Multichannel sampling expansion in the linear canonical transform domain and its application to superresolution

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space [Formula: see text] for convenience. As a generation of FT, it is necessary to detect the linear canonical transform (LCT) on a new space for obtaining the similar properties like FT on [Formula: see text]. Therefore, a space [Formula: see text] generalized from [Formula: see text] is introduced firstly, and further we prove that LCT is a homeomorphism from [Formula: see text] onto itself. The linear canonical wavelet transform (LCWT) is a newly proposed transform based on the convolution theorem in LCT domain. Moreover, we propose an equivalent definition of LCWT associated with LCT and further study some properties of LCWT on [Formula: see text]. Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of [Formula: see text] and [Formula: see text].

Lattices sampling and sampling rate conversion of multidimensional bandlimited signals in the linear canonical transform domain

Analysis of A-stationary random signals in the linear canonical transform domain

The aliased polyphase sampling in the linear canonical transform (LCT) domain has been proposed in this study. The result shows that the sampling theorem can be achieved by using parallel samplers to obtain a higher sampling rate in the LCT domain. Further analysis shows that the LCT spectrum of x ( nM + m ) replicates | X ( a , b , c , d )( u )| with a period bMF s along with linear phase modulation in the LCT domain, where b is one parameter of the LCT, M is the number of samplers and F s is the sampling frequency at which these samplers are operated in the Fourier transform domain. Finally, the simulations are also proposed to verify the correctness of the derived results.

The problem of signal reconstruction using the samples of a signal f(t) and/or signals related to f(t) has been widely investigated in the literature. However, none of the existing sampling expansions deals with the signal reconstruction in the linear canonical transform (LCT) domain using undersampled samples taken in several LCT domains. In this paper, we present some novel sampling expansions involving signal reconstruction using samples of the linear canonical transform (LCT) of the signal in some specific LCT domains.

The linear canonical transform (LCT), which is a generalized form of the classical Fourier transform (FT), the fractional Fourier transform (FRFT), and other transforms, has been shown to be a powerful tool in optics and signal processing. Many results of this transform are already known, including its convolution theorem. However, the formulation of the convolution theorem for the LCT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, we first propose a generalized convolution theorem for the LCT and then derive a corresponding product theorem associated with the LCT. The ordinary convolution theorem for the FT, the fractional convolution theorem for the FRFT, and some existing convolution theorems for the LCT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented.

Uncertainty principle plays an important role in signal processing, physics and mathematics, and it represents the relations between time spread and frequency spread (or position and velocity). Linear canonical transform (LCT) is one generalisation of Fresnel transform, fractional Fourier transform and others. The LCT has been used in physical optics and signal processing. Three novel results of uncertainty principle in the LCT domains are obtained here, in which one is connected with parameters a and b and the other one is connected with c and d; the last one is connected with the four transformation parameters a, b, c and d. Their physical meanings are given as well. These results disclose the inequalities' relations between two spreads, between two group delays and between one spread and one group delay in the LCT domains. It also shows that any one of the three cases can reduce to classical uncertainty principle in time/frequency domain. The effects of time scaling on these results' bounds are also involved.

The novel Hausdorff–Young inequalities associated with the linear canonical transform (LCT) are derived based on the relation between the Fourier transform and the LCT in p-norm space (0<p<∞). Uncertainty relations for Shannon entropy and Renyi entropy based on the derived Hausdorff–Young inequality are yielded. It shows that these relations are functions of the transform parameters (a, b, c, d). Meanwhile, from the uncertainty relation for Shannon entropy the Heisenberg's uncertainty relation in LCT domains is derived, which holds for both real and complex signals. Moreover, the Heisenberg's uncertainty principle for the windowed fractional Fourier transform is obtained. Finally, one review of the uncertainty relations for the LCT and other transforms is listed in tables systematically for the first time.

Speech recovery based on the linear canonical transform

In this paper, we mainly investigate the nonuniform sampling for random signals which are bandlimited in the linear canonical transform (LCT) domain. We show that the nonuniform sampling for a random signal bandlimited in the LCT domain is equal to the uniform sampling in the sense of second order statistic characters after a pre-filter in the LCT domain. Moreover, we propose an approximate recovery approach for nonuniform sampling of random signals bandlimited in the LCT domain. Furthermore, we study the mean square error of the nonuniform sampling. Finally, we do some simulations to verify the correctness of our theoretical results.

In this paper, we investigate sampling expansion for the linear canonical transform (LCT) in function spaces. First, some properties of the function spaces related to the LCT are obtained. Then, a sampling theorem for the LCT in function spaces with a single-frame generator is derived by using the Zak Transform and its generalization to the LCT domain. Some examples are also presented.

Generalized Heisenberg uncertainty relations for complex-valued signals out of Hilbert transform associated with LCT

The main objective of the paper is to generalize and enrich the Weyl transform by introducing the Weyl correspondence in the linear canonical transform (LCT) domain. In this paper, we propose the linear canonical-Wigner transform in harmonic analysis of phase space along with the admissible Wigner-Ville distribution (WVD) and Weyl transform in the LCT domain and discuss some useful results. Further we establish the relationship between the Wigner-Ville distribution and the Weyl transform in the LCT domain.

The linear canonical transform (LCT) has been shown to be useful and powerful in signal processing, optics, etc. Many results of this transform are already known, including sampling theory. Most existing sampling theories of the LCT consider the class of bandlimited signals. However, in the real world, many analog signals arising in engineering applications are non-bandlimited. In this correspondence, we propose a sampling and reconstruction strategy for a class of function spaces associated with the LCT, which can provide a suitable and realistic model for real applications. First, we introduce definitions of semi- and fully-discrete convolutions for the LCT. Then, we derive necessary and sufficient conditions pertaining to the LCT, under which integer shifts of a chirp-modulated function generate a Riesz basis for the function spaces. By applying the results, we present a more comprehensive sampling theory for the LCT in the function spaces, and further, a sampling theorem which recovers a signal from its own samples in the function spaces is established. Moreover, some sampling theorems for shift-invariant spaces and some existing sampling theories for bandlimited signals associated with the Fourier transform (FT), the fractional FT, or the LCT are noted as special cases of the derived results. Finally, some potential applications of the derived theory are presented.

Discrete quaternion linear canonical transform