On One-Dimensional Quaternion Fourier Transform

Abstract

There have been several efforts in the literature to extend the traditional Fourier transformation by using the quaternion algebra. This paper presents the one-dimensional quaternion Fourier transform. We derive its properties which are the extensions of corresponding properties of the one-dimensional Fourier transformation. Finally, the convolution theorem related to the one-dimensional quaternion Fourier transform is discussed.

One-Dimensional Quaternion Discrete Fourier Transform and an Approach to Its Fast Computation

“…Although the quaternion Fourier transform is mainly used for two-dimensional data, i.e., images, it has been specified for one-dimensional data as well [19,21]. We will only deal with the discrete version of the one-dimensional quaternion Fourier transform.…”

“…As for the one-dimensional QDFT, this issue has not been studied in detail, since many computer scientists simply did not see the use of such a transform. That is why there are relatively few research studies on this topic [16,[19][20][21][22][23][24]. In this article, we would like to fill this gap and give a detailed definition of 1-D QDFT, as well as propose a fast method for its implementation.…”

Convolution, Correlation and Uncertainty Principle in the One-Dimensional Quaternion Quadratic-Phase Fourier Transform Domain

“…In this section, we will introduce the definition of quaternion one-dimensional quadraticphase Fourier transform (1D-QQPFT) by using [34][35][36][37]. Prior to that we note e 1 , e 2 and e 3 (or equivalently i, j, k) denote the three imaginary ,units in the quaternion algebra [38].…”

“…• For µ = (0, 1, 0, 0, 0), the 1D-QQPFT (2) boils down to the quaternion one-dimensional Fourier Transform [34] • For µ = (A/2B, −1/B, C/2B, 0, 0) and multiplying the right side of ( 9) by 1/ √ e 2 B the 1D-QQPFT(2) reduces to the quaternion one-dimensional linear canonical transform [37].…”

Review of Quaternion Differential Equations: Historical Development, Applications, and Future Direction

“…Several studies on the Fibonacci sequence from the perspective of number theory include [4][5][6][7][8][9], while Karmano (2013) [10] examined the Riemann Zeta quaternion function. Studies on quaternions can also be viewed from the transformation perspective, one of which is the quaternion Fourier transformation [11][12][13][14][15][16]. From the analytical aspect, [17,18] investigated quaternions based on function differentiation, derivatives [19], Hilbert space [20], as well as boundary problems and integrals [21].…”