Algebraic signal processing theory: Cooley-tukey-type algorithms for polynomial transforms based on induction

SIAM Journal on Matrix Analysis and Applications

Volumn 32, Issue 2, 2011, Pages 364-384

  Sandryhaila, Aliaksei ^a   Kovačević, Jelena ^b   Püschel, Markus ^c  

^a CARNEGIE MELLON UNIVERSITY   (United States)

^b CARNEGIE MELLON UNIVERSITY   (United States)

^c ETH ZURICH   (Switzerland)

Author keywords

Algebra; Discrete cosine transform; Discrete Fourier transform; Discrete sine transform; Fast algorithm; Fast Fourier transform; Matrix factorization; Module; Polynomial transform

Indexed keywords

DISCRETE COSINES; DISCRETE SINE TRANSFORMS; FAST ALGORITHMS; MATRIX FACTORIZATIONS; MODULE; POLYNOMIAL TRANSFORM;

ALGORITHMS; DATA COMPRESSION; DATA HANDLING; DISCRETE COSINE TRANSFORMS; DISCRETE FOURIER TRANSFORMS; SIGNAL PROCESSING;

POLYNOMIALS;

EID: 80051513363     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/100805777     Document Type: Article

References (44)

1. L. AUSLANDER, E. FEIG, AND S. WINOGRAD, Abelian semi-simple algebras and algorithms for the discrete Fourier transform, Adv. in Appl. Math., 5 (1984) pp. 31-55.
2. L. AUSLANDER, E. FEIG, AND S. WINOGRAD, The multiplicative complexity of the discrete Fourier transform, Adv. in Appl. Math., 5 (1984) pp. 87-109.
3. G. D. BERGLAND, Numerical analysis: A fast Fourier transform algorithm for real-valued series, Comm. ACM, 11 (1968) pp. 703-710.
4. TH. BETH, Verfahren der Schnellen Fouriertransformation [Methods for the Fast Fourier Transform], Teubner, 1984.
5. TH. BETH, On the computational complexity of the general discrete Fourier transform, Theoret. Comput. Sci., 51(1987), pp. 331-339.
6. J. P. BOYD, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover, New York, 2001.
7. R. N. BRACEWELL, The fast Hartley transform, Proc. IEEE, 72(1984), pp. 1010-1018.
8. V. BRITANAK AND K. R. RAO, The fast generalized discrete Fourier transforms: A unified approach to the discrete sinusoidal transforms computation, Signal Process., 79(1999), pp. 135-150. (Pubitemid 30509185)
9. M. CLAUSEN, Beiträge zumEntwurf schneller Spektraltransformationen (Habilitationsschrift), Ph. D. thesis, Univ. Karlsruhe, 1988.
10. M. CLAUSEN AND U. BAUM, Fast Fourier Transforms, BI-Wissenschaftsverlag, Mannheim, 1993.
11. J. W. COOLEY AND J. W. TUKEY, An algorithm for the machine calculation of complex Fourier series, Math. Comp., 19(1965), pp. 297-301.
12. W. C. CURTIS AND I. REINER, Representation Theory of Finite Groups, Interscience, New York, 1962.
13. P. DIACONIS AND D. ROCKMORE, Efficient computation of the Fourier transform on finite groups, J. Amer. Math. Soc, 3(1990), pp. 297-332.
14. J. R. DRISCOLL, D. M. HEALY, JR., AND D. ROCKMORE, Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs, SIAM J. Comput., 26(1997), pp. 1066-1099. (Pubitemid 127599137)
15. P. DUHAMEL, Implementation of "split-radix"FFT algorithms for complex, real, and real-symmetric data, IEEE Trans. Acoust. Speech Signal Process., 34(1986), pp. 285-295. (Pubitemid 16576060)
16. D. S. DUMMIT AND R. M. FOOTE, Abstract Algebra, 3rd ed., Wiley, New York, 2003.
17. P. A. FUHRMAN, A Polynomial Approach to Linear Algebra, Springer, New York, 1996.
18. M. T. HEIDEMAN AND C. S. BURRUS, On the number of multiplications necessary to compute a length-2n DFT, IEEE Trans. Acoust. Speech Signal Process., 34(1986), pp. 91-95. (Pubitemid 16575391)
19. H. W. JOHNSON AND C. S. BURRUS, On the structure of efficient DFT algorithms, IEEE Trans. Acoust. Speech Signal Process., 33(1985), pp. 248-254.
20. TH. KAILATH AND V. OLSHEVSKY, Displacement structure approach to polynomial Vandermonde and related matrices, Linear Algebra Appl., 261(1997), pp. 49-90.
21. S. MALLAT, A Wavelet Tour of Signal Processing, Academic Press, New York, 1999.
22. G. MANDYAM AND N. AHMED, The discrete Laguerre transform: Derivation and applications, IEEE Trans. Signal Process., 44(1996), pp. 2925-2931. (Pubitemid 126778040)
23. J.-B. MARTENS, The Hermite transform-applications, IEEE Trans. Acoust. Speech Signal Process., 38(1990), pp. 1607-1618.
24. J.-B. MARTENS, The Hermite transform-theory, IEEE Trans. Acoust. Speech Signal Process., 38(1990), pp. 1595-1605.
25. J. C. MASON AND D. C. HANDSCOMB, Chebyshev Polynomials, Chapman and Hall/CRC, London, 2002.
26. P. J. NICHOLSON, Algebraic theory of finite Fourier transforms, J. Comput. System. Sci., 5(1971), pp. 524-547.
27. H. J. NUSSBAUMER, Fast Fourier Transformation and Convolution Algorithms, 2nd ed., Springer, New York, 1982.
28. D. POTTS, G. STEIDL, AND M. TASCHE, Fast algorithms for discrete polynomial transforms, Math. Comput., 67(1998), pp. 1577-1590.
29. M. PÜSCHEL, Decomposing monomial representations of solvable groups, J. Symbolic Comput., 34(2002), pp. 561-596.
30. M. PÜSCHEL AND J. M. F. MOURA, Algebraic Signal Processing Theory. Available at http://arxiv.org/abs/cs. IT/0612077. Parts of this manuscript appeared as [34] and [32].
31. M. PÜSCHEL AND J. M. F. MOURA, The algebraic approach to the discrete cosine and sine transforms and their fast algorithms, SIAM J. Comput., 32(2003), pp. 1280-1316.
32. M. PÜSCHEL AND J. M. F. MOURA, Algebraic signal processing theory: 1-D space, IEEE Trans. Signal Process., 56(2008), pp. 3586-3599.
33. M. PÜSCHEL AND J. M. F. MOURA, Algebraic signal processing theory: Cooley-Tukey type algorithms for DCTs and DSTs, IEEE Trans. Signal Process., 56(2008), pp. 1502-1521. (Pubitemid 351493845)
34. M. PÜSCHEL AND J. M. F. MOURA, Algebraic signal processing theory: Foundation and 1-D time, IEEE Trans. Signal Process., 56(2008), pp. 3572-3585.
35. K. R. RAO AND P. YIP, Discrete Cosine Transform: Algorithms, Advantages, Applications, Academic Press, New York, 1990.
36. D. ROCKMORE, Efficient computation of Fourier inversion for finite groups, J. ACM, 41(1994), pp. 31-66.
37. H. V. SORENSEN, D. L. JONES, C. S. BURRUS, AND M. T. HEIDEMAN, On computing the discrete Hartley transform, IEEE Trans. Acoust. Speech Signal Process., 33(1985), pp. 1231-1238.
38. G. STEIDL, Fast radix-p discrete cosine transform, Appl. Algebra Engrg. Comm. Comp., 3(1992), pp. 39-46.
39. G. STEIDL AND M. TASCHE, A polynomial approach to fast algorithms for discrete Fourier-cosine and Fourier-sine transforms, Math. Comp., 56(1991), pp. 281-296.
40. R. TOLIMIERI, M. AN, AND C. LU, Algorithms for Discrete Fourier Transforms and Convolution, 2nd ed., Springer, New York, 1997.
41. C. VAN LOAN, Computational Framework of the Fast Fourier Transform, SIAM, Philadelphia, 1992.
42. Y. VORONENKO AND M. PÜSCHEL, Algebraic signal processing theory: Cooley-Tukey type algorithms for real DFTs, IEEE Trans. Signal Process., 57(2009), pp. 205-222.
43. Z. WANG, Fast algorithms for the discrete W transform and for the discrete Fourier transform, IEEE Trans. Acoust. Speech Signal Process., 32(1984), pp. 803-816. (Pubitemid 15463745)
44. S. WINOGRAD, On the multiplicative complexity of the discrete Fourier transform, Adv. Math., 32(1979), pp. 83-117.