Question

# Compute y = F₈c by the three steps of the Fast Fourier Transform if c = (1, 0, 1, 0, 1, 0, 1, 0). Repeat the computation with c = (0, 1, 0, 1, 0, 1, 0, 1).

Solution

Verified
Step 1
1 of 3

$\textbf{The first step}$ is to separate $c=(1,0,1,0,1,0,1,0)$ into $c^{'}$ and $c^{''}$.

$c^{'}=(1,1,1,1)\quad\land\quad c^{''}=(0,0,0,0)$

$\textbf{The second step}$ is to compute the values of $y^{'}$ and $y^{''}$.

\begin{align*} y^{'}&=F_{4}c^{'}\\ &=\begin{bmatrix}1 & 1 & 1 & 1\\1 & i & i^2 & i^3\\1 & i^2 & i^4 & i^6\\1 & i^3 & i^6 & i^9\end{bmatrix}\begin{bmatrix}1\\1\\1\\1\end{bmatrix}\\ &=\begin{bmatrix}1 & 1 & 1 & 1\\1 & i & -1 & -i\\1 & -1 & 1 & -1\\1 & -i & -1 & i\end{bmatrix}\begin{bmatrix}1\\1\\1\\1\end{bmatrix}\\ &=\begin{bmatrix}4\\0\\0\\0\end{bmatrix}\\\\ y^{''}&=F_{4}c^{''}\\ &=\begin{bmatrix}1 & 1 & 1 & 1\\1 & i & i^2 & i^3\\1 & i^2 & i^4 & i^6\\1 & i^3 & i^6 & i^9\end{bmatrix}\begin{bmatrix}0\\0\\0\\0\end{bmatrix}\\ &=\begin{bmatrix}0\\0\\0\\0\end{bmatrix} \end{align*}

$\textbf{The third step}$ is to combine $y^{'}$ and $y^{''}$ into the vector $y$.

$y=(y^{'},y^{''})=(4,0,0,0,0,0,0,0)$