Inference of Parameterized Fourier Transforms

This paper presents new theoretical and application aspects on inferring the parameters in integral transforms. An integral transform, e.g., the linear canonical transform (LCT), usually has many parameters. It is a difficult issue to find the optimal parameters of the integer transform. In this work, we propose a generalized inference method to determine the optimal parameters of the LCT and the fractional Fourier transform (FrFT). With proper design of object functions, optimal parameter estimation can be performed effectively and automatically. The LCT with the optimal parameter set can well represent a complex signal and minimize the energy outside of the low-frequency band in terms of the ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> or the ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> norm. It is applicable in signal processing, filter design, system modeling, and machine learning.