Problem 84

Consider the following system

The spectrum of the input signal x₁(t) is

$$X_1(f) = \left\{ \begin{array} {cl} 1 & \mbox{for $\vert f... ...} \\ 0 & \mbox{for $\vert f\vert \gt f_0$} \end{array}\right.$$

and the transfer function of the linear system S₁ is given by

$$H_1(f) = \cos(\frac{\pi}{4} \frac{f}{f_0}) \quad \mbox{for $\vert f\vert \leq 2f_0$.}$$

1.

Compute and sketch the spectrum of the signal y₁(t).

2.

By multiplying y₁(t) with the pulse train

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

we generate the signal x₂(t). Find a condition on T such that y₁(t) can be reconstructed completely.

3.

Assume now that $T=\frac{1}{2.5f_0}$. Compute and sketch the spectrum of x₂(t).

4.

The system S₂ is used to recover the input signal x₁(t). Find the transfer function H₂(f) of S₂ such that y₂(t)=x₁(t).