Convolution theorems for the linear canonical transform and their applications

Abstract

As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.

Fast linear canonical transforms

“…If the LCT of the continuous signal has compact support at least approximately over a finite range, and if the sampling rate used is sufficient to prevent the replicas from overlapping, it is possible to extract one copy from among the replicas using an appropriate filter. Deng et al [35] published a derivation of the same theorem almost simultaneously with that of Stern (the two papers being submitted less than three weeks apart). Their approach was to derive the convolution and multiplication theorems for the LCT and then calculate the LCT of the product of an arbitrary function and a train of delta functions.…”

Exact Relation Between Continuous and Discrete Linear Canonical Transforms

“…Then, apply the LCT operator to the equivalent expressions for in (10) to obtain (11) where . This result is the generalization of the Poisson sum formula [5], and is related to the LCT sampling theorem [9]- [11].…”

Relationship between sampling and multirate filterbanks in the linear canonical transform domain

“…The well-known operations such as Fourier transform (FT), the fractional Fourier transform (FRFT) [11], the Fresnel transform [13], and the scaling operation are all special cases of the LCT. The LCT is also applied in filter design, signal synthesis, pattern recognition, timefrequency analysis, holographic three-dimensional television, and many others [14][15][16][17]. Understanding the LCT can help gain more insight into its special cases and carry the knowledge gained from one subject to others [12].…”