Quadratic-phase scaled Wigner distribution: convolution and correlation

Abstract

In this paper, we propose the novel integral transform coined as the Quadratic-phase Scaled Wigner Distribution (QSWD) by extending the Wigner distribution associated with quadratic-phase Fourier transform(QWD) to the novel one inspired by the definition of fractional bispectrum. A natural magnification effect 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 inverse, Moyal's, convolution and correlation. Finally, the applications of the newly defined QSWD for the detection of single-component and bi-component linearfrequency-modulated (LFM) signal are also performed to show the advantage of the theory.

Introductory Chapter: The Generalizations of the Fourier Transform

“…The most neoteric generalization of the classical Fourier transform (FT) with five real parameters appeared via the theory of reproducing kernels is known as the quadratic-phase Fourier transform (QPFT) [17]. It treats both the stationary and nonstationary signals in a simple and insightful way that are involved in radar, signal processing, and other communication systems [18][19][20][21][22][23][24][25]. Here, we gave the notation and definition of the quadratic-phase Fourier transform and study some of its properties.…”

Wigner-Ville distribution and ambiguity function of QPFT signals

“…Convolution and correlation results are very significant in signal and image processing. In [29] authors introduced the convolution theorem for WVD-QPFT, but in this section we establish convolution for the proposed transform by using the two parameter convolution operator, thus it is more flexible and accurate. We then establish the new convolution and correlation for the WVD-QPFT, these theorems will open new gates to investigate the sampling and filtering theorems of the WVD-QPFT.…”

“…For more results on convolution and correlation, we refer to [14]- [15]. In [29], the authors proposed Winger distribution associated with quadratic phase transform (WVD-QPFT) as a generalization of classical WVD by substituting the kernel with the QPFT kernel. They established some vital properties like the marginal, shifting, conjugate-symmetry, anti-derivative, Moyal's and inversion formulae.…”

Wigner–Ville Distribution Associated with Clifford Geometric Algebra Cln,0, n=3(mod 4) Based on Clifford–Fourier Transform

“…Hahn and Snopek developed Fourier-Wigner distributions of 2D quaternion signals [15], and then Bahri thoroughly discussed the 2D WVD associated with QFT [16]. Since then, tremendous work has been done on WVD [17][18][19][20]. The idea of associating the WVD with the Clifford algebra of n-dimensional Euclidean space R n has not been explored yet.…”