Algebraic signal processing theory: 1-D space

IEEE Transactions on Signal Processing

Volumn 56, Issue 8 I, 2008, Pages 3586-3599

  Püschel, Markus ^a   Moura, José M F ^a  

^a CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Algebra, boundary condition; Chebyshev polynomials; Convolution; Discrete cosine transform (DCT); Discrete sine transform (DST); Fourier transform; Module; Representation theory; Shift; Signal extension; Signal model

Indexed keywords

ALGEBRA; BOOLEAN ALGEBRA; COSINE TRANSFORMS; DISCRETE COSINE TRANSFORMS; DISCRETE FOURIER TRANSFORMS; MATHEMATICAL OPERATORS; MATHEMATICAL TRANSFORMATIONS; MATRIX ALGEBRA; SIGNAL PROCESSING; STANDARDS; TRANSIENTS;

ALGEBRA, BOUNDARY CONDITION; ALGEBRAIC THEORY; CHEBYSHEV POLYNOMIALS; CONVOLUTION; D-SPACE; DISCRETE COSINE TRANSFORM (DCT); DISCRETE COSINES; DISCRETE SINE TRANSFORM (DST); FAST ALGORITHMS; FINITE DISCRETE TIME; FINITE LENGTH; FOURIER; FOURIER TRANSFORM; GENERAL FRAMEWORK; LINEAR SIGNAL PROCESSING; MISSING SIGNAL; MODULE; REPRESENTATION THEORY; SHIFT; SIGNAL EXTENSION; SIGNAL MODEL; SIGNAL MODELING; SIGNAL PROCESSING THEORIES; SINE TRANSFORMS; SPACE MODELS; SYMMETRIC SPACES; TIME SHIFTS; Z TRANSFORMS;

FOURIER TRANSFORMS;

EID: 44449162315     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2008.925259     Document Type: Article

References (25)

1. M. Püschel and J. M. F. Moura, "Algebraic signal processing theory: Foundation and 1-D time," IEEE Trans. Signal Process., vol. 56, no. 8, Aug. 2008.
2. N. Ahmed, T. Natarajan, and K. R. Rao, "Discrete cosine transform," IEEE Trans. Comput., vol. C-23, no. 1, pp. 90-93, Jan. 1974.
3. K. R. Rao and P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications. New York: Academic Press, 1990.
4. Z. Wang and B. R. Hunt, "The discrete W transform," Appl. Math. Comput., vol. 16, pp. 19-48, 1985.
5. R. M. Gray, Probability, Random Processes, and Ergodic Properties. New York: Springer-Verlag, 1988.
6. M. Püschel and J. M. F. Moura, "Algebraic Signal Processing Theory," [Online]. Available: http://arxiv.org/abs/cs.IT/0612077
7. M. Püschel and J. M. F. Moura, "Algebraic signal processing theory: Cooley-Tukey type algorithms for DCTs and DSTs," IEEE Trans. Signal Process., vol. 56, no. 4, pp. 1502-1521, Apr. 2008.
8. Y. Voronenko and M. Püschel, "Algebraic signal processing theory: Cooley-Tukey type algorithms for real DFTs," IEEE Trans. Signal Process., to be published.
9. M. Püschel and J. M. F. Moura, "The algebraic approach to the discrete cosine and sine transforms and their fast algorithms," SIAM J. Comput., vol. 32, no. 5, pp. 1280-1316, 2003.
10. G. Steidl and M. Tasche, "A polynomial approach to fast algorithms for discrete Fourier-cosine and Fourier-sine transforms," Math. Comput. vol. 56, no. 193, pp. 281-296, 1991.
11. T. Kailath and V. Olshevsky, "Displacement structure approach to discrete trigonometric transform based preconditioners of G. Strang and T. Chan type," Calcolo, vol. 33, pp. 191-208, 1996.
12. E. Feig and M. Ben-Or, "On algebras related to the discrete cosine transform," Linear Algebra Appl., vol. 266, pp. 81-106, 1997.
13. N. Jacobson, Basic Algebra I. San Francisco, CA: Freeman, 1974.
14. P. A. Fuhrman, A Polynomial Approach to Linear Algebra. New York: Springer-Verlag, 1996.
15. J. M. F. Moura and N. Balram, "Recursive structure of noncausal Gauss Markov random fields," IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 334-354, 1992.
16. G. Strang, "The discrete cosine transform," SIAM Rev., vol. 41, no. 1, pp. 135-147, 1999.
17. S. A. Martucci, "Symmetric convolution and the discrete sine and cosine transforms," IEEE Trans. Signal Process., vol. 42, no. 5, pp. 1038-1051, May 1994.
18. V. Sánchez, P. García, A. M. Peinado, J. C. Segura, and A. J. Rubio, "Diagonalizing properties of the discrete cosine transforms," IEEE Trans. Signal Process., vol. 43, no. 11, pp. 2631-2641, Nov. 1995.
19. M. Püschel, "Cooley-Tukey FFT like algorithms for the DCT," in Proc. Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), 2003, vol. 2, pp. 501-504.
20. J. M. F. Moura and M. G. S. Bruno, "DCT/DST and Gauss-Markov fields: Conditions for equivalence," IEEE Trans. Signal Process., vol. 46, no. 9, pp. 2571-2574, Sep. 1998.
21. M. Püschel and M. Rötteler, "Fourier transform for the spatial quincunx lattice," in Proc. Int. Conf. Image Processing (ICIP), 2005, vol. 2, pp. 494-497.
22. M. Püschel and M. Rötteler, "Algebraic signal processing theory: 2-D hexagonal spatial lattice," IEEE Trans. Image Process., vol. 16, no. 6, pp. 1506-1521, 2007.
23. T. S. Chihara, An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, 1978.
24. G. Szegö, Orthogonal Polynomials, 3rd ed. Providence, RI: Amer. Math. Soc. Colloq. Publ., 1967.
25. T. J. Rivlin, The Chebyshev Polynomials. New York: Wiley Interscience, 1974.