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## 3.3. Integer Discrete Cosine Transform–Based Schemes

### 3.3.1. Integer Discrete Cosine Transform

#### 3.3.1.1. One-Dimensional Integer Discrete Cosine Transform and Its Fast Algorithm

Let x(n) (n = 0, 1, …, N1) be a real input sequence. We assume that N = 2t, where t > 0. The scaled DCT of x(n) is defined as follows:

$X(k)=∑n=0N−1x(n)cosπ(2n+1)k2N,k=0,1,…,N−1$

(3.10)

Let CN be the transform matrix of the DCT, that is

$CN={cosπ(2n+1)k2N}k,n=0,1,…,N−1$

(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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