Sampling of Band-Pass Signals

Consider the following system, in which S denotes a linear system to be determined below and the signal m(t) is given by

$$m(t) = \sum_{n=-\infty}^{\infty} \delta(t-nT).$$

The input signal x(t) is a band-pass signal, i.e.,

$$x(t) \longleftrightarrow X(f) \mbox{ with } X(f) = 0 \mbox{ for $f_0-f_m < \vert f\vert < f_0+f_m$}$$

for some f₀ and f_m.

1.

Sketch the signal m(t) and then show that m(t) can be expanded into the Fourier series

$$m(t) = \sum_{k=-\infty}^{\infty} \frac{1}{T} \exp(j 2 \pi \frac{t}{T}).$$

2.

Using the result from part (a), compute the Fourier transform Y(f) of the signal y(t) in terms of the Fourier transform X(f) of the input signal.

3.

Assume that X(f) is given by

$$X(f) = \left\{ \begin{array} {cl} 1 & \mbox{for $4.5 < \vert f\vert < 5.5$,}\\ 0 & \mbox{else.} \end{array} \right.$$

and that the sampling rate $\frac{1}{T}=4$. Sketch and accurately label the Fourier transform Y(f).

4.

Is it possible to reconstruct the original signal x(t) from the sampled signal y(t)? If yes, determine the linear system S that will recover the signal x(t). If no, explain why not.

5.

Are there any other values for the sampling rate $\frac{1}{T}$ that would allow reconstruction of x(t) from the sampled signal y(t). Justify your answer!