|
| DC(0th) |
| 1st | 2nd |
| 3rd | 4th |
|
| \(\large{ H_8= \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ \end{bmatrix} } \qquad (1)\) |
| \(\large{H_n\left(r, c\right)=\left(-1\right)^{d\left(r, c\right)}} \qquad (2)\) |
| \(\large{r, c}\): | 行, 列のインデックス |
| \(\large{d\left(r, c\right)= \displaystyle \sum_{k=0}^{p-1} b\left(k, r\right)f\left(k, c\right)}\) | \(\large{\left(p=\log_{2} n\right)}\) |
| \(\large{f\left(k, c\right)= \begin{eqnarray} \left\{ \begin{array}{l} b\left(p-1, c\right) \qquad \qquad \qquad \qquad \qquad (k=0) \\ b\left(p-k, c\right) + b\left(p-k-1, c\right) \qquad (0 \lt k \lt p) \\ \end{array} \right. \end{eqnarray} }\) |
| \(\large{b\left(j, i\right)= \left(i \gg j \right) \& 1}\) |
| \(\large{i \gg j}\): | \(i\)を\(j\)回だけ論理右シフト | \(\large{\&}\): | ビット毎の論理積 |
| \(\large{S=\frac{1}{n}H_n Y} \qquad (3)\) |
| \(\large{ Y= \begin{bmatrix} y\left(0\right) \\ y\left(1\right) \\ y\left(2\right) \\ \vdots \\ y\left(n-1\right) \\ \end{bmatrix} }\) | \(\large{ S= \begin{bmatrix} DC(0th) \\ 1st[sin] \\ 1st[cos] \\ \vdots \\ n/2-1th[sin] \\ n/2-1th[cos] \\ n/2th \\ \end{bmatrix} }\) |