Vladimir Britanak

A new fast algorithm for the unified forward and inverse MDCT/MDST computation

The modified discrete cosine transform (MDCT) and modified discrete sine transform (MDST) are employed in subband/transform coding schemes as the analysis/synthesis filter banks based on the concept of time domain aliasing cancellation (TDAC). Princen, Bradley and Johnson defined two types of the MDCT, specifically, for an evenly stacked and oddly stacked analysis/synthesis systems. The MDCT is the basic processing component in the international audio coding standards and commercial products for high-quality audio compression. Almost all existing audio coding systems have used the complex-valued or real-valued FFT algorithms, and the DCT/DST of type IV (DCT-IV/DST-IV) for the fast MDCT computation. New fast and efficient algorithm for a unified forward and inverse MDCT/MDST computation in the oddly stacked system is proposed. It is based on the DCT/DST of types II and III (DCT-II/DST-II, DCT-III/DST-III), and the real arithmetic is used only. Corresponding generalized signal flow graph is regular, structurally simple and enables to compute MDCT/MDST and their inverses in general for any N divisible by 4 (N being length of a data sequence). Consequently, the new fast algorithm can be adopted for the MDCT computation in the current audio coding standards such as MPEG family (MPEG-1, MPEG-2, MPEG-2 Advanced Audio Coding and MPEG-4 audio), and in commercial products (proprietary audio coding algorithms) such as Sony MiniDisc/ATRAC/ATRAC2/SDDS digital audio coding systems, the AT & T Perceptual Audio Coder (PAC) or Lucent Technologies PAC/Enhanced PAC/Multichannel PAC, and Dolby Labs AC-3 digital audio compression algorithm. Besides the new fast algorithm has some interesting properties, it provides an efficient implementation of the forward and inverse MDCT computation for layer III in MPEG audio coding, where the length of data blocks N ≠ 2n, Especially, for the AC-3 algorithm, it is shown how both the proposed new MDCT/MDST algorithm and existing fast algorithms/computational architectures for the discrete sinusoidal transforms computation of real data sequences such as the DCT-IV/DST-IV, generalized discrete Fourier transform of type IV (DFT-IV) and generalized discrete Hartley transform of type IV (DHT-IV) can be used for the fast alternate or simultaneous (on-line) MDCT/MDST computation by simple pre-and post-processing of data sequences.

The fast generalized discrete Fourier transforms: a unified approach to the discrete sinusoidal transforms computation

Abstract Special forms of the generalized discrete Fourier transform (GDFT) matrices are investigated and their sparse matrix factorizations are presented to complete Wangs set of real sparse matrix factorizations for the family of discrete sinusoidal transforms. Different versions of the GDFT, different versions of the generalized discrete Hartley transform (GDHT) or equivalently of the discrete W transform (DWT), various versions of the discrete cosine transform (DCT) and discrete sine transform (DST) are members of the discrete sinusoidal transform family. There are intrinsic relationships among corresponding versions of the GDFT, GDHT (DWT), DCT and DST for real data sequences. A real sparse matrix factorization of GDFT matrices leads to simple fast algorithms for their computation, where only real arithmetic is involved. The resulting generalized signal flow graphs for the computation of different versions of the GDFT represent simple and compact unified approach to the fast discrete sinusoidal transforms computation. It is also shown that all algorithms are based on the universal DCT-II/DST-II (DCT-III/DST-III) computational structure which is used as the basic processing component.

CHAPTER 1 – Discrete Cosine and Sine Transforms

This chapter provides an overview of this book. The book presents the complete set of discrete cosine transforms (DCTs) and discrete sine transforms (DSTs) constituting the entire class of discrete sinusoidal unitary transforms, including their definitions, general mathematical properties, relations to the Karhunen-Loeve transform (KLT), with the emphasis on fast algorithms and integer approximations for their efficient implementations in the integer domain. The book covers various latest developments in DCTs and DSTs in a unified way, and it is essentially a detailed excursion on orthogonal/orthonormal DCT and DST matrices, their matrix factorizations, and integer approximations. For the DCT and DST to be viable, feasible, and practical, the fast algorithms are essential for their efficient implementation in terms of reduced memory, implementation complexity, and recursivity. Extensive definitions, principles, properties, signal flow graphs, derivations, proofs, and examples are provided throughout the book for proper understanding of the strengths and shortcomings of the spectrum of cosine/sine transforms and their application in diverse disciplines.

Short communication: the fast DCT-IV/DST-IV computation via the MDCT

The discrete cosine transform of type IV (DCT-IV) and corresponding discrete sine transform of type IV (DST-IV) have played key role in the efficient implementation of orthogonal lapped transforms and perfect reconstruction cosine-modulated filter banks such as the oddly stacked modified discrete cosine transform (MDCT) or equivalently, the modulated lapped transform (MLT). However, the DCT-IV and DST-IV of double sizes are related to two variants of filter banks defined by Dolby Labs AC-3 digital audio compression algorithm. Since these two variants of filter banks are efficiently computed by recently proposed new fast algorithm for the oddly stacked MDCT (Signal Processing 82 (2002) 433), it is shown that the efficient DCT-IV and DST-IV computation can be realized via the MDCT of double size. The careful analysis of regular structure of the new fast MDCT algorithm allows to extract a new DCT-IV/DST-IV computational structure and to suggest a new sparse matrix factorization of the DCT-IV matrix. Finally, the new DCT-IV/DST-IV computational structure provides an alternative efficient implementation of the forward and inverse MDCT in layer III of MPEG (MP3) audio coding.

Fast computational structures for an efficient implementation of the complete TDAC analysis/synthesis MDCT/MDST filter banks

A new fast computational structure identical both for the forward and backward modified discrete cosine/sine transform (MDCT/MDST) computation is described. It is the result of a systematic construction of a fast algorithm for an efficient implementation of the complete time domain aliasing cancellation (TDAC) analysis/synthesis MDCT/MDST filter banks. It is shown that the same computational structure can be used both for the encoder and the decoder, thus significantly reducing design time and resources. The corresponding generalized signal flow graph is regular and defines new sparse matrix factorizations of the discrete cosine transform of type IV (DCT-IV) and MDCT/MDST matrices. The identical fast MDCT computational structure provides an efficient implementation of the MDCT in MPEG layer III (MP3) audio coding and the Dolby Labs AC-3 codec. All steps to derive the computational structure are described in detail, and to put them into perspective a comprehensive list of references classified into categories is provided covering new research results achieved in the time period 1999-2008 in theoretical and practical developments of TDAC analysis/synthesis MDCT/MDST filter banks (general mathematical, symmetry and special properties, fast MDCT/MDST algorithms and efficient software/hardware implementations of the MDCT in MP3).

An efficient computing of oddly stacked MDCT/MDST via evenly stacked MDCT/MDST and vice versa

New fast algorithms for the evenly stacked and oddly stacked modified discrete cosine transform (MDCT) and modified discrete sine transform (MDST) computation are proposed. Although the evenly and oddly stacked MDCT/ MDST are quite different filter banks based on time domain aliasing cancellation (TDAC), there exists an intimate relation between them, and consequently, the efficient computing of oddly stacked MDCT/MDST can be realized via the evenly stacked MDCT/MDST and vice versa only by simple pre-and post-processing of input and output data sequences. This fact allows to handle the evenly and oddly stacked MDCT/MDST in a unified framework. In particular, it is shown that the transposed evenly and oddly stacked MDCT and MDST matrices are actually the pseudoinverses of their corresponding forward transform matrices. The regular generalized signal flow graphs define interrelated sparse matrix factorizations of the evenly and oddly stacked MDCT/MDST matrices. The proposed new fast algorithms provide efficient implementations of the oddly stacked MDCT in layer III of MPEG (MP3) audio coding. Complete signal flow graphs for the efficient implementation of the MDCT in MP3 and comparison with existing efficient modified/corrected implementations are also presented.

Review: A survey of efficient MDCT implementations in MP3 audio coding standard: Retrospective and state-of-the-art

This tutorial paper describes various efficient implementations (published and new unpublished) of the forward and backward modified discrete cosine transform (MDCT) in the MPEG layer III (MP3) audio coding standard developed in the time period 1990-2010, including the efficient implementation of polyphase filter banks for completeness. The efficient MDCT implementations are discussed in the context of (fast) complete analysis/synthesis MDCT filter banks in the MP3 encoder and decoder. In general, for each efficient forward/backward MDCT block transforms implementation are presented: complete formulas or sparse matrix factorizations of the algorithm, the corresponding signal flow graph for the short audio block and the total arithmetic complexity as well as the useful comments related to improving the arithmetic complexity and a possible structural simplification of the algorithm. Finally, all efficient forward/backward MDCT implementations are compared both in terms of the arithmetic complexity and structural simplicity. It is important to note that almost all presented algorithms can be also used for the 2^n-length data blocks in others MPEG audio coding standards and proprietary audio compression algorithms.

New generalized conversion method of the MDCT to MDST coefficients in the frequency domain for arbitrary symmetric windowing function

A new generalized conversion method of the MDCT to MDST coefficients directly in the frequency domain is proposed for arbitrary symmetric windowing function. Based on the compact block matrix representation of the MDCT and MDST filter banks, on their properties and on relations among transform sub-matrices, a relation in the matrix-vector form between the MDCT and MDST coefficients in the frequency domain is derived. Given MDCT coefficients of three consecutive data blocks at a decoder, the MDST coefficients of the current data block can be obtained by combining the MDCT coefficients of the previous, current and next blocks via conversion matrices. Since the forms of conversion matrices depend on the employed windowing function, a specific solution for each windowing function is derived. Because the conversion matrices have a very regular structure, the matrix-vector products are reduced to simple analytical formulas. The new generalized conversion method is more efficient and structurally simpler both in terms of arithmetic complexity and memory requirements compared to existing exact frequency domain-based conversion methods. Although the new generalized conversion method enables us to compute the exact MDST coefficients only in specified one or more frequency ranges, the computation of complete set of MDST coefficients still requires a high number of arithmetic operations. As an alternative, an efficient and flexible approximate conversion method is constructed. With properly selected parameters it can produce acceptable approximated results with much lower computational complexity. Therefore, the approximate conversion method has a potential to be used in many MDCT-based audio decoders, and particularly at resource-limited and low-cost decoders for spectral analysis to obtain the magnitude and phase information.

New fast algorithms for the low delay MDCT computation in the MPEG-4 AAC enhanced low delay audio coding standard

Abstract Recently, the MPEG committee has completed the development of a new audio coding standard, the MPEG-4 Advanced Audio Coding-Enhanced Low Delay (AAC-ELD). The state-of-the-art MPEG audio coding standards, such as MPEG-4 AAC Low Complexity (AAC-LC), High Efficiency AAC (HE-AAC) and AAC Low Delay (AAC-LD), utilize for the time-to-frequency transformation of an audio block and vice versa, the well known time domain aliasing cancellation modified discrete cosine transform (TDAC-MDCT). In order to achieve low algorithmic delay, the AAC-ELD has adopted a perfect reconstruction low delay filter bank, called the low delay MDCT (LD-MDCT). New fast algorithms for the LD-MDCT computation in the AAC-ELD are proposed. Further, a simple relation between the LD-MDCT and the TDAC-MDCT is derived. Exploiting this relation, as an alternative, the improved fast LD-MDCT algorithms based on the TDAC-MDCT are presented. They do not require reverse operations and sign changes with respect to both the time and frequency indices compared to existing TDAC-MDCT-based fast algorithms. Since the AAC-LC, HE-AAC and AAC-LD audio codecs use the TDAC-MDCT, the new proposed and improved TDAC-MDCT-based fast LD-MDCT algorithms provide the unified efficient implementation of LD-MDCT and TDAC-MDCT filter banks in all four codecs: AAC-ELD, AAC-LD, HE-AAC and AAC-LC.

MDCT/MDST, MLT, ELT, and MCLT Filter Banks: Definitions, General Properties, and Matrix Representations

The perfect reconstruction cosine/sine-modulated filter banks belonging to the class of modulated filter banks have been studied extensively due to their attractive features (simple structure, analysis and synthesis filters are of equal length, low computational complexity), and consequently, they have received a great interest in audio coding applications. In fact, they are employed in the international speech and audio coding standards and proprietary audio compression algorithms. The oddly and evenly stacked modified discrete cosine transform (MDCT) and the corresponding modified discrete sine transform (MDST), the modulated lapped transform (MLT), the extended lapped transforms (ELTs), and their biorthogonal versions are real-valued cosine/sine-modulated filter banks satisfying the perfect reconstruction property. The modulated complex lapped transform (MCLT) is the complex-valued filter bank whose real part is the MLT or equivalently, the oddly stacked MDCT, and the imaginary part is the oddly stacked MDST. In this chapter, definitions, general properties, and matrix representations of the MDCT/MDST, MLT, ELT, and MCLT filter banks are presented. In order to an analysis/synthesis filter bank be perfect reconstruction, the necessary and sufficient conditions imposed on the analysis and synthesis windowing functions play an important role. Therefore, additionally the windowing procedure and perfect reconstruction (biorthogonal) conditions in the case of identical and (nonidentical) analysis and synthesis windowing functions, design of a windowing function including definitions of commonly windowing functions used in audio coding applications, adaptive switching of transform block sizes and windowing functions, and general perfect reconstruction conditions for the ELT filter bank with multiple overlapping factor both for the orthogonal and biorthogonal cases are derived and/or discussed in detail.