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Patera , Anti symmetric multivariate exponential functions and corresponding Fourier transforms , J. A 40 , no. Patera , Anti symmetric multivariate trigonometric functions and corresponding Fourier transforms , J. I Math. Fourier Anal. Louis Pasteur, Strasbourg, , pp. B45a, Moody , L. Patera , Gaussian cubature arising from hybrid characters of simple Lie groups , J. IMRN 16 , — Sergeev and A.

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Veselov , Jacobi-Trudy formula for generalized Schur polynomials , Mosc. Theory and Applications, vol. MR Bogoliubov, Boxed plane partitions as an exactly solvable boson model , J.

Yip, and K. Rao, Discrete cosine and sine transforms , General properties, fast algorithms and integer approximations.

## Discrete cosine transform

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## Discrete sine transform

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In general, audio coding or audio compression algorithms are used to obtain compact digital representation of high-quality audio signals for their efficient transmission and storage. The central objective in audio coding is to represent the signal with a minimum number of bits while achieving its transparent reproduction. Besides speech coding schemes based on linear prediction methods especially tailored for efficient speech compression, the developed perceptual transform-based audio coding schemes gained a greater attention, particularly for applications in consumer electronics.

## Discrete cosine transform - Wikipedia

The discussion is concentrated especially on adopted near-perfect QMF and perfect reconstruction cosine-modulated filter banks, processing methods, and specified transform block sizes. In fact, they are employed in the international speech and audio coding standards and proprietary audio compression algorithms.

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.

The MPEG-1 and MPEG-2 audio have been the first international audio coding standards specifying the digital format for high quality compression of audio signals. In layer III known as MP3 it has additionally adopted the perfect reconstruction modulated lapped transform MLT which is actually the modified discrete cosine transform MDCT associated with the sine windowing function.

This chapter describes and compares various efficient implementations of the forward and backward MLT MDCT in the MP3 audio coding standard developed in the time period —, including the efficient implementation of pseudo-QMF banks for completeness. The AC-3 besides the MDCT defines additional two variants of cosine-modulated filter banks called the first and second short transforms. Matrix representations of AC-3 filter banks, [54] their properties and relations among transform sub- matrices provide the basis to derive relations between the frequency coefficients and the time domain aliasing data sequences of AC-3 transforms, and in particular, the basis for derivation of a fast algorithm for conversion of frequency coefficients of AC-3 transforms directly in the frequency domain.

The SBR is a bandwidth extension method which significantly improves the compression efficiency of perceptual audio and speech coding schemes.