Gaussian Pulses

1.

Show that the Fourier transform of the signal

$$x(t) = \frac{1}{\sqrt{2\pi}w_0} \exp \left( -\frac{1}{2} \left( \frac{t-t_0}{w_0} \right)^2 \right)$$

is given by

$$X(f) = \exp(-j 2 \pi f t_0 - 2(w_0 \pi f)^2).$$

2.

Let x(t) be the input to a linear system with impulse response

$$h(t) = \frac{1}{\sqrt{2\pi}w_1} \exp \left( -\frac{1}{2} \left( \frac{t-t_1}{w_1} \right)^2 \right).$$

Find the resulting output signal y(t).

3.

Is the linear system in part (b) causal? Explain.

4.

Now let x(t) be input to a linear system with impulse response h(t) = u(t) where u(t) denotes the unit step function. (I.e., u(t)=1 for $t \geq 0$ and u(t)=0 for t < 0). Find the resulting output signal.

Hint: One of parts (b) and (d) is best solved in the frequency domain while the other is easier in the time domain.