Single Sideband Modulation

The message signal m(t) is input to the system shown in the block diagram below. The message signal is strictly band-limited, i.e., M(f)=0 for $f\gt\frac{\omega_0}{2\pi}$. Furthermore, the frequency $f_c=\frac{\omega_c}{2\pi}$ is much larger than the highest signal frequency: $\omega_c \gg \omega_0$.The lowpass filters are assumed to be ideal and have cut-off frequencies equal to the highest signal frequency.

1.

Compute the Fourier transforms of the signals x₁(t) and x₂(t) as a function of the Fourier transform M(f) of the message signal. Sketch the two resulting Fourier transforms assuming a typical spectrum M(f). Be careful and consider that the Fourier transforms are complex valued!

2.

Describe and quantify the effect of the low pass filters: give an expression for the Fourier transforms of the signals y₁(t) and y₂(t) and sketch the two resulting Fourier transforms assuming a typical spectrum M(f).

3.

Compute the Fourier transforms of the signals z₁(t) and z₂(t). Then, sketch the two resulting Fourier transforms assuming a typical spectrum M(f).

4.

Compute the spectrum of the signal s(t) and sketch it assuming a typical spectrum M(f).

5.

What happens if s(t) is multiplied with $\cos(\omega_c t)$ and then passed through an ideal lowpass filter with cut-off frequency $\omega_0$?

6.

Explain how the circuit above can be used in a communication system.