Linear canonical transforms play an important role in many fields of optics and signal processing. Well-known transforms such as the Fourier transform, the fractional Fourier transform, and the Fresnel transform can be seen as special cases of the linear canonical transform. In this paper we develop a sampling theorem for linear canonical transformed signals. The well-known Shannon sampling the...

We survey a new application of the Weil representation to construct a canonical basis Φ of eigenvectors for the discrete Fourier transform (DFT). The transition matrix Θ from the standard basis to Φ defines a novel transform which we call the discrete oscillator transform (DOT for short). In addition, we describe a fast algorithm for computing Θ in certain cases.

The offset linear canonical transform (OLCT) is the name of a parameterized continuum of transforms which include, as particular cases, the most widely used linear transforms in engineering such as the Fourier transform (FT), fractional Fourier transform (FRFT), Fresnel transform (FRST), frequency modulation, time shifting, time scaling, chirping and others. Therefore the OLCT provides a unifie...

The generalization of the classical Poisson sum formula, by replacing the ordinary Fourier transform by the canonical transformation, has been derived in the linear canonical transform sense. Firstly, a new sum formula of Chirp-periodic property has been introduced, and then the relationship between this new sum and the original signal is derived. Secondly, the generalization of the classical P...

A signal may have compact support, be band-limited (i.e., its Fourier transform has compact support), or neither ("unbounded"). We determine conditions for the linear canonical transform of a signal having these properties. We examine the significance of these conditions for special cases of the linear canonical transform and consider the physical significance of our results.

Nowadays, almost all images acquired are in color format. Traditional methods process color images by either transforming them into gray scale or dividing them into red, green, and blue components for independent processing, which is definitely not effective in representing color information. Recently, a novel Polar Linear Canonical Transform (PLCT) with parameters in SL(2,<) has been reported,...

This paper investigates the interpolation formulae and the sampling theorem for bandpass signals in the linear canonical transform LCT domain. Firstly, one of the important relationships between the bandpass signals in the Fourier domain and the bandpass signals in the LCT domain is derived. Secondly, two interpolation formulae from uniformly sampled points at half of the sampling rate associat...

In [M. K. Vemuri, Realizations of the canonical representation, submitted], it was shown that the spectral synthesis problem for the Alpha transform is closely related to the problem of classifying realizations of the canonical representation (of the Heisenberg group). In this paper, we show that discrete sets are sets of spectral synthesis for the Alpha transform.

The linear canonical transform provides a mathematical model of paraxial propagation though quadratic phase systems. We review the literature on numerical approximation of this transform, including discretization, sampling, and fast algorithms, and identify key results. We then propose a frequency-division fast linear canonical transform algorithm comparable to the Sande-Tukey fast Fourier tran...