Quadratic-phase scaled Wigner distribution: convolution and correlation

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 linear frequency-modulated (LFM) signal are also performed to show the advantage of the theory.


Introduction
In modern day communication systems, medical imaging, sonar, radar and wireless communications, we have to deal with linear frequency-modulated (LFM) signals which are chirp-like non-transient signals. Thus, the analysis of time-frequency characteristics of non-transient signals is very important. The most important distribution of all the time-frequency distributions which serves as the versatile frequency analysis tool particularly for the detection of chirplike signals is the Wigner distribution (WD) [1][2][3][4][5][6][7][8][9][10].
Denoting f t + ρ 2 f * t − ρ 2 (superscript * denotes complex conjugate) by the instantaneous auto-correlation function of any finite energy signal f (t), the WD is defined  [11,12] [16], pseudo-affine Wigner distribution [15,17], polynomial Wigner distribution [21,22] and S-method [19,20] came into existence. Although these classes of WD's were able to reduce the interference components, some of them have obvious drawbacks as they are either highly dependent on the localization window width or have to sacrifice the timefrequency resolution to gain the cross-term reduction.
In [23], authors presented a novel way for the improvement of the of the cross-term reduction time-frequency resolution and angle resolution when dealing with the multicomponent LFM signals by introducing the scaled Wigner distribution (SWD) which is parameterized by a constant k ∈ Q. Later authors in [24] extended the WD associated with offset linear canonical transform to the novel one. Recently, Dar and Bhat [25] introduced the scaled version of ambiguity function and scaled Wigner distribution in the linear canonical domain. The extension of SWD to different transforms is still in its infancy.
A superlative generalized version of the Fourier transform(FT) called quadratic-phase Fourier transform(QPFT) has been introduced by [26][27][28]. The QPFT generalizes several well known integral transforms, viz. Fourier, fractional Fourier and linear canonical transforms, and provides a unified analysis of both transient and non-transient signals in an easy and insightful fashion (see [29,30]). Since the QPFT has five free parameters, it has proved to be a reliable tool for an efficient representation of signals demanding several controllable parameters arising in diverse branches of science and engineering, including theory of reproducing kernels, harmonic analysis, image processing , sampling and signal processing especially for detection of LFM signals and filter design. A notable consideration has been given in the extension of the WDs to the classical QPFT and its generalizations see [31].
So motivated and inspired by the merits of SWD and QPFT, we in this paper propose the novel scaled Wigner distribution in the quadratic-phase setting coined as the quadratic-phase scaled Wigner distribution (QSWD), which provides a unified treatment for all existing classes of WD's. Therefore, it is worthwhile to rigorously study the QSWD which can be productive for signal processing theory and applications, especially for detection and estimation of LFM signals.
The highlights of the paper are pointed out below: -To introduce a novel integral transform coined as the quadratic-phase scaled Wigner distribution. -To study the general properties of the proposed distribution, including the time marginal, conjugate symmetry, non-linearity, time shift, frequency shift, frequency marginal and scaling.
-To establish inverse formula, Moyal formula and introduce convolution and correlation theorems for the proposed QSWD. -To present the applications of the proposed QSWD in the detection of single-component and bi-component linear frequency-modulated (LFM) signals.
The rest of the paper is organized as: Sect. 2 is devoted to the definition of SWD and QPFT. In Sect. 3, we propose the definition of QSWD and establish some of its properties. The applications of the proposed distribution for the detection of single-component and bi-component LMF signals are provided in 4. Finally, a conclusion is drawn in Sect. 4.

Preliminary
In this section, we gave the definitions of the scaled Wigner distribution (SWD), the quadratic-phase Fourier transform(QPFT) and the Wigner distribution in the quadraticphase setting which will be needed throughout the paper.

Scaled Wigner distribution
Zhang et al. [23] in 2021 introduced the novel Scaled Wigner distribution (SWD) by replacing the classical instantaneous auto-correlation f t + ρ 2 f * t − ρ 2 found in (1) with the fractional instantaneous autocorrelation f t + k ρ 2 f * t − k ρ 2 in the definition of classical Wigner distribution. For a finite energy signal, the SWD is defined as where k ∈ Q + the set of positive rational numbers.

Quadratic-phase Fourier transform (QPFT)
For a given set of parameters of Λ = (A, B, C, D, E), B = 0, the quadratic-phase Fourier transform any signal f (t) is defined by [26][27][28] Q where the quadratic-phase Fourier kernel Q Λ (t, w) is given by

Wigner distribution in the quadratic-phase Fourier domain
The Wigner distribution associated with quadratic-phase Fourier transform is defined as [31] where Q Λ (t, u) is given in (5) 3 Quadratic-phase scaled Wigner distribution (QSWD) In this section, we shall introduce the notion of the quadraticphase scaled Wigner distribution(QSWD) by virtue of generalizing the Fourier transform kernels to the QPFT kernels in a different way for finite energy signals followed by some of its basic properties.

Definition of the QSWD
Thanks to the SWD, we obtain different expressions for the SWD as follows where On substituting the Fourier kernel in (8) with QPFT kernel, we obtain Thus, we obtain a new kind of SWD associated with the QPFT by replacing (7), i.e., With the virtue of the above equation, we have following definition where k ∈ Q + that is a set of positive rational numbers.
It can be seen that the QSWD is a scaled and modulated version of the classical WD as Therefore, the proposed transform (11) inherits most advantages of SWD and WD and also preserves the characteristics of quadratic-phase Fourier transform. Moreover, the prolificacy of the QSWD given in Definition 1 can be ascertained from the following important deductions: (i) Similar to (1), the quadratic-phase scaled cross-Wigner distribution of the finite energy signals f and g can be given by (ii) For the parameter Λ = (cot θ/2, − csc θ, cot θ, 0, 0), the QSWD (11) yields the novel scaled Wigner distribution associated with fractional Fourier transform: (iii) When the parameter Λ = (0, −1, 0, 0, 0), is chosen, the QSWD (11) boils down to the classical scaled Wigner distribution given in (7). In addition of the above, if we take k = 1, it reduces to classical Wigner distribution.

General properties of QSWD
In this subsection, we investigate some general properties of the QSWD with their detailed proofs. These properties play vital role in signal processing. We shall see the differences between the classical SWD and QSWD. where Proof We omit proof as it follows from Definition 1 Proof We omit proof as it follows from definition OF QSWD.

Theorem 3 (Time shift)
The QSWD of a signal f (t − λ) can be expressed as: Proof We have from definition of QSWD Which completes the proof.

Theorem 4 (Frequency shift)
The QSWD of a signal f (t)e iu 1 t can be expressed as: Proof Using Definition 1, we have which completes the proof Theorem 5 (Scaling property) The QSWD of a signalf (t) = √ λ f (λt) can be expressed as: where Setting λρ = β, the above equation yields Theorem 6 (Time marginal property) The time marginal property of QSWD has the following form: Proof By using Definition 1, we have

Theorem 7 (Frequency marginal property) The frequency marginal property of QSWD is given by
Proof From Definition 1, we have Making change of variable t + k ρ 2 = x such that dρ = 2 dx k , the above equation yields which completes the proof.

Main properties of QSWD
In this subsection, we describe the main properties of the QSWD such as inverse property, Moyal formula, convolution and correlation.

Theorem 8 (Inverse property)
For every signal f ∈ L 2 (R), we have the following inversion formula of the QSWD: Proof It is clear that the QSWD has a following relationship with the FT Now by applying the application of the inverse FT in (23), we have On setting k ρ 2 = t, (24) yields Now replacing 2t by s, we have from (25) f (s) f * (0)e i(As 2 +Ds) = 1 On further simplifying, (26) yields Which completes the proof.

Theorem 9 (Moyal formula)
The Moyal formula for QSWD has the following form: Proof From Definition 1, we have By making the change of variable which completes the proof.
Theorem 10 (Convolution) For f , g ∈ L 2 (R), the QSWD holds the following result

)dρ represents classical convolution of f and g.
Proof From Definition 1, we have Setting ρ = w + k p 2 and η = w − k p 2 , so that dρ = k dp 2 and dη = dw, the above yields which completes the proof.
Theorem 11 (Correlation) For f , g ∈ L 2 (R), the QSWD holds the following result . This completes the proof.
-One component LFM signal: Consider a one-component LFM signal as where λ 1 and λ 2 represent the initial frequency and frequency rate of f (t), respectively. Now by Definition 1, we have From (30), we observe that the proposed QSWD of a single component signal f (t) given in (29) where f 1 (t) = e i(α 1 t+β 1 t 2 ) (β 1 = 0), f 2 (t) = e i(α 2 t+β 2 t 2 ) (β 2 = 0) and β 1 = β 2 . Now using the non-linearity property (2), the QSWD of the signal f (t) given in (31) can be computed as follows: The first two terms in last equation stand for the autoterms of one-component signals, whereas the rest represent the cross terms that are given by Since the first two auto-terms W Λ,k f 1 and W Λ,k f 2 are able to generate impulses which the cross terms W Λ,k f 1 , f 2 , W Λ,k f 2 , f 1 cannot generate, and therefore, although the existence of cross terms has a certain influence on the detection performance, the bi-component LFM signal still can be detected. This indicates that the QSWD is also useful and powerful for detecting bi-component LFM signals.
Moreover, for an adequate value of k and parameter Λ = (A, B, C, D, E), the QSWD benefits in cross-term reduction while maintaining a perfect time-frequency resolution with clear auto terms angle resolution. Thus, it is clear that the QSWD applied to LFM signals is useful and exhibits better detection performance than the classical scaled Wigner distribution mainly due to the presence of factor k more degrees of freedom.

Conclusion
In the study, we have accomplished three major objectives: first, we have introduced the notion of quadratic-phase scaled Wigner distribution in the context of time-frequency analysis and investigate all of its basic properties by means of the quadratic-phase Fourier transforms. Second, we establish the fundamental properties of the proposed transform, including the marginal, conjugate-symmetry, shifting, scaling, inverse and Moyal's formulae by using the machinery of QSWD and operator theory. Moreover, the convolution and correlation properties for QSWD are studied by using the classical convolution and correlation. Third, the applications of the newly defined QSWD for the detection of single-component and bi-component linear frequency-modulated (LFM) signal are also performed to show the advantage of the theory.