AM Stereo

Two carriers of the same frequency but with a phase difference of $\pi/2$ may be used to transmit two signals simultaneously. This is called quadrature amplitude modulation and provides the basis for the generation of AM stereo signals. Specifically, the DSB-SC variety of such a signal may be described by

$$s(t) = A_c (m_l(t) \cos(2 \pi f_c t) + m_r(t) \sin(2 \pi f_c t)),$$

where f_c is the carrier frequency and m_l(t) and m_r(t) are the left-hand and right-hand message signals, respectively.

You may assume throughout that m_l(t) and m_r(t) are lowpass signals with bandwidth f_m much smaller than f_c.

1.

Draw a block diagram of a quadrature amplitude modulator, i.e., a system with inputs m_l(t) and m_r(t) that generates the signal s(t).

2.

Compute the Fourier transform of the signal s(t) in terms of M_r(f) and M_l(f).

3.

A ``mono'' receiver can be implemented as in the following block diagram:

Derive an expression for the signal c(t) and then compute the Fourier transform of the signal c(t).

4.

Assume the lowpass filter to be ideal with cut-off frequency f_p satisfying $f_m < f_p \ll f_c$, find an expression for the output signal $\hat{m}(t)$. Explain how the mono receiver processes the received signal containing stereo signals.

5.

Propose a receiver that is capable of extracting both m_l(t) and m_r(t) from s(t). Draw a block diagram of your proposed receiver and explain in sufficient detail how and why it works.

Hint:

$$\begin{array}{l} \cos(x \pm y) = \cos x \cos y \mp \sin x \si... ... \cos x \sin y = \frac{1}{2}(\sin(x+y) - \sin(x-y)) \end{array}$$