Wigner - Ville distribution and ambiguity function associated with the quaternion o ﬀ set linear canonical transform

: Wigner - Ville transform or Wigner - Ville distribution ( WVD ) associated with quaternion o ﬀ set linear canonical transform ( QOLCT ) was proposed by Bhat and Dar. This work is devoted to the develop - ment 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 WVD and ambiguity function ( AF ) associated with the QOLCT ( WVD - QOLCT/AF - QOLCT ) . First, the de ﬁ nition of the WVD - QOLCT is proposed, and then several important properties such as dilation, nonlinearity, and boundedness are derived. Second, we derived the AF for the proposed transform. A bunch of important properties, including the reconstruction formula associated with the AF, are studied.


Introduction
In the modern era of information theory, Wigner-Ville distribution (WVD), or more commonly Wigner-Ville transform (WVT), is an important and effective tool. It was the great Eugene Wigner who introduced the concept of WVD during the calculation of quantum corrections. Moving on the same road in 1948, Ville derived it independently by using the quadratic representation of a signal in the local time-frequency domain. The survey on this novel transform continued. This transform can be generalized to a linear canonical transform (LCT) by changing the kernel of the Fourier transform (FT) to that of an LCT in the WVT domain. For more about this transform, we refer to [1][2][3][4][5][6][7][8][9][10][11].
On the one hand, the quaternion Fourier transform (QFT) is worthwhile to explore in the era of communications. A bunch of needed properties like energy conservation, uncertainty principle, shift, modulation, differentiation, convolution, and correlation of QFT have been explored. Various number of transforms generalized from QFTs are closely related, for instance, the fractional QFT, quaternion wavelet transform, quaternion linear canonical transform (QLCT), and many more. Like the case of QFTs, one can also extend the WVD to the quaternion algebra while having similar properties to that in the classical sense. Many researchers generalized the WVD to the novel quaternion algebra, termed the quaternion WVD. One can see [12][13][14][15][16][17][18][19].
On the other hand, the ambiguity function (AF) that was introduced by Woodward in 1953 plays an important role in the Information theory. It has been applied in various fields, such as radar signal processing, sonar technology, and optical information theory (see [20][21][22][23][24][25][26]). In recent times, a lot of research work has been carried out by coupling the classical AF with other transforms. Other notable works are found in [27][28][29][30]. The WVD and AF associated with the LCT have great advantages and flexibility over the classical WVD and AF, respectively. The generalized WVD and AF have achieved better detection performance. Therefore, it is worthwhile to study a new generalized WVD and AF.
Recently, Bhat and Dar [31] have studied convolution and correlation theorems for WVD for the quaternion offset linear canonical transform (QOLCT). However, due to the noncommutativity of the multiplications in quaternion signals, only a special condition for the convolution and correlation theorems can be established. To be precise, we are bound to take real-valued signals in the function space over the Hamiltonian group. We continued their studies and established some more properties like dilation, nonlinearity, and boundedness. Nevertheless, to extend the scope of study, we introduced an ambiguity function for the QOLCT and were at ease to obtain the classical properties associated with it.
The article is organized as follows. In Section 2, we look at some basics about the quaternion algebra and the QOLCT. Section 3 is devoted to the definition of WVD associated with the QOLCT (WVD-QOLCT). Various properties of the WVD-QOLCT are investigated. In Section 4, we first introduce the AF for the QOLCT. We continued the study and obtained various properties like nonlinearity, boundedness, and a reconstruction formula for the proposed transform.

Preliminary
Here, we look at some basics needed for the rest of the article.

Quaternion algebra
The quaternion algebra is an extended version of complex numbers. Hamilton invented it in 1843, and in his honour, it is denoted by . The elements of have the Cartesian form as where i, j, and k are imaginary units obeying the Hamilton's multiplication rules (see [18]), Looking at (1), is obviously noncommutative. So, we cannot directly extend a number of results on complex numbers to the quaternion. For the sake of convenience, we represent a quaternion q to be the sum of scalar q 1 and purely D 3 quaternion q. Thus, we can write explicitly every quaternion as follows: q q iq jq kq q q q q , , , The conjugation is obtained by changing the sign of the pure part, i.e., q q iq jq kq .
It is evident that quaternion conjugation is linear but anti-involution, Modulus of a quaternion q can be defined as follows: ) ∈ × . The inner product of the functions f g , in can be defined as follows: If we let f g = , then , then the Cauchy-Schwarz inequality [15] holds where t t t , , then the inversion of the two-sided QLCT of f is given as follows: , and its QLCT are related to the Placherel identity in the following way:

QOLCT
Amid the noncommutative nature in the multiplication of quaternions, we have three types of the QOLCT. The first one is left-sided QOLCT, the second one is right-sided QOLCT, and the third one is two-sided QOLCT.
The two-sided QOLCT of any quaternion-valued function f L , is given by , d , , where t t t ,  , then the inversion of two-sided QOLCT is given by ( ) ∈ and its two-sided QOLCT are related to the Plancherel identity in the following way:

WVD-QOLCT
Recently, authors in [31] defined WVD-QOLCT and studied some properties associated with it and also established a special condition for the convolution and correlation theorems for the WVD-QQPFT, which are important for signal processing. In this section, we continue to study the properties of WVD-QOLCT, and prior to that, we first define the WVD-QOLCT. ( ) ∈ is given as follows: , and the quaternion kernels K z w , In this article, we always assume that b 0 i ≠ , i 1, 2, = otherwise the proposed transform reduces to a chirp multiplication. Hence, for any f g L , , 2 2 ( ) ∈ , we have, is known as quaternion correlation product.
Applying the inverse QOLCT to (16), we obtain h t z t w , , , We now study several important properties of the WVD-QOLCT defined by (2).
, , , m m where     , , , which completes the theorem. . Then, with the changing of variables w t z 2 = + and y t z 2 = − in last second step, we obtain