DCTⅣをDFTで表す。

DCTⅣの定義式はつぎのとおり

\begin{equation}  S(r)= \sum_{k=0}^{N-1}s(k) {\rm cos}( \frac{2π}{2N}(k+ \frac{1}{2})(r+ \frac{1}{2})) \end{equation}

この式を変形します。

\begin{equation} S(r) = \frac{1}{2} \sum_{k=0}^{N-1}s(k) exp( \frac{i 2π}{2N}(k+ \frac{1}{2})(r+ \frac{1}{2})) + \frac{1}{2} \sum_{k=0}^{N-1}s(k) exp( \frac{-i 2π}{2N}(k+ \frac{1}{2})(r+ \frac{1}{2})) \end{equation}

\begin{equation} =  \frac{1}{2} \sum_{k=0}^{N-1}s(k) exp( \frac{i 2π}{2N}(k+ \frac{1}{2})(r+ \frac{1}{2})) + \frac{1}{2} \sum_{k=N}^{2N-1} s(2N-1-k) exp( \frac{-i 2π}{2N}(2N-1-k+ \frac{1}{2})(r+ \frac{1}{2})) \end{equation}

\begin{equation} =  \frac{1}{2} \sum_{k=0}^{N-1}s(k) exp( \frac{i 2π}{2N}(k+ \frac{1}{2})(r+ \frac{1}{2})) - \frac{1}{2} \sum_{k=N}^{2N-1} s(2N-1-k) exp( \frac{-i 2π}{2N}(-1-k+ \frac{1}{2})(r+ \frac{1}{2})) \end{equation}

\begin{equation} = \frac{1}{2} \sum_{k=0}^{N-1}s(k) exp( \frac{i 2π}{2N}(k+ \frac{1}{2})(r+ \frac{1}{2})) - \frac{1}{2} \sum_{k=N}^{2N-1} s(2N-1-k) exp( \frac{i 2π}{2N}(k+ \frac{1}{2})(r+ \frac{1}{2})) \end{equation}

\begin{equation} = \frac{1}{2} \sum_{k=0}^{2N-1}p(k)exp( \frac{i 2π}{2N}(k+ \frac{1}{2})(r+ \frac{1}{2})) \end{equation}

 $  p(k) = s(k)     (k=0～N-1) $

 　　　$ -s(2N-1-k)       (k=N～2N-1) $

 $ p(k) $ は、 $ s(k) $ を奇関数展開したものです。

さらにDFTで表します。

\begin{equation} S(r) = \frac{1}{2} \sum_{k=0}^{2N-1}p(k)exp(iπ \frac{k+ \frac{1}{2}}{2N})exp( \frac{i2π}{2N}(k+ \frac{1}{2})r) \end{equation}

\begin{equation} =  \frac{1}{2}exp( \frac{iπr}{2N}) \sum_{k=0}^{2N-1}p(k)exp(iπ \frac{k+ \frac{1}{2}}{2N})exp( \frac{i2π}{2N}kr) \end{equation}