Application and graphical interpretation of a new two-dimensional quaternion fractional Fourier transform

Main Article Content

Khinal Parmar
V. R. Lakshmi Gorty


In this paper, a new two-dimensional quaternion fractional Fourier transform is developed. The properties such as linearity, shifting and derivatives of the quaternion-valued function are studied. The convolution theorem and inversion formula are also established. An example with graphical representation is solved. An application related to two-dimensional quaternion Fourier transform is also demonstrated.

Article Details


  1. M. Bahri, A.K. Amir, C. Lande, The quaternion domain Fourier transform and its application in mathematical statistics, IAENG Int. J. Appl. Math. 48 (2018), 1-7.
  2. M. Bahri, R. Ashino, R. Vaillancourt, Continuous quaternion Fourier and wavelet transforms, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014), 1460003.
  3. T. B ¨ulow, M. Felsberg, G. Sommer, Non-commutative hyper complex Fourier transforms of multidimensional signals, In: G. Sommer (eds) Geometric Computing with Clifford Algebras, Springer, Berlin, Heidelberg, 2001.
  4. D. Cheng, K.I. Kou, Plancherel theorem and quaternion Fourier transform for square integrable functions, Complex Var. Elliptic Equ. 64 (2019), 223-242.
  5. B. Davies, Integral transforms and their Applications, Springer, New York, 1978.
  6. T.A. Ell, Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems, in: Proceedings of 32nd IEEE Conference on Decision and Control, IEEE, San Antonio, TX, USA, 1993: pp. 1830-1841.
  7. T.A. Ell, S.J. Sangwine, Hypercomplex Fourier Transforms of Color Images, IEEE Trans. Image Process. 16 (2007), 22-35.
  8. X. Guanlei, W. Xiaotong, X. Xiaogang, Fractional quaternion Fourier transform, convolution and correlation, Signal Processing. 88 (2008), 2511-2517.
  9. E.M.S. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations, Adv. Appl. Clifford Alg. 17 (2007), 497-517.
  10. A.C. Lewis, William Rown Hamilton, Lectures on quaternions (1853), in: Landmark Writings in Western Mathematics 1640-1940, Elsevier, 2005: pp. 460-469.
  11. P. Lian, Uncertainty principle for the quaternion Fourier transform, J. Math. Anal. Appl. 467 (2018), 1258-1269.
  12. L. Romero, R. Cerutti, L. Luque, A new Fractional Fourier Transform and convolutions products, Int. J. Pure Appl. Math. 66 (2011), 397-408.
  13. R. Roopkumar, Quaternionic one-dimensional fractional Fourier transform, Optik, 127 (2016), 11657-11661.
  14. A.H. Zemanian, Generalized integral transformations, John Wiley Sons Inc., New York, 1968.