An Integer Approximation Method for Discrete Sinusoidal Transforms

  • R. J. Cintra


Approximate methods have been considered as a means to the evaluation of discrete transforms. In this work, we propose and analyze a class of integer transforms for the discrete Fourier, Hartley, and cosine transforms (DFT, DHT, and DCT), based on simple dyadic rational approximation methods. The introduced method is general, applicable to several blocklengths, whereas existing approaches are usually dedicated to specific transform sizes. The suggested approximate transforms enjoy low multiplicative complexity and the orthogonality property is achievable via matrix polar decomposition. We show that the obtained transforms are competitive with archived methods in literature. New 8-point square wave approximate transforms for the DFT, DHT, and DCT are also introduced as particular cases of the introduced methodology.


Approximate transforms discrete sinusoidal transforms Low-complexity transforms Nonorthogonal transforms Orthogonalization 


  1. 1.
    J.D. Allen, S.M. Blonstein, The multiply-free Chen transform—a rational approach to JPEG, in Proceedings of the Picture Coding Symposium, Tokyo, Japan (1991), pp. 237–240 Google Scholar
  2. 2.
    D. Bellan, DFT-based detection of sinusoids in real-time applications, in Proceedings of the IEEE International Symposium on Intelligent Signal Processing, Aug. 2009, pp. 104–108 CrossRefGoogle Scholar
  3. 3.
    N. Bhatnagar, A binary friendly algorithm for computing discrete Hartley transform, in Proceedings of the 13th International Conference on Digital Signal Processing, Santorini, Greece, July 1997, pp. 353–356 CrossRefGoogle Scholar
  4. 4.
    R.E. Blahut, Fast Algorithms for Digital Signal Processing (Addison-Wesley, Reading, 1985) zbMATHGoogle Scholar
  5. 5.
    S. Bouguezel, M.O. Ahmad, M.N.S. Swamy, Low-complexity 8×8 transform for image compression. Electron. Lett. 44, 1249–1250 (2008) CrossRefGoogle Scholar
  6. 6.
    R.N. Bracewell, The Hartley Transform (Oxford University Press, London, 1986) zbMATHGoogle Scholar
  7. 7.
    W.L. Briggs, V.E. Henson, The DFT: An Owner’s Manual for the Discrete Fourier Transform (SIAM, Philadelphia, 1995) zbMATHGoogle Scholar
  8. 8.
    V. Britanak, P. Yip, K.R. Rao, Discrete Cosine and Sine Transforms (Academic Press, New York, 2007) Google Scholar
  9. 9.
    C.S. Burrus, T. Parks, DFT/FFT and Convolution Algorithms (Wiley, New York, 1985) Google Scholar
  10. 10.
    W.K. Cham, Y.T. Chan, Integer discrete cosine transforms, in Proceedings of the International Symposium on Signal Processing, Theories, Implementations, and Applications, Brisbane, Australia (1987), pp. 674–676 Google Scholar
  11. 11.
    R.K.W. Chan, M.-C. Lee, Multiplierless fast DCT algorithms with minimal approximation errors, in International Conference on Pattern Recognition, Los Alamitos, CA, USA, vol. 3 (IEEE Comput. Soc., Los Alamitos, 2006), pp. 921–925 Google Scholar
  12. 12.
    R.J. Cintra, V.S. Dimitrov, The arithmetic cosine transform: Exact and approximate algorithms. IEEE Trans. Signal Process. 58, 3076–3085 (2010) MathSciNetCrossRefGoogle Scholar
  13. 13.
    R.J. Cintra, H.M. Oliveira, How to interpolate in arithmetic transform algorithms, in Proceedings of the IEEE 27th International Conference on Acoustics, Speech, and Signal Processing, Orlando, FL, vol. 4, May 2002, p. IV CrossRefGoogle Scholar
  14. 14.
    H.S. Dee, V. Jeoti, Computing DFT using approximate fast Hartley transform, in Proceedings of the International Symposium on Signal Processing and its Applications (ISSPA), Kuala Lumpur, Malaysia, Aug. 2001, pp. 100–103 Google Scholar
  15. 15.
    C. Ding, X. He, H.D. Simon, On the equivalence of nonnegative matrix factorization and spectral clustering, in Proceedings of SIAM Data Mining Conference (2005), pp. 606–610 Google Scholar
  16. 16.
    P.S.R. Diniz, Adaptive Filtering: Algorithms and Practical Implementation, 3rd edn. (Springer, Berlin, 2008) zbMATHGoogle Scholar
  17. 17.
    J.J. Du Croz, N.J. Higham, Stability of methods for matrix inversion. IMA J. Numer. Anal. 12, 1–19 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    E.M. Fadaili, N.T. Moreau, E. Moreau, Nonorthogonal joint diagonalization/zero diagonalization for source separation based on time-frequency distributions. IEEE Trans. Signal Process. 55, 1673–1687 (2007) MathSciNetCrossRefGoogle Scholar
  19. 19.
    E. Feig, S. Winograd, On the multiplicative complexity of discrete cosine transforms. IEEE Trans. Inf. Theory 38, 1387–1391 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    R.C. French, R.F. Mitchell, Class of nonorthogonal transformations for signal processing. Electron. Lett. 10, 78–79 (1974) CrossRefGoogle Scholar
  21. 21.
    G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd edn. (Johns Hopkins University Press, Baltimore, 1996) zbMATHGoogle Scholar
  22. 22.
    V.K. Goyal, Theoretical foundations of transform coding. IEEE Signal Process. Mag. 18, 9–21 (2001) CrossRefGoogle Scholar
  23. 23.
    T.I. Haweel, A new square wave transform based on the DCT. Signal Process. 82, 2309–2319 (2001) CrossRefGoogle Scholar
  24. 24.
    M.T. Heideman, Multiplicative Complexity, Convolution, and the DFT (Springer, Berlin, 1988) zbMATHGoogle Scholar
  25. 25.
    D.J. Higham, Condition numbers and their condition numbers. Linear Algebra Appl. 214, 193–213 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    N.J. Higham, Computing the polar decomposition—with applications. SIAM J. Sci. Stat. Comput. 7, 1160–1174 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    A. Hossen, U. Heute, Fast approximate DCT: basic-idea, error analysis, applications, in Proceedings of the 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 3, Apr. 1997, pp. 2005–2008 CrossRefGoogle Scholar
  28. 28.
    X.-Y. Jing, D. Zhang, A face and palmprint recognition approach based on discriminant DCT feature extraction. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 34, 2405–2415 (2004) CrossRefGoogle Scholar
  29. 29.
    C.R. Johnson, Matrix Theory and Applications (Am. Math. Soc., Providence, 1990) zbMATHGoogle Scholar
  30. 30.
    H.W. Jones, D.N. Hein, S.C. Knauer, The Karhunen-Loève, discrete cosine and related transforms obtained via the Hadamard transform, in Proceedings of the International Telemetering Conference, Los Angeles, CA, Nov. 1978, pp. 14–16 Google Scholar
  31. 31.
    M. Koyuturk, A. Grama, N. Ramakrishnan, Nonorthogonal decomposition of binary matrices for bounded-error data compression and analysis. ACM Trans. Math. Softw. 32, 33–69 (2006) MathSciNetCrossRefGoogle Scholar
  32. 32.
    S.G. Krantz, Real Analysis and Foundations (Chapman & Hall/CRC, London, 2005) zbMATHGoogle Scholar
  33. 33.
    K. Lengwehasatit, A. Ortega, DCT computation based on variable complexity fast approximations, in Proceedings of the 1998 International Conference on Image Processing, vol. 3, Oct. 1998, pp. 95–99 Google Scholar
  34. 34.
    K. Lengwehasatit, A. Ortega, Scalable variable complexity approximate forward DCT. IEEE Trans. Circuits Syst. Video Technol. 14, 1236–1248 (2004) CrossRefGoogle Scholar
  35. 35.
    J. Liang, T.D. Tran, Fast multiplierless approximations of the DCT with the lifting scheme. IEEE Trans. Signal Process. 49, 3032–3044 (2001) CrossRefGoogle Scholar
  36. 36.
    Y.-W. Lin, C.-Y. Lee, Design of an FFT/IFFT processor for MIMO OFDM systems. IEEE Trans. Circuits Syst. I, Regul. Pap. 54, 807–815 (2007) MathSciNetCrossRefGoogle Scholar
  37. 37.
    H.S. Malvar, A. Hallapuro, M. Karczewicz, L. Kerofsky, Low-complexity transform and quantization in H.264/AVC. IEEE Trans. Circuits Syst. Video Technol. 13, 598–603 (2003) CrossRefGoogle Scholar
  38. 38.
    N. Merhav, B. Vasudev, A multiplication-free approximate algorithm for the inverse discrete cosine transform, in Proceedings of the 1999 International Conference on Image Processing, vol. 2 (1999), pp. 759–763 Google Scholar
  39. 39.
    G.J. Miao, M.A. Clements, Digital Signal Processing and Statistical Classification (Artech House, Norwood, 2002) Google Scholar
  40. 40.
    W.B. Mikhael, A.P. Berg, Image representation using nonorthogonal basis images with adaptive weight optimization. IEEE Signal Process. Lett. 3, 165–167 (1996) CrossRefGoogle Scholar
  41. 41.
    B.K. Moser, Linear Models: a Mean Model Approach. Probability and Mathematical Statistics (Academic Press, New York, 1996) zbMATHGoogle Scholar
  42. 42.
    B.K. Natarajan, B. Vasudev, A fast approximate algorithm for scaling down digital images in the DCT domain, in Proceedings of the International Conference on Image Processing, vol. 2, Oct. 1995, pp. 241–243 CrossRefGoogle Scholar
  43. 43.
    S. Oraintara, Y.-J. Chen, T.Q. Nguyen, Integer fast Fourier transform. IEEE Trans. Signal Process. 50, 607–618 (2002) MathSciNetCrossRefGoogle Scholar
  44. 44.
    R.-H. Park, K.S. Yoon, W.Y. Choi, Eight-point discrete Hartley transform as an edge operator and its interpretation in the frequency domain. Pattern Recognit. Lett. 19, 569–574 (1998) zbMATHCrossRefGoogle Scholar
  45. 45.
    S.K. Raikar, A. Makur, Noise feedback structure for non-orthogonal transform coding, in Proceedings of the 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 6, Apr. 2003, pp. 497–500 Google Scholar
  46. 46.
    I.S. Reed, M.-T. Shih, T.K. Truong, E. Hendon, D.W. Tufts, A VLSI architecture for simplified arithmetic Fourier transform algorithm. IEEE Trans. Signal Process. 40, 1122–1133 (1992) zbMATHCrossRefGoogle Scholar
  47. 47.
    N. Roma, L. Sousa, Efficient hybrid DCT-domain algorithm for video spatial downscaling. EURASIP J. Adv. Signal Process. 2007, 30–30 (2007) CrossRefGoogle Scholar
  48. 48.
    G.A.F. Seber, A Matrix Handbook for Statisticians (Wiley, New York, 2007) CrossRefGoogle Scholar
  49. 49.
    G. Strang, The discrete cosine transform. SIAM Rev. 41, 135–147 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    T.D. Tran, The binDCT: fast multiplierless approximation of the DCT. IEEE Signal Process. Lett. 7, 141–144 (2000) MathSciNetCrossRefGoogle Scholar
  51. 51.
    C. Van Loan, Computational Frameworks for the Fast Fourier Transform (SIAM, Philadelphia, 1992) zbMATHGoogle Scholar
  52. 52.
    K.A. Wahid, V.S. Dimitrov, W. Badawy, G.A. Jullien, Error-free arithmetic and architecture for H.264, in Conference Record of the Thirty-Ninth Asilomar Conference on Signals, Systems and Computers (2005), pp. 703–707 CrossRefGoogle Scholar
  53. 53.
    S. Winograd, Arithmetic Complexity of Computations, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 33 (SIAM, Philadelphia, 1980) zbMATHGoogle Scholar
  54. 54.
    W.-T. Wonga, F.Y. Shihb, J. Liua, Shape-based image retrieval using support vector machines, Fourier descriptors and self-organizing maps. Inf. Sci. 177, 1878–1891 (2007) CrossRefGoogle Scholar
  55. 55.
    F. Xu, C.-H. Chang, C.-C. Jong, Hamming weight pyramid—a new insight into canonical signed digit representation and its applications. Comput. Electr. Eng. 33, 195–207 (2007) zbMATHCrossRefGoogle Scholar
  56. 56.
    X. Yu, J. Wang, N.K. Loh, G.A. Jullien, W.C. Miller, Method for generating a new optimal nonorthogonal base in signal representation. Electron. Lett. 28, 2191–2193 (1992) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Departamento de EstatísticaUniversidade Federal de PernambucoRecifeBrazil

Personalised recommendations