_{N}be the transform matrix of the DCT, that is

Let x(n) (n = 0, 1, …, N−1) be a real input sequence. We assume that N = 2^{t}, where t > 0. The scaled DCT of x(n) is defined as follows:

$X\left(k\right)=\sum _{n=0}^{N-1}x\left(n\right)\mathrm{cos}\frac{\text{\pi}\left(2n+1\right)k}{2N},k=\mathrm{0,1},\dots ,N-1$

(3.10)

Let C_{N} be the transform matrix of the DCT, that is

${C}_{N}={\left\{\mathrm{cos}\frac{\text{\pi}\left(2n+1\right)k}{2N}\right\}}_{k,n=\mathrm{0,1},\dots ,N-1}$

(3.11)

To derive the fast algorithm, we first get a factorization of the transform matrix based on the following lemma ...

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