Deyun Wei

Generalized Sampling Expansion for Bandlimited Signals Associated With the Fractional Fourier Transform

The aim of the generalized sampling expansion (GSE) is the reconstruction of an unknown continuously defined function f(t), from the samples of the responses of M linear time invariant (LTI) systems, each sampled by the 1/M th Nyquist rate. In this letter, we investigate the GSE in the fractional Fourier transform (FRFT) domain. Firstly, the GSE for fractional bandlimited signals with FRFT is proposed based on new linear fractional systems, which is the generalization of classical generalized Papoulis sampling expansion. Then, by designing fractional Fourier filters, we obtain reconstruction method for sampling from the signal and its derivative based on the derived GSE and the property of FRFT. Last, the potential application of the GSE is presented to show the advantage of the theory.

A Convolution and Correlation Theorem for the Linear Canonical Transform and Its Application

As a generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) plays an important role in many fields of optics and signal processing. Many properties for this transform are already known, but the correlation theorem, similar to the version of the Fourier transform (FT), is still to be determined. In this paper, firstly, we introduce a new convolution structure for the LCT, which is expressed by a one dimensional integral and easy to implement in filter design. The convolution theorem in FT domain is shown to be a special case of our achieved results. Then, based on the new convolution structure, the correlation theorem is derived, which is also a one dimensional integral expression. Last, as an application, utilizing the new convolution theorem, we investigate the sampling theorem for the band limited signal in the LCT domain. In particular, the formulas of uniform sampling and low pass reconstruction are obtained.

Sampling of bandlimited signals in the linear canonical transform domain

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This paper investigates the sampling of bandlimited signals in LCT domain. First, we propose the linear canonical series (LCS) based on the LCT, which is a generalized pattern of Fourier series. Moreover, the LCS inherits all the nice properties from the LCT. Especially, the Parseval’s relation is presented for the LCS, which is used to derive the sampling theorem of LCT. Then, utilizing the generalized form of Parseval’s relation for the complex LCS, we obtain the sampling expansion for bandlimited signals in LCT domain. The advantage of this reconstruction method is that the sampling expansion can be deduced directly not based on the Shannon theorem.

Multiplicative filtering in the fractional Fourier domain

In this article, we investigate the multiplicative filtering in the fractional Fourier transform (FRFT) domain based on the generalized convolution theorem which states that the convolution of two signals in time domain results in simple multiplication of their FRFTs in the FRFT domain. In order to efficiently implement multiplicative filtering, we express the generalized convolution structure by the conventional convolution operation. Utilizing the generalized convolution structure, we convert the multiplicative filtering in the FRFT domain easily to the time domain. Based on the model of multiplicative filtering in the FRFT domain, a practical method is proposed to achieve the multiplicative filtering through convolution in the time domain. This method can be realized by classical Fast Fourier transform (FFT) and has the same capability compared with the method achieved in the FRFT domain. As convolution can be performed by FFT, this method is more useful from practical engineering perspective.

Reply to “Comments on ‘A Convolution and Product Theorem for the Linear Canonical Transform’”

In this letter, we reply to the comment regarding our recent report on filtering. The authors found that two kinds of filtering methods are essentially the same . However, we will demonstrate that the multiplicative filter in the linear canonical transform (LCT) domain is carried out easily using our convolution structure. Considering the designing of filter, firstly, we can easily convert the multiplicative filter in the LCT domain to the time domain using our formula. Secondly, we can easily implement with hardware unit though our filtering model. Lastly, we present a simple example to illustrate the advantage of our convolution structure.

Multi-channel sampling expansion in fractional Fourier domain and its application to superresolution

This paper addresses the problem of multi-channel sampling (MS) expansion for fractional bandlimited signal in the fractional Fourier domain. Firstly, the MS expansion for fractional bandlimited signals with fractional Fourier transform (FRFT) is proposed based on new multi-channel system equations, which is the generalization of classical generalized Papoulis sampling expansion. The MS expansion which is constructed by the ordinary convolution in the time domain is easy to implement. Then, by designing a type of fractional Fourier filters, the reconstruction expression for the recurrent nonuniformly sampled signal has been obtained by using the derived MS expansion. Last, the application of MS in the context of the image superresolution is also discussed. The simulation results of image superresolution are also presented.