Advertisement

Fast MDCT/MDST, MLT, ELT, and MCLT Algorithms

  • Vladimir Britanak
  • K. R. Rao
Chapter

Abstract

The perfect reconstruction cosine/sine-modulated analysis/synthesis filter banks such as the MDCT, MLT, MDST, ELT, and MCLT are fundamental processing components for the time-to-frequency transformation of an audio data block in many international audio coding standards, proprietary audio compression algorithms, broadcasting/speech/data communication codecs, as well as open-source royalty free audio/speech codecs for high quality audio/speech compression. Since the computations of cosine/sine-modulated filter banks are the most time-consuming operations in audio coding schemes, the crucial aspect for their applicability is the existence of fast algorithms that allow their efficient software/hardware implementation compared to the direct implementation via their corresponding analytical forms. In this chapter, fast algorithms for the efficient implementation of the forward/backward evenly stacked MDCT/MDST, oddly stacked MDCT/MDST, MLT, ELT, and MCLT block transforms are presented. The emphasis is imposed particularly on basic steps, various tricks (trigonometric and algebraic), and approaches leading to the derivation of final formulae of a fast algorithm. For each fast algorithm complete formulae or a sparse block matrix factorization, a corresponding generalized signal flow graph, the total computational complexity, and a possible structural simplification of the algorithm are presented.

References

  1. 1.
    M. Bosi, R.E. Goldberg, Introduction to Digital Audio Coding and Standards (Springer Science+Business Media, New York, 2003)Google Scholar
  2. 2.
    H.S. Malvar, Signal Processing with Lapped Transforms (Artech House, Norwood, MA, 1992)zbMATHGoogle Scholar
  3. 3.
    H. Malvar, A modulated complex lapped transform and its applications to audio processing, in Proceedings of the IEEE ICASSP’99, Phoenix, AR, May 1999, pp. 1421–1424Google Scholar
  4. 4.
    J.P. Princen, A.B. Bradley, Analysis/synthesis filter bank design based on time domain aliasing cancellation. IEEE Trans. Acoust. Speech Signal Process. ASSP-34(5), 1153–1161 (1986)Google Scholar
  5. 5.
    J.P. Princen, A.W. Johnson, A.B. Bradley, Subband/transform coding using filter bank designs based on time domain aliasing cancellation, in Proceedings of the IEEE ICASSP’87, Dallas, TX, April 1987, pp. 2161–2164Google Scholar
  6. 6.
    A. Spanias, T. Painter, V. Atti, Audio Signal Processing and Coding (Wiley-Interscience, Hoboken, NJ, 2007)Google Scholar

Fast Algorithms for the Evenly Stacked MDCT/MDST

  1. 7.
    V. Britanak, An efficient computing of oddly stacked MDCT/MDST computation via evenly stacked MDCT/MDST and vice versa. Signal Process. 85(7), 1353–1374 (2005)zbMATHGoogle Scholar
  2. 8.
    V. Britanak, K.R. Rao, A unified fast MDCT/MDST computation in the evenly stacked analysis/synthesis system. Circuits Syst. Signal Process. 21(4), 415–426 (2002)MathSciNetzbMATHGoogle Scholar
  3. 9.
    G. Davidson, L. Fielder, M. Antil, High-quality audio transform coding at 128 kbits/s, in Proceedings of the IEEE ICASSP’90, Alberquerque, NM, April 1990, pp. 1117–1120Google Scholar
  4. 10.
    G. Davidson, L. Fielder, M. Antil, Low-complexity transform coder for satellite link applications, in 89th AES Convention, Los Angeles, CA, September 1990. Preprint #2966Google Scholar
  5. 11.
    G. Davidson, W. Anderson, A. Lovrich, A low-cost adaptive transform decoder implementation for high-quality audio, in Proceedings of the IEEE ICASSP’92 vol. II, San Francisco, CA, April 1992, pp. 93–196Google Scholar
  6. 12.
    A.G. Elder, S.G. Turner, A real-time PC based implementation of AC-2 digital audio compression, in 95th AES Convention, New York, NY, October 1993. Preprint #3773Google Scholar
  7. 13.
    T.D. Lookabaugh, M.G. Perkins, Application of the Princen–Bradley filter bank to speech and image compression. IEEE Trans. Acoust. Speech Signal Process. 38(11), 1914–1926 (1990)Google Scholar
  8. 14.
    D. Ševič, M. Popovič, Improved implementation of the Princen–Bradley filter bank. IEEE Trans. Signal Process. 42(11), 3260–3261 (1994)Google Scholar

Fast Algorithms for the Oddly Stacked MDCT/MDST

  1. 15.
    K.C. Anup, B.A. Kumar, A new efficient implementation of TDAC synthesis filterbanks based on radix-2 FFT, in Proceedings of the 14th EUSIPCO Signal Processing Conference, Florence, September 2006Google Scholar
  2. 16.
    M. Bosi-Goldberg, Analysis/synthesis filtering system with efficient oddly stacked single sideband filter bank using time domain aliasing cancellation. US Patent Application No. 5890106, Dolby Labs, San Francisco, CA, March 1999Google Scholar
  3. 17.
    V. Britanak, Improved and extended mixed-radix decimation in frequency fast MDCT algorithm. Comput. Inform. 29(6), 1001–1012 (2010)MathSciNetzbMATHGoogle Scholar
  4. 18.
    V. Britanak, K.R. Rao, A new fast algorithm for the unified forward and inverse MDCT/MDST computation. Signal Process. 82(3), 433–459 (2002)zbMATHGoogle Scholar
  5. 19.
    D.-Y. Chan, J.-F. Yang, S.-Y. Chen, Regular implementation algorithms of time domain aliasing cancellation. IEE Proc. Vision Image Signal Process. 143(6), 387–392 (1996)Google Scholar
  6. 20.
    M.-H. Cheng, Y.-H. Hsu, Fast IMDCT and MDCT algorithms – a matrix approach. IEEE Trans. Signal Process. 51(1), 221–229 (2003)MathSciNetzbMATHGoogle Scholar
  7. 21.
    R.K. Chivukula, Y.A. Reznik, Efficient implementation of a class of MDCT/IMDCT filter banks for speech and audio coding applications, in Proceedings of the IEEE ICASSP’2008, Las Vegas, NV, March–April 2008, pp. 213–216Google Scholar
  8. 22.
    Y.-K. Cho, T.-H. Song, H.-S. Kim, An optimized algorithm for computing the modified discrete cosine transform and its inverse transform, in Proceedings of the IEEE Region 10 Conference TENCON’2004, Chiang Mai, November 2004, pp. 626–628Google Scholar
  9. 23.
    S. Cramer, R. Gluth, Computationally efficient real-valued filter banks based on a modified O 2 DFT, in Proceedings of EUSIPCO’90, Signal Processing V: Theories and Applications, Elsevier Science Publishers B.V., Barcelona, September 1990, pp. 585–588Google Scholar
  10. 24.
    P. Duhamel, Y. Mahieux, J.P. Petit, A fast algorithm for the implementation of filter banks based on time domain aliasing cancellation, in Proceedings of the IEEE ICASSP’91, Toronto, May 1991, pp. 2209–2212Google Scholar
  11. 25.
    R. Gluth, Regular FFT-related transform kernels for DCT/DST-based polyphase filter banks, in Proceedings of the IEEE ICASSP’91, Toronto, May 1991, pp. 2205–2208Google Scholar
  12. 26.
    Z.G. Gui, Y. Ge, D.Y. Zhang, J.S. Wu, Generalized fast mixed-radix algorithm for the computation of forward and inverse MDCTs. Signal Process. 92(2), 363–373 (2012)Google Scholar
  13. 27.
    Y.-T. Hwang, S.-C. Lai, A novel MDCT/IMDCT computing kernel design, in Proceedings of the IEEE Workshop on Signal Processing Systems Design and Implementation, Athens, November 2005, pp. 526–531Google Scholar
  14. 28.
    M. Iwadare, A. Sugiyama, F. Hazu, A. Hirano, T. Nishitani, A 128 kb/s hi-fi CODEC based on adaptive transform coding with adaptive block size MDCT. IEEE J. Sel. Areas Commun. 10(1), 138–144 (1992)Google Scholar
  15. 29.
    H.-S. Kim, Y.-K. Cho, W.-P. Lee, A new optimized algorithm for computation of MDCT and its inverse transform, in Proceedings of the IEEE International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS’2004), Seoul, November 2004, pp. 528–530Google Scholar
  16. 30.
    S.-W. Lee, Improved algorithm for efficient computation of the forward and backward MDCT in MPEG audio coder. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 48(10), 990–994 (2001)Google Scholar
  17. 31.
    T. Li, R. Zhang, R. Yang, H. Huang, F. Lin, A unified computing kernel for MDCT/IMDCT in modern audio coding standards, in Proceedings of the IEEE International Symposium on Communication and Information Technologies (ISCIT’2007), Sydney, October 2007, pp. 546–550Google Scholar
  18. 32.
    B. Lincoln, An experimental high fidelity perceptual audio coder, Project in MUS420 Win97, Center for Computer Research in Music and Acoustics, Stanford University, CA 94305-8180 (1998). Web site: http://ccrma-www.stanford.edu/~bosse/proj/proj.html
  19. 33.
    C.-M. Liu, W.-C. Lee, A unified fast algorithm for cosine modulated filter banks in current audio coding standards. J. Audio Eng. Soc. 47(12), 1061–1075 (1999)Google Scholar
  20. 34.
    Y. Mahieux, J.P. Petit, High-quality audio transform coding at 64 kbps. IEEE Trans. Commun. 42(11), 3010–3019 (1994)Google Scholar
  21. 35.
    V. Nikolajevič, G. Fettweis, Improved implementation of MDCT in MP3 audio coding, in Proceedings of the IEEE 10th Asian–Pacific Conference on Communications and 5th International Symposium on Multi-Dimensional Mobile Communications (APCC/MDMC’2004), vol. 1, Tsinghua University, Beijing, August–September 2004, pp. 309–312Google Scholar
  22. 36.
    D. Ševič, M. Popovič, A new efficient implementation of the oddly stacked Princen–Bradley filter bank. IEEE Signal Process Lett. 1(11), 166–168 (1994)Google Scholar
  23. 37.
    X. Shao, S.G. Johnson, Type-IV DCT, DST and MDCT algorithms with reduced number of arithmetic operations. Signal Process. 88(6), 1313–1326 (2008)zbMATHGoogle Scholar
  24. 38.
    H. Shu, X. Bao, C. Toumoulin, L. Luo, Radix-3 algorithm for the fast computation of forward and inverse MDCT. IEEE Signal Process. Lett. 14(2), 93–96 (2007)Google Scholar
  25. 39.
    T. Sporer, K. Brandenburg, B. Edler, The use of multirate filter banks for coding of high quality digital audio, in Proceedings of the 6th European Signal Processing Conference (EUSIPCO), vol. 1, Amsterdam, June 1992, pp. 211–214Google Scholar
  26. 40.
    T.K. Truong, P.D. Chen, T.C. Cheng, Fast algorithm for computing the forward and inverse MDCT in MPEG audio coding. Signal Process. 86(5), 1055–1060 (2006)zbMATHGoogle Scholar
  27. 41.
    P.-S. Wu, Y.-T. Hwang, Efficient IMDCT core designs for audio signal processing, in Proceedings of the IEEE Workshop on Signal Processing Systems (SiPS’2003), Seoul, August 2003, pp. 275–280Google Scholar
  28. 42.
    J.-S. Wu, H.-Z. Shu, L. Senhadji, L.-M. Luo, Mixed-radix algorithm for the computation of forward and inverse MDCTs. IEEE Trans. Circuits Syst. Regul. Pap. 56(4), 784–794 (2009)MathSciNetGoogle Scholar
  29. 43.
    G. Wu, E.-H. Yang, A new efficient method of computing MDCT in MP3 audio coding, in Proceedings of the IEEE International Conference on Multimedia Information Networking and Security (MINES’2010), Nanjing Yiangsu, November 2010, pp. 63–67Google Scholar
  30. 44.
    J. Wu, L. Wang, L. Senhadji, H. Shu, Improved radix-3 decimation-in-frequency algorithm for the fast computation of forward and inverse MDCT, in Proceedings of the IEEE International Conference on Audio, Language and Image Processing (ICALIP’2010), Shanghai, November 2010, pp. 694–699Google Scholar

Fast MLT and ELT Algorithms

  1. 45.
    C.Y. Jing, H.-M. Tai, Fast algorithm for computing modulated lapped transform. Electron. Lett. 37(12), 796–797 (2001)Google Scholar
  2. 46.
    C.Y. Jing, H.-M. Tai, Design and implementation of a fast algorithm for modulated lapped transform. IEE Proc. Image Signal Process. 149(1), 27–32 (2002)Google Scholar
  3. 47.
    H.S. Malvar, Lapped transforms for efficient transform/sub-band coding. IEEE Trans. Acoust. Speech Signal Process. 38(6), 969–978 (1990)Google Scholar
  4. 48.
    H.S. Malvar, Extended lapped transforms: fast algorithms and applications, in Proceedings of the IEEE ICASSP’91, Toronto, May 1991, pp. 1797–1800Google Scholar
  5. 49.
    H.S. Malvar, Fast algorithm for modulated lapped transform. Electron. Lett. 27(9), 775–776 (1991)Google Scholar
  6. 50.
    H.S. Malvar, Extended lapped transforms: properties, applications, and fast algorithms. IEEE Trans. Signal Process. 40(11), 2703–2714 (1992)zbMATHGoogle Scholar
  7. 51.
    H.S. Malvar, Fast algorithms for orthogonal and biorthogonal modulated lapped transforms, in Proceedings of IEEE Symposium on Advances in Digital Filtering and Signal Processing, Victoria, June 1998, pp. 159–163Google Scholar

Recursive/Regressive MDCT/MDST Filter Structures (oddly Stacked System)

  1. 52.
    C.-H. Chen, B.-D. Liu, J.-F. Yang, Recursive architectures for realizing modified discrete cosine transform and its inverse. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process. 50(1), 38–45 (2003)Google Scholar
  2. 53.
    H.C. Chiang, J.C. Liu, Regressive implementations for the forward and inverse MDCT in MPEG audio coding. IEEE Signal Process. Lett. 3(4), 116–118 (1996)Google Scholar
  3. 54.
    T.W. Fox, A. Carriera, Goertzel implementations of the forward and inverse modified discrete cosine transform, in Proceedings of the IEEE Canadian Conference on Electrical and Computer Engineering (CCECE’2004), vol. 4, Niagara Falls, May 2004, pp. 2371–2374Google Scholar
  4. 55.
    R. Koenig, T. Stripf, J. Becker, A novel recursive algorithm for bit-efficient realization of arbitrary length inverse modified cosine transforms, in Proceedings of the International Conference on Design, Automation and Test in Europe (DATE’2008), Munich, March 2008, pp. 604–609Google Scholar
  5. 56.
    S.-C. Lai, S.-F. Lei, C.-H. Luo, Common architecture design of novel recursive MDCT and IMDCT algorithms for application to AAC, AAC in DRM, and MP3 codecs. IEEE Trans. Circuits Syst. Express Briefs 56(10), 793–797 (2009)Google Scholar
  6. 57.
    S.-C. Lai, Y.-P. Yeh W.-C. Tseng, S.-F. Lei, Low-cost and high-accuracy design of fast recursive MDCT/MDST/IMDCT/IMDST algorithms and their realization. IEEE Trans. Circuits Syst. Express Briefs 59(1), 65–69 (2012)Google Scholar
  7. 58.
    S.-F. Lei, S.-C. Lai, Y.-T. Hwang, C.-H. Luo, A high-precision algorithm for the forward and inverse MDCT using the unified recursive architecture, in Proceedings of the IEEE International Symposium on Consumer Electronics, Algarve, April 2008, pp. 1–4Google Scholar
  8. 59.
    S.-F. Lei, S.-C. Lai, P.-Y. Cheng, C.-H. Luo, Low complexity and fast computation for recursive MDCT and IMDCT algorithms. IEEE Trans. Circuits Syst. Express Briefs 57(7), 571–575 (2010)Google Scholar
  9. 60.
    H. Li, P. Li, Y. Wang, An efficient hardware accelerator architecture for implementing fast IMDCT computation. Signal Process. 90(8), 2540–2545 (2010)zbMATHGoogle Scholar
  10. 61.
    V. Nikolajevic, G. Fettweis, New recursive algorithms for the forward and inverse MDCT, in Proceedings of the IEEE Workshop on Signal Processing Systems: Design and Implementation (SiPS’2001), Antwerp, September 2001, pp. 51–57Google Scholar
  11. 62.
    V. Nikolajevič, G. Fettweis, Computation of forward and inverse MDCT using Clenshaw’s recurrence formula. IEEE Trans. Signal Process. 51(5), 1439–1444 (2003)MathSciNetzbMATHGoogle Scholar
  12. 63.
    V. Nikolajevič, G. Fettweis, New recursive algorithms for the unified forward and inverse MDCT/MDST. J. VLSI Sig. Proc. Systems Signal Image Video Technol. 34(3), 203–208 (2003)zbMATHGoogle Scholar

Fast MCLT Algorithms

  1. 64.
    V. Britanak, New recursive fast radix-2 algorithm for the modulated complex lapped transform. IEEE Trans. Signal Process. 60(12), 6703–6708 (2012)MathSciNetzbMATHGoogle Scholar
  2. 65.
    X. Chen, Q. Dai, A novel DCT-based algorithm for computing the modulated complex lapped transform. IEEE Trans. Signal Process. 54(11), 4480–4484 (2006)zbMATHGoogle Scholar
  3. 66.
    Q. Dai, X. Chen, New algorithm for modulated complex lapped transform with symmetrical window function. IEEE Signal Process. Lett. 11(12), 925–928 (2004)Google Scholar
  4. 67.
    X. Dai, M.D. Wagh, Fast algorithm for modulated complex lapped transform. IEEE Signal Process Lett. 16(1), 30–33 (2009)Google Scholar
  5. 68.
    H. Malvar, Fast algorithm for the modulated complex lapped transform. IEEE Signal Process. Lett. 10(1), 8–10 (2003)Google Scholar
  6. 69.
    H. Shu, J. Wu, L. Senhadji, L. Luo, New fast algorithm for modulated complex lapped transform with sine windowing function. IEEE Signal Process Lett. 16(2), 93–96 (2009)Google Scholar
  7. 70.
    H.-M. Tai, C. Jing, Design and efficient implementation of a modulated complex lapped transform using pipeline technique. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E84-A(5), 1280–1287 (2001)Google Scholar

Supporting Literature

  1. 71.
    M.F. Aburdene, J. Zheng, R.J. Kozick, Computation of discrete cosine transform using Clenshaw’s recurrence formula. IEEE Signal Process Lett. 2(8), 155–156 (1995)Google Scholar
  2. 72.
    G. Bi, Y. Zeng, Transforms and Fast Algorithms for Signal Analysis and Representation, chap. 6 (Birkhäuser, Boston, 2004), pp. 207–245zbMATHGoogle Scholar
  3. 73.
    E.O. Brigham, FFT transform applications (Chapter 9), in The Fast Fourier Transform and its Applications (Prentice-Hall, Englewood Cliffs, NJ, 1988), pp. 188–191Google Scholar
  4. 74.
    V. Britanak, K.R. Rao, The fast generalized discrete Fourier transforms: a unified approach to the discrete sinusoidal transforms computation. Signal Process. 79(12), 135–150 (1999)zbMATHGoogle Scholar
  5. 75.
    V. Britanak, P. Yip, K.R. Rao, Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations (Academic, Elsevier Science, Amsterdam, 2007)Google Scholar
  6. 76.
    L.-P. Chau, W.-C. Siu, Recursive algorithm for the discrete cosine transform with general lengths. Electron. Lett. 30(3), 197–198 (1994). Errata, Electron. Lett. 30(7), 608 (1994)Google Scholar
  7. 77.
    L.-P. Chau, W.-C. Siu, Efficient recursive algorithm for the inverse discrete cosine transform. IEEE Signal Process Lett. 7(10), 276–277 (2000)Google Scholar
  8. 78.
    R.E. Crochiere, L.E. Rabiner, Multirate techniques in filter banks and spectrum analyzers and synthesizers (Chapter 7), in Multirate Digital Signal Pocessing (Prentice-Hall, Englewood Cliffs, NJ, 1983), pp. 289–400Google Scholar
  9. 79.
    P. Duhamel, Implementation of ‘Split-Radix’ FFT algorithms for complex, real, and real-symmetric data. IEEE Trans. Acoust. Speech Signal Process. ASSP-34(2), 285–295 (1986)MathSciNetGoogle Scholar
  10. 80.
    P. Duhamel, B. Piron, J.M. Etcheto, On computing the inverse DFT. IEEE Trans. Acoust. Speech Signal Process. ASSP-36(2), 285–286 (1988)zbMATHGoogle Scholar
  11. 81.
    G. Goertzel, An algorithm for the evaluation of finite trigonometric series. Am. Math. Mon. 65, 34–35 (1958)MathSciNetGoogle Scholar
  12. 82.
    S.G. Johnson, M. Frigo, A modified split-radix FFT with fewer arithmetic operations. IEEE Trans. Signal Process. 55(1), 111–119 (2007)MathSciNetzbMATHGoogle Scholar
  13. 83.
    C.W. Kok, Fast algorithm for computing discrete cosine transform. IEEE Trans. Signal Process. 45(3), 757–760 (1997)Google Scholar
  14. 84.
    K.R. Rao, J.J. Hwang, Techniques and Standards for Image, Video, and Audio Coding (Prentice Hall, Upper Saddle River, NJ, 1996)Google Scholar
  15. 85.
    H.V. Sorensen, M.T. Heideman, C.S. Burrus, On computing the split-radix FFT, IEEE Trans. Acoust. Speech Signal Process. ASSP-34(1), 152–156 (1986)Google Scholar
  16. 86.
    H.V. Sorensen, D.L. Jones, M.T. Heideman, Real-valued fast Fourier transform algorithms. IEEE Transactions on Acoustics, Speech and Signal Processing ASSP-35(6), 849–863 (1987)Google Scholar
  17. 87.
    Z. Wang, G.A. Julien, W.C. Miller, Recursive algorithm for the forward and inverse discrete cosine transform with arbitrary length. IEEE Signal Process. Lett. 1(7), 101–102 (1994)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Vladimir Britanak
    • 1
  • K. R. Rao
    • 2
  1. 1.Institute of InformaticsSlovak Academy of SciencesBratislavaSlovakia
  2. 2.The University of Texas at ArlingtonArlingtonUSA

Personalised recommendations