The New Kind of Convolution and Correlation Theorems Associated with the Linear Canonical Wavelet Transform

Linear canonical transform (LCT), distinguished by its three free parameters, provides remarkable flexibility, establishing itself as a fundamental tool for time-frequency analysis and the examination of non-stationary signals. In recent years, wavelet transform (WT) has gained substantial attention as a potent signal analysis technique. Nevertheless, researchers in the domains of signal processing and image processing are continuously striving to develop innovative techniques for a better understanding and analysis of diverse signals. This thesis focuses on the exploration of a novel transformation, namely linear canonical wavelet transform (LCWT), which seamlessly integrates the strengths of both LCT and WT while addressing their inherent limitations. LCWT has emerged as a robust tool for signal processing. However, the theoretical framework for certain aspects of this transformation, such as convolution and its correlation theorems, remains imperfect. In response, we propose a novel convolution method to enhance the understanding of LCWT. This paper begins with a concise introduction to the fundamental theory of LCWT. Subsequently, we introduce a pioneering convolution and correlation operator and derive the convolution and correlation theorem by amalgamating LCWT. Finally, leveraging the derived theorem, we have proposed the theory for a novel filtering design approach within the domain of LCWT.