Amplitude Modulation

Consider the following amplitude modulated signal,

$$x(t) = (A + m(t)) \cdot \cos(2\pi f_ct).$$

The constant A has been chosen such that (A+m(t))>0 for all t. Furthermore, the spectrum of the message signal m(t) is

$$M(f) = \Pi(f/2f_m) = \left\{ \begin{array} {cl} 1 & \mbox{for $\vert f\vert < f_m$} \\ 0 & \mbox{otherwise.}\end{array}\right.$$

The carrier frequency f_c is much larger than f_m.

1.

Sketch the spectrum of x(t).

2.

To demodulate x(t) we use the following system.

The subsystems labeled $(\cdot)^2$ and $\sqrt{\cdot}$ take the square and square-root of their respective inputs. Sketch the spectrum of the signal x(t)².

3.

Assume the transfer function of the lowpass is $H(f)=\Pi(f/4f_m)$.Find an expression for the signal x_L(t).

4.

Give an expression for the output signal y(t).

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