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Towards Quaternion Quadratic Phase Fourier Transform

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, known as the quaternion QPFT QQPFT. We initiate our investigation by studying the QPFT of 2D quaternionic signals, then we introduce the QQPFT of 2D quaternionic signals. Using the fundamental relationship between the QQPFT and quaternion Fourier transform QFT, we derive the inverse transform and Parseval and Plancherel formulas associated with the QQPFT. Some other properties including linearity, shift and modulation of the QQPFT are also studied. Finally, we formulate several classes of uncertainty principles UPs for the QQPFT, which including Heisenberg type UP, logarithmic UP, Hardys UP, Beurlings UP and Donohon Starks UP. It can be regarded as the first step in the applications of the QQPFT in the real world.


Introduction
In time-frequency analysis, the most recent signal processing tool is the quadratic-phase Fourier transform (QPFT) introduced by Castro et al. [1] which provides a unified treatment of both the transient and non-transient signals in a simple and insightful fashion. The QPFT has five real parameters with exponential kernel. With a slight modification in [1], we define the QPFT as where Λ µ (x, w) is a quadratic-phase kernel and is given by e −i(ax 2 +bxw+cw 2 +dx+ew) (1.2) and the corresponding inversion formula is given by where a, b, c, d, e ∈ R, b = 0. These arbitrary real parameters present in (1.2) are of great importance as their choice sense of rotation as well as shift can be inculcated in both the axis of time and frequency domain. Hence can be used in better analysis of non-transient signals which are employed in radar and other communication systems. Due to its global kernel and extra degrees of freedom, the QPFT has arrived an efficient tool in solving several problems arising in diverse branches of science and engineering, including harmonic analysis, image processing, sampling, reproducing kernel Hilbert spaces and so on 1 [2,3,4,5].
The generalization of integral transforms from real and complex numbers to the quaternion setting is popular nowadays for the study of higher dimension viz: the quaternion Fourier transform (QFT) [6,7], the quaternion linear canonical transform (QLCT) [8,9], the fractional quaternion Fourier transform (Fr-QFT) [10,11], the quaternion offset linear canonical transform (QOLCT) [12,13,14,15]. In past decades, quaternion algebra has become a leading area of research with its applications in color image processing, image filtering, watermarking, edge detection and pattern recognition(see [16,17,18,19,20,21,22]). The Fourier transform (FT) in quaternion setting i.e.the quaternion Fourier transform (QFT) [23] plays a significant role in the representation of hyper-complex signals in signal processing which is believed to be the substitute of the commonly used two-dimensional Complex Fourier Transform (CFT). The QFT has wide range of applications see( [24,25]). On the other hand the uncertainty principle (UP) plays a vital role in various scientific fields such as mathematics, quantum physics, signal processing and information theory [26,27,28]. The UPs like Heisenberg's, Hardy's, Beurling's associated with QFT are given in [29,30,31,32] and the extension of UPs in the domains of QLCT,QOLCT are given in [33,34,35,36,37]. These UPs have many applications in the analysis of optical systems, signal recovery and so on see( [38,39,40,41]). Therefore modern era of information processing is in dire need of quaternionic valued signals and therefore is a very hot area of research. Since the QPFT is a five parameter class of linear integral transform and has more degrees of freedom and is more flexible than the FT, the FRFT, the LCT but with similar computation cost as the conventional FT. Due to the mentioned advantages, it is natural to generalize the classical QPFT to the quaternionic algebra.
To the best of our knowledge, the generalization of the QPFT to quaternion algebra, and the study of the properties and UPs associated with Q-QPFT have not been carried out yet. So motivated and inspired by the merits of QPFT and QFT, we in this paper propose the novel integral transform coined as the quaternion quadratic-phase Fourier transform (Q-QPFT), which provides a unified treatment for several existing classes of signal processing tools. Therefore it is worthwhile to rigorously study the Q-QPFT and associated UPs which can be productive for signal processing theory and applications.

Paper Contributions.
The contributions of this paper are summarized below: • To introduce a novel integral transform coined as the quaternion quadratic-phase Fourier transform.
• To establish the fundamental relationship between the proposed transform (Q-QPFT) and the quaternion Fourier transform (QFT).
• To study the fundamental properties of the proposed transform, including the parseval's formula, inversion formula, shift and modulation.
• To formulate several classes of uncertainty principles, such as the Heisenberg UP and the logarithmic UP associated with the quaternion quadratic-phase Fourier transform.
• To formulate the Hardy's, Beurling's and Donoho-Stark's uncertainty principles for the Q-QPFT.

Paper Outlines.
The paper is organized as follows: In Section 2, we give a brief review to the quaternion algebra and summarize some definitions and results of two-sided QFT useful in the sequel. The definition and the properties of the novel Q-QPFT are studied in Section 3. In Section 4, we establish some different forms of uncertainty principles (UPs) for the two-sided Q-QPFT which including Heisenberg-type UP, logarithmic UP, Hardy's UP, Beurling's UP, and Donoho-Stark's UP . Finally, a conclusion is drawn in Section 5.

Preliminary
In this section, we give a brief review to the quaternion algebra and summarize some definitions and results of two-sided QFT which will be needed throughout the paper.

Quaternion.
Hamilton introduced the 4-D quaternion algebra in 1843 denoted by H in his honor, which has three imaginary units {i, j, k} and obey Hamilton's multiplication rules: i 2 = j 2 = k 2 = ijk = −1, ij = −ji = k. Let [q] 0 and q = i[q] 1 + j[q] 2 + k[q] 3 be the real scalar part and the vector part of quaternion number q ∈ H. The conjugate of quaternion number q ∈ H is given by and its norm is defined as Also it is easy to check that |pq| = |p||q|, p, q ∈ H.
Moreover the real scalar part has a cyclic multiplication symmetry The inner product of quaternion functions f, g on R 2 with values in H is defined as follows: with symmetric real scalar part And for f = g, we obtain the L 2 (R 2 , H)−norm:

Quaternion Fourier transform.
The QFT plays a vital role in signal processing and color imaging. There are three different types of QFT, the left-sided QFT, the two-sided QFT and the right-sided QFT.
Here our focus will be on two-sided QFT (in the rest of paper QFT means two-sided QFT).
Lemma 2.1 (QFT Parseval [7]). The quaternion product of f, g ∈ L 1 (R 2 , H) ∩ L 2 (R 2 , H) and its QFT are related by In particular if f = g we get the quaternion version of the Plancherel formula; that is,

Quaternion Quadratic-Phase Fourier Transform
In this section we extend the definition of QPFT (1.1) to the 2D quaternionic signals and then based on the definition of the QFT (2.1), we define the novel quaternion QPFT of 2D quaternionic signals and establish several fundamental properties associated with the proposed Q-QPFT.
where the kernels are given by respectively.
(3.5) And for f = g, we get the Parseval theorem as Proof. Proof of the above theorem follows from the one-dimensional case. Now, we introduce the quaternionic quadratic-phase Fourier transforms (Q-QPFTs). Due to the noncommutative property of multiplication of quaternions, there are many different types of Q-QPFTs: two-sided Q-QPFTs, left-sided Q-QPFTs, and right-sided Q-QPFTs.
The following lemma gives the relationship between various types of Q-QPFTs.
, the two-sided Q-QPFT of a signal f can be written as sum of the two right-sided or left-sided Q-QPFT as: Proof. Proof of above lemma follows by the procedure of Lemma 2.3 in [42].
Remark 3.2. The Lemma 3.1 assures that it is sufficient to study two-sided Q-QPFT, as the analogue results for left-sided Q-QPFT or right-sided Q-QPFT can be deduced from (3.10) and (3.10).

3.2.
Two-sided Q-QPFT. In this subsection we study the two-sided Q-QPFT (for simplicity of notation we write the Q-QPFT instead of the two-sided Q-QPFT). First we recall the definition of Q-QPFT (3.7) and present an example for the lucid illustration of the proposed transform, then we show that Q-QPFT can be reduced to the two-sided QFT. Finally, we conclude this subsection with the properties of the Q-QPFT which are vital for signal processing.
are quaternion kernel signals given by (3.3)and (3.4), respectively. Where a s , b s , c s , d s , e s ∈ R, b s = 0 and s = 1, 2.
We now present an example for the lucid illustration of the proposed quaternion quadratic-phase Fourier transform (3.12) Then by definition of Q-QPFT, we have Using R e −z(x+z ′ ) 2 dx = π z , for z, z ′ ∈ C(Gaussian integral) in above equation ,we immediately obtain Now we gave the fundamental relationship between the proposed Q-QPFT and the QFT.

13)
where Proof. From Definition 3.3, we obtain Then, multiplying both sides of the above equation by e i(c 1 w 2 1 +e 1 w 1 ) e j(c 2 w 2 2 +e 2 w 2 ) , yields . This leads to the desired result. H), then every signal f ∈ L 1 (R 2 , H) can be reconstructed back by the formula Proof. By the application of the inversion formula of QFT, we have On setting w = bw, above equation yields On further simplifying, we have Which completes the proof.
Proof. We omit proof as it follows from Definition 3.3.

Theorem 3.9 (Modulation property). The quaternion quadratic-phase Fourier transform (3.12) of a modulated quaternion signal
Proof. From Definition 3.3, we get Which completes the proof.
We omit properties like Reflection, Conjugation and Scaling as they directly follows from Definition 3.3.

26)
Proof. From Lemma 2.2, we have With the help of equations present in Theorem 3.4, above yields On substituting w = bw, we get Again using Theorem 3.4, we obtain

Further simplification yields
Which completes the proof.

Uncertainty Principles Associated with the Quaternion-QPFT
In this section based on the fundamental relationship between Q-QPFT and QFT, we investigate some different forms of UPs associated with Q-QPFT including Heisenberg UP, logarithmic UPs, Hardy's UP, Beurling's UP, and Donoho-Stark's UP.
Lets begin with the Heisenberg type uncertainty principle for the proposed transform (Q-QPFT), which is a generalization of the corresponding Heisenberg's uncertainty principle for the QFT.
The following inequality holds: Proof. The classical Heisenberg uncertainty principle in the QFT domain is given by [43] R 2 Replacing f by G f in (4.2), we have On substituting w = bw, (4.3) yields (4.4) Using equations present in Theorem 3.4 in (4.4), we get Hence, Equivalently Simplifying (4.6), we obtain Which completes the proof.
The directional uncertainty principle for the Q-QPFT takes the following form H), we have, the following inequality: (4.7) Proof. The directional uncertainty principle in the QFT domain reads [Theorem 16 [42]] (4.8) Now using the machinery of previous theorem in (4.8), we will get the desired result (4.7).
Next, using Logarithmic uncertainty principle for the QFT, we establish Logarithmic uncertainty principle for the proposed Q-QPFT.
In continuation, we shall derive the Hardy's uncertainty principle for the quaternion quadratic-phase Fourier transform (3.12). We first recall Hardy's uncertainty principle for the QFT.
In particular,f = 0 when d ≤ 2. Proof. If we take f = G f as defined in Theorem 3.4, then it follows that Hence by Lemma 4.2, we must have G f (x) = P (x)e −k|x| 2 . Now, As b 1 , b 2 are finite real numbers, therefore we can write In particular,f = 0 on account G f (x) = 0 when d ≤ 2.
Which completes the proof.
Towards the end of this section, we establish Donoho-Stark's uncertainty principle for the Q-QPFT by considering relationship between the proposed transform (Q-QPFT) and QFT. Let us begin with the definition.   [44,45]). Let f ∈ L 2 (R 2 , H) Theorem 4.6 (Donoho-Stark's UP for the Q-QPFT). Assuming that non-zero signal f in L 2 (R 2 , H) is a ε E 1 −concentrated on E 1 ⊆ R 2 and Q H µ 1 ,µ 2 [f ](w) is ε E 2 −concentrated on E 2 ⊆ R 2 . Then Proof. By Theorem 3.4, we have Also from (3.14), we have |G f (x)| = b 1 b 2 |f (x)|. (4.20) By the given condition f is ε E 1 −concentrated on E 1 ⊆ R 2 ,i.e. (4.21) From (4.20) and (4.21), we obtain Which implies G f ∈ L 2 (R 2 , H) is ε E 1 −concentrated on E 1 ⊆ R 2 . Hence we proved that the function G f (x) and its QFT F H [G f ](w) are ε E 1 −concentrated on E 1 ⊆ R 2 and ε E 2 −concentrated on bE 2 ⊆ R 2 , respectively. Therefore by Lemma 4.3, we have Which completes the proof.

Conclusion
In the study, we have accomplished three major objectives: first, we have introduced the notion of quaternion quadratic-phase Fourier transform (Q-QPFT). Second, we establish the fundamental properties of the proposed transform, including the parseval's formula, inversion formula, shift and modulation by using the fundamental relationship between Q-QPFT and QFT. Third, we investigate some different forms of UPs associated with Q-QPFT including Heisenberg UP, logarithmic UPs, Hardy's UP, Beurling's UP, and Donoho-Stark's UP. In our future works we shall study the short-time quadratic-phase Fourier transform in the quaternion setting.

Declarations
• Availability of data and materials: The data is provided on the request to the authors.