Signal Energy

Consider an ideal lowpass filter with cut-off frequency f_c, i.e., a linear, time-invariant system with frequency response

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

Assume that the following signal is input to the ideal lowpass:

$$x(t) = \exp(-t/\tau) \cdot u(t),$$

where $\tau \gt$ and u(t) denotes the unit step function. The resulting output signal is denoted by y(t)

1.

Compute the energy E_x of the input signal x(t).

2.

Find the Fourier transform Y(f) of the output signal y(t).

3.

Compute the energy E_y of the output signal y(t).

4.

Determine the cut-off frequency f_c such that $E_y = \frac{1}{2}E_x$.

Hint: You may need $\int \frac{1}{1+x^2} dx = \arctan(x)$.