New universal rotation-based fast computational structures for an efficient implementation of the DCT-IV/DST-IV and analysis/synthesis MDCT/MDST filter banks

New universal rotation-based fast computational structures identical both for the discrete cosine/sine transform of type IV (DCT-IV/DST-IV) and the forward/backward modified discrete cosine/sine transform (MDCT/MDST) computation are described. They are the result of a systematic construction of a fast algorithm for an efficient implementation of the time domain aliasing cancellation (TDAC) analysis/synthesis MDCT/MDST filter banks employed in various audio compression schemes. New fast algorithms provide novelty computational structures based exclusively on the computation of Givens-Jacobi rotations, and thus, the need of any discrete sinusoidal unitary transform such as the discrete Fourier transform (DFT), DCT-IV/DST-IV or discrete cosine/sine transforms of type II (DCT-II/DST-II) of reduced size is completely eliminated, so simplifying the computational structure of the algorithms. The rotators and summators are used only as the basic computational modules (in a hardware implementation they are simple hardware blocks). The simple and regular Givens-Jacobi rotation-based fast computational structures valid for any N divisible by 4 (N being the length of data sequence) define new sparse matrix factorizations of the DCT-IV and MDCT matrices and in particular, generate an efficient implementation of the MDCT in MP3 audio coding standard. For a given N=2^n they can be easily reconfigurable for a specific audio coding scheme. Finally, since Givens-Jacobi rotation can be factored into a product of Gauss elementary matrices being unit lower and unit upper triangular matrices, the new fast rotation-based computational structures are suitable for an integer approximation of the DCT-IV/DST-IV (integer DCT-IV/DST-IV) and MDCT/MDST (integer MDCT/MDST) which are currently modern transform technologies for lossless audio coding.

[1]  Gerald Schuller,et al.  Improved integer transforms using multi-dimensional lifting [audio coding examples] , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[2]  Henrique S. Malvar,et al.  Fast algorithms for orthogonal and biorthogonal modulated lapped transforms , 1998, 1998 IEEE Symposium on Advances in Digital Filtering and Signal Processing. Symposium Proceedings (Cat. No.98EX185).

[3]  Steven G. Johnson,et al.  Type-IV DCT, DST, and MDCT algorithms with reduced numbers of arithmetic operations , 2007, Signal Process..

[4]  R. Geiger,et al.  Integer low delay and MDCT filter banks , 2002, Conference Record of the Thirty-Sixth Asilomar Conference on Signals, Systems and Computers, 2002..

[5]  Gerald Schuller,et al.  Improved integer transforms for lossless audio coding , 2003, The Thrity-Seventh Asilomar Conference on Signals, Systems & Computers, 2003.

[6]  Meghanad D. Wagh,et al.  Fast Algorithm for Modulated Complex Lapped Transform , 2009, IEEE Signal Processing Letters.

[7]  Soontorn Oraintara,et al.  Approximation error analysis for transform-based lossless audio coding , 2004, IEEE Global Telecommunications Conference, 2004. GLOBECOM '04..

[8]  Jin Li.  A progressive to lossless embedded audio coder (PLEAC) with reversible modulated lapped transform , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[9]  Gerlind Plonka,et al.  A global method for invertible integer DCT and integer wavelet algorithms , 2004 .

[10]  Szu-Wei Lee.  Improved algorithm for efficient computation of the forward and backward MDCT in MPEG audio coder , 2001 .

[11]  R. Gluth.  Regular FFT-related transform kernels for DCT/DST-based polyphase filter banks , 1991, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing.

[12]  Pengwei Hao,et al.  On the necessity and sufficiency of PLUS factorizations , 2005 .

[13]  D. Faddeev,et al.  Computational methods of linear algebra , 1959 .

[14]  Tao Zhang,et al.  On the relationship of MDCT transform kernels in Dolby AC-3 , 2008, 2008 International Conference on Audio, Language and Image Processing.

[15]  Susanto Rahardja,et al.  A fast algorithm of integer MDCT for lossless audio coding , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[16]  Manfred Tasche,et al.  Invertible integer DCT algorithms , 2003 .

[17]  G. Plonka,et al.  Fast and numerically stable algorithms for discrete cosine transforms , 2005 .

[18]  Heng-Ming Tai,et al.  Design and implementation of a fast algorithm for modulated lapped transform , 2002 .

[19]  Miriam Primbs.  Worst-case error analysis of lifting-based fast DCT-algorithms , 2005, IEEE Transactions on Signal Processing.

[20]  K.R. Rao,et al.  Improved lossless audio coding using the noise-shaped IntMDCT , 2004, 3rd IEEE Signal Processing Education Workshop. 2004 IEEE 11th Digital Signal Processing Workshop, 2004..

[21]  Meghanad D. Wagh,et al.  An MDCT Hardware Accelerator for MP3 Audio , 2008, 2008 Symposium on Application Specific Processors.

[22]  Xin Yang,et al.  Tradeoffs in modified discrete cosine transform implementations , 2001, ASICON 2001. 2001 4th International Conference on ASIC Proceedings (Cat. No.01TH8549).

[23]  Yuriy A. Reznik,et al.  Efficient implementation of a class of MDCT/IMDCT filterbanks for speech and audio coding applications , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[24]  Hi-Seok Kim,et al.  A new optimized algorithm for computation of MDCT and its inverse transform , 2004, Proceedings of 2004 International Symposium on Intelligent Signal Processing and Communication Systems, 2004. ISPACS 2004..

[25]  Jürgen Herre,et al.  IMPROVED INTEGER TRANSFORMS USING MULTI-DIMENSIONAL LIFTING , 2004 .

[26]  Ajay Kumar Bangla,et al.  A new efficient implementation of TDAC synthesis filterbank based on Radix-2 FFT , 2006, 2006 14th European Signal Processing Conference.

[27]  Gerhard Fettweis,et al.  New recursive algorithms for the forward and inverse MDCT , 2001, 2001 IEEE Workshop on Signal Processing Systems. SiPS 2001. Design and Implementation (Cat. No.01TH8578).

[28]  H.S. Malvar,et al.  A nonuniform modulated complex lapped transform , 2001, IEEE Signal Processing Letters.

[29]  Hi-Seok Kim,et al.  An optimized algorithm for computing the modified discrete cosine transform and its inverse transform , 2004, 2004 IEEE Region 10 Conference TENCON 2004..

[30]  John Princen,et al.  Subband/Transform coding using filter bank designs based on time domain aliasing cancellation , 1987, ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[31]  H. Tai,et al.  Fast algorithm for computing modulated lapped transform , 2001 .

[32]  Ralf Geiger,et al.  Audio Coding based on Integer Transforms , 2001 .

[33]  T.W. Fox,et al.  Goertzel implementations of the forward and inverse modified discrete cosine transform , 2004, Canadian Conference on Electrical and Computer Engineering 2004 (IEEE Cat. No.04CH37513).

[34]  Jin Li.  Low noise reversible MDCT (RMDCT) and its application in progressive-to-lossless embedded audio coding , 2005, IEEE Transactions on Signal Processing.

[35]  G. Bioul,et al.  Synthesis of Arithmetic Circuits: FPGA, ASIC and Embedded Systems , 2006 .

[36]  Soontorn Oraintara,et al.  The integer MDCT and its application in the MPEG layer III audio , 2003, Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03..

[37]  K.R. Rao,et al.  An efficient implementation of the forward and inverse MDCT in MPEG audio coding , 2001, IEEE Signal Processing Letters.

[38]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[39]  Henrique S. Malvar,et al.  Fast algorithm for the modulated complex lapped transform , 2003, IEEE Signal Processing Letters.

[40]  Heng-Ming Tai,et al.  Design and Efficient Implementation of a Modulated Complex Lapped Transform Processor Using Pipelining Technique , 2001 .

[41]  Jiasong Wu,et al.  Mixed-Radix Algorithm for the Computation of Forward and Inverse MDCTs , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[42]  Lotfi Senhadji,et al.  A fast algorithm for the computation of 2-D forward and inverse MDCT , 2008, Signal Process..

[43]  K. Rao,et al.  Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations , 2006 .

[44]  RuiMin Hu,et al.  Scalable Audio Coding Based on Integer Transform , 2006, 2006 First International Conference on Communications and Networking in China.

[45]  Susanto Rahardja,et al.  Integer MDCT with enhanced approximation of the DCT-IV , 2006, IEEE Transactions on Signal Processing.

[46]  Songyu Yu,et al.  1-D and 2-D transforms from integers to integers , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[47]  Xinjian Chen,et al.  A Novel DCT-Based Algorithm for Computing the Modulated Complex Lapped Transform , 2006, IEEE Transactions on Signal Processing.

[48]  Xinjian Chen,et al.  New algorithm for modulated complex lapped transform with symmetrical window function , 2004, IEEE Signal Processing Letters.

[49]  Jürgen Herre,et al.  IntMDCT - A link between perceptual and lossless audio coding , 2002, 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[50]  Vladimir Britanak.  An efficient computing of oddly stacked MDCT/MDST via evenly stacked MDCT/MDST and vice versa , 2005, Signal Process..

[51]  Gerhard Fettweis,et al.  Computation of forward and inverse MDCT using Clenshaw's recurrence formula , 2003, IEEE Trans. Signal Process..

[52]  Vladimir Britanak.  The fast DCT-IV/DST-IV computation via the MDCT , 2003, Signal Process..

[53]  Tao Zhang,et al.  A new algorithm on short window MDCT for Dolby AC3 , 2007, 2007 International Symposium on Intelligent Signal Processing and Communication Systems.

[54]  G. Fettweis,et al.  Improved implementation of MDCT in MP3 audio coding , 2004, APCC/MDMC '04. The 2004 Joint Conference of the 10th Asia-Pacific Conference on Communications and the 5th International Symposium on Multi-Dimensional Mobile Communications Proceeding.

[55]  Sridhar Srinivasan,et al.  Modulo transforms - an alternative to lifting , 2006, IEEE Transactions on Signal Processing.

[56]  Po-sheng Wu,et al.  Efficient IMDCT core designs for audio signal processing , 2003, 2003 IEEE Workshop on Signal Processing Systems (IEEE Cat. No.03TH8682).

[57]  Mu-Huo Cheng,et al.  Fast IMDCT and MDCT algorithms - a matrix approach , 2003, IEEE Trans. Signal Process..

[58]  Che-Hong Chen,et al.  Recursive architectures for realizing modified discrete cosine transform and its inverse , 2003, IEEE Trans. Circuits Syst. II Express Briefs.

[59]  Trieu-Kien Truong,et al.  Fast algorithm for computing the forward and inverse MDCT in MPEG audio coding , 2006, Signal Process..

[60]  Pengwei Hao,et al.  Customizable triangular factorizations of matrices , 2004 .

[61]  Soontorn Oraintara,et al.  Fast and lossless implementation of the forward and inverse MDCT computation in MPEG audio coding , 2002, 2002 IEEE International Symposium on Circuits and Systems. Proceedings (Cat. No.02CH37353).

[62]  Soontorn Oraintara,et al.  Lossless Audio Coding Using the IntMDCT and Rounding Error Shaping , 2006, IEEE Transactions on Audio, Speech, and Language Processing.

[63]  Ted Painter,et al.  Audio Signal Processing and Coding , 2007 .

[64]  Marina Bosi,et al.  Introduction to Digital Audio Coding and Standards , 2004, J. Electronic Imaging.

[65]  Gene H. Golub,et al.  Matrix computations , 1983 .

[66]  Gerhard Fettweis,et al.  New Recursive Algorithms for the Unified Forward and Inverse MDCT/MDST , 2003, J. VLSI Signal Process..

[67]  Susanto Rahardja,et al.  Method for realising reversible integer type-IV discrete cosine transform , 2004 .

[68]  D. Faddeev,et al.  Computational Methods of Linear Algebra , 1959 .

[69]  Yin-Tsung Hwang,et al.  A novel MDCT/IMDCT computing kernel design , 2005, IEEE Workshop on Signal Processing Systems Design and Implementation, 2005..

[70]  Henrique S. Malvar.  Lossless and Near-Lossless Audio Compression Using Integer-Reversible Modulated Lapped Transforms , 2007, 2007 Data Compression Conference (DCC'07).

[71]  Chi-Min Liu,et al.  A Unified Fast Algorithm for Cosine Modulated Filter Banks in Current Audio Coding Standards , 1999 .

[72]  M. Dietz,et al.  MPEG-4 high-efficiency AAC coding [Standards in a Nutshell] , 2008 .

[73]  Meghanad D. Wagh,et al.  Bilinear algorithms for discrete cosine transforms of prime lengths , 2006, Signal Process..

[74]  Vladimir Britanak,et al.  Fast computational structures for an efficient implementation of the complete TDAC analysis/synthesis MDCT/MDST filter banks , 2009, Signal Process..

[75]  Susanto Rahardja,et al.  On Integer MDCT for Perceptual Audio Coding , 2007, IEEE Transactions on Audio, Speech, and Language Processing.

[76]  N. Rama Murthy,et al.  A parallel/pipelined algorithm for the computation of MDCT and IMDCT , 2003, Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03..

[77]  K.R. Rao,et al.  A comparison of integer fast Fourier transforms for lossless coding , 2004, IEEE International Symposium on Communications and Information Technology, 2004. ISCIT 2004..

[78]  Vladimir Britanak,et al.  A new fast algorithm for the unified forward and inverse MDCT/MDST computation , 2002, Signal Process..

[79]  Awaji Island,et al.  LOSSLESS AUDIO COMPRESSION USING INTEGER MODIFIED DISCRETE COSINE TRANSFORM , 2003 .

[80]  Jar-Ferr Yang,et al.  Recursive architectures for the forward and inverse modified discrete cosine transforms , 2000, 2000 IEEE Workshop on SiGNAL PROCESSING SYSTEMS. SiPS 2000. Design and Implementation (Cat. No.00TH8528).

[81]  Huazhong Shu,et al.  Radix-3 Algorithm for the Fast Computation of Forward and Inverse MDCT , 2007, IEEE Signal Processing Letters.

[82]  Henrique S. Malvar.  A modulated complex lapped transform and its applications to audio processing , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[83]  Jin Li.  Reversible FFT and MDCT via matrix lifting , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[84]  Fuhuei Lin,et al.  A unified computing kernel for MDCT/IMDCT in modern audio coding standards , 2007, 2007 International Symposium on Communications and Information Technologies.

[85]  Jar-Ferr Yang,et al.  Unified selectable fixed-coefficient recursive structures for computing DCT, IMDCT and subband synthesis filtering , 2004, 2004 IEEE International Symposium on Circuits and Systems (IEEE Cat. No.04CH37512).

[86]  Jiasong Wu,et al.  New Fast Algorithm for Modulated Complex Lapped Transform With Sine Windowing Function , 2009, IEEE Signal Processing Letters.

[87]  Henrique S. Malvar,et al.  Signal processing with lapped transforms , 1992 .