Question

# Find the transform of a general signal$f=[f_1, f_2, f_3, f_4]^T$of four values.

Solution

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Step 1
1 of 2

Using $\pmb{F_4}$ from Example 4 and a general $\pmb{f}$ we have

\begin{align*} \pmb{\hat{f}}&=\pmb{F_4}\pmb{f}\\[7pt] &=\mqty[w^0 & w^0 & w^0 & w^0 \\ w^0 & w^1 & w^2 & w^3 \\ w^0 & w^2 & w^4 & w^6 \\ w^0 & w^3 & w^6 & w^9]\pmb{f}\\[7pt] &=\mqty[1 & 1 & 1 & 1 \\ 1 & -i & -1 & i \\ 1 & -1 & 1 & -1\\ 1 & i & -1 & -i]\mqty[f_1\\f_2\\f_3\\f_4]\\[7pt] &=\mqty[f_1 + f_2 + f_3 + f_4 \\ f_1 - if_2 -f_3 +if_4 \\ f_1 - f_2 + f_3 - f_4 \\ f_1 + if_2 -f_3 -if_4 ] \end{align*}

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