Linear Canonical Transforms’ Discretization Formula Based on the Frequency-Domain Convolution Theory

Abstract

The Linear Canonical Transforms’ (LCT) domain not only contains the informations of time and frequency, but also includes the informations of varying frequency with time. Because of these characteristics which makes it have a unique advantage in dealing with nonstationary signals, it attracts more and more scientists and the engineers’ attention. Fourier Transform (FT), Fractional Fourier Transform (FrFT), Fresnel Transform and Scale Transforms can be seen as a special LCT’s form. And in the combination of time-frequency analysis method (Wigner distribution, short-time Fourier transform (STFT), Fuzzy function) [2] on the basis of Linear Canonical Transforms will have more development space. However, the discretization analysis of Linear Canonical Transforms is not clear enough, especially the discretization formula is not suitable for computer’s calculation. Therefore, based on the sampling theorem of bandpass signal, the discretization formula is deduced based on the frequency-domain discretization formula and the convolution formula, and the correctness and reversibility are verified on the computer’s calculation.