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Question 5.E.8.4: Use the following vectors to perform the indicated computati......

Use the following vectors to perform the indicated computations:

a = \begin{pmatrix}α_0\\α_1\end{pmatrix},   b = \begin{pmatrix}β_0\\β_1\end{pmatrix},   \hat{a} = \begin{pmatrix}α_0\\α_1\\0\\0\end{pmatrix},   \hat{b} = \begin{pmatrix}β_0\\β_1\\0\\0\end{pmatrix}.

(a) Compute a \odot b,  F_4(a \odot b),   and   (F_4\hat{a}) × (F_4 \hat{b}).

(b) By using F^{−1}_4 as given in Example 5.8.1, compute

F^{−1}_4 [(F_4\hat{a}) × (F_4\hat{b})].

Compare this with the results guaranteed by the convolution theorem.

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(a) a \odot b = \begin{pmatrix}α_0β_0\\α_0β_1 + α_1β_0\\α_1β_1\\0 \end{pmatrix}

F_4(a \odot b) = \begin{pmatrix}α_0β_0 + α_0β_1 + α_1β_0 + α_1β_1\\ α_0β_0  −  iα_0β_1  −  iα_1β_0  −  α_1β_1\\ α_0β_0  −  α_0β_1  −  α_1β_0 + α_1β_1\\ α_0β_0 + iα_0β_1 + iα_1β_0  −  α_1β_1\end{pmatrix} = (F_4\hat{a}) × (F_4 \hat{b})

(b) F^{−1}_4 [(F_4\hat{a}) × (F_4 \hat{b})] = a \odot b

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