The fractional Fourier transform has found applications in a variety of topics linked with science and engineering. In this context, sampling theory is one of the most well-studied subjects. Since the fractional Fourier transform or the FrFT generalizes the notion of bandlimitedness, extension of Shannon's sampling theorem to the FrFT domain generalizes the classical result for the Fourier domain. These ideas have further been extended to the class of non-bandlimited functions via shift-invariant subspaces and sparse models. In this paper, we discuss a different approach to sampling theory in the FrFT domain. For the first time, we propose sampling and recovery of bandlimited functions in the FrFT domain that is based on one-bit samples. Our work is inspired by the Sigma-Delta quantization scheme. In particular, we capitalize on the idea of noise shaping and develop a one-bit sampling architecture that allows for recovery of bandlimited functions in the FrFT domain by pushing quantization noise to the higher frequencies. Since the FrFT generalizes the Fourier transform, our work results in a generalized Sigma-Delta architecture. We validate our theoretical concepts through computer experiments and provide an approximation theoretic error bound.

We derive sampling formula for signals that are bandlimited to a disc of radius R in the linear canonical transform (LCT) domain. By bandlimitedness in a disc Din the LCT domain, we mean that the LCT F (ω) of a signal f(t) vanishes outside the disc D. We first express the signal in polar coordinates and then obtain the sampling formula. The samples of the angle θ are taken at 2N + 1 uniformly distributed points on the unit circle while the samples of the radial distance r are taken at the zeros of the Bessel function. As a special case, we obtain sampling formula for signals that are bandlimited to a disc in the fractional Fourier transform domain.

The fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has many applications in several areas, including signal processing and optics. Almeida (see ibid., vol.4, p.15-17, 1997) and Mendlovic et al. (see Appl. Opt., vol.34, p.303-9, 1995) derived fractional Fourier transforms of a product and of a convolution of two functions. Unfortunately, their convolution formulas do not generalize very well the classical result for the Fourier transform, which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. This paper introduces a new convolution structure for the FRFT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters.

The analytic part of a signal f(t) is obtained by suppressing the negative frequency content of f, or in other words, by suppressing the negative portion of the Fourier transform, f/spl circ/, of f. In the time domain, the construction of the analytic part is based on the Hilbert transform f/spl circ/ of f(t). We generalize the definition of the Hilbert transform in order to obtain the analytic part of a signal that is associated with its fractional Fourier transform, i.e., that part of the signal f(t) that is obtained by suppressing the negative frequency content of the fractional Fourier transform of f(t). We also show that the generalized Hilbert transform has similar properties to those of the ordinary Hilbert transform, but it lacks the semigroup property of the fractional Fourier transform.

We derive a sampling expansion for bandlimited signals with polynomial growth on the real axis. The sampling expansion uses nonuniformly spaced sampling points. But unlike other known sampling expansions for such signals, ours converge uniformly to the signal on any compact set. An estimate of the truncation error of such a series is also obtained.

In recent years, the fractional Fourier transform has been the focus of many research papers. In this letter, we show that the fractional Fourier transform is nothing more than a variation of the standard Fourier transform and, as such, many of its properties, such as its inversion formula and sampling theorems, can be deduced from those of the Fourier transform by a simple change of variable.