Optimal design of orders of DFrFTs for sparse representations

• Published in: IET signal processing Vol. 12; no. 8; pp. 1023 - 1033
• Main Authors: Zhang, Xiao-Zhi, Ling, Bingo Wing-Kuen, Tao, Ran, Yang, Zhi-Jing, Woo, Wai-Lok, Sanei, Saeid, Teo, Kok L
• Format: Journal Article
• Language: English
• Published: The Institution of Engineering and Technology 01.10.2018
• Subjects: concave programming; constrained optimisation problem; DFrFT orders; discrete Fourier transforms; discrete fractional Fourier transforms; harmonic functions; L1 ‐norm nonconvex objective function; optimal design; optimal frequency sampling problem; overcomplete transform; Research Article; signal representation; signal sampling; sparse representations; time‐frequency plane; discrete Fourier transforms; discrete fractional Fourier transforms; sparse representations; constrained optimisation problem; DFrFT orders; signal representation; L1-norm nonconvex objective function; optimal design; overcomplete transform; time-frequency plane; optimal frequency sampling problem; signal sampling; harmonic functions; concave programming
• ISSN: 1751-9675; 1751-9683; 1751-9683
• DOI: 10.1049/iet-spr.2017.0283

This study proposes an optimal design of the orders of the discrete fractional Fourier transforms (DFrFTs) and construct an overcomplete transform using the DFrFTs with these orders for performing the sparse representations. The design problem is formulated as an optimisation problem with an $L_1$L1-norm non-convex objective function. To avoid all the orders of the DFrFTs to be the same, the exclusive OR of two constraints are imposed. The constrained optimisation problem is further reformulated to an optimal frequency sampling problem. A method based on solving the roots of a set of harmonic functions is employed for finding the optimal sampling frequencies. As the designed overcomplete transform can exploit the physical meanings of the signals in terms of representing the signals as the sums of the components in the time–frequency plane, the designed overcomplete transform can be applied to many applications.