Throughout this problem consider the following signal

$$s(t) = f_2 \cdot \mbox{sinc}(2 \pi f_1 t) \cdot \mbox{sinc}(2 \pi f_2 t),$$

where $\mbox{sinc}(x) = \sin(x)/x$. Assume that f₂ > f₁. Also, let $\Pi(x)$ denote the rectangular pulse defined in class, i.e.,

$$\Pi(x) = \left\{ \begin{array}{cl} 1 & \mbox{for $\vert x\vert < \frac{1}{2}$ }\\ 0 & \mbox{else.} \end{array}\right.$$

1.

Show that the inverse Fourier transform of $\Pi(f/f_0)$ is given by $f_0 \cdot \mbox{sinc}(\pi f_0 t)$.

2.

Use the convolution rule, to find the Fourier transform S(f) of s(t). Plot the magnitude of S(f). Be very accurate!

3.

Is it possible to completely reconstruct the signal s(t) from samples taken at rate 1/T_s? Justify your answer! If you answered ``yes'', give the largest sampling period T_s which allows for perfect reconstruction.

4.

Now, consider the signal $r(t)= s(t) \cdot \cos(2 \pi f_2 t)$. Compute the Fourier transform R(f) of r(t). Plot the magnitude of R(f). Be very accurate!

5.

Conclude that r(t) can be written in the form $A \cdot \mbox{sinc}(2 \pi f_a t) \cdot \mbox{sinc}(2 \pi f_b t)$ with suitably selected constants A, f_a, and f_b. You must provide values for all three constants and explain how you obtained them.