Single Sideband Signals

1.

Verify that the inverse Fourier transform of

$$G(f) = \left\{ \begin{array} {cl} \mbox{e}^{-\alpha f} & \... ...mbox{e}^{\alpha f} & \mbox{ for $f < 0$} \end{array} \right.$$

is given by

$$g(t) = \frac{j 4 \pi t}{\alpha^2 + (2 \pi t)^2}.$$

2.

By considering the limit as $\alpha$ tends to of the result in part (a), show that the following is a Fourier transform pair

$$\frac{1}{\pi t} \leftrightarrow -j \cdot \mbox{sign}(f) = \l... ...for $f = 0$}\\ j & \mbox{ for $f < 0$}. \end{array} \right.$$

3.

For the remainder of the problem, the signal x(t) has the Fourier transform

$$X(f) = \left\{ \begin{array} {cl} 1 - \frac{\vert f\vert}{... ... f\vert \leq f_M$} \\ 0 & \mbox{ else.} \end{array} \right.$$

Let x(t) be input to the following system in which $H(f)=-j\cdot\mbox{sign}(f)$:

Derive an expression for the Fourier transform $\hat{X}(f)$ of the signal $\hat{x}(t)$ and sketch $\hat{X}(f)$.

4.

Assuming that f_c is much larger than f_M, compute expressions for the Fourier transforms of the signals x_c,1(t), x_c,2(t), and x_c(t). Graph and accurately label these Fourier transforms.

5.

Indicate an application where the above system would be useful.