Fourier Transform Example Matrix

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Fourier Transform Matrix | Download Scientific Diagram

Fourier Transform Example

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The Fourier Transform can be computed as a matrix multiplication:

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The Fourier Transform can be computed as a matrix multiplication:

$D_1 = \begin{bmatrix} 1&0&0&0&\cdots&0&0\\0&0&1&0&\cdots &0&0 \\\vdots& \vdots &\vdots & \vdots & \ddots & \vdots & \vdots \\0&0&0&0&\cdots&1&0 \\ 0&1&0&0&\cdots&0&0 \\ 0&0&0&1&\cdots&0&0 \\ \vdots&\vdots &\vdots & \vdots & \ddots & \vdots & \vdots \\ 0&0&0&0&\cdots&0&1\end{bmatrix} \in \mathbb{C}^{n \times n}.$

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$D_3 = \begin{bmatrix} 1&0&\cdots & 0 & \omega_n^0 & 0 & \cdots & 0\\ 0&1&\cdots&0 &0& \omega_n^1&\cdots & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots& \vdots &\ddots&\vdots\\ 0&0&\cdots & 1 & 0 & 0 & \cdots & \omega_n^{n/2-1}\\1&0&\cdots & 0 & \omega_n^{n/2} & 0 & \cdots & 0\\ 0&1&\cdots&0 &0& \omega_n^{n/2+1}&\cdots & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots& \vdots &\ddots&\vdots\\ 0&0&\cdots & 1 & 0 & 0 & \cdots & \omega_n^{n-1}\end{bmatrix}\in \mathbb{C}^{n \times n}$

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Fourier Transform Example Matrix

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