Linear, time-invariant systems

Let the response of a linear time-invariant (LTI) system to $x(t)=\Pi(t)$ be $y(t)=\Lambda(t)$. Here $\Pi(t)$ denotes the rectangular pulse:

$$\Pi(t) = \left\{ \begin{array} {cl} 1 & \vert t\vert \leq \frac{1}{2}\\ 0 & \mbox{else} \end{array}\right.$$

and $\Lambda(t)$ denotes the triangular pulse:

$$\Lambda(t) = \left\{ \begin{array} {cl} 1+t & -1 \leq t \l... ... 1-t & 0 \leq t \leq 1\\ 0 & \mbox{else.} \end{array}\right.$$

1.

Compute the Fourier transform of x(t) and y(t).

2.

If H(f) denotes the frequency response of the linear system, then the relationship $Y(f)=H(f) \cdot X(f)$ holds. Can you determine the frequency responseH(f) of the system for all frequencies f? In particular can you determine the response of the the system if $x(t)=\cos(2 \pi t)$?

3.

Show that $h_1(t)=\Pi(t)$ and $h_2(t)=\Pi(t)+\cos(2 \pi t)$ can both be impulse responses of this system. Hence, having the response of a system to $\Pi(t)$ does not uniquely determine the system.

4.

Does the response of an LTI system to $e^{-\alpha t} u(t)$, with u(t) denoting the unit step function, uniquely determine a system?

5.

In general, what conditions must the input x(t) satisfy so that the system can be uniquely determined from the output y(t)?