convolution theorem in FT domain is shown to be a special case of our achieved results. Then, a practical multichannel sampling expansion for band limited signal with the LCT is introduced. This sampling expansion which is constructed by the new convolution structure can reduce the effect of spectral leakage and is easy to implement. Last, the potential application of the multichannel sampling is presented to show the advantage of the theory. Especially, the application of multichannel sampling in the context of the image superresolution is also discussed. The simulation results of superresolution are also presented.

Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. This paper addresses the problem of signal reconstruction from multichannel and periodic nonuniform samples in the LCT domain. Firstly, the multichannel sampling theorem (MST) for band-limited signals with the LCT is proposed based on multichannel system equations, which is the generalization of the well-known sampling theorem for the LCT. We consider the problem of reconstructing the signal from its samples which are acquired using a multichannel sampling scheme. For this purpose, we propose two alternatives. The first scheme is based on the conventional Fourier series and inverse LCT operation. The second is based on the conventional Fourier series and inverse Fourier transform (FT) operation. Moreover, the classical Papoulis MST in FT domain is shown to be special case of the achieved results. Since the periodic nonuniformly sampled signal in the LCT has valuable applications, the reconstruction expression for the periodic nonuniformly sampled signal has been then obtained by using the derived MST and the specific space-shifting property of the LCT. Last, the potential applications of the MST are presented to show the advantage of the theory.