In Sec. 3.1.1, we used timelimited pulses, each with width less than the sampling interval T , to achieve practical sampling. Show that it is not necessary to restrict the sampling pulse width. We can...

In Sec. 3.1.1, we used timelimited pulses, each with width less than the sampling interval T , to achieve practical sampling. Show that it is not necessary to restrict the sampling pulse width. We can use sampling pulses of arbitrarily large duration and still be able to reconstruct the signal x(t) as long as the pulse rate is no less than the Nyquist rate for x(t).

Next, consider signal x(t), which is bandlimited to B Hz. Define a sampling pulse as p(t) = e^−atu(t). Find the spectrum Xp˜(ω) of the sampled signal xp˜(t), which is formed as the product of x(t) and ˜p(t), the T -periodic replication of p(t). Show that x(t) can be reconstructed from xp˜(t) provided that the sampling rate is no less than 2B Hz. Explain how you would reconstruct x(t) from the sampled signal.