Discrete cosine and sine transforms: general properties, by Vladimir Britanak

By Vladimir Britanak

The Discrete Cosine remodel (DCT) is utilized in many functions through the clinical, engineering and study groups and in facts compression particularly. quick algorithms and functions of the DCT kind II (DCT-II) became the guts of many validated foreign image/video coding criteria. considering that then different kinds of the DCT and Discrete Sine remodel (DST) were investigated intimately. This re-creation offers the whole set of DCT and DST discrete trigonometric transforms, together with their definitions, normal mathematical homes, and relatives to the optimum Karhunen-Loéve rework (KLT), with the emphasis on quick algorithms (one-dimensional and two-dimensional) and integer approximations of DCTs and DSTs for his or her effective implementations within the integer area. DCTs and DSTs are real-valued transforms that map integer-valued indications to floating-point coefficients. To cast off the floating-point operations, a number of equipment of integer approximations were proposed to build and flexibly generate a relations of integer DCT and DST transforms with arbitrary accuracy and function. The integer DCTs/DSTs with reasonably cheap and low-powered implementation can exchange the corresponding real-valued transforms in instant and satellite tv for pc communique structures in addition to transportable computing functions. The booklet is largely a close day trip on orthogonal/orthonormal DCT and DST matrices, their matrix factorizations and integer aproximations. it really is was hoping that the booklet will function a beneficial reference for undefined, academia and study institutes in constructing integer DCTs and DSTs in addition to an suggestion resource for additional complicated study. Key positive aspects - Presentation of the entire set of DCTs and DSTs in context of whole classification of discrete unitary sinusoidal transforms: the starting place, definitions, normal mathematical houses, mutual relationships and family to the optimum Karhunen-Loéve rework (KLT). - Unified remedy with the short implementations of DCTs and DSTs: the quick rotation-based algorithms derived within the kind of recursive sparse matrix factorizations of a remodel matrix together with one- and two-dimensional instances. - distinctive presentation of varied equipment and layout methods to integer approximation of DCTs and DSTs using the elemental innovations of linear algebra, matrix thought and matrix computations resulting in their effective multiplierless real-time implementations, or commonly reversible integer-to-integer implementations. - accomplished checklist of extra references reflecting recent/latest advancements within the effective implementations of DCTs and DSTs typically one-, two-, 3- and multi-dimensional quick DCT/DST algorithms together with the new lively learn issues for the period of time from 1990 during the past.

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Some obvious properties follow: (a) Inversion: Fs = Fs−1 . 25) meaning that, Fs {Fs [x(t)]} = x(t), for t ≥ 0. 27) where α, β are constants. The fact that FST involves integration makes it obvious that it is a linear operator. (c) Scaling in time: Fs [x(at)] = a−1 Xs ω , a for real a > 0. 7). (d) Shift in time: Fs [x(t − α)] = Fs [x(t)] cos(αω) + Fc [x(t)] sin(αω). 8a), the result is obtained using the compound angle expansion. By defining an odd extension xo of the function x(t) such that xo (t) = t x(|t|), |t| it is possible to get Fs [xo (t + α) + xo (t − α)] = 2Xs (ω) cos (αω).

These do not make a unitary matrix until we include the normalization factor as well as the scaling factor for the l = 0 component. In terms of the usual integer index, the matrix elements for DCT-V are: (CV N )lk = √ 2 2N − 1 σl σk cos kl 2π 2N − 1 , l, k = 0, 1, . . , N − 1. , x = 0 applied at l = −1/2 and x = π at l = N − 1, the following matrix is obtained:   1 −1 −1 2 −1    • • • A6 =  .  −1 2 −1 −2 2 As can be seen, this matrix again requires a similarity transformation to make √ it real and symmetric.

47, No. 5, May 2000, pp. 779–783. [9] B. G. Sherlock and Y. P. Kakad, “Windowed discrete cosine and sine transforms for shifting data”, Signal Processing, Vol. 81, No. 7, July 2001, pp. 1465–1478. [10] B. G. Sherlock and Y. P. Kakad, “Transform domain technique for windowing the DCT and DST”, Journal of the Franklin Institute, Vol. 339, No. 1, January 2002, pp. 111–120. [11] S. A. Martucci, “Convolution-multiplication properties of the entire family of discrete sine and cosine transforms”, Proceedings of the 26th Annual Conference on Information Sciences and Systems (CISS’92), Princeton, NJ, March 1992, pp.

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