Aamir Hamid Dar

Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS), China

There has been a significant increase in the use of the Fractional Fourier transform (FrFT) in recent years due to its numerous applications in signal and image processing, among other fields. At the same time, the applications of Wigner distributions (WD) and ambiguity functions (AF) in signal analysis and image processing cannot be excluded. This...

Quadratic-phase Fourier transform (QPFT) as a general integration transform has been considered into Wigner distribution (WD) and ambiguity function (AF) to show more powerful ability for non-stationary signal processing. In this paper, an advanced WD and AF associated with quadratic-phase Fourier transform (AQWD/AQAF) are presented, which can be r...

The aim of this article is to study Slepian Pollak Inequality (Bell Labs inequality) for Multi band wavelets (M-band wavelets). A version of the already known energy inequality for the wavelets that help to zoom in onto narrow band high frequency components of a signal.

Wigner–Ville distribution (WVD) in the frame work of quaternion linear canonical transform (QLCT) (WVD–QLCT) is the generalized version of WVD in the quaternion algebra. Recently, some properties and applications in detection of linear frequency modulated (LFM) signals have been studied for the WVD–QLCT. In this paper, we first establish a relatio...

The traditional scaled Wigner distribution (SWD) is extended to a novel one inspired by merits of fractional instantaneous autocorrelation present in the definition of fractional bi-spectrum and the fractional Fourier transform (FrFT). We begin by examining the basic characteristics of the novel fractional scaled Wigner distribution (Fr-SWD), such...

The most well-known time–frequency tools for assessing non-transient signals are the Wigner distribution (WD) and ambiguity function (AF), which are used extensively in signal processing and related disciplines. In this article, a new kind of WD and AF associated with the quadratic phase Fourier transform (QPFT) is proposed; this new quadratic phas...

The present thesis entitled “The linear canonical transform and its generalizations ” illustrates the results of researches carried out by the author. In this work a unified functional analytic approach to the treatment of certain integral transforms associated with linear canonical transform and its generalizations is provided. In this work, we co...

In this paper, we propose a novel integral transform coined as quaternion quadratic-phase wavelet transform (QQPWLT) by invoking the elegant convolution structure associated with the quaternion quadratic-phase Fourier transform. Firstly, we explore some mathematical properties of the QQPWLT, including the orthogonality relation, inversion formula,...

The quadratic phase Fourier transform(QPFT) has received my attention in recent years because of its applications in signal processing. At the same time the applications of Wigner-Ville distribution (WVD) and ambiguity function (AF) in signal analysis and image processing can not be excluded. In this paper we investigated the Wigner-Ville Distribut...

The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various application in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inade- quate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique lo...

The free metaplectic transformation (FMT) is an n$$ n $$‐dimensional linear canonical transform. This transform is much useful, especially in multidimensional signal processing and applications. In this paper, our aim is to achieve an efficient time‐frequency representation of higher‐dimensional nonstationary signals by introducing the novel free m...

In this paper, we combine the benefits of the well-known special affine Fourier and Stockwell transforms into a novel integral transform dubbed as special affine Stockwell transform and investigate the associated constant Q-property in the joint time-frequency domain. We do this by using the convolution structure of the special affine Fourier trans...

In this paper, we study the convolution structure in the special affine Fourier transform domain to combine the advantages of the well known special affine Fourier and Stockwell transforms into a novel integral transform coined as special affine Stockwell transform and investigate the associated constant Q-property in the joint time-frequency domai...

In the field of optics and signal processing, the novel quadratic-phase Fourier transform (QPFT) has emerged as a powerful tool. However, it has a drawback as it fails in locating the quadratic-phase domain frequency contents which is much needed in numerous applications, where a joint information of time and quadratic-phase Fourier domain frequenc...

Due to the extra degrees of freedom and simple geometrical manifestation, the linear canonical transform (LCT) has being broadly employed across several disciplines of science and engineering including signal processing, optical and radar systems, electrical and communication systems, quantum physics etc. The main objective of this paper is to stud...

Linear canonical Hankel domain based Stockwell transform (LCHST) is the generalization of Hankel-Stockwell transform. In this paper, we propose the definition of LCHST and then obtain the classical results associated with the proposed transform. The crux of the paper lies in proving a sharp version of Heisenberg’s uncertainty principle for LCHST.

In this paper, we present a novel integral transform known as the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT). We first define the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT) of integrable (and square integrable) functions on R. Later on, we show that 1D-QQPFT satisfies all the respective prop...

In this paper, we introduce the two-dimensional quaternion short-time offset lin- ear canonical transform (ST-QOLCT), which is a generalization of the classical short-time offset linear canonical transform (ST-OLCT) in quaternion algebra set- ting. Several useful properties of the ST-QOLCT are obtained from the properties of the ST-QOLCT kernel. Ba...

In this paper, we introduce quaternion offset linear canonical transform of integrable and square integrable functions. Moreover, we show that the proposed transform satisfies all the respective properties like inversion formula, linearity, Moyal’s formula , product theorem and the convolution theorem

In this paper, a novel quadratic phase S-transform (QPST) is proposed, by generalizing the S-transform (ST) with five parameters a, b, c ,d and e. QPST displays the time and quadratic phase domain-frequency information jointly in the time- frequency plane. Firstly, we define the novel QPST and give its relation with quadratic phase Fourier transfor...

The special affine Fourier transform (SAFT) is an extended version of the classical Fourier transform and incorporates various signal processing tools which include the Fourier transforms, the fractional Fourier transform, the linear canonical transform, and other related transforms. This paper aims to introduce a novel octonion special affine Four...

A generalization of Mallat’s classical multiresolution analysis, based on the theory of spectral pairs, was considered in two articles by Gabardo and Nashed. In this setting, the associated translation set is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an e...

The quaternion offset linear canonical transform (QOLCT) which is time- shifted and frequency-modulated version of the quaternion linear canonical transform (QLCT) provides a more general framework of most existing signal processing tools. For the generalized QOLCT, the classical Heisenberg’s and Lieb’s uncertainty principles have been studied rece...

The quadratic-phase Fourier transform (QPFT) is a neoteric addition to the class of integral transforms and embodies a variety of signal processing tools like the Fourier, fractional Fourier, linear canonical and special affine Fourier transform. In this paper, we generalize the quadratic-phase Fourier transform to quaternion-valued signals, known...

A new version of ambiguity function (AF) associated with the offset linear canonical transform (OLCT) is considered in this paper. This new version of AF coined as the k−ambiguity function associated with the OLCT (k−AFOL) is defined based on the OLCT and the fractional instantaneous auto-correlation. A natural magnification effect characterized by...

Two-dimensional hyper-complex (Quaternion) quadratic-phase Fourier transforms (Q-QPFT) have gained much popularity in recent years because of their applications in many areas, including color image and signal processing. At the same time, the applications of Wigner–Ville distribution (WVD) in signal analysis and image processing cannot be ruled out...

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 trans- form(QWD) to the novel one inspired by the definition of fractional bispectrum. A natural magnification effect characterized by the extra degrees o...

A generalization of Mallatâs classical multiresolution analysis, based on the theory of spectral pairs, was considered in two articles by Gabardo and Nashed. In this setting, the associated translation set is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an e...

Quadratic-phase Fourier transform (QPFT) as a general integral transform has been considered into Wigner distribution (WD) and Ambiguity function (AF) to show more powerful ability for non-stationary signal processing. In this article, a new version of ambiguity function (AF) coined as scaled ambiguity function associated with the Quadratic-phase F...

In this paper, we investigate the (two sided) quater- nion windowed quadratic-phase Fourier transform (QWQPFT) and study the uncertainty principles associated with the QWQPFT. We first propose the definition of QWQPFT and establish its re- lation with quaternion Fourier transform (QFT), then we investi- gate several properties of QWQPFT which inclu...

Wigner-Ville transform or Wigner-Ville distribution (WVD) associated with quaternion offset linear canonical transform (QOLCT) was proposed by Bhat and Dar [6]. This work is devoted to the development of the theory proposed by them which is an emerging tool in the scenario of signal processing. The main contribution of this work is to introduce Wig...

The most recent generalization of octonion Fourier transform (OFT) is the octonion linear canonical transform (OLCT) that has become popular in present era due to its applications in color image and signal processing. On the other hand the applications of Wigner distribution (WD) in signal and image analysis cannot be excluded. In this paper, we in...

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 degre...

The most recent generalization of octonion Fourier transform (OFT) is the octonion linear canonical transform (OLCT) that has become popular in present era due to its applications in color image and signal processing. On the other hand the applications of Wigner distribution (WD) in signal and image analysis cannot be excluded. In this paper, we in...

In this paper, we introduce the notion of quaternion linear canonical S- transform(Q-LCST) which is an extension of the linear canonical S-transform and study the uncertainty principles associated with the Q-LCST. Firstly, we propose the definition of Q-LCST and then study the fundamental properties of quaternion linear canonical S-transform(Q-LCST...

In this article, a new version of ambiguity function (AF) and Wigner distri- bution (WD) based on the linear canonical transform (LCT) and the fractional instanta- neous auto-correlation are proposed which are coined as scaled ambiguity function and scaled Wigner distribution (SAFL/SWDL). We initiate our investigation by establishing the fundamenta...

A linear canonical S transform (LCST) is considered a generalization of the Stockwell transform (ST). It analyzes signals and has multi-angle, multi-scale, multiresolution, and temporal localization abilities. The LCST is mostly suitable to deal with chirp-like signals. It aims to possess the characteristics lacking in a classical transform. Our ai...

The quaternion offset linear canonical transform(QOLCT) has gained much popularity in recent years because of its applications in many areas,including color image and signal processing.At the same time we can not exclude the applications of Wigner-Ville distribution (WVD) in signal analysis and image processing.In this paper we investigate the Wign...

The quadratic-phase Fourier transform (QPFT) is a neoteric addition to the class of Fourier transforms and embodies a variety of signal processing tools including the Fourier, fractional Fourier, linear canonical, and special affine Fourier transform. In this paper, we generalize the quadratic-phase Fourier transform to quaternion-valued signals, k...

In this paper, we present a novel integral transform known as the 2-D hyper-complex(quaternion) Gabor quadratic-phase Fourier transform (Q-GQPFT), which is embodiment of several well known sig- nal processing tools. We first define the 2-D hyper-complex(quaternion) quadratic-phase Fourier transform (Q-QPFT) and then we propose the definition of nov...

The scaled Wigner distribution is obtained from the classical Wigner dis- tribution by replacing the instantaneous auto-correlation with fractional instantaneous auto-correlation which is parameterized by a constant k ∈ Q+. In this paper, we introduce the scaled Wigner distribution in the offset linear canonical domain (SWDOLC). A natural magnifica...

This work is devoted to the development of the octonion linear canonical transform (OLCT) theory proposed by Gao and Li in 2021 that has been designated as an emerging tool in the scenario of signal processing. The purpose of this work is to introduce octonion linear canonical transform of real-valued functions. Further more keeping in mind the var...

A generalization of Mallat’s classical multiresolution analysis, based on the theory of spectral pairs, was considered in two articles by Gabardo and Nashed. In this setting, the associated translation set is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an e...

The quadratic-phase Fourier transform (QPFT) has gained much popularity in recent years because of its applications in image and signal processing. However, the QPFT is inadequate for localizing the quadratic-phase spectrum, which is required in some applications. In this paper, the quadratic-phase wave packet transform (QP-WPT) is proposed to addr...

The free metaplectic transformation (FMT) or the nonseparable linear canonical transformation (NSLCT) has gained much popularity in recent times because of its various application in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the NSLCT is inadequate for localized analysis of non-transient...

The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various application in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inadequate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique loca...

The quadratic phase Fourier transform(QPFT) has gained much popularity in recent years because of its applications in image and signal processing. However, the QPFT is inadequate for localizing the quadratic-phase spectrum which is required in some applications. In this paper, the quadratic-phase wave packet transform (QP-WPT) is proposed to addres...

A multiresolution analysis associated with linear canonical transform was defined by Shah and Waseem for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis associated with linear canonical transform (...

This work is devoted to the development of the octonion linear canonical transform (OLCT) theory proposed by Gao and Li in 2021 that has been designated as an emerging tool in the scenario of signal processing. The purpose of this work is to introduce octonion linear canonical transform of real-valued functions. Further more keeping in mind the var...

In this paper, we introduce quaternion offset linear canonical transform of integrable and square integrable functions. Moreover, we show that the proposed transform satisfies all the respective properties like inversion formula, linearity, Moyal's formula , product theorem and the convolution theorem.

The octonion offset linear canonical transform (O−OLCT) can be defined as a time-shifted and frequency-modulated version of the octonion linear canonical transform (O−LCT), a more general framework of most existing signal processing tools. In this paper, we first define the (O−OLCT) and provide its closed-form representation. Based on this fact, we...

The quaternion offset linear canonical transform (QOLCT) which is time-shifted and frequency-modulated version of the quaternion linear canonical transform (QLCT) provides a more general framework of most existing signal processing tools. For the generalized QOLCT, the classical Heisenberg's and Lieb's uncertainty principles have been studied recen...

In this paper, we introduce the notion of Quaternion Linear Canonical Stockwell Transform which is an extension of the Linear Canonical Transform. We establish some inequalities like Heisenberg's Inequality and logarithmic inequality for LCST.

The Gabor quaternionic offset linear canonical transform (GQOLCT) can be defined as a generalization of the quaternionic offset linear canoni- cal transform (QOLCT). In this paper, we investigate the 2D Gabor quater- nion offset linear canonical transform (GQOLCT). A new definition of the GQOLCT is provided along with its several important properti...

To deal with the time-varying signals, linear canonical S transform (LCST) is introduced to possess some desirable characteristics that are absent in conventional time–frequency transforms. Inspired by LCST, we in this paper developed an idea of novel MRA associated with LCST. Moreover, the construction method of orthogonal wavelets is developed. F...

In this paper, we present a novel integral transform known as the quaternion Gabor quadratic-phase Fourier transform (Q-GQPFT), which is embodiment of several well known signal processing tools. We first define the quaternion quadratic-phase Fourier transform (Q-QPFT) and then we propose the definition of novel Q-GQPFT, which is a modified version...

In this paper, we introduce the notion of quaternion linear canonical S-transform(Q-LCST) which is an extension of the linear canonical S-transform. Firstly, we study the fundamental properties of quaternion linear canonical S-transform(Q-LCST) and then establish some basic results including orthogonality relation and reconstruction formula . Final...

Fractional Fourier transforms has wide application in optical, acoustical, electromagnetic, and other wave propagation problems. In this paper, a new kind of wave packet transform (WPT) associated with the QFrFT is proposed, this new WPT (QFrWPT) is defined by using the machinery of the QFrFT and the WPT, Some properties of QFrWPT are investigated...

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wa...

A generalization of Mallat’s classical multiresolution analysis, based on the theory of spectral pairs, was considered in two articles by Gabardo and Nashed. In this setting, the associated translation set is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional spec- tral pair and the associated dilation is an...

A multiresolution analysis associated with linear canonical transform was defined by Shah and Waseem for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstan-dard setting and introduce vector-valued nonuniform multiresolution analysis associated with linear canonical transform...