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The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting

  • Ameni Gargouri EMAIL logo
Published/Copyright: November 28, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

In this article, we propose a novel integral transform coined as quaternion quadratic phase S-transform (Q-QPST), which is an extension of the quadratic phase S-transform and study the uncertainty principles associated with the Q-QPST. The Q-QPST possesses some desirable characteristics that are absent in conventional time-frequency transforms, especially for dealing with the time-varying quaternion-valued signals. First, we propose the definition of Q-QPST and then we explore some mathematical properties of the of quaternion Q-QPST, including the linearity, modulation, shift, orthogonality relation, and reconstruction formula. Second, we derive the associated Heisenberg’s uncertainty inequality and the corresponding logarithmic version for Q-QPST. Finally, an illustrative example and some potential applications of the Q-QPST are introduced.

Keywords: quadratic-phase transform; quaternion quadratic-phase S-transform; modulation; uncertainty principle
MSC 2010: 42A05; 42B10; 46S10; 42C20; 44A35

1 Introduction

Several popular parametric time-frequency analysis tools have revolutionized signal, image, and video processing [1–3] in the modern era. These include the fractional Fourier transform [4], linear canonical transform [5], short-time Fourier transform [6], and Wigner distributions [7]. The quadratic-phase Fourier transform (QPFT), a superior expanded version of the classical Fourier transform, is a new tool added to this set [8]. It is used in radar and other communication systems and handles both stationary and non-stationary signals in an understandable and straightforward manner. Formally, for the arbitrary real parameter set ϒ = ( A , B , C , D , E ) , B ≠ 0 , the QPFT of any signal f ∈ L 2 ( R ) is defined by

(1) Q ϒ [ f ] ( u ) = ∫ R f ( x ) K ϒ ( x , u ) d x ,

where K ϒ ( x , u ) is a quadratic-phase kernel and is given by

(2) K ϒ ( x , u ) = 1 2 π e i ( A x 2 + B x u + C u 2 + D x + E u ) .

The inversion and Parseval’s formulae corresponding to (1) are given by

(3) f ( x ) = Q ϒ − 1 { Q ϒ [ f ] } ( x ) = ∫ R Q ϒ [ f ] ( u ) K − ϒ ( x , u ) d u ,

(4) ⟨ f , g ⟩ = ∣ B ∣ ⟨ Q μ [ f ] , Q μ [ g ] ⟩ .

The QPFT has demonstrated its significance in addressing numerous issues in scientific and engineering fields, such as harmonic analysis, sampling, image processing, and many more, thanks to its additional degrees of freedom and global kernel [9,10]. Because the QPFT is insufficient to localize the non-stationary signals’ quadratic-phase spectrum content. Various authors have extended the QPFT to create new integral transforms, including the quadratic-phase S-transform (ST), quadratic-phase Wigner distribution, quadratic-phase wavelet and wave packet transforms, and many more, which can be found in [11–14].

The quaternion Fourier transform (QFT) [15–18] is a non-trivial generalization of the classical real and complex Fourier transform. For a long time, mathematicians and physicists have been very interested in this generalization utilizing quaternion algebra. Since QFT has found numerous applications in color image processing, image filtering, watermarking, edge detection, and pattern recognition (see [19–25]), the following extensions of QFT came into existence: the quaternion windowed Fourier transform (QWFT) [26–28]; the fractional quaternion Fourier transform (Fr-QFT) [29,30]; the quaternion linear canonical transform (QLCT) [31,32]; the quaternion offset linear canonical transform [33–36]. Bhat and Dar [37] introduced the extension of the QPFT into the domain of quaternion algebra (Q-QPFT). Later, they introduced several extensions of the Q-QPFT and also studied its vital properties and establish some uncertainty principles. In comparison to QFT, FR-QFT, and QLCT, Q-QPFT is thought to be a crucial tool for non-stationary signal processing due to its extra degrees of freedom and free parameters [38–40].

Authors in [41] introduced the ST, which has garnered significant interest in a number of scientific and engineering domains, including optics, geophysics, oceanology, biomedical imaging, bioinformatics, and signal processing [42–51]. If τ ( x , w ) be a window function, then the ST of function f ( x ) ∈ L 2 ( R ) with respect to ξ is defined by [52]:

(5) S τ [ f ] ( u , w ) = ∫ R ξ ( u − x , w ) e − i 2 π w x f ( x ) d x ,

where the window function ψ satisfies the following condition:

(6) ∫ R ξ ( x , w ) d x = 1 ∀ x ∈ R \ 0 .

To provide a time-linear quadratic phase domain representation, authors derived the idea of QPST in [53]. It is proposed that the QPST has several advantageous properties not found in traditional time-frequency transforms.

In the course of our investigation into QPFT extensions, we presented the idea of the novel quaternion quadratic phase Fourier S-transform (Q-QPST), which is itself an extension of the QPST. The Q-QPST is believed to analyze quaternion-valued signals with abilities of multi-angle, multi-resolution, multi-scale and temporal localization. It is particularly suitable for dealing with chirp-like quaternion-valued signals. First, we propose the definition of the novel Q-QPST. The essential characteristics of the suggested transform are then examined, after which certain key conclusions are established, such as the reconstruction formula and the orthogonality relation. Finally, we obtain the related logarithmic uncertainty inequality and the associated Heisenberg’s uncertainty inequality for the Q-QPST.

The rest of the article is organized as follows:

We go over some preliminary findings and definitions in Section 2, which are used in the parts that follow. The Q-QPST is introduced in Section 3 and its features, including Modulation, orthogonality, and the reconstruction formula, are studied thereafter. The well-known Heisenberg’s uncertainty inequality for the Q-QPST is presented in Section 4. In Section 5, the Q-QPST’s possible uses are highlighted. The article’s conclusions are presented in Section 6.

2 Preliminary

In this section, we review QPST and Q-QPFT.

2.1 QPST

The QPST is known as a hybrid form of the ST. This transform is an extension of the QPFT and ST, which displays the time and quadratic phase domain-frequency information jointly in the time-frequency plane. It is defined as:

Definition 1

The QPST of any signal f ∈ L 2 ( R ) with respect to any non-zero window function ξ ∈ L 2 ( R ) is defined by [53]

(7) S ξ ϒ [ f ] ( u , w ) = Q ϒ { ξ ( u − x , w ) ¯ f ( x ) } ( w , u ) = ∫ R ξ ( u − x , w ) ¯ f ( x ) K ϒ ( x , u ) d x ,

and its inverse is given by

(8) f ( x ) ξ ( u − x , w ) ¯ = Q − ϒ { S ξ ϒ f ( u , w ) } = ∫ R S ξ ϒ f ( u , w ) K − ϒ ( x , u ) d w ,

where K ϒ ( x , u ) is given by (2) for B ≠ 0 .

It is worth mentioning that when ϒ = ( A ⁄ 2 B , − 1 ⁄ B , C ⁄ 2 B , 0 , 0 ) and multiplying the RHS of (7) the QPST yields linear canonical ST [54]. For ϒ = ( cot α ⁄ 2 , − csc α ⁄ 2 , cot α ⁄ 2 , 0 , 0 ) , α ≠ n π (7) yields the fractional ST [42] when multiplied its RHS by 1 − ι cot α . QPST defined in (7) boils down to classical ST when we take ϒ = ( 0 , 1 , − 1 , 0 , 0 ) .

2.2 Q-QPFT

The Q-QPFT is a recent addition to the class of integral transforms which is generalization of the QPFT in the frame of quaternion algebra. Let ϒ s = ( A s , B s , C s , D s , E s ) , s = 1 , 2 , be the real parameter sets satisfying B s ≠ 0 , then the Q-QPFT of a signal f ∈ L 2 ( R 2 , H ) is defined as [37,39]

(9) Q ϒ 1 , ϒ 2 H { f } ( u ) = ∫ R 2 K ϒ 1 ι ( x 1 , u 1 ) f ( x ) K ϒ 2 j ( x 2 , u 2 ) d x ,

and its inverse is given by

(10) f ( x ) = Q ϒ 1 , ϒ 2 − 1 { Q ϒ 1 , ϒ 2 H [ f ] } ( x ) = ∫ R 2 K ϒ 1 ι ( u 1 , x 1 ) ¯ Q ϒ 1 , ϒ 2 H [ f ] ( u ) K ϒ 2 j ( u 2 , x 2 ) ¯ d u ,

where u = ( u 1 , u 2 ) , x = ( x 1 , x 2 ) ∈ R 2 and the kernel signals

(11) K ϒ 1 ι ( x 1 , u 1 ) = 1 2 π e ι ( A 1 x 1 2 + B 1 x 1 u 1 + C 1 u 1 2 + D 1 u 1 + E 1 u 1 )

and

(12) K ϒ 2 j ( x 2 , u 2 ) = 1 2 π e j ( A 2 x 2 2 + B 2 x 2 u 2 + C 2 u 2 2 + D 2 u 2 + E 2 u 2 ) ,

respectively.

3 The novel Q-QPST

The QPST is a five-parameter linear integral transformation and includes the traditional Stockwell spectrum, the linear canonical Stockwell spectrum, and the fractional Stockwell spectrum, as its special cases. Many domains, including detection, parameter estimation, filter design, and non-stationary signal representation, use QPST extensively. In this section, we will look at applying the quaternion algebra to generalize the QPST. The Q-QPST is the name given to this extension. Additionally, a number of their fundamental qualities are looked into.

By substituting the kernel of the Q-QPFT for the kernel of the QPFT in the classical definition of QPST, we can derive a definition of the Q-QPST based on the definition of the quadratic phase S-transform (QPST).

Definition 2

For a non-zero quaternion window function ξ ∈ L 1 ( R 2 , H ) ∩ L 2 ( R 2 , H ) satisfying

∫ R 2 ξ ( x,w ) d x = 1

the 2D Q-QPST of quaternion signal f ∈ L 2 ( R 2 , H ) with respect to ξ is defined by

(13) S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) = ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) f ( x ) ξ ( u − x , w ) ¯ K ϒ 2 j ( x 2 , w 2 ) d x ,

where u = ( u 1 , u 2 ) ∈ R 2 , w = ( w 1 , w 2 ) ∈ R 2 , x = ( x 1 , x 2 ) , ϒ s = ( A s , B s , C s , D s , E s ) ∈ R 2 be a parameter satisfying B s ≠ 0 and the kernel signals

(14) K ϒ 1 ι ( x 1 , w 1 ) = 1 2 π e ι ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 )

and

(15) K ϒ 2 j ( x 2 , w 2 ) = 1 2 π e j ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 ) ,

respectively.

It is noteworthy that certain new time-frequency transformations that are not yet documented in the public literature are created by the Q-QPST:

  1. For ϒ = ( A ⁄ 2 B , − 1 ⁄ B , C ⁄ 2 B , 0 , 0 ) , it yields quaternion linear canonical ST.

  2. For ϒ = ( cot α ⁄ 2 , − csc α ⁄ 2 , cot α ⁄ 2 , 0 , 0 ) , α ≠ n π , we can obtain a novel quaternion fractional ST.

  3. For ϒ s = ( 0 , 1 , − 1 , 0 , 0 ) , the Q-QPST boils down to quaternion ST.

From Definition 2, we have

(16) S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) = Q ϒ 1 , ϒ 2 H { f ( x ) ξ ( u − x , w ) ¯ } ( w , u ) .

Applying inverse Q-QPT to (16), we have

(17) f ( x ) ξ ( u − x , w ) ¯ = Q ϒ 1 , ϒ 2 − 1 { S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) } = ∫ R 2 K ϒ 1 − ι ( x 1 , w 1 ) S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) K ϒ 2 − j ( x 2 , w 2 ) d w .

Our goal in the follow-up is to examine the essential characteristics of the suggested Q-QPST as stated in (13).

3.1 Properties of the novel Q-QPST

We cover a number of the Q-QPST’s fundamental characteristics in this subsection. These characteristics are crucial to the representation of signals. With a few adjustments in the quaternion windowed QPFT, the majority of the quaternion ST’s characteristics can be determined in the Q-QPST domain. Nonetheless, it is evident that the Q-QPST’s attributes, such as shift, modulation, and so forth, differ from those of the quaternion short-time quadratic phase Fourier analysis.

Property 1

S ξ H ϒ 1 , ϒ 2 acts as a multiplication operator if ξ ( u − x , w ) independent of x .

Proof

Since ξ ( u − x , w ) independent of x therefore let ξ ( u − x , w ) = H ( w ) then from (13), we have

(18) S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) = ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) f ( x ) H ( w ) K ϒ 2 j ( x 2 , w 2 ) d x = H ( w ) ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) f ( x ) K ϒ 2 j ( x 2 , w 2 ) d x = H ( w ) Q ϒ 1 , ϒ 2 H [ f ] ( w ) .

So S ξ H ϒ 1 , ϒ 2 is a multiplication operator.□

Note: If we take ξ ( u − x , w ) = H ( w ) = 1 then (18) reduces to Q-QPFT as

S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) = Q ϒ 1 , ϒ 2 H [ f ] ( w ) .

Remark 1

If ξ ( u − x , w ) = H ( x ) , i.e ξ is dependent on x alone, then from (18), we have

S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) = ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) f ( x ) H ( x ) K ϒ 2 j ( x 2 , w 2 ) d x = ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) G ( x ) K ϒ 2 j ( x 2 , w 2 ) d x = Q ϒ 1 , ϒ 2 H [ G ] ( w ) ,

where G ( x ) = f ( x ) H ( x ) ; thus, we see Q-QPST reduces to Q-QPFT.

Property 2

For ξ ∈ L 2 ( R 2 , H ) satisfying ∫ R 2 ξ ( u − x , w ) d u = 1 then

∫ R 2 S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) d u = Q ϒ 1 , ϒ 2 H [ f ] ( w ) .

Proof

From (13), we have

∫ R 2 S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) d u = ∫ R 2 ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) f ( x ) ξ ( u − x , w ) ¯ K ϒ 2 j ( x 2 , w 2 ) d u d x = ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) f ( x ) K ϒ 2 j ( x 2 , w 2 ) ∫ R 2 ξ ( u − x , w ) ¯ d u d x = ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) f ( x ) K ϒ 2 j ( x 2 , w 2 ) d x = Q ϒ 1 , ϒ 2 H [ f ] ( w ) ,

which completes proof.□

Property 3

(Linearity) Let ξ be a non-zero quaternion window function in L 2 ( R 2 , H ) and f n ∈ L 2 ( R 2 , H ) , n ∈ N , then the following holds:

(19) S ξ H ϒ 1 , ϒ 2 ∑ n ∈ N α n f n ( u , w ) = ∑ n ∈ N α n S ξ H ϒ 1 , ϒ 2 [ f n ] ( u , w ) , α n ∈ H

Proof

We avoid proof as it follows directly from Definition 1.□

Property 4

(Parity) Let ξ , f ∈ L 2 ( R 2 , H ) where ξ is a non-zero quaternion window function, then we have

(20) S P ξ H ϒ 1 , ϒ 2 [ P f ] ( u , w ) = S P ξ H ϒ 1 ′ , ϒ 2 ′ [ f ] ( − u , − w ) ,

where P ξ ( x ) = ξ ( − x ) and ϒ s ′ = ( A s , B s , C s , − D s , − E s ) , s = 1 , 2 .

Proof

From (13), we obtain

(21)□ S P ξ H ϒ 1 , ϒ 2 [ P f ] ( u , w ) [ P f ] ( u , w ) = ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) P f ( x ) P ξ ( u − x , w ) ¯ K ϒ 2 j ( x 2 , w 2 ) d x = ∫ R 2 1 2 π e ι ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 ) f ( − x ) ξ ( − u − ( − x ) , − w ) ¯ 1 2 π e j ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 ) = ∫ R 2 1 2 π e ι [ A 1 ( − x 1 ) 2 + B 1 ( − x 1 ) ( − w 1 ) + C 1 ( − w 1 ) 2 + ( − D 1 ) ( − x 1 ) + ( − E 1 ) ( − w 1 ) ] f ( − x ) ξ ( − u − ( − x ) , − w ) ¯ × 1 2 π e j [ A 2 ( − x 2 ) 2 + B 2 ( − x 2 ) ( − w 2 ) + C 2 ( − w 2 ) 2 + ( − D 2 ) ( − x 2 ) + ( − E 2 ) ( − w 2 ) ] = S ξ H ϒ 1 ′ , ϒ 2 ′ [ f ] ( − u , − w ) , where ϒ s ′ = ( A s , B s , C s , − D s , − E s ) , s = 1 , 2 .

Property 5

(Shift) Let f , ξ ∈ L 2 ( R 2 , H ) where ξ is a non-zero quaternion window function, then the following equation follows:

(22) S ξ H ϒ 1 , ϒ 2 [ f ( x - α ) ] ( u , w ) = e ι ( A 1 α 1 2 + B 1 α 1 w 1 + D 1 α 1 ) S ξ H ϒ 1 , ϒ 2 [ F ] ( u − α , w ) e j ( A 2 α 2 2 + B 2 α 2 w 2 + D 2 α 2 ) ,

where F ( t ) = e ι 2 A 1 α 1 t 1 f ( t ) e j 2 A 2 α 2 t 2

Proof

With the help of (13), we have

(23) S ξ H ϒ 1 , ϒ 2 [ f ( x - α ) ] ( u , w ) = ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) f ( x - α ) ξ ( u − x , w ) ¯ K ϒ 2 j ( x 2 , w 2 ) d x .

Taking the change of variable t = x - α in (23), we obtain

S ξ H ϒ 1 , ϒ 2 [ f ( x − α ) ] ( u , w ) = ∫ R 2 K ϒ 1 ι ( t 1 + α 1 , w 1 ) f ( t ) ξ ( u − ( t + α ) , w ) ¯ K ϒ 2 j ( t 2 + α 2 , w 2 ) d t = ∫ R 2 1 2 π e ι [ A 1 ( t 1 + α 1 ) 2 + B 1 ( t 1 + α 1 ) w 1 + C 1 w 1 2 + D 1 ( t 1 + α 1 ) + E 1 w 1 ] f ( t ) ξ ( ( u − α ) − t , w ) ¯ × 1 2 π e j [ A 2 ( t 2 + α 2 ) 2 + B 2 ( t 2 + α 2 ) w 2 + C 2 w 2 2 + D 2 ( t 2 + α 2 ) + E 2 w 2 ] d t = ∫ R 2 1 2 π e ι ( A 1 t 1 2 + B 1 t 1 w 1 + C 1 w 1 2 + D 1 t 1 + E 1 w 1 ) e ι ( A 1 α 1 2 + 2 A 1 α 1 t 1 + B 1 α 1 w 1 + D 1 α 1 ) f ( t ) ξ ( ( u − α ) − t , w ) ¯ × 1 2 π e j ( A 2 t 2 2 + B 2 t 2 w 2 + C 2 w 2 2 + D 2 t 2 + E 2 w 2 ) e j ( A 2 α 2 2 + 2 A 2 α 2 t 2 + B 2 α 2 w 2 + D 2 α 2 ) d t = e ι ( A 1 α 1 2 + B 1 α 1 w 1 + D 1 α 1 ) ∫ R 2 1 2 π e ι ( A 1 t 1 2 + B 1 t 1 w 1 + C 1 w 1 2 + D 1 t 1 + E 1 w 1 ) × { e ι 2 A 1 α 1 t 1 f ( t ) e j 2 A 2 α 2 t 2 } ξ ( ( u − α ) − t , w ) ¯ × 1 2 π e j ( A 2 t 2 2 + B 2 t 2 w 2 + C 2 w 2 2 + D 2 t 2 + E 2 w 2 ) d t e j ( A 2 α 2 2 + B 2 α 2 w 2 + D 2 α 2 ) = e ι ( A 1 α 1 2 + B 1 α 1 w 1 + D 1 α 1 ) S ξ H ϒ 1 , ϒ 2 [ F ] ( u − α , w ) e j ( A 2 α 2 2 + B 2 α 2 w 2 + D 2 α 2 ) .

This completes the proof.□

Property 6

(Modulation) Let ξ ∈ L 2 ( R 2 , H ) be a non-zero quaternion window function and f ∈ L 2 ( R 2 , H ) , then we have

(24) S ξ H ϒ 1 , ϒ 2 [ ℳ y f ] ( u , w ) = e − ι C 1 B 1 2 y 1 2 + 2 C 1 B 1 y 1 w 1 + E 1 B 1 y 1 ∫ R 2 K ϒ 1 ι x 1 , w 1 + y 1 B 1 f ( x ) ξ ( ( u − x ) , w ) ¯ × K ϒ 2 j x 2 , w 2 + y 2 B 2 d x e − j C 2 B 2 2 y 2 2 + 2 C 2 B 2 y 2 w 2 + E 2 B 2 y 2 ,

where ℳ y f ( x ) = e ι y 1 x 1 f ( x ) e j y 2 x 2

Proof

We have from (13) that,

S ξ H ϒ 1 , ϒ 2 [ ℳ y f ] ( u , w ) = ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) e ι y 1 x 1 f ( x ) e j y 2 x 2 ξ ( u − x , w ) ¯ K ϒ 2 j ( x 2 , w 2 ) d x = ∫ R 2 1 2 π e ι ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 ) e ι y 1 x 1 f ( x ) e j y 2 x 2 ξ ( ( u − x ) , w ) ¯ × 1 2 π e j ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 ) d x = ∫ R 2 1 2 π e ι ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 + y 1 x 1 ) f ( x ) ξ ( ( u − x ) , w ) ¯ × 1 2 π e j ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 + y 2 x 2 ) d x = ∫ R 2 1 2 π e ι A 1 x 1 2 + B 1 x 1 w 1 + y 1 B 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 f ( x ) ξ ( ( u − x ) , w ) ¯ × 1 2 π e j A 2 x 2 2 + B 2 x 2 w 2 + y 2 B 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 d x = ∫ R 2 1 2 π e ι A 1 x 1 2 + B 1 x 1 w 1 + y 1 B 1 + C 1 w 1 + y 1 B 1 2 + D 1 x 1 + E 1 w 1 + y 1 B 1 × e − ι C 1 B 1 2 y 1 2 + 2 C 1 B 1 y 1 w 1 + E 1 B 1 y 1 f ( x ) ξ ( ( u − x ) , w ) ¯ e − j C 2 B 2 2 y 2 2 + 2 C 2 B 2 y 2 w 2 + E 2 B 2 y 2 × 1 2 π e j A 2 x 2 2 + B 2 x 2 w 2 + y 2 B 2 + C 2 w 2 + y 2 B 2 2 + D 2 x 2 + E 2 w 2 + y 2 B 2 d x = e − ι C 1 B 1 2 y 1 2 + 2 C 1 B 1 y 1 w 1 + E 1 B 1 y 1 ∫ R 2 K ϒ 1 ι x 1 , w 1 + y 1 B 1 f ( x ) ξ ( ( u − x ) , w ) ¯ × K ϒ 2 j x 2 , w 2 + y 2 B 2 d x e − j C 2 B 2 2 y 2 2 + 2 C 2 B 2 y 2 w 2 + E 2 B 2 y 2 .

Hence, this completes the proof.□

The orthogonality relation for Q-QPST will now be developed, and from this, we will derive the reconstruction formula related to Q-QPST, both of which are essential characteristics for signal analysis. Prior to delving into the orthogonality relation, it is important to understand that for the remainder of the study, we shall assume that

(25) ∫ R 2 ∣ ξ ( u,w ) ∣ 2 d u = Φ ξ , 0 < Φ ξ < ∞ .

NOTE: The next two theorems require us to take the scalar component of the quaternions due to their non-commutativity, which we denote by [ q ] 0 for any quaternion q .

Theorem 1

(Orthogonality relation) Let S ξ H ϒ 1 , ϒ 2 [ f ] be the Q-QPST of any signal f ∈ L 2 ( R 2 , H ) with respect to a non-zero quaternion window function ξ ∈ L 2 ( R 2 , H ) , then we have

(26) ∫ R 2 ∫ R 2 [ S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) S ξ H ϒ 1 , ϒ 2 [ g ] ( u , w ) ¯ ] 0 d w d u = 1 B 1 B 2 [ ⟨ Φ ξ f , g ⟩ ] 0 .

Proof

By the definition of Q-QPST, we obtain

∫ R 2 ∫ R 2 [ S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) S ξ H ϒ 1 , ϒ 2 [ g ] ( u , w ) ¯ ] 0 d w d u = ∫ R 2 ∫ R 2 ∫ R 2 1 2 π e ι ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 ) f ( x ) ξ ( u − x , w ) ¯ × 1 2 π e j ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 ) ∫ R 2 1 2 π e − ι ( A 1 t 1 2 + B 1 t 1 w 1 + C 1 w 1 2 + D 1 t 1 + E 1 w 1 ) g ( t ) ¯ × ξ ( u − t , w ) 1 2 π e − j ( A 2 t 2 2 + B 2 t 2 w 2 + C 2 w 2 2 + D 2 t 2 + E 2 w 2 ) d t 0 d x d u d w = ∫ R 2 ∫ R 2 ∫ R 2 ∫ R 2 1 2 π e ι [ A 1 ( x 1 2 − t 1 2 ) + D 1 ( x 1 − t 1 ) ] f ( x ) Ψ ( u − x , w ) ¯ 1 2 π e j [ A 2 ( x 2 2 − t 2 2 ) + D 1 ( x 2 − t 2 ) ] × 1 2 π e ι B 1 w 1 ( x 1 − t 1 ) g ( t ) ¯ ξ ( u − x , w ) 1 2 π e j B 2 w 2 ( x 2 − t 2 ) d w 0 d t d x d u = 1 B 1 B 2 ∫ R 2 ∫ R 2 ∫ R 2 e ι [ A 1 ( x 1 2 − t 1 2 ) + D 1 ( x 1 − t 1 ) ] f ( x ) ξ ( u − x , w ) ¯ e j [ A 2 ( x 2 2 − t 2 2 ) + D 1 ( x 2 − t 2 ) ] × g ( t ) ¯ ξ ( u − t , w ) δ ( t - x ) d x ] 0 d t d u . = 1 B 1 B 2 ∫ R 4 [ f ( x ) g ( t ) ¯ ξ ( u − t , w ) ¯ ξ ( u − t , w ) ] 0 d t d u = 1 B 1 B 2 ∫ R 2 f ( x ) g ( t ) ¯ d t ∫ R 2 ∣ ξ ( u − t , w ) ∣ 2 d u 0 = 1 B 1 B 2 ∫ R 2 Φ ξ f ( x ) g ( t ) ¯ d t 0 = 1 B 1 B 2 [ ⟨ Φ ξ f , g ⟩ ] 0 .

Thus, the proof is completed.□

Remark 2

If we take f = g in (26), Theorem 1 takes the form

(27) ∫ R 2 ∫ R 2 ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) ∣ 2 d w d u = Φ ξ B 1 B 2 ‖ f ‖ 2 .

Theorem 2

(Reconstruction formula) Every signal f ∈ L 2 ( R 2 , H ) can be reconstructed from the transformed signal S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) by the formula

(28) f ( x ) = B 1 B 2 Φ ξ ∫ R 2 ∫ R 2 1 2 π e − ι ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 ) S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) × ξ ( u − x , w ) 1 2 π e − j ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 ) 0 d w d u ,

where ξ ∈ L 2 ( R 2 , H ) is a quaternion window function that satisfies (25).

Proof

By the virtue of Theorem 1, we can write

1 B 1 B 2 [ ⟨ f Φ ξ , g ⟩ ] 0 = ∫ R 2 ∫ R 2 [ S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) S ξ H ϒ 1 , ϒ 2 [ g ] ( u , w ) ¯ ] 0 d w d u = ∫ R 2 ∫ R 2 S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) ∫ R 2 1 2 π e − ι ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 ) g ( x ) ¯ × ξ ( u − x , w ) 1 2 π e − j ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 ) d x 0 d w d u = ∫ R 2 ∫ R 2 ∫ R 2 S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) 1 2 π e − ι ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 ) × ξ ( u − x , w ) 1 2 π e − j ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 ) g ( x ) ¯ 0 d w d u d x = ∫ R 2 ∫ R 2 S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) 1 2 π e − ι ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 ) × ξ ( u − x , w ) 1 2 π e − j ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 ) d w d u , g 0

Since the above equation is valid for every g ∈ L 2 ( R 2 , H ) , we can write

f ( x ) Φ ξ = B 1 B 2 ∫ R 2 ∫ R 2 1 2 π e − ι ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 ) S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) × ξ ( u − x , w ) 1 2 π e − j ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 ) 0 d w d u ,

which yields

f ( x ) = B 1 B 2 Φ ξ ∫ R 2 ∫ R 2 1 2 π e − ι ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 2 + D 1 x 1 + E 1 w 1 ) S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) × ξ ( u − x , w ) 1 2 π e − j ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 2 + D 2 x 2 + E 2 w 2 ) 0 d w d u .

Thus, the proof is completed.□

4 Uncertainty principles for the novel Q-QPST

The harmonic analysis’s classical uncertainty principle asserts that a non-trivial function cannot be sharply localized concurrently with its Fourier transform. An uncertainty principle in quantum mechanics states that it is impossible to know an electron’s position and velocity at the same time. Stated differently, an increase in positional knowledge results in a decrease in electron velocity or momentum. Harmonic analysis relies heavily on uncertainty principles because they offer a lower bound on the best simultaneous resolution in both the time and frequency domains. For the Q-QPST, as defined by (13), we will create an analogue of the well-known Heisenberg’s uncertainty inequality and the accompanying logarithmic uncertainty principle in this section. First, we prove the following lemma.

Lemma 1

Let f , ξ ∈ L 2 ( R 2 , H ) , where ξ is a non-zero quaternion window function, then we have f ∈ L 2 ( R 2 , H ) , we have

(29) Φ ξ ∫ R 2 x s 2 ∣ f ( x ) ∣ 2 d x = ∫ R 2 ∫ R 2 x s 2 ∣ Q ϒ 1 , ϒ 2 − 1 { S ξ H ϒ 1 , ϒ 2 [ f ] ( u , w ) } ( x ) ∣ 2 d x d u ,

where s = 1 , 2 .

Proof

We avoided the proof as it follows by Theorems 1 and 2.□

Now, we can establish the Heisenberg-type inequalities for the proposed Q-QPST as defined by (13).

Theorem 3

Let S ξ H ϒ 1 , ϒ 2 [ f ] be the Q-QPST of any signal f ∈ L 2 ( R 2 , H ) , of a signal f with respect to the non-zero quaternion window function Ψ ∈ L 2 ( R 2 , H ) , then we have

(30) ∫ R 2 ∫ R 2 w s 2 ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) ∣ 2 d u d w 1 ⁄ 2 ∫ R 2 x s 2 ∣ f ( x ) ∣ 2 d x 1 ⁄ 2 ≥ Φ ξ 2 B s 2 ‖ f ‖ 2

where s = 1 , 2 .

Proof

Heisenberg’s inequality for the Q-QPFT can be written as [37]

(31) ∫ R 2 x s 2 ∣ f ( x ) ∣ 2 d x ∫ R 2 w s 2 ∣ Q ϒ 1 , ϒ 2 H [ f ] ( w ) ∣ 2 d w ≥ 1 4 B s 2 ∫ R 2 ∣ f ( x ) ∣ 2 d x 2 .

Equation (31) can be rewritten as

(32) ∫ R 2 x s 2 ∣ Q ϒ 1 , ϒ 2 − 1 { Q ϒ 1 , ϒ 2 H [ f ] } ∣ 2 d x ∫ R 2 w s 2 ∣ Q ϒ 1 , ϒ 2 H [ f ] ( w ) ∣ 2 d w ≥ 1 4 B s 2 ∫ R 2 ∣ f ( x ) ∣ 2 d x 2 .

Now, by virtue of Plancherel’s theorem for the Q-QPFT, we have from (32)

(33) ∫ R 2 x s 2 ∣ Q ϒ 1 , ϒ 2 − 1 { Q ϒ 1 , ϒ 2 H [ f ] } ∣ 2 d x ∫ R 2 w s 2 ∣ Q ϒ 1 , ϒ 2 H [ f ] ( w ) ∣ 2 d w ≥ 1 2 B s ∫ R 2 ∣ Q ϒ 1 , ϒ 2 H [ f ] ( w ) ∣ 2 d w 2 .

Since S ξ H ϒ 1 , ϒ 2 [ f ] ∈ L 2 ( R 2 , H ) , replacing Q ϒ 1 , ϒ 2 H [ f ] by S ξ H ϒ 1 , ϒ 2 [ f ] , (33) yields

(34) ∫ R 2 x s 2 ∣ Q ϒ 1 , ϒ 2 − 1 { S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) } ∣ 2 d x ∫ R 2 w s 2 ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) ∣ 2 d w ≥ 1 2 B s ∫ R 2 ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( w ) ∣ 2 d w 2 .

On taking square root to both sides of (34) and integrating with respect to d u , we have

(35) ∫ R 2 ∫ R 2 x s 2 ∣ Q ϒ 1 , ϒ 2 − 1 { S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) } ∣ 2 d x 1 ⁄ 2 ∫ R 2 w s 2 ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( w ) ∣ 2 d w 1 ⁄ 2 d u ≥ 1 2 B s ∫ R 2 ∫ R 2 ∣ S ξ H ϒ 1 , ϒ 2 [ f ] w ∣ 2 d w d u .

Applying the Cauchy-Schwarz inequality, (35) becomes

(36) ∫ R 2 ∫ R 2 x s 2 ∣ Q ϒ 1 , ϒ 2 − 1 { S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) } ∣ 2 d x d u 1 ⁄ 2 ∫ R 2 ∫ R 2 w s 2 ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( w ) ∣ 2 d w d u 1 ⁄ 2 ≥ 1 2 B s ∫ R 2 ∫ R 2 ∣ S ξ H ϒ 1 , ϒ 2 [ f ] w ∣ 2 d w d u .

Applying Lemma 1 and Remark 2 to the LHS and RHS, respectively of the above inequality, it gives

(37) Φ ξ ∫ R 2 x s 2 ∣ f ( x ) ∣ 2 d x 1 ⁄ 2 ∫ R 2 ∫ R 2 w s 2 ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) ∣ 2 d w d u 1 ⁄ 2 ≥ Φ ξ 2 B s 2 ‖ f ‖ 2 .

On further simplifying (37), we obtain

(38) ∫ R 2 ∫ R 2 w s 2 ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) ∣ 2 d w d u 1 ⁄ 2 ∫ R 2 x s 2 ∣ f ( x ) ∣ 2 d x 1 ⁄ 2 ≥ Φ ξ 2 B s 2 ‖ f ‖ 2 ,

which completes the proof.□

We now establish the logarithmic uncertainty principle for the Q-QPST as defined by (13).

Theorem 4

For f , ξ ∈ S ( R 2 , H ) , the Q-QPST [ S ξ H ϒ 1 , ϒ 2 ] satisfies the following logarithmic estimate of the uncertainty inequality:

(39) ∣ B 1 B 2 ∣ ∫ R 2 ∫ R 2 ln ∣ w ∣ ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) ∣ 2 d w d u + ∣ B 1 B 2 ∣ Φ ξ ∫ R 2 ln ∣ x ∣ ∣ f ( x ) ∣ 2 d x ≥ ( D − ln ∣ ∣ B 1 B 2 ∣ ∣ ) Φ ξ ‖ f ‖ 2 ,

where D = ln ( 2 π 2 ) − 2 ψ ( 1 ⁄ 2 ) and ψ = d d t ( ln ( Γ ( x ) ) ) and Γ ( x ) is a Gamma function.

Proof

The logarithmic inequality for the Q-QPFT [37] states

(40) ∫ R 2 ln ∣ w ∣ ∣ Q ϒ 1 , ϒ 2 H [ f ] ( w ) ∣ 2 d w + ∫ R 2 ln ∣ x ∣ ∣ f ( x ) ∣ 2 d x ≥ ( D − ln ∣ B 1 B 2 ∣ ) ∫ R 2 ∣ f ( x ) ∣ 2 d x .

Applying the inversion formula of Q-QPFT to the LHS and Parseval’s formula for Q-QPFT to RHS, we obtain from above inequality

(41) ∫ R 2 ln ∣ x ∣ ∣ Q ϒ 1 , ϒ 2 − 1 { Q ϒ 1 , ϒ 2 H [ f ] } ( x ) ∣ 2 d x + ∫ R 2 ln ∣ w ∣ ∣ Q ϒ 1 , ϒ 2 H [ f ] ( w ) ∣ 2 d w ≥ ( D − ln ∣ B 1 B 2 ∣ ) ∫ R 2 ∣ Q ϒ 1 , ϒ 2 H [ f ] ( x ) ∣ 2 d x .

Since Q ϒ 1 , ϒ 2 H [ f ] and S ξ H ϒ 1 , ϒ 2 [ f ] are in S ( R 2 , H ) , we can replace Q ϒ 1 , ϒ 2 H [ f ] by S Ψ , ϒ 1 , ϒ 2 Ψ , H [ f ] on both the sides of (41) to obtain

(42) ∫ R 2 ln ∣ w ∣ ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) ∣ 2 d w + ∫ R 2 ln ∣ x ∣ ∣ Q ϒ 1 , ϒ 2 − 1 { S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) } ( x ) ∣ 2 d x ≥ ( D − ln ∣ B 1 B 2 ∣ ) ∫ R 2 ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) ∣ 2 d w .

First integrating (42) with respect to d u and then applying the Fubini theorem, we obtain

(43) ∫ R 2 ∫ R 2 ln ∣ w ∣ ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) ∣ 2 d w d u + ∫ R 2 ∫ R 2 ln ∣ x ∣ ∣ Q ϒ 1 , ϒ 2 − 1 { S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) } ( x ) ∣ 2 d x d u ≥ ( D − ln ∣ B 1 B 2 ∣ ) ∫ R 2 ∫ R 2 ∣ ϒ 1 , ϒ 2 S ξ H [ f ] ( u,w ) ∣ 2 d w d u .

Now, applying Lemma 1 on LHS and Remark 2 on RHS of (43), we obtain

∫ R 2 ∫ R 2 ln ∣ w ∣ ∣ S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) ∣ 2 d w d u + Φ ξ ∫ R 2 ln ∣ x ∣ f ( x ) ∣ 2 d x ≥ ( D − ln ∣ B 1 B 2 ∣ ) Φ ξ ‖ f ‖ 2 ∣ B 1 B 2 ∣ ,

which completes the proof.□

5 Example and possible applications of the novel Q-QPST

For lucid illustration of the proposed Q-QPST, we present an example:

Example 1

Let us consider a two-dimensional Gaussian quaternion function of the form f ( x ) = e − ( τ 1 x 1 2 + τ 2 x 2 2 ) , for τ 1 , τ 2 ∈ R are positive real constants.

Also, the rectangular window function is

ξ ( x ) = 1 , if ∣ x 1 ∣ < 1 2 , ∣ x 2 ∣ < 1 2 , 0 , elsewhere .

Then, the Q-QPST of f with respect to the above window function is given by

(44) S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) = ∫ R 2 K ϒ 1 ι ( x 1 , w 1 ) f ( x ) ξ ( u − x , w ) ¯ K ϒ 2 j ( x 2 , w 2 ) d x = 1 2 π ∫ x 1 − 1 ⁄ 2 x 1 + 1 ⁄ 2 ∫ x 2 − 1 ⁄ 2 x 2 + 1 ⁄ 2 e i ( A 1 x 1 2 + B 1 x 1 w 1 + C 1 w 1 + D 1 x 1 + E 1 w 1 ) − τ 1 x 1 2 e i ( A 2 x 2 2 + B 2 x 2 w 2 + C 2 w 2 + D 2 x 2 + E 2 w 2 ) − τ 2 x 2 2 d t = 1 2 π e i ( C 1 w 1 2 + E 1 w 1 ) ∫ x 1 − 1 ⁄ 2 x 1 + 1 ⁄ 2 e i ( A 1 + i τ 1 ) x 1 2 + B 1 x 1 w 1 + C 1 w 1 + D 1 x 1 d x 1 × ∫ x 2 − 1 ⁄ 2 x 2 + 1 ⁄ 2 e j ( A 2 + j τ 2 ) x 2 2 + B 2 x 2 w 2 + C 2 w 2 + D 2 x 2 d x 2 × e j ( C 2 w 2 2 + E 2 w 2 ) .

For simplicity, we choose τ 1 = ι A 1 and τ 2 = j A 2 , we obtain from (44)

S ξ H ϒ 1 , ϒ 2 [ f ] ( u,w ) = 1 2 π e i ( C 1 w 1 2 + E 1 w 1 ) ∫ x 1 − 1 ⁄ 2 x 1 + 1 ⁄ 2 e i ( B 1 w 1 + D 1 ) x 1 d x 1 ∫ x 2 − 1 ⁄ 2 x 2 + 1 ⁄ 2 e j ( B 2 w 2 + D 2 x 2 ) x 2 d x 2 × e j ( C 2 w 2 2 + E 2 w 2 ) = e i ( B 1 w 1 x 1 + C 1 w 1 2 + D x 1 + E 1 w 1 ) 2 π ( B 1 w 1 + D 1 ) e i B 1 w 1 + D 1 2 − e − i B 1 w 1 + D 1 2 e j B 2 w 2 + D 2 2 − e − j B 2 w 2 + D 2 2 e j ( B 2 w 2 x 2 + C 2 w 2 2 + D x 2 + E 2 w 2 ) ( B 2 w 2 + D 2 ) .

The Q-QPST has multiple free parameters and includes the quaternion ST, the quaternion fractional ST, and the quaternion ST as its special cases. The significance of the real parameters ϒ s , s = 1 , 2 , used in the construction of the proposed Q-QPST lies in the fact that an appropriate real parameter can be employed to maximize the concentration of the Q-QPST spectrum [12]. The Q-QPST can achieve a reasonable improvement in performance over the current signal processing tools, which include the ST, the linear canonical transform, the fractional ST, and their quaternion counterparts, for representing the LFM signal, which is a crucial non-stationary signal often utilized in radar and sonar systems. This is because the Q-QPST has more degrees of freedom than these existing tools. The Q-QPST can be extremely important in the fields of parameter estimation, filter design, and signal detection.

6 Conclusions

The concept of the Q-QPST, an expansion of the quadratic phase Fourier ST, has been presented, which is believed to analyze quaternion-valued signals with abilities of multi-resolution, multi-angle, multi-scale, and temporal localization. It is particularly suitable for dealing with chirp-like quaternion-valued signals. Prior to establishing certain important conclusions, such as the orthogonality relation and reconstruction formula, we first discussed the fundamental features of the Q-QPST. Most significantly, we obtained the equivalent logarithmic version for the Q-QPST and the stronger version of the accompanying Heisenberg’s uncertainty inequality. This will provide room for more research in this field. Our work will contribute to the discovery of stronger and more generalized forms of the inequality, which will transform signal and image processing.

Acknowledgements

This project was supported via funding by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University under the research project number PSAU/2022/01/20314.

  1. Funding information: This project was supported via funding by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University under the research project number PSAU/2022/01/20314.

  2. Author contribution: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.

  3. Conflict of interest: The author declares that there are no conflicts of interest related to this present study.

  4. Data availability statement: The data are provided on the request to the author.

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Received: 2023-12-26
Revised: 2024-09-25
Accepted: 2024-10-01
Published Online: 2024-11-28

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
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  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
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  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2024-0077
Keywords for this article
quadratic-phase transform; quaternion quadratic-phase S-transform; modulation; uncertainty principle
Creative Commons
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Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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