Yang's Construction of 3D Hadamard Matrices

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3D Hadamard matrices (cubes), are hypermatrices in which the rows, columns and slices are all mutually orthogonal. They have found application in optics [1, 2]. They add an additional dimension, thus one more degree of freedom, which can be exploited in modern orthogonal frequency-division multiplexing (OFDM/5G+RF) systems. Channel coding involves parameters such as frequency, time, phase and polarization. Hypermatrices are equivalent to tensors. See [3] for some of the original definitions of hyperdeterminants by Cayley. Multilinear algebras and high-order tensor algebras have not yet been exploited by the wireless signal processing community. I think this will change. In 1996, Yang developed a construction method for 3D Hadamard matrices based on 2D Hadamard matrices [4]. This Demonstration illustrates Yang's construction method. To extend this code to complex cubical Hadamard matrices, see [5].

Contributed by: Phil Neumiller (August 2022)
Open content licensed under CC BY-NC-SA




[1] K. Latham, C. Samson, J. Woodacre, E. Simpson, R. Zemp and J. A. Brown, "A New 3D Imaging Technique Integrating Ultrafast Compounding, Hadamard Encoding, and Reconfigurable Fresnel Lensing, Demonstrated on a 128-Element, Crossed Electrode Endoscope," in 2019 IEEE International Ultrasonics Symposium (IUS), Glasgow, UK, Piscataway, NJ: IEEE, 2019 pp. 2052–2055. doi:10.1109/ULTSYM.2019.8926257.

[2] N. S. Rao, S. H. Shruthi, D. Achutha, M. K. Dileep, R. Sandeep and D. L. Girijamba, "Perceptual Video Hashing Using 3D Hadamard Transformation," in 2017 International Conference on Current Trends in Computer, Electrical, Electronics and Communication (CTCEEC), Mysore, India, Piscataway, NJ: IEEE, 2017 pp. 477–480, doi:10.1109/CTCEEC.2017.8455022.

[3] Wikipedia. "Hyperdeterminant." (Nov 10, 2021) en.wikipedia.org/wiki/Hyperdeterminant.

[4] Y. Yang, Theory and Applications of Higher-Dimensional Hadamard Matrices, New York: Science Press, 2001.

[5] B. Lantz and M. Zowada, (2012) "An Overview of Complex Hadamard Cubes," Rose-Hulman Undergraduate Mathematics Journal, 13(2), 2012 Article 3. scholar.rose-hulman.edu/rhumj/vol13/iss2/3.

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