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Special Affine Fourier Transform for Space-Time Algebra Signals

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Part of the Lecture Notes in Computer Science book series (LNIP,volume 13002)

Abstract

We generalize the space-time Fourier transform (SFT) [9] 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.

Keywords

  • Clifford’s geometric algebra
  • Space-time
  • Space-time algebra
  • Special affine Fourier transform
  • Uncertainty principle

This work is dedicated to 70 year old London Pastor John Sherwood who was arrested on 23 April 2021 by police in Uxbridge for preaching publicly about these verses in Genesis: So God created man in His own image; in the image of God He created him; male and female He created them. Then God blessed them, and God said to them, ’Be fruitful and multiply; fill the earth and subdue it; have dominion over the fish of the sea, over the birds of the air, and over every living thing that moves on the earth.’, Genesis 1:27+28, NKJV, Biblegateway [18]. Please note that this research is subject to the Creative Peace License [13].

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Notes

  1. 1.

    The signature \((+---)\) chosen in [8] would also be possible, but then the important quaternionic subalgebra (4) would be absent. The possibility of our \((-+++)\) is also indicated in Footnote 15 on page 22 of [8].

  2. 2.

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

  3. 3.

    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].

  4. 4.

    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 \(\overrightarrow{M}=\overrightarrow{N}=\overrightarrow{0}\).

  5. 5.

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

  6. 6.

    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

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Acknowledgments

The author wishes to thank God: In the beginning God created the heavens and the earth. The earth was without form, and void; and darkness was on the face of the deep. And the Spirit of God was hovering over the face of the waters. Then God said, “Let there be light”; and there was light. (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.

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Hitzer, E. (2021). Special Affine Fourier Transform for Space-Time Algebra Signals. In: , et al. Advances in Computer Graphics. CGI 2021. Lecture Notes in Computer Science(), vol 13002. Springer, Cham. https://doi.org/10.1007/978-3-030-89029-2_49

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  • DOI: https://doi.org/10.1007/978-3-030-89029-2_49

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