In this paper, we propose the novel integral transform coined as the Quadratic-phase Scaled Wigner Distribution (QSWD) by extending the Wigner distri- bution associated with quadratic-phase Fourier trans- form(QWD) to the novel one inspired by the definition of fractional bispectrum. A natural magnification ef- fect characterized by the extra degrees of freedom of the quadratic-phase Fourier transform (QPFT) and by a factor k on the frequency axis enables the QSWD to have flexibility to be used in cross-term reduction. By using the machinery of QSWD and operator theory, we first establish the general properties of the proposed transform, including the conjugate-symmetry, non-linearity, shifting, scaling and marginal. Then, we study the main properties of the proposed transform, including the in- verse, Moyal’s, convolution and correlation. Finally, the applications of the newly defined QSWD for the de- tection of single-component and bi-component linear- frequency-modulated (LFM) signal are also performed to show the advantage of the theory.
Mathematics Subject Classification (2020) 42C40 · 81S30 · 11R52 · 44A35.