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

Abstract

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.

A modified convolution and product theorem for the linear canonical transform derived by representation transformation in quantum mechanics

“…It is seen from the results of Figs. 2 and 3 that the proposed weighted convolution theorem gives better results than the convolution theorem by Deng et al (2006) and Wei et al (2012;2009), as the results given by the proposed theorem resemble maximally the shape of the LCT of the triangular function. Further, an application of filtering was presented with the proposed convolution theorem and it was found that with the help of proposed theorem the signal is recovered with a minimum mean square error.…”

“…The convolution theorem in the LCT domain given by Deng et al (2006) and Wei et al (2012;2009) is compared with the proposed convolution theorem by simulating on system with an Intel core TM i3-330M 2.13 GHz processor with 3 GB RAM. The convolution operation of a rectangular function x(t) of unit amplitude is performed with itself, i.e., (x ⊗ x)(t).…”

“…1. Then the LCT of convolution operations defined by Deng et al (2006) and Wei et al (2012;2009) is compared with the LCT of the proposed convolution operation for (a, b, c, d) = (0.707, 0.707, −0.707, 0.707) and (a, b, c, d) = (0.5, 0.866, −0.866, 0.5) as shown in Figs. 2 and 3, respectively.…”

“…Many properties of the LCT and FRFT are already known (Alieva and Bastiaans, 1999;Pei and Ding, 2002;Ozaktas et al, 2000), including the product and convolution theorem (Deng et al, 2006;Ozaktas et al, 1994;Almeida,1997;Zayed,1998;Sharma and Joshi, 2007;Wei et al, 2012;2009;Singh and Saxena, 2011), but none have received acclamation because either their definitions do not generalize very nicely to the classical definition for the FT or result in a bigger hardware complexity due to a large number of chirp signals.…”

“…In other words, the Fourier transform of the convolution of two signals is the point-wise product of the Fourier transform of their respective signals. In the literature, many definitions of the convolution theorem for the LCT (Deng et al, 2006;Wei et al, 2012;2009) are proposed. The classical definition of the convolution and product theorem of the FT for the signals f (x) and g(x) is given by Convolution:…”