5.1 Prove the convolution property of the Fourier Transform (10 points)
For any FT pairs $f_1(x)$ / $\hat{f}_1(u)$, and $f_2(x)$ / $\hat{f}_2(u)$, the convolution of $f_1$ and $f_2$, $f_3(x) = f_1(x) *$
f2(x) has FT given by:
$\hat{f}_3(u) = \hat{f}_1(u) \cdot \hat{f}_2(u)$
In other words, convolution in the spatial domain becomes multiplication in the frequency
domain. Prove this property using the definition of the Fourier Transform.
5.2 Convolution of sinc functions (10 points)
Calculate the following convolution:
sinc($a_1x$) * sinc($a_2x$) * sinc($a_3x$) * ... * sinc($a_nx$)
where 0 < $a_1$ < $a_2$ < ... < $a_n$.
5.3 Do we need the phase? (5 points)
For an unknown arbitrary signal f(x), if we know the magnitude of its FT, ie: $|\hat{f}(u)|$, do we
have enough information to uniquely recover f(x)? Why or why not?
