Discrete Cosine Transform (DCT)

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A DCT-I of n=5 real numbers abcde is exactly equivalent to a DFT of 8 real numbers abcdedcb (even symmetry), here divided by 2. (In contrast, DCT-II ~ IV involve a halfsample shift in the equivalent DFT.) Note that DCT-I is not defined for n less than 2. (All other DCT types are defined for any positive n.) 6
DCT-II: f j = ∑ − n 1 k = 0 x k π cos[ j( k + n 1 )] 2 Some authors further multiply the f o term by 1/ 2 (see below for the corresponding change in DCT-III) . This makes the DCT-II matrix orthogonal (up to a scale factor), but breaks the direct correspondence with a realeven DFT of half-shifted input. 7
Page 1 and 2: Discrete Cosine Transform (DCT) 1
Page 3 and 4: Two related transform are the discr
Page 5: Formulation: DCT is a linear and i
Page 9 and 10: DCT-IV: f j = ∑ − n 1 k = 0 x k
Page 11 and 12: Inverse Transforms: IDCT-I is DCT-
Page 13 and 14: References: 1. Rao and Yip, Discre
Page 15 and 16: Then the 1-D DCT is applied to the
Page 17 and 18: 4 C u πu cos S 16 8 ( u) = 7 ∑ x
Page 19 and 20: We have the following DCT v.s. DFT
Page 21 and 22: Let X M = (f(0), f(1), f(2), f(3),
Page 23 and 24: F M = X M x (1/8)T M Because cosα
Page 25 and 26: 25 X M = F M x (8T M -1 ) = FM x L
Page 27 and 28: 27 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Page 29 and 30: 29 So the IDFT can be performed by
Page 31 and 32: 31 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪
Page 33: If you count the operations you wil

Discrete Cosine Transform (DCT)

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