Skip to main content
SpringerLink
Log in

Special Affine Fourier Transform for Space-Time Algebra Signals in Detail

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

We generalize the space-time Fourier transform (SFT) (Hitzer in Adv Appl Clifford Algebras 17(3):497–517, 2007) to a special affine Fourier transform (SASFT, also known as offset linear canonical transform) for 16-dimensional space-time multivector Cl(3, 1)-valued signals over the domain of space-time (Minkowski space) \(\mathbb {R}^{3,1}.\) We establish how it can be computed in terms of the SFT, and introduce its properties of multivector coefficient linearity, shift and modulation, inversion, Rayleigh (Parseval) energy theorem, partial derivative identities, a directional uncertainty principle and its specialization to coordinates. All important results are proven in full detail.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. The signature \((+---)\) chosen in [9] would also be possible, but then the important quaternionic subalgebra (2.4) would be absent. The possibility of our \((-+++)\) is also indicated in Footnote 15 on page 22 of [9].

  2. Note that this four-dimensional subalgebra of STA is spatially isotropic, i.e. invariant under spatial rotations.

  3. Note that as for bivectors, the space-time duality map (2.6) exchanges relative vectors \(e_{t1}, e_{t2}, e_{t3},\) with pure space bivectors \(e_{12}, e_{23}, e_{31},\) and vice versa. For Maxwell’s theory [5, 9] this means to exchange electric and magnetic fields.

  4. Regarding physics, the split is determined by the time vector and its dual trivector (the three-dimensional space volume element). Applied to any space-time signal, it naturally generates two wave packages, one traveling to the left and one to the right, classical solutions of relativistic wave equations [10].

  5. Note that Abe and Sheridan adopt in their 1994 papers that introduce the SAFT slightly different sign conventions in (61) of [1] and in (3) of [2]. For consistency, we use the conventions specified in (3) of [2].

  6. The SASFT is therefore more general than the linear canonical SFT, obtained by setting for the SASFT the translation offsets to zero: \(m=n=0\) and \(\overrightarrow{M}=\overrightarrow{N}=\overrightarrow{0}.\)

  7. As [2] points out on page 1802, for the lens transformation a degenerate version of the SAFT is required, see also [1].

  8. As pointed out related to equation (13) on page 1802 of [2], a special limit for \(b\rightarrow 0\) formula will need to be used in this case.

References

  1. Abe, S., Sheridan, J.T.: Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach. J. Phys. A Math. Gen. 27(12), 4179–4187 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Abe, S., Sheridan, J.T.: Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation. Opt. Lett. 19(22), 1801–1803 (1994)

    Article  ADS  Google Scholar 

  3. Bracewell, R.: The Fourier Transform and Its Applications, 3rd edn. Mc Graw Hill India, Gautam Buddha Nagar (2014)

    MATH  Google Scholar 

  4. Chen, L.-P., Kou, K.I., Liu, M.-S.: Pitt’s inequality and the uncertainty principle associated with the quaternion Fourier transform. J. Math. Anal. Appl. 423(1), 681–700 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  6. El Haoui, Y., Hitzer, E., Fahlaoui, S.: Heisenberg’s and Hardy’s uncertainty principles for special relativistic space-time Fourier transformation. Adv. Appl. Clifford Algebras 30, 69 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. El Haoui, Y., Hitzer, E.: Generalized uncertainty principles associated with the quaternionic offset linear canonical transform. Complex Var. Elliptic Equ. (2021). https://doi.org/10.1080/17476933.2021.1916919

    Article  MATH  Google Scholar 

  8. Folland, G.B.: Fourier Analysis and Its Applications. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  9. Hestenes, D.: Space-Time Algebra, 2nd edn. Birkhäuser, Basel (2015)

    Book  MATH  Google Scholar 

  10. Hitzer, E.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebras 17(3), 497–517 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hitzer, E.: Directional uncertainty principle for quaternion Fourier transform. Adv. Appl. Clifford Algebras 20(2), 271–284 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hitzer, E.: Introduction to Clifford’s geometric algebra. SICE J. Control Meas. Syst. Integr. 51(4), 338–350 (2012). Preprint: arXiv:1306.1660. Last Accessed 22 May 2020

  13. Hitzer, E., Sangwine, S.J. (eds.): Quaternion and Clifford Fourier Transforms and Wavelets. Trends in Mathematics, vol. 27. Birkhäuser, Basel (2013)

  14. Hitzer, E.: Creative Peace License. https://gaupdate.wordpress.com/2011/12/14/the-creative-peace-license-14-dec-2011/. Accessed 31 May 2021

  15. Hitzer, E.: New developments in Clifford Fourier transforms. In: Mastorakis, N.E., Pardalos, P.M., Agarwal, R.P., Kocinac, L. (eds.) Advances in Applied and Pure Mathematics, Proceedings of the 2014 International Conference on Pure Mathematics, Applied Mathematics, Computational Methods (PMAMCM 2014), Santorini Island, Greece, July 17–21, 2014. Mathematics and Computers in Science and Engineering Series, vol. 29, pp. 19–25 (2014). http://inase.org/library/2014/santorini/bypaper/MATH/MATH-01.pdf. Preprint: http://viXra.org/abs/1407.0169

  16. Hitzer, E.: The orthogonal planes split of quaternions and its relation to quaternion geometry of rotations. In: Brackx, F., De Schepper, H., Van der Jeugt, J. (eds.) Proceedings of the 30th International Colloquium on Group Theoretical Methods in Physics (group30), 14–18 July 2014, Ghent, Belgium, IOP Journal of Physics: Conference Series (JPCS), vol. 597, p. 012042 (2015). https://doi.org/10.1088/1742-6596/597/1/012042. Open Access: http://iopscience.iop.org/1742-6596/597/1/012042/pdf/1742-6596_597_1_012042.pdf, Preprint: http://viXra.org/abs/1411.0362

  17. Hitzer, E.: Special affine Fourier transform for space-time algebra signals. In: Magnenat-Thalmann, N., et al. (eds.) Advances in Computer Graphics. CGI 2021. Lecture Notes in Computer Science, vol. 13002, pp. 658–669. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-89029-2_49

  18. Hitzer, E.: Quaternion and Clifford Fourier Transforms. Chapman and Hall/CRC, Boca Raton (2022)

    MATH  Google Scholar 

  19. Kennedy, R.F., Jr.: The Real Anthony Fauci—Bill Gates, Big Pharma, and the Global War on Democracy and Public Health. Skyhorse Publishing, New York (2021)

    Google Scholar 

  20. Laville, G., Ramadanoff, I.: Stone–Weierstrass Theorem. Preprint: https://arxiv.org/pdf/math/0411090.pdf. Accessed 27 May 2021

  21. Macdonald, A.: Vector and Geometric Calculus, May 2020 printing. CreateSpace Independent Publishing Platform, Scotts Valley, California, United States (2020)

Download references

Acknowledgements

The author wishes to thank God: Arise, O Lord! O God, lift up Your hand! Do not forget the humble. Why do the wicked renounce God? He has said in his heart,“You will not require an account.”But You have seen, for You observe trouble and grief, To repay it by Your hand. The helpless commits himself to You; You are the helper of the fatherless. Break the arm of the wicked and the evil man; Seek out his wickedness until You find none. (Psalm 10:12-15, NKJV, Biblegateway). He further thanks his colleagues B. Mawardi. Y. El Haoui, and S.J. Sangwine, as well as the organizers of the ENGAGE 2021 workshop at CGI 2021, and the organizers of CGI 2021.

Author information

Authors and Affiliations

  1. International Christian University, Mitaka, Tokyo, 181-8585, Japan

    Eckhard Hitzer

Authors
  1. Eckhard Hitzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eckhard Hitzer.

Ethics declarations

Conflict of Interest

The author declares that he has no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Soli Deo Gloria. This work is dedicated to Robert F. Kennedy Jr. of Children’s Health Defense, who recently published The Real Anthony Fauci – Bill Gates, Big Pharma, and the Global War on Democracy and Public Health [19], writing: ... [on] the historical underpinnings of the bewildering cataclysm that began in 2020. ... Across Western nations, shell-shocked citizens experienced all the well-known tactics of rising totalitarianism – mass propaganda and censorship, the orchestrated promotion of terror, the manipulation of science, the suppression of debate, the vilification of dissent, and use of force to prevent protest. Conscientious objectors who resisted these unwanted, experimental, zero-liability medical interventions faced orchestrated gaslighting, marginalization, and scapegoating. [19], p. XIV. Please note that this research is subject to the Creative Peace License [14].

This article is part of the Topical Collection on ENGAGE 2021 edited by Andreas Aristidou.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hitzer, E. Special Affine Fourier Transform for Space-Time Algebra Signals in Detail. Adv. Appl. Clifford Algebras 32, 60 (2022). https://doi.org/10.1007/s00006-022-01228-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-022-01228-w

Keywords

Search

Navigation