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 bandlimited, such that

$$M(f) = 0 \;\;\; \mbox{for $\vert f\vert \gt f_m$}$$

The carrier frequency f_c is much larger than f_m.

1.

For a typical spectrum M(f), sketch the magnitude of the spectrum X(f) of the signal x(t).

2.

The envelope detector discussed in class can be modeled as a nonlinear device followed by a lowpass filter as shown in the following block diagram:

The nonlinear device NL is described by the relationship between its input x(t) and its output r(t) as

$$r(t) = \mbox{NL}(x(t)) =\left\{ \begin{array} {cl} x(t) & \mbox{if $x(t) \geq 0$} \\ 0 & \mbox{otherwise}\end{array}\right.$$

Assume the signal $g(t) = \cos(2\pi f_c t)$ is passed through the above nonlinearity. Sketch the resulting signal $\mbox{NL}(g(t))$ and show that this signal can be represented as a Fourier series by

$$\mbox{NL} (g(t)) = \sum_{n=-\infty}^{\infty} \frac{1}{2\pi}... ...{\sin(\frac{\pi}{2}(n+1))}{n+1}\right) \mbox{e}^{-jn2\pi f_ct}.$$

3.

For a typical message signal m(t), sketch the signals x(t) and r(t). Further, sketch the product of the signals (A + m(t)) and $\mbox{NL}(g(t))$ to verify that $r(t) = (A + m(t))\cdot\mbox{NL}(g(t))$.

4.

Using the results from parts (b) and (c), compute the Fourier transform of the signal r(t). Sketch and accurately label the magnitude of the Fourier transform of r(t).

5.

How would you choose the cutoff frequency of the lowpass filter in the demodulator above?