Linear, Time-invariant Systems

Consider the following system in which h(t) denotes the impulse response of a linear, time-invariant system.

Throughout this problem, h(t) is given by

$$h(t) = \Pi(\frac{t-\frac{\tau}{2}}{\tau}) = \left\{ \begin{ar... ... $0 \leq t \leq \tau$ }\\ 0 & \mbox{else.} \end{array}\right.$$

1.

Compute the Fourier transform H(f) of h(t).

2.

Assume for the remainder of the problem, that the input signal x(t) equals $\Pi(\frac{t-\frac{t_0}{2}}{t_0})$. Assume $\tau > t_0$. Determine the signal s₁(t) and the Fourier transform S₁(f). Plot s₁(t).

3.

Determine the signal s₂(t) and the Fourier transform S₂(f). Plot s₂(t).

4.

Determine the signal s₃(t) and the Fourier transform S₃(f). Plot s₃(t).

5.

Compute the output signal y(t) when $x(t)=\Pi(\frac{t-\frac{t_0}{2}}{t_0})$ is the input. Also, compute the Fourier transform Y(f). Plot y(t).

6.

Determine the impulse response and frequency response of the overall system, i.e., the system with input x(t) and output y(t).