Special Affine Fourier Transformation Research Articles

Octonion Special Affine Fourier Transform: Pitt’s Inequality and the Uncertainty Principles

The special affine Fourier transform (SAFT) is an extended version of the classical Fourier transform and incorporates various signal processing tools which include the Fourier transforms, the fractional Fourier transform, the linear canonical transform, and other related transforms. This paper aims to introduce a novel octonion special affine Fourier transform (O−SAFT) and establish several classes of uncertainty inequalities for the proposed transform. We begin by studying the norm split and energy conservation properties of the proposed (O−SAFT). Afterwards, we generalize several uncertainty relations for the (O−SAFT) which include Pitt’s inequality, Heisenberg–Weyl inequality, logarithmic uncertainty inequality, Hausdorff–Young inequality, and local uncertainty inequalities. Finally, we provide an illustrative example and some possible applications of the proposed transform.

Modulation Spaces, Multipliers Associated with the Special Affine Fourier Transform

We study some fundamental properties of the special affine Fourier transform (SAFT) in connection with the Fourier analysis and time-frequency analysis. We introduce the modulation space $${{\varvec{M}}}^{r,s}_A$$ in connection with SAFT and prove that if a bounded linear operator between new modulation spaces commutes with A-translation, then it is a A-convolution operator. We also establish Hörmander multiplier theorem and Littlewood-Paley theorem associated with the SAFT.

Uncertainty Principle for the Short-time Special Affine Fourier Transform

The special affine Fourier transform (SAFT) generalizes a number of well known unitary transformations, signal processing transformations and optics related mathematical operations. For example, Fourier transform, fractional Fourier transform, linear canonical transform, etc. The short-time special affine Fourier transform (STSAFT) is a novel time-frequency analysis tool, which solve the limitation of SAFT in time-frequency signal description. In this paper, we generalize some different uncertainty principles (UPs) for the SAFT and STSAFT of complex signals. The UPs for the STSAFT of complex signals are obtained in both local and global cases. Then the uncertainty principle for two STSAFT time-frequency domains is discussed. Finally, we provide a class of signals to illustrate our results.

Novel special affine wavelet transform and associated uncertainty principles

Due to the extra degrees of freedom, special affine Fourier transform (SAFT) has achieved a respectable status within a short span and got versatile applicability in the areas of signal processing, image processing, sampling theory, quantum mechanics. However, due to its global kernel, SAFT fails to obtain local information of non-transient signals. To overcome this, in this paper, we introduce the concept of novel special affine wavelet transform (NSAWT) and extend key harmonic analysis results to NSAWT analogous to those for the wavelet transform. We first establish some fundamental properties including Moyal’s principle, Inversion formula and the range theorem. Some Heisenberg type inequalities and Pitt’s inequality are established for SAFT and consequently Heisenberg uncertainty principle is derived for NSAWT.

Sampling and Super Resolution of Sparse Signals Beyond the Fourier Domain

Recovering a sparse signal from its low-pass projections in the Fourier domain is a problem of broad interest in science and engineering and is commonly referred to as super-resolution. In many cases, however, Fourier domain may not be the natural choice. For example, in holography, low-pass projections of sparse signals are obtained in the Fresnel domain. Similarly, time-varying system identification relies on low-pass projections on the space of linear frequency modulated signals. In this paper, we study the recovery of sparse signals from low-pass projections in the Special Affine Fourier Transform domain (SAFT). The SAFT parametrically generalizes a number of well known unitary transformations that are used in signal processing and optics. In analogy to the Shannon's sampling framework, we specify sampling theorems for recovery of sparse signals considering three specific cases: (1) sampling with arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels and, (3) recovery from Gabor transform measurements linked with the SAFT domain. Our work offers a unifying perspective on the sparse sampling problem which is compatible with the Fourier, Fresnel and Fractional Fourier domain based results. In deriving our results, we introduce the SAFT series (analogous to the Fourier series) and the short time SAFT, and study convolution theorems that establish a convolution--multiplication property in the SAFT domain.

Shift-invariant and sampling spaces associated with the special affine Fourier transform

The Special Affine Fourier Transformation or the SAFT generalizes a number of well known unitary transformations as well as signal processing and optics related mathematical operations. Shift-invariant spaces also play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. Shannon's sampling theorem, which is at the heart of modern digital communications, is a special case of sampling in shift-invariant spaces. Furthermore, it is well known that the Poisson summation formula is equivalent to the sampling theorem and that the Zak transform is closely connected to the sampling theorem and the Poisson summation formula. These results have been known to hold in the Fourier transform domain for decades and were recently shown to hold in the Fractional Fourier transform domain by A. Bhandari and A. Zayed. The main goal of this article is to show that these results also hold true in the SAFT domain. We provide a short, self-contained proof of Shannon's theorem for functions bandlimited in the SAFT domain and then show that sampling in the SAFT domain is equivalent to orthogonal projection of functions onto a subspace of bandlimited basis associated with the SAFT domain. This interpretation of sampling leads to least-squares optimal sampling theorem. Furthermore, we show that this approximation procedure is linked with convolution and semi-discrete convolution operators that are associated with the SAFT domain. We conclude the article with an application of fractional delay filtering of SAFT bandlimited functions.

Poisson Summation Formulae Associated with the Special Affine Fourier Transform and Offset Hilbert Transform

This paper investigates the generalized pattern of Poisson summation formulae from the special affine Fourier transform (SAFT) and offset Hilbert transform (OHT) points of view. Several novel summation formulae are derived accordingly. Firstly, the relationship between SAFT (or OHT) and Fourier transform (FT) is obtained. Then, the generalized Poisson sum formulae are obtained based on above relationships. The novel results can be regarded as the generalizations of the classical results in several transform domains such as FT, fractional Fourier transform, and the linear canonical transform.

A generalized convolution theorem for the special affine Fourier transform and its application to filtering

Abstract The special affine Fourier transform (SAFT), which is a time-shifted and frequency-modulated version of the linear canonical transform (LCT), has been shown to be a powerful tool for signal processing and optics. Many properties for this transform are already known, but an extension of convolution theorem of Fourier transform (FT) is still not having a widely accepted closed form expression. The purpose of this paper is to introduce a new convolution structure for the SAFT that preserves the convolution theorem for the FT, which states that the FT of the convolution of two functions is the product of their Fourier transforms. Moreover, some of well-known results about the convolution theorem in FT domain, fractional Fourier transform (FRFT) domain, LCT domain are shown to be special cases of our achieved results. Last, as an application, utilizing the new convolution theorem, we investigate the multiplicative filter in the SAFT domain. The new convolution structure is easy to implement in the designing of filters.

Solution of the Energy Concentration Problem in Reproducing-Kernel Hilbert Space

The problem of maximizing the energy of a signal bandlimited to $[-\sigma, \sigma]$ on an interval $[-\tau, \tau]$ in the time domain is one of the important classical problems in signal processing. This problem was solved by a group of mathematicians, Slepian, Landau, and Pollak, at Bell Labs in the 1960s. The solution involved the prolate spheroidal wave functions. More recently, Pei and Ding solved the energy problem for more general transforms than the Fourier transform, such as the fractional Fourier transform and the linear canonical transform. Their solution involved what they called the generalized prolate spheroidal wave functions. The goals of this article are (1) to show that the problem of finding a bandlimited signal with maximum energy concentration in a given interval is a special case of a more general problem in reproducing-kernel Hilbert spaces, (2) to solve the problem in the setting of a Hilbert space and then obtain the solutions for the Fourier, fractional Fourier and linear canonical transforms, as well as the special affine Fourier transformation, as special cases, and (3) to show that the solution in the Hilbert space setting leads naturally to the solution of the energy problem for multidimensional signals.

Special affine Fourier transformation in frequency-domain

Special affine Fourier transformation (SAFT) can be considered as an extension of the fractional Fourier transformation (FRT) and the most general linear mapping in phase space. A general formula for SAFT in frequency-domain is derived, which gives a direct relationship between the input and output spatial frequency spectra of a light field. It shows that the SAFT has similar and symmetric feature in both space- and frequency-domains. As its special cases, Collins formula in frequency-domain, the spatial frequency representations of the almost-FRT, almost-Fresnel and almost-Fourier transformations are explicitly obtained. These formulae may provide a tool for investigating the performance of a lossless optical system including small deformations in both domains in a unified way within the framework of linear theory.

Comment on ‘The Fractional Fourier Transform in Optical Propagation Problems’

Although their mathematical forms apparently resemble each other, the diffraction integral and fractional Fourier transformation (FRT) have completely different physical meanings. We point out that an interpretation of the FRT given recently in a paper by Alieva et al. is not physically appropriate. We then show how those integral transformations can be treated in a unified way within the framework of the special affine Fourier transformation. Finally the multidimensional FRT presented in the above paper is further generalized to allow n independent fractional degrees.

Comment on 'The Fractional Fourier Transform in Optical Propagation Problems'

Although their mathematical forms apparently resemble each other, the diffraction integral and fractional Fourier transformation (FRT) have completely different physical meanings. We point out that an interpretation of the FRT given recently in a paper by Alieva et al. is not physically appropriate. We then show how those integral transformations can be treated in a unified way within the framework of the special affine Fourier transformation. Finally the multidimensional FRT presented in the above paper is further generalized to allow n independent fractional degrees.

Almost-Fourier and almost-Fresnel transformations

Based on the special affine Fourier transformation (SAFT), which is an extension of the fractional Fourier transformation, the small deviations from the perfect optical operations on wave functions are treated in a unified way. As simple and instructive examples, the almost-Fourier and almost-Fresnel transformations are constructed explicitly. The present investigation shows transformations which include the small imperfections of optical instruments can be developed within the framework of linear theory.

Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation

The special affine Fourier transformation (SAFT) is a generalization of the fractional Fourier transformation (FRT) and represents the most general lossless inhomogeneous linear mapping, in phase space, as the integral transformation of a wave function. Here we first summarize the most well-known optical operations on light-wave functions (i.e., the FRT, lens transformation, free-space propagation, and magnification), in a unified way, from the viewpoint of the one-parameter Abelian subgroups of the SAFT. Then we present a new operation, which is the Lorentz-type hyperbolic transformation in phase space and exhibits squeezing. We also show that the SAFT including these five operations can be generated from any two independent operations.

Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach

We develop the operator formalism to show how systematically the fractional Fourier transformation of a wavefunction, recently introduced by Namias (1980), can be derived from the rotation of the corresponding Wigner distribution function in phase space. In this formalism, the phase factor obtained by McBride and Kerr (1987) is seen to come from the caustics of the harmonic oscillator Green function. Then the idea is generalized to the case of an arbitrary area-preserving linear transformation in phase space, and a concept of the special affine Fourier transformation (SAFT) is introduced. An explicit form of the integral representation of the SAFT is given, and some simple examples are presented.