On the spectra of finite group presentations in signal processing

Publication: 22/11/2021

Page: 37-48

Volume 1 Issue 2

How to cite 

D. Samaila., Shu’aibu, G. N., Madu, B. A. (2021). On the spectra of finite group presentations in signal processing. IRESPUB Journal of Natural & Applied Sciences, 1(2), 37-48. 

D. Samaila1*, G. N. Shu’aibu2, B. A. Madu2

1Department of Mathematics, Adamawa State University, Mubi, Nigeria

2Mathematical Sciences Department, University of Maiduguri, Borno State, Nigeria

 
Abstract

Spectral analysis plays an important role in the study or investigation of signal representation. This paper aimed at investigating the signal representation as decimation from algebraic structure using spectral decomposition of modules. The aim is achieved by considering mapping of linear transformations of a group G to a signal space X(n) using different techniques for spectral analysis.

 
Keywords

finite group; representation; Fourier analysis; spectral analysis.

References
  1. Knapp A. W.: Group Representations and Harmonic Analysis, Part II, American Mathematical Society, 1996; (43) pp.537-549.
  2. Hamermesh M.: Group Theory and Its Application to Physical Problems, New York, Dover Publications; 1989.
  3. Lenz R., and Thanh B.: Invariants: A Group Theoretical Foundation and some Applications in Signal Processing and Pattern Recognition. International Symposium on Nonlinear Theory and its Applications (NOLTA2005), Bruges, Belgium, 2005; pp.433-436.
  4. Rajathilagam B., Murali R. and Soman K.P.: “G-Lets: A New Signal Processing Algorithm”, J. Comp. App., 2012; Vol. 37, No. 6, pp.0975-0987.
  5. Riley, K. F., Hobson, M. P., and Bence, S. J.: Mathematical Methods for Physics and Engineering, Cambridge University Press; 2002.
  6. Duzhin, S. V., and Chebotarevsky, B. D.: Transformation Groups for Beginners, Student Mathematical Library 25; 2004.
  7. Mallat S.: Geometrical grouplets, Comp. Harmonic Analysis, 2009; (26) pp.161-180.
  8. William J. D.: Topics in Nonabelian Harmonic Analysis and DSP Application, Proceedings of the International Symposium on Musical Acoustics, Nara, Japan; 2004.
  9. Benjamin S.: Representation theory of Finite Groups. Carleton University, Carleton; 2009.
  10. Dresselhaus, M. S.: Group Theory, Springer; 2008.
  11. Lenz R.: Group Theoretical Methods in Image Processing, Springer-Verlag New York, Inc.; 1990.
  12. Viana, M., and Lakshminarayanan, V.: Dihedral Fourier analysis, Lecture Notes in Statistics, Springer, New York, NY.; 2010.
  13. Candes, E.J.: Ridgelets: theory and applications, PhD thesis, Stanford University; 1998.
  14. Candes, E. J., and Donoho, D. L.: Ridgelets: The key to high dimensional intermittency, Trans. R. Soc. London A. 1999; (357) pp.2495-2509.
  15. Matus F. and Flusser J.: Image representations via a finite Radon transform, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1993; (15) pp.996-1006.
  16. Vale, R., and Waldron, S.: Tight frames generated by finite non-abelian groups, Numerical Algorithms, 2008; (48) pp.11-27.
  17. Starck, J. L, Elad, M. and Donoho, D.: Image decomposition via die combination of sparse representation and a variational approach, IEEE Trans. Image Processing, 2005; (14) pp.1570-1582.
  18. Aharon, M., Elad, M., and Bruckstein, A.M.: The K-SVD: An algorithm for designing over complete dictionaries for sparse representation, IEEE Trans. Signal Processing, 2006; (54) pp.4311-4322, November.
  19. Puschel M., Rotteler M.: Algebraic Signal Processing Theory: 2-D hexagonal spatial lattice, IEEE Transactions on Image Processing, 2007; 16(6), pp.1506-1521.
  20. Puschel M., Mura J. F.: Algebraic Signal Processing Theory: 1-D Space, IEEE Transactions on Signal Processing, 2008; 56(8), pp.3586-3599.
  21. Zhihai Z., Ning Z. and Meng L.: Signal Decimation Representation Associate with the Algebraic Signal Processing, Journal of Computers, 2018; Vol. 29 No. 2, pp.96-103.