In this problem you will use the fast fourier transform on
$\displaystyle {\ }\quad \vec{\mathbf{f}} = \bigl( -2, 1, -2, 2, 3, -2, 2, -1 \bigr)$

(A) Split $\vec{\mathbf{f}}$ into its even and odd components:

$\vec{\mathbf{f}}_{\mathrm{even}} = \Bigl($ $,\$ $,\$ $,\$ $\Bigr)$
$\vec{\mathbf{f}}_{\mathrm{odd}}\; = \Bigl($ $,\$ $,\$ $,\$ $\Bigr)$

(B) Compute the Fourier transforms of the even and odd components:

$\mathcal{F}\Bigl\lbrace \vec{\mathbf{f}}_{\mathrm{even}} \Bigr\rbrace = \Bigl($ $,\$ $,\$ $,\$ $\Bigr)$
$\mathcal{F}\Bigl\lbrace \vec{\mathbf{f}}_{\mathrm{odd}}\; \Bigr\rbrace = \Bigl($ $,\$ $,\$ $,\$ $\Bigr)$

(C) Combine the Fourier transforms of the even and odd components to get the transform of $\vec{\mathbf{f}}$

$\mathcal{F}_0\Bigl\lbrace \vec{\mathbf{f}} \Bigr\rbrace = \frac{1}{2} \Bigl($ $+\ \ \phantom{\omega^2}$ $\Bigr)\ =\$
$\mathcal{F}_1\Bigl\lbrace \vec{\mathbf{f}} \Bigr\rbrace = \frac{1}{2} \Bigl($ $+\ \ \overline{\omega}\phantom{^1}$ $\Bigr)\ =\$
$\mathcal{F}_2\Bigl\lbrace \vec{\mathbf{f}} \Bigr\rbrace = \frac{1}{2} \Bigl($ $+\ \ \overline{\omega}^2$ $\Bigr)\ =\$
$\mathcal{F}_3\Bigl\lbrace \vec{\mathbf{f}} \Bigr\rbrace = \frac{1}{2} \Bigl($ $+\ \ \overline{\omega}^3$ $\Bigr)\ =\$
$\mathcal{F}_4\Bigl\lbrace \vec{\mathbf{f}} \Bigr\rbrace = \frac{1}{2} \Bigl($ $-\ \ \phantom{\omega^2}$ $\Bigr)\ =\$
$\mathcal{F}_5\Bigl\lbrace \vec{\mathbf{f}} \Bigr\rbrace = \frac{1}{2} \Bigl($ $-\ \ \overline{\omega}\phantom{^1}$ $\Bigr)\ =\$
$\mathcal{F}_6\Bigl\lbrace \vec{\mathbf{f}} \Bigr\rbrace = \frac{1}{2} \Bigl($ $-\ \ \overline{\omega}^2$ $\Bigr)\ =\$
$\mathcal{F}_7\Bigl\lbrace \vec{\mathbf{f}} \Bigr\rbrace = \frac{1}{2} \Bigl($ $-\ \ \overline{\omega}^3$ $\Bigr)\ =\$

You can earn partial credit on this problem.