Problem 37

Throughout this problem consider the following two signals

$$s_1(t) = \cos(2\pi f_0 t) \cdot \Pi(t/T_0)$$

and

$$s_2(t) = (f_1+f_2) \cdot \mbox{sinc}(\frac{2 \pi (f_2 - f_1)t}{2}) \cdot \mbox{sinc}(\frac{2 \pi (f_2 + f_1)t}{2}),$$

where $\mbox{sinc}(x) = \sin(x)/x$ and $\Pi(x)$ denotes 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.

Compute the Fourier transform of s₁(t).

3.

Using the convolution rule, find the Fourier transform S₂(f) of s₂(t). Plot the magnitude of S₂(f).

4.

Is it possible to completely reconstruct either of the two signals s₁(t) and s₂(t) from samples taken at rate 1/T_s? Justify your answer for each signal separately.

5.

If you answered ``yes'' for either or both of the two signals, give the smallest sampling period T_s which allows for perfect reconstruction.