Download Discrete Cosine and Sine Transforms: General Properties, by Vladimir Britanak PDF

By Vladimir Britanak

Because the publication, "Discrete Cosine rework" by way of okay. R. Rao and P. Yip, (Academic Press, Boston) was once released in 1990, the DCT has more and more attracted the eye of clinical, engineering and study groups. The DCT is utilized in many purposes and in facts compression specifically. this is often on account that the DCT has first-class energy-packing strength and likewise methods the statistically optimum Karhunen-LoшЕ? rework (KLT) in decorrrelating a sign. the advance of varied speedy algorithms for the effective implementation of the DCT related to actual mathematics in basic terms, extra contributed to its reputation. within the final a number of years there were major advances and advancements in either idea and purposes on the subject of rework processing of indications. specifically, electronic processing stimulated the research of different varieties of discrete cosine transforms (DCTs) for his or her integer approximations. foreign criteria corporations (ISO/IEC and ITU-T) have followed using a variety of varieties of the integer DCT. whilst, the research of alternative sorts of discrete sine transforms (DSTs) has made an analogous effect. there's accordingly a necessity to increase the assurance to incorporate those innovations. This ebook is aimed toward doing simply that.

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Extra info for Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations

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20) ... 15). 6, where the unified derivation of the DCTs is presented. 20) is regarded as the sum of two matrices so that A−1 = B + R, where      2 −1  B = (1 − ρ ) 2 1+ √ρ − 2ρ 0 ... √ − 2ρ 1 + ρ2 −ρ ... 0 −ρ 1 + ρ2 ... ... √. − 2ρ ... √. − 2ρ 1 + ρ2    ,   and  √ −ρ  ( 2 − 1)ρ R = (1 − ρ2 )−1   ... 2 √ ( 2 − 1)ρ 0 ... 0 ... 0 ... √ 0 . . ( 2 − 1)ρ  ...   . 21) √ ... ( 2 − 1)ρ  −ρ2 We note that R is close to being a null matrix except at the corners of the main diagonal.

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. This is provided by the diagonal matrix D = diag[1, 1, . . tex 7/8/2006 12: 59 Page 35 Definitions and General Properties 35 √ the last element of each of the eigenvectors must be scaled by 1/ 2. The elements of the DCT-VI are then given by 2 (CVI σk εl cos N )lk = √ 2N − 1 l+ 1 2π k , 2 2N − 1 l, k = 0, 1, .

From these studies have come not only DCTs and DSTs but also other discrete unitary transforms. We examine the derivations for DCT-I and DCT-II here. Although the derivations lack the elegance of the unified treatment given by Strang [13] presented in the previous chapter, they do relate quite directly to the reasons why these transforms are close to being optimal in decorrelation operations. 1 DCT-I To begin, we recall that if a nonsingular matrix A is diagonalized by a matrix S in a similarity transformation, then its inverse A−1 is also diagonalized by the same transformation.

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