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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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