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The $\text{DFT}$ of a vector $\begin{bmatrix} a & b & c & d \end{bmatrix}$ is the vector $\begin{bmatrix} \alpha & \beta & \gamma & \delta \end{bmatrix}.$ consider the product $\begin{bmatrix} p & q & r & s \end{bmatrix}=\begin{bmatrix} a & b & c & d \end{bmatrix}\begin{bmatrix} a& b&c &d \\ d&a &b &c \\c & d& a&b \\ b& c&d &a \end{bmatrix}.$

The $\text{DFT}$ of a vector $\begin{bmatrix} p & q & r & s \end{bmatrix}$ is a scaled version of

1. $\begin{bmatrix} \alpha^{2} & \beta^{2} & \gamma^{2} & \delta^{2} \end{bmatrix}$
2. $\begin{bmatrix} \sqrt{\alpha} & \sqrt{\beta} & \sqrt{\gamma} & \sqrt{\delta} \end{bmatrix}$
3. $\begin{bmatrix} \alpha + \beta & \beta + \delta & \delta + \gamma & \gamma + \alpha \end{bmatrix}$
4. $\begin{bmatrix} \alpha & \beta & \gamma & \delta \end{bmatrix}$

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