Amplitude Modulation and Sampling

Consider the system in the block diagram below.

The input signal x(t) is an amplitude modulated signal given by

$$x(t) = (A +m(t)) \cos(2 \pi f_0 t).$$

The constant A is large enough that A + m(t) is positive for all t. Further, the message bearing signal m(t) is strictly band-limited, i.e., M(f)=0 for |f|>f_m and $f_m \ll f_0$.

1.

Compute the Fourier transform X(f) of the input signal x(t).

2.

Show that the following is a Fourier transform pair

$$\sum_{n=-\infty}^{\infty} \delta(t-nT_0) \leftrightarrow \frac{1}{T_0} \sum_{m=-\infty}^{\infty} \delta(t-\frac{m}{T_0}).$$

3.

The input signal is sampled by the impulse train. Are the conditions of the sampling theorem discussed in class satisfied? Explain!

4.

Assume that the sampling rate equals the carrier frequency, i.e., $\frac{1}{T_0}=f_0$. Compute the Fourier transform Y(f) of the signal y(t).

5.

Compute the output z(t) under the assumptions that the lowpass filter is ideal and it's cut-off frequency f_c satisfies $f_m < f_c \ll f_0$.